Cosmology.pdf

MODERN COSMOLOGY

The Universe in its Evolution and Structure

Max Camenzind

Landessternwarte Konigstuhl HeidelbergJuly 19, 2009

The big bang appears to the casual observer as just another myth, albeitwithout some of the more obvious anthropocentric characteristics. The

difference, however, is that modern cosmology is based uponthe scientificmethod. The scientific method has very specific rules.

Preface

In the history of science few developments have been more important than the advent of thenew heliocentric cosmology in the sixteenth and seventeenth centuries. Whereas most of theancient Greeks and European medievals had believed the earth was at the center of the universewith the sun, moon, planets, and stars orbiting around us - in the sixteenth century a newidea began to emerge. According to this new way of thinking, it was not the earth but the sunthat was at the center of the cosmic system. This revolutionary idea was famously proposedby Nicholas Copernicus in his bookOn the Revolution of the Heavenly Spheres, published in1543.

The progress of modern cosmology has been guided by both observational and theoreticaladvances. The subject really took off in 1917 with a paper by Albert Einstein that attemptedthe ambitious task of describing the universe by means of a simplified mathematical model.Five years later Alexander Friedmann constructed models ofthe expanding universe that hadtheir origin in a big bang. These theoretical investigations were followed in 1929 by the pi-oneering work on nebular redshifts by Edwin Hubble and Milton Humason, who providedthe observational foundations of present–day cosmology. In 1948 the steady state theory ofHermann Bondi, Thomas Gold, and Fred Hoye added a spice of controversy that led to manyobservational tests, essential for the healthy growth of the subject as a branch of science. Thenin 1965 Arno Penzias and Robert Wilson discovered the microwave background, which notonly revived George Gamow’s concept of the hot big bang proposed nearly two decades be-fore, but also prompted even more daring speculations aboutthe early history of the universe.

Present models of the universe hold two fundamental premises: the cosmological principleand the dominant role of gravitation. Derived by Hubble, thecosmological principle holdsthat if a large enough sample of galaxies is considered, the universe looks the same from allpositions and in all directions in space. The second point ofagreement is that gravitationis the most important force in shaping the universe. According to Einstein’s general theoryof relativity, which is a geometric interpretation of gravitation, matter produces gravitationaleffects by actually distorting the space about it; the curvature of space is described by a formof non-Euclidean geometry. A number of cosmological theories satisfy both the cosmologicalprinciple and general relativity. The two main theories arethe big-bang hypothesis and thesteady-state hypothesis, with many variations on each basic approach.

The material here is therefore of two kinds: relevant piecesof physics and astronomy thatare often not found in undergraduate courses, and applications of these methods to more recentresearch results. The former category is dominated by general relativity and quantum fields.Relativistic gravitation has always been important in the large-scale issues of cosmology, butthe application of modern particle physics to the very earlyuniverse is a more recent develop-ment. Many excellent texts exist on these subjects, but I want to focus on those aspects thatare particularly important in cosmology. At times, I have digressed into topics that are strictly“unnecessary”, but which were just too interesting to ignore. These are, after all, the crown

iv Preface

jewels of 20th Century physics, and I firmly believe that their main features should be a stan-dard part of a graduate education in physics and astronomy. Despite this selective approach,which aims to concentrate of the essential core of the subject, the course will cover a widerange of topics. My original plan was in fact focused more specifically on matters to do withparticle physics, the early universe, and structure formation. However, as I wrote, the subjectimposed its own logic: it became clear that additional topics simply had to be added in orderto tell a consistent story. This tendency for different parts of cosmology to reveal unexpectedconnections is one of the joys of the subject, and is also a mark of the maturity of the field.

According to big-bang theories, at the beginning of time, all of the matter and energy inthe universe was concentrated in a very dense state, from which it exploded, with the resultingexpansion continuing until the present. Thisbig bangis dated between 12 and 15 billion yearsago. In this initial state, the universe was very hot and contained a thermal soup of quarks,electrons, photons, and other elementary particles. The temperature rapidly decreased, fallingfrom 1013 degrees Kelvin after the first microsecond to about one billion degrees after threeminutes. As the universe cooled, the quarks condensed into protons and neutrons, the buildingblocks of atomic nuclei. Some of these were converted into helium nuclei by fusion; therelative abundance of hydrogen and helium is used as a test ofthe theory. After many millionsof years the expanding universe, at first a very hot gas, thinned and cooled enough to condenseinto individual stars and galaxies, and even Black Holes.

Several spectacular discoveries since 1960 have shed new light on the problem. Opticaland radio astronomy complemented each other in the discovery of the quasars and the radiogalaxies. It is believed that the energy reaching us now fromsome of these objects was emittednot long after the creation of the universe. Further evidence for the big-bang theory was thediscovery in 1965 that a cosmic background noise is receivedfrom every part of the sky. Thisbackground radiation has the same intensity and distribution of frequencies in all directionsand is not associated with any individual celestial object.This radiation filling space has ablack body temperature of 2.73 K and is interpreted as the electromagnetic remnant of theprimordial fireball, stretched to long wavelengths by the expansion of the universe. Morerecently, the analysis of radiation from distant celestialobjects detected by artificial satelliteshas given additional evidence for the big-bang theory and the small scale structure imprintedby early fluctuations in the density distribution.

Max Camenzind

Heidelberg, July 17, 2009

Contents

I The Observable Universe 1

1 History and Overview 31.1 History of the Universe – Myths and Science . . . . . . . . . . . .. . . . . 31.2 Steps towards Modern Cosmology . . . . . . . . . . . . . . . . . . . . .. . 61.3 Key Facts of Modern Cosmology . . . . . . . . . . . . . . . . . . . . . . .. 71.4 Problems and Mysteries . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91.5 The Edge of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 The Observable Universe 132.1 The Universe is Expanding and Accelerating . . . . . . . . . . .. . . . . . . 152.2 Cosmic Background Radiation as a Relic of the Early Universe . . . . . . . . 222.3 Galaxies are not Uniformly Distributed . . . . . . . . . . . . . .. . . . . . 302.4 Gravity is Dominated by Dark Matter . . . . . . . . . . . . . . . . . .. . . 332.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

II Relativistic World Models 43

3 The Relativistic Cosmos 453.1 Relativity and Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . .. 453.2 Einstein’s Vision of Gravity . . . . . . . . . . . . . . . . . . . . . . .. . . . 47

3.2.1 The Concept of SpaceTime . . . . . . . . . . . . . . . . . . . . . . . 473.2.2 Gravity is an Affine Connection on SpaceTime . . . . . . . . .. . . 513.2.3 Differential Forms on SpaceTime . . . . . . . . . . . . . . . . . .. 553.2.4 Curvature of SpaceTime . . . . . . . . . . . . . . . . . . . . . . . . 573.2.5 Curvature and Einstein’s Equations . . . . . . . . . . . . . . .. . . 58

3.3 General Relativity is the Correct Theory of Gravity . . . .. . . . . . . . . . 623.4 Isotropic SpaceTimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65

3.4.1 Slicing of SpaceTime . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4.2 Isotropic Riemannian Spaces . . . . . . . . . . . . . . . . . . . . .. 683.4.3 Spaces of Constant Curvature . . . . . . . . . . . . . . . . . . . . .693.4.4 Friedmann–Robertson–Walker (FRW) SpaceTimes . . . . . .. . . . 70

vi Contents

3.5 The FRW World Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.6 The Origin of the Cosmological Redshift . . . . . . . . . . . . . .. . . . . . 733.7 The Luminosity Distance and the Hubble–Law . . . . . . . . . . .. . . . . 743.8 The Hubble Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.9 The Apparent Angular Width of Galaxies . . . . . . . . . . . . . . .. . . . 793.10 Number Counts in the Expanding Universe . . . . . . . . . . . . .. . . . . 813.11 Slightly Inhomogeneous World Models . . . . . . . . . . . . . . .. . . . . 833.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 The Universe with Matter and Dark Energy 874.1 Description of Matter in a Relativistic Cosmos . . . . . . . .. . . . . . . . . 87

4.1.1 Fluid Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.1.2 Kinetic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1.3 EOS for Vacuum Energy . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Einstein’s Equations for FRW Models . . . . . . . . . . . . . . . . .. . . . 924.2.1 Derivation of Friedmann’s Equations . . . . . . . . . . . . . .. . . 924.2.2 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.3 Density Parameters for Friedman Models . . . . . . . . . . . .. . . 96

4.3 FRW Models without Vacuum Energy . . . . . . . . . . . . . . . . . . . .. 974.3.1 The Euclidean Universe, k=0 . . . . . . . . . . . . . . . . . . . . . .984.3.2 The Closed Universe, k=1 . . . . . . . . . . . . . . . . . . . . . . . 984.3.3 The Open Universe, k=–1 . . . . . . . . . . . . . . . . . . . . . . . 994.3.4 The Singularity at t=0: Big–Bang . . . . . . . . . . . . . . . . . .. 1004.3.5 The Mattig Formula for the Luminosity Distance . . . . . .. . . . . 100

4.4 The Present Universe with Dark Energy . . . . . . . . . . . . . . . .. . . . 1034.4.1 Cosmological Parameters . . . . . . . . . . . . . . . . . . . . . . . .1044.4.2 Solutions for Inflationary Universes . . . . . . . . . . . . . .. . . . 1054.4.3 Age of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4.4 The Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.4.5 The Particle Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.4.6 Conformal Maps of the Present Universe . . . . . . . . . . . . .. . 112

4.5 Measuring the Acceleration of the Present Universe . . . .. . . . . . . . . . 1224.5.1 Luminosity Distance and Hubble–Diagrams forΛCDM . . . . . . . 1234.5.2 Measuring Cosmology with Supernovae . . . . . . . . . . . . . .. . 129

4.6 Angular Width in FRW Models . . . . . . . . . . . . . . . . . . . . . . . . .1474.7 Redshift Distribution of Cosmological Objects . . . . . . .. . . . . . . . . . 1494.8 The Cosmological Fundamental Plane . . . . . . . . . . . . . . . . .. . . . 1534.9 On the Origin of the Dark Energy . . . . . . . . . . . . . . . . . . . . . .. 154

4.9.1 The Vacuum is not Empty . . . . . . . . . . . . . . . . . . . . . . . 1564.9.2 Quintessence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1564.9.3 Braneworld Models – Higher Dimensions . . . . . . . . . . . . .. . 1594.9.4 Effects from Inhomogeneous Universe . . . . . . . . . . . . . .. . . 166

4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Contents vii

List of Figures 171

List of Tables 173

Part I

The Observable Universe

1 History and Overview

Our exploration of cosmology begins with a brief history of the human desire to understand thecosmos. Mythology, humanity’s first attempt to grapple withcosmological questions, consistsof narrative tales that describe the universe in understandable terms. The text briefly discussesthe Tanzanian myth The Word as an example of a creation myth. Cosmology, particularly asexpressed by a mythology, can influence a culture’s or an individual’s actions.1

1.1 History of the Universe – Myths and Science

The big bang appears to the casual observer as just another myth, albeit without some of themore obvious anthropocentric characteristics. The difference, however, is that modern cos-mology is based upon the scientific method. The scientific method has very specific rules. Itis based on objective data, observations that are independent of who made those observations.Once sufficient data are collected, a hypothesis is framed toexplain and unify them. In orderto be regarded as scientific, the hypothesis must meet at minimum five characteristics: it mustbe relevant, testable, consistent, simple, and have explanatory power. Of these, the qualityof testability particularly defines the scientific method. Ahypothesis that does not containthe potential to be falsified is not scientific. Once a hypothesis has met success at explain-ing data and has proven itself useful in predicting new phenomena, it is generally called atheory. Some particularly well established theories, especially those pertaining to a limitedphenomenon or forming the foundation for a broader theory, are called laws. Hence we referto the law of gravity, even though scientific ”laws” are subject to modification as our under-standing improves. Newton’s law of gravity is extremely well verified for the regime in whichit is applicable (weak gravitational fields and speeds smallcompared to the speed of light);but Newton’s law must be superseded by Einstein’s general theory of relativity.

The first attempt to construct a systematic cosmology that was grounded in physical theorywas the model of Aristotle. Aristotle developed a theory of motion, and defined the conceptsof ”natural motion” and ”force.” In Aristotle’s view, the Earth was the center of the universeand the center of all natural motions. Motions on the Earth were linear and finite, while theheavenly bodies executed perfect circles forever. The stars and planets were composed of aperfect element called ”ether,” whereas Earthly objects were made up of varying combinationsof the four ancient elements of earth, air, fire and water; a body’s motion was a consequenceof its composition. Although our modern definitions of theseconcepts are quite different fromAristotle’s, natural motion and force remain fundamental to our understanding of the structureand evolution of the universe. Aristotle’s Earth-centeredworldview (Fig. 1.1) was embodiedin the detailed model of Ptolemy, with its deferents, epicycles, and eccentrics, all designed

1We follow here John F. Hawley’s book onFoundations of Modern Cosmolgy, Oxford University Press

4 1 History and Overview

to predict the complicated celestial motions of the planetswhile still requiring motion in theheavens to be built upon circles.

Figure 1.1: Aristotle’s crystalline sphere cosmos is the first cosmological model. Theillustration derivesfrom the Cosmographia seu descriptio totius orbis (1524) of Petrus Apianus (Peter Apian, 1495-1552).Earth is at the center of the cosmos, fixed and stable, followed by the Moon, Mercury, Venus, theSun, Mars, Jupiter, Saturn, and finally, three medieval ether regions that culminated with the EmpyreanSphere. The ordering of the planets differed, so far as we know, between Aristotle and Ptolemy. Therewas much debate throughout antiquity and the middle ages about the precise relation of the planets andthe Sun. The number of spheres given by Eudoxus (ca. 390 B.C.) is27 (including the sphere of the FixedStars). Callippus (ca. 370 B.C.) increases this to 34. Aristotle (384-322BC) improves on Callippus byincluding additional spheres to counteract some of the motions of the planetary spheres. These additionalspheres are placed between the outermost sphere of a given planet and the innermost sphere of the nextplanet and are one less than the number of spheres of the latter.

Aristotle (384-322 BC) surveyed the whole of human knowledge of his time, and wrote alot of works on philosophy and science. Most of the survivingworks are not from the manyvolumes Aristotle wrote personally but are thought to be lecture notes, perhaps recorded bya student at the Lyceum. Thus it may not be too surprising thatmany of them are difficult tofollow and sometimes even self-contradictory! Aristotle’s cosmos is divided into the perfect,unchanging heavens, and the imperfect “sublunary sphere” (everything on Earth, within theorbit of the Moon). Everything was thought to have a natural place within this scheme, andnatural motion was the result of objects’ natural tendency to attain their places. The passagequoted here begins with Aristotle’s reasons for believing in a fifth element, sometimes called

1.1 History of the Universe – Myths and Science 5

“quintessence” (watch for this to show up much later in the course!) or “ether”, apart from theusual four (earth, fire, air, and water). He argues that bodies have natural motions (as opposedto unnatural ones). He thinks fire and air naturally move upwards in a line, while earth andwater move down. But he argues that there is a simple body (or element) for each simplemotion, and circular motion is also a simple motion, so theremust be a new element whosemotion is naturally circular.

During the Renaissance, humanity’s cosmological model changed dramatically. The firstblow in the ”Scientific Revolution” was struck by Copernicus, whose Sun-centered model ofthe heavens gained rapid ascendency in Renaissance Europe.Tycho Brahe’s detailed nakedeye observations of the heavens provided the data that Kepler used to derive his laws of plan-etary motion. Kepler’s laws of planetary made it possible for the first time for humans tounderstand the paths of the ”wanderers” across the sky.

A consequence of Kepler’s second law is that planets orbit more slowly the more distantthey are from the Sun. The third law enables the period of a planet, comet, or asteroid to becomputed once observations establish the length of the semimajor axis of its orbit. These lawswere among the greatest quantitative achievements of the Renaissance.

Kepler and Galileo were contemporaries, though Kepler was more of a theorist and Galileowas primarily an observer. Galileo was the first to make serious scientific use of the telescope,an instrument which provided observations that challengedthe Ptolemaic model of the heav-ens. (Kepler was unable to afford to purchase a telescope, a prohibitively expensive device atthe time, though he was able to borrow one for a summer from a visiting nobleman. Galileopromised for several years to make a telescope for Kepler, but never got around to fulfillinghis promise.) Galileo observed craters on the Moon, demonstrating that it was not a perfect,smooth sphere; he also gave the large lunar plains the name of”maria” (seas) because hethought they might be filled with water. He also found that theMilky Way was not a solidband of light but was filled with myriad stars, too small to be resolved by the unaided eye.Another key observation by Galileo was that Venus went through a full cycle of phases, justlike the Moon; this was impossible in the Ptolemaic model butwas required by the Coperni-can model, since Venus is between the Earth and the Sun in the latter. But one of Galileo’smost important discoveries was of the four largest satellites of Jupiter, now called the Galileanmoons. These bodies demonstrated that the Earth was not the only center of motion in theuniverse, thus refuting one of the important tenets of Ptolemaic-Aristotelian cosmology andphysics.

Galileo also studied mechanics. From direct observation and careful reasoning, he wasable to arrive at the conclusion that all bodies fall at the same rate, if air resistance is negligible.This principle, now called theequivalence principle, is one of the foundations of the generaltheory of relativity, though Galileo could not have appreciated this at the time. Galileo alsorealized that motion might not be easily detectable by observers partaking of that motion, i.e.that motion is relative, though he never succeeded in working out the laws of motion. But afew months after Galileo’s death, Isaac Newton was born on a farm in Lincolnshire, England,beginning a life that would complete the Copernican Revolution with the fundamental laws ofphysics and gravitation that govern the universe under mostconditions.

The Aristotelian cosmological system was consistent with his physics. Although Aristo-tle’s cosmology is quite different from the modern point of view, in what ways is it consistentwith modern ideas? How did it differ? What were some of the motivations for Copernicus topropose such a grand change to the prevailing concepts of theuniverse?

Tycho Brahe’s observations of a supernova that appeared in 1572 helped to end the belief


in the Aristotelean unchanging perfection of the celestialrealm. How would you feel if youobserved something that so challenged a basic tenet of your world view? In Tycho’s ownwords:

Amazed, and as if astonished and stupefied, I stood still, gazing for a certain length oftime with my eyes fixed intently upon it and noticing that samestar placed close to the starswhich antiquity attributed to Cassiopeia. When I had satisfied myself that no star of that kindhad ever shone forth before, I was led into such perplexity bythe unbelievability of the thingthat I began to doubt the faith of my own eyes.

Kepler too was to observe a supernova, only 32 years later in 1604. The next supernovavisible to the naked eye did not occur until 1987 when a star exploded in the nearby irregulargalaxy known as the Large Magellanic Cloud.

1.2 Steps towards Modern Cosmology

Controversy over the nature ofspiral nebulaehad persisted since the late 18th century, withone camp insisting they were external universes, while their opponents were equally con-vinced that the spiral nebulae were localized clusters of stars within our Galaxy. An importantearly discovery was Shapley’s determination of the size of the Milky Way Galaxy, and of ourlocation within it. Shapley found the Milky Way to be much larger than previously believed,and on this basis he erroneously concluded that the spiral nebulae must be relatively nearbyclusters. Shapley and Curtis participated in a famous debate in 1920 over the nature of thespiral nebulae, but insufficient data prevented a resolution of the puzzle. Finally, Hubble de-termined that the Andromeda Nebula (now known as the Andromeda Galaxy) was much toofar to lie within the confines of the Milky Way; Hubble had discovered external galaxies. Inthe first quarter of the twentieth century, humanity’s view of the cosmos leaped from a fairlylimited realm of the Sun surrounded by an amorphous groupingof stars, to one in which theMilky Way is just a typical spiral galaxy in a vast universe filled with galaxies.

Not long after Hubble’s discovery of external galaxies camehis discovery of a linear rela-tionship between their redshifts and their distances, a relationship known today as theHubbleLaw. Determining the value of the constant of proportionality,the Hubble constant, remainsan important research goal of modern astronomy. The Hubble ”constant” is not really constant,because it can change with time, though at any given instant of cosmic time in a homogeneous,isotropic universe, it is the same at all spatial locations.The inverse of the Hubble constant,called theHubble time, gives an estimate of the age of the universe.

The development of the theory of general relativity provided the framework in which Hub-ble’s discovery could be understood. Einstein found that his equations would not admit astatic, stable model of the universe, even with the additionof the ”cosmological constant”.The timely discovery of the redshift-distance relationship provided evidence that the universewas not static, but was expanding. The Robertson-Walker metric is the most general metricfor an isotropic, homogeneous universe that is also dynamic; i.e. it changes with time. An im-portant parameter in this metric is the scale factor, the quantity which describes how lengths inthe universe change with cosmic time. The scale factor can beused to derive the cosmologicalredshift, the change in wavelength of light as it traverses the universe.

Measuring Hubble’s constant requires accurate distances to increasingly remote galaxies.One of the best distance measures is the Cepheid variable star. The HST has now been able todetect Cepheid variable stars in the galaxy M100 in the Virgogalaxy cluster. Several Cepheids

1.3 Key Facts of Modern Cosmology 7

have been found such as this one. These new data give us a distance to M100 of 17 Mpc andis consistent with a rather large Hubble constant of about 70km/sec/Mpc. This value has nowbeen confirmed by Supernovae type Ia measurments and a detailed analysis of the cosmicmicrowave anisotropies by the WMAP satellite. It is the first time in the history of the Hubbleconstant that a converging value has been acchieved by various methods.

1.3 Key Facts of Modern Cosmology

The relativistic cosmos is expanding. The history of the universe can be divided into manyeras, depending on which constituent was most important. Today the matter density of the uni-verse dominates its gravity completely; thus we say that we live in thematter era. However,conditions were not always as they are today. The relationship between temperature and red-shift of the CMBR demonstrates that as we look backward in time towardt = 0, the photonsin the CMBR become increasingly hot. Density also changes with time, becoming ever higheras we look toward earlier times; but the variation with scalefactor is different for matter den-sity and for radiation energy density, with matter varying as the cube of the scale factor, whileradiation energy density varies as the fourth power ofR. Hence there was a time in the pastwhen radiation was more important than ordinary matter in determining the evolution of theuniverse. This high-temperature epoch defines the early universe. Temperature is a measureof energy, and Einstein’s equationE = mc2 tells us that energy and matter are equivalent.At sufficiently high temperatures, particles with large mass can be created, along with theirantiparticles, from pure energy. The temperature also influenced how the fundamental forcesof nature behaved during the earliest intervals of the universe’s history.

The interval of domination by radiation is called theradiation era. As we approach t=0in this era, we encounter increasingly unfamiliar epochs, dominated by different physics anddifferent particles. The earliest was thePlanck epoch, during which all four fundamentalforces were unified and ”particles” as we know them could not have existed. Next followedthe GUT epoch, when gravity had decoupled but the other threeforces remained unified.The small excess of matter that makes up the universe today must have been created duringthis epoch, by a process still not completely understood. Asthe temperature dropped, theuniverse traversed the quark epoch, the hadron epoch, the lepton epoch, and the epoch ofnucleosynthesis.

The first atomic nuclei formed during the nucleosynthesis epoch. Most of the helium in theuniverse was created from the primordial neutrons and protons by the time the nucleosynthesisepoch ended scarcely three minutes after the big bang. A few other trace isotopes, specificallydeuterium (heavy hydrogen) and lithium 7, were also created, and their density depends sen-sitively upon the density of the universe during this time. If the universe were too dense,then most of the deuterium would have fused into helium. Onlyin a low-density universecan the deuterium survive. The major factor controlling theultimate densities of helium anddeuterium is the abundance of neutrons. The more neutrons that decay before combining withprotons, the smaller the abundances of heavier elements. The availability of neutrons dependson the expansion rate as well as the cosmic matter density. Comparing the observed densitiesof the primordial isotopes to those computed from models andtranslating the results intoΩB ,the density parameter, givesΩB = 0.015/h2 whereh is the Hubble parameter divided by100 km/sec/Mpc. The smallerH0, the largerΩ; if H0=50,ΩB is approximately 0.06, whereasH0 = 100 givesΩB of only 0.015. This range is still much less thanΩ = 1, but nucleosynthe-


sis limits can indicate only the density of baryons, becauseonly baryons participate in nuclearreactions. Hence we must conclude that the universe contains less than the critical density ofbaryons.

After nucleosynthesis, nothing much happened for roughly amillion years as the universecontinued to cool. The ordinary matter consisted of a hot plasma of nuclei and electrons. Thefree electrons made the plasma opaque; a photon of radiationcould not have traveled far beforebeing scattered. However, once the universe cooled to approximately 3000 K, the electrons nolonger moved fast enough to escape the attraction of the nuclei, and atoms formed. Althoughthere had been no previous combination, this event is still known asrecombination. The lastmoment at which the universe was opaque forms the surface of last scattering; it representsthe effective edge of the universe that is even theoretically visible with an optical telescope,since no optical telescope could ever penetrate the dense, opaque plasma that existed prior torecombination. Once the radiation was able to stream freelythrough the universe, matter andradiation lost the tight coupling that had bound them since the beginning. Henceforth matterand radiation evolved almost entirely independently. The photons that filled the universe atthe surface of last scattering make up the CMBR today, but nowtheir energy is mostly inthe microwave band. At some point before or near recombination, the matter density andthe energy density were equally important. This is the epochin which structure formationbegan to occur. The seeds of structure formation may have been planted much earlier, duringthe GUT epoch, but the tight coupling between radiation and matter prevented the densityperturbations from doing much. Once matter and radiation went their separate ways, densitypertubations could evolve on their own. The most overdense areas collapsed gravitationally,forming galaxies and clusters of galaxies. Less dense areasprobably led to voids, the largeunderdense areas we see on the sky today. The process by whichstructure formed in the earlyuniverse is still very poorly understood; better data from instruments such as the plannedsuccessors to COBE (WMAP and Planck) will help to elucidate the mystery of the galaxies.

The matter density is one of the fundamental parameters of the universe; in the standard(Λ = 0) models, the matter density determines the geometry of the cosmos. An accuratedetermination of this quantity is thus of great cosmological importance. Many methods havebeen developed to try to measure the matter density. Most rely upon detecting the orbits ofvisible matter and using Kepler’s laws to compute the mass; these dynamical methods areprobably the most widely employed. At all scales, we have found that the amount of matterrevealed through gravitational interactions is greater than can be explained by the mass of thevisible stars. The nature of this unseen dark matter is one ofthe most important outstandingcosmological problems.

At the scale of individual galaxy disks, much of the dark matter can be attributed to stellarashes such as white dwarfs, neutron stars, and black holes, as well as to extremely faint objectssuch as very low-mass stars and brown dwarfs. However, at larger scales, the total aggregationof known baryonic matter cannot account for the mass that orchestrates gravitational interac-tions. Galaxies, including the Milky Way, appear to be surrounded by huge spheroidal darkhalos. The composition of the dark halos is unknown, but observations have indicated thatat least some of the mass must take the form of compact objectscalled MACHOs. Since webelieve that only baryonic (i.e. ordinary) matter can form compact objects, this suggests thatgalaxies are surrounded by an invisible cloud of objects such as neutron stars or small blackholes. How and when these bodies formed is still a mystery.

At larger scales, measurements ofΩ rise until they typically level off within a range of0.1 to 0.3, with the middle of this range (about 0.25) currently seeming to be most likely.

1.4 Problems and Mysteries 9

This is far greater than can be accommodated by the abundances of primordial elements suchas helium and deuterium. Thus we conclude that some 90% of thematter of the universeis not only invisible, but is nonbaryonic. If it is not baryons, what is it? It must consistof some type of WIMP (weakly interacting massive particle). What this particle might beremains unknown, though particle physicists can provide plenty of possibilities. One obviouscandidate is the neutrino. Evidence has grown over the last decade that the neutrino actuallyhas an extremely tiny mass. There is no theoretical reason that the neutrino must be massless;indeed, one hypothesis for the dearth of solar neutrinos observed over the past 30 years isthat neutrinos are massive, and the kind that can be observedis converted into a kind thatis invisible to most neutrino detectors on the journey from the Sun. Laboratory evidenceis beginning to support this hypothesis. If neutrinos have even a small mass per particle,they could add considerably to the cosmic matter density because they are approximately asabundant as photons. However, the experimental limits on the neutrino mass preclude it fromproviding enough matter density to close the universe. Plenty of more exotic WIMPS havebeen suggested, but as yet none has been detected. A great deal of experimental and theoreticalingenuity and effort is being devoted to identifying the elusivemissing matter.

The dark matter is also inextricably connected with the formation of galaxies and galaxyclusters, since it must dominate the gravity of the universe. It is well known that galaxies tendto occur in clusters, such as the Hercules cluster. Galaxy clusters are gravitationally bound;that is, the galaxies within a cluster orbit one another. TheMilky Way is a member of a smallcluster of a few dozen members, dominated by itself and the Andromeda galaxy; this clusteris called the Local Group. The nearest large cluster to the Local Group is the Virgo cluster,about 16 Mpc distant.

1.4 Problems and Mysteries

Despite its successes, the standard big bang cosmology has some problems that are difficult toresolve. Among these are:

• The horizon problem

• The flatness problem

• The structure problem

• The relic problem

These can be summarized as follows: (1) The universe is observed to be highly homogeneousand isotropic, but how did it become so when all regions of theobservable universe were notin mutual causal contact at early times? (2) The universe is nearly flat today, but this impliesthat it must have had anΩ very nearly equal to one at early times. Unless the universe isexactly flat, this seems to require fine tuning. Why is the universe so flat? (3) What formedthe perturbations that lead to the structure we see around us? Why is structure the sameeverywhere, even though different parts of the universe were not causally connected early inthe big bang model? (4) GUTs predict massive particles that are not observed. What happenedto these ”relics” of the GUT epoch?

The inflationary model addresses all these issues by presuming that what we call the ob-servable universe is actually a very small portion of the initial universe that underwent a deSitter phase of exponential expansion around the time of theGUT epoch. This model posits


that what became our observable universe was small enough tobe in causal contact at thebig bang; it then grew at an exponential rate during the inflationary epoch. The exponentialgrowth had the effect of flattening out any curvature, stretching the geometry of the universeso much that it became flat. Any massive GUT particles were diluted, spread out over thisnow fantastically huge domain to an extremely tiny density,so that they no longer are observ-able. Quantum fluctuations in the vacuum are preserved and ”blown up” to large scales by theexpansion, providing the seeds for structure formation.

The source of this exponential growth was a negative pressure produced by a nonzerovacuum energy. A nonzero vacuum energy could result from quantum processes in the earlyuniverse. In quantum field theory, a field is associated with each particle, and the field in turnis related to a potential, the latter being a function which describes the energy density of thefield. The right potential would result in a false vacuum, a situation in which the field was zerobut the corresponding potential was not zero. The false vacuum state could have provided avacuum energy that would behave exactly like a positive (repulsive) cosmological constant,resulting in a temporary de Sitter phase during which the patch of universe grew by a factorof perhaps10100 or even more. Eventually, however, this vacuum energy was converted intoreal particles and the field found its way to the true vacuum, bringing the inflation to a halt.The universe then continued to evolve from this point as in our standard model.

The inflationary model is an area of active research. It makessome predictions aboutthe structures in the universe which are consistent with theCMBR data, although not yetproven, and it predicts that the present universe should be flat. This may present a problemfor the model since most measurements giveΩ < 1. Furthermore,Ω = 1 models are tooyoung for the larger Hubble values that tend to be measured byrecent techniques. Moreover,the particle that might have provided the vacuum energy density is still unidentified, eventheoretically; it is sometimes called theinflatonbecause its sole purpose seems to be to haveproduced inflation. Despite these outstanding questions, it seems difficult to understand howthe horizon problem could be explained unless something like inflation occurred. Researchcontinues along all these lines of investigation.

1.5 The Edge of Time

Completely unknown is the beginning of time, the point in thebig bang known as the Planckepoch. During this time the universe was sufficiently dense and energetic that all the funda-mental forces, including gravity, were merged into one grand force. Quantum mechanics isgenerally associated only with the world of the very small, but during the Planck epoch theentire observable universe was tiny. Under such conditions, quantum mechanics and gravitymust merge into quantum gravity. Unfortunately, at this time physicists do not have a theoryof quantum gravity. So we can speculate on what such a theory might be like, and what itmight tell us.

The basis for quantum mechanics is the recognition that everything has a wavelike nature,even those things we normally consider particles. By the same token, those things that weusually consider waves (e.g., light) also have a particle nature. The evolution of quantumsystems is governed by the Schroedinger equation. However,the Schroedinger equation givesonly the evolution of the probabilities associated with a system. Interpreting what this meansis somewhat difficult, given our usual expectations regarding the nature of reality. Accordingto the Copenhagen interpretation of quantum mechanics, a system exists in a superposition of

1.5 The Edge of Time 11

states so long as it remains unobserved. An ”observation,” which it must be understood refersto any interaction that requires a variable to take a value and need not imply a conscious agent,collapses the wavefunction. The collapse of the wave function means that the system abruptlyceases to obey Schroedinger’s equation; what was previously probabilities becomes a knownquantity. The story of “Schroedinger’s cat” illustrates. Acat is locked in a box containing avial of poison, which will be broken by a quantum process suchas the decay of a radioactiveatom. This event has some probability, which may be calculated, of occurring during anyparticular time interval. While the cat is in the box, unobserved, we cannot know whether theevent has taken place or not and thus whether the cat is alive or dead. A strict application ofthe Copenhagen interpretation demands that the cat be neither alive nor dead, or perhaps bothalive and dead, in some superposition of states according tothe probability that the atom hasdecayed; in this view, the cat becomes alive or dead only whenthe experimenter opens the boxto investigate. Schroedinger’s cat illustrates difficulties with ”standard” quantum mechanicsas applied to complex systems such as living beings. Schroedinger himself meant this thoughtexperiment to show that quantum mechanics, a young science at the time, did not apply tosuch systems. Yet when we seek to apply quantum mechanics to cosmology, we know that itmust apply to the universe as a whole, and thus there is no suchthing as ”classical” behavior;everything is ultimately quantum.

When we ponder the Planck era, we are led to questions about thenature of space andtime themselves. What is it that provides the “arrow of time,”the perception that we moveinto the uncertain future and leave behind the unchangeablepast. The laws of physics aretime symmetric, meaning that they work the same whether timeruns forward or backward.(A substitution of -t for t everywhere gives the same equations). The one exception is thesecond law of thermodynamics which states that entropy mustincrease with time. This meansthat a complicated system will tend to evolve toward its mostprobable state, which is a stateof equilibrium (and maximum disorder). If the sense of the arrow of time comes from thesecond law this means that the big bang had to start in a state of low entropy (high order),and the arrow of time results from the universal evolution from this initial state to the finaldisordered state, be it big crunch or the heat death of the ever expanding universe. We are ledto ask whether the theory of quantum gravity explains why theinitial big bang was in a lowentropy state. Is quantum gravity a theory that is not time symmetric? Does the second lawof thermodynamics, an empirical relationship first discovered by engineers in the nineteenthcentury, tell us something about the most profound secrets of the universe?

Although we have no established theory of quantum gravity, some promising starts havebeen made. One of the most studied is string theory, in which reality at the Planck scaleof distance and time is described by the quantum oscillations of strings and loops. Stringtheories require that many more spatial dimensions exist than our familiar three. At leastten spatial dimensions exist in these theories, but only three are of cosmic scale; the restare compactified into ”coils” the size of the Planck distance. Thus the very early universemay have undergone a ”proto-inflation” in which three spatial dimensions grew into thosethat make up the observable universe. String theories are not yet well understood and manydetails remain to be worked out, but so far they can at least unify gravitation and the otherfundamental forces in a natural way. They are not the only candidates, however; anothertheory envisions that the exotic level of the Planck scale consists of a foam of quantized spaceand time. Perhaps someday we shall have a better understanding of the mysteries of the Planckscales. Such a discovery would be at least as momentous as general relativity itself.


1.6 Textbooks

• Cosmological Physics, by John Peacock (CUP 1998)

• Principles of Physical Cosmology, by P. J. E. Peebles (Princeton University Press,Princeton, 1993)

• The Early Universe, by Kolb and Turner (Addison-Wesley, New York, 1990)

• The Early Universe, by G. Borner (Springer-Verlag 2003)

• Modern Cosmology, by S. Dodelson (Academic Press 2003)

• An Introduction to Modern Cosmology, by A. Liddle (John Wiley 2003)

• Physical Foundations of Cosmology, by V. Mukhanov (CUP Nov. 2005)

1.7 Units

Physical Units: The speed of light isc = 299 792 458 m/s, one light second is therefore299,792 km. Velocity is usually taken in dimensionless units, i.e. in units of the speed oflight. With the Boltzmann constantkB = 1.3805 × 10−23 J/K one can convert temperatureinto energy units

1 eV = 11, 600K = 1.78 × 10−36kg = 1.60 × 10−19 J . (1.1)

Astronomical Units: The natural unit of mass is the solar massM⊙ = 1.99 × 1030 kg,and the common unit of length is one parsec, 1 pc = 3.26 light years =3.09 × 1016 m. Morecommon in cosmology is 1 Mpc =106 pc, which is a typical distance between galaxies. Wealso use 1 Gpc = 1000 Mpc for the scale of the Universe.

The Hubble constantH0 is usually given in units of 100 km/s/Mpc in the form ofH0 =100h km/s/Mpc. Since the Hubble constant is a velocity per distance, the inverse of theHubble constant defines then theHubble time tH = 1/H0, which is equal to 13.97 billionyears forH0 = 72. Similarly,RH = c/H0 defines a scale of the Universe, which is called theHubble radius. RH = 4300 Mpc for the standard value of the Hubble constant.

2 The Observable Universe

The observable universe is the space around us bounded by theevent horizon – the distance towhich light can have traveled since the universe originated. This space is huge but finite witha radius of 4200 Mpc. There are definite total numbers of everything in this volume: about1011 galaxies,1021 stars,1078 atoms,1088 photons. There is a hierarchy of structure: Every-thing is composed of smaller things and is a part of somethinglarger as shown in Figure 2.1.The character of structures with different scale changes according to the interplay of variousphysical forces. Quantum phenomena control the small scales, while gravity dominates onlarge scales, and both come into play at the beginning of the universe.

Note that according toinflationary cosmology, the entire universe is much bigger thanthe observable one, and the confine of observable universe depends on the location. Observersliving in the Andromeda galaxy and beyond have their own observable universes that aredifferent from but overlap with ours.

In a certain sense, we live in the centre of the universe that we observe, somewhat incontradiction to the Copernican principle, which says thatthe Universe is more or less uniformand it has no distinguished centre. This is simply because light does not travel infinitely fastand we must make observations of the past. As we look further and further away, we seethings from epochs (times) closer and closer to the limit of time zero in the Big bang model.Because light travels at the same speed in any direction towards us, we live at the geometricalcentre of our own observable universe.

In 1922,Alexander Friedmann predicted the Big Bang cosmology, which portrays theuniverse as expanding space from a point where the matter-energy density was extremely high.The expansion can be visualized by a two dimensional analogy. As the balloon expands, all thepoints on the surface recede from each other, and the wavelength on the surface is stretched. Itis similar to the shift to longer wavelength when the source and receiver are moving away fromeach other. This phenomenon is called red shift of the spectrum because in visible light theshift to longer wavelength is toward the red colour. It playsa prominent role in discovering thecosmic expansion through the detection of the spectral lineshift from distant galaxies. Notethat contrary to the balloon analogy, it is the space itself that is expanding. It needs neither acenter to expand away from nor empty space on the ouside to expand into.

There are four key observational facts which require the Universe to be described in termsof a relativistic model:

• the expansion of the Universe (also called Hubble–expansion);

• cosmic microwave radiation (CMBR) which is a relic from the recombination era and,

• the distribution of galaxies in the Universe which results from the gravitational action ofsmall perturbations rooted in the very early epoch of the Universe;

14 2 The Observable Universe

Figure 2.1: The observable Universe as seen from the Earth. This extends in a waythe Aris-totelian cosmos of Fig. 1.1. More realistically, one should center the observable Universe on thesolar system barycenter. The farther out we look, the farhter back in time we see. Light takes 50million years to arrive from M 87. The limit of our view is the time when the Universe emergedfrom a state of hot plasma and became transparent, some 300,000 years after the big bang.

• the presence of dark matter whose gravity dominates the motion in galactic halos andgalaxy clusters. Though the nature of dark matter is still completely obscure, its existencecan no longer be avoided.

When Einstein developed his theory of gravity in the general theory of relativity, hethought he ran into the same problem that Newton did: his equations said that the universeshould be either expanding or collapsing, yet he assumed that the universe was static. Hisoriginal solution contained a constant term, called the cosmological constant, which cancelledthe effects of gravity on very large scales, and led to a static universe. After Hubble discov-ered that the universe was expanding, Einstein called the cosmological constant hisgreatestblunder.

At around the same time, larger telescopes were being built that were able to accurately

2.1 The Universe is Expanding and Accelerating 15

Figure 2.2: The observable Universe of Superclusters. Superclusters are large groupings ofsmaller galaxy groups and clusters, and are among the largest structures of the cosmos. The ex-istence of superclusters indicates that the galaxies in our Universe are not uniformly distributed;most of them are grouped together in groups and clusters, with groups containing up to 50galaxies and clusters up to several thousand. Those groups and clusters and additional isolatedgalaxies in turn form even larger structures called superclusters. The typical distance betweensuperclusters is about 100 Mpc. No clusters of superclusters are known, but the existence ofstructures larger than superclusters is debated (Filaments).

measure the spectra, or the intensity of light as a function of wavelength, of faint objects. Us-ing these new data, astronomers tried to understand the plethora of faint, nebulous objects theywere observing. Between 1912 and 1922, astronomer Vesto Slipher at the Lowell Observatoryin Arizona discovered that the spectra of light from many of these objects was systematicallyshifted to longer wavelengths, or redshifted. A short time later, other astronomers showed thatthese nebulous objects were distant galaxies.

2.1 The Universe is Expanding and Accelerating

Until 1929, the Universe of galaxies was thought to be static. Even Einstein originally searchedfor a static solution of his euqations. For that reason, he had to introduce the cosmologicalconstantΛ which is nowadays fundamental for the understanding of the dynamics of the Uni-


verse.In 1929 Edwin Hubble, working at the Carnegie Observatoriesin Pasadena, California,

measured the redshifts of a number of distant galaxies. He also measured their relative dis-tances by measuring the apparent brightness of a class of variable stars called Cepheids ineach galaxy. When he plotted redshift against relative distance, he found that the redshiftz ofdistant galaxies increased as a linear function of their distanced

cz = H0 d . (2.1)

The only explanation for this observation is that the Universe of galaxies is expanding.

Figure 2.3: The mean apparent magnitude of Cepheids in a local galaxy as a functionof the period.[Data: Labhardt]

For the past 70 years astronomers have sought a precise measurement of Hubble’s constantH0, ever since astronomer Edwin Hubble realized that galaxieswere rushing away from eachother at a rate proportional to their distance, i.e. the farther away, the faster the recession. Formany years, right up until the launch of the Hubble telescope– the range of measured valuesfor the expansion rate was from 50 to 100 kilometers per second per megaparsec. Localgalaxies, i.e. galaxies between us and the Virgo and Hercules cluster, are not very suitable totest the assumption of expansion (see Fig. 2.4). Since Cepheids can only be used as standardcandles upto distances of the Virgo galaxies, Supernovae oftype Ia turned out to be much moresuitable. They become as bright as normal galaxies and are therefore visible upto cosmologicaldistances (Fig. 2.5). The main goal of theHST Key Project on the Extragalactic DistanceScalewas to determine the Hubble Constant,H0, to an accuracy of +/- 10%. This goal hasbeen achieved by the systematic observations of Cepheid variable stars in several galaxies


Figure 2.4: The recession velocities as a function of distances determined by Cepeheid distances forlocal galaxies. The observed redshift of the galaxies is corrected by means of the motion against thecosmic microwave background. [HST Key Project]

Figure 2.5: The distance module as a function of redshiftz, as tested with SN Ia.

using the Hubble Space Telescope. The Key Project team currently had 28 members [5].Cepheid distances lie at the heart of the HST Key Project on the Extragalactic Distance Scale.The Key Project has been designed to use Cepheid variables todetermine primary distances


to a representative sample of galaxies in the field, in small groups, and in major clusters. Thegalaxies were chosen so that each of the secondary distance indicators with measured highinternal precisions can be accurately calibrated in zero point, and then intercompared on anabsolute basis. The Cepheid distances can then be used for secondary calibrations and appliedto independent galaxy samples at cosmologically significant distances. Cepheid distances tothe Virgo and Fornax clusters provide a consistency check ofthe secondary calibrations.

Type Ia Supernova are the explosions of white dwarfs. This isa pinnacle that only a fewstars like our sun are able to achieve. Unfortunately we are not sure exactly how these eventsoccur. We think they are related to white dwarf stars which are near another star in a binarysystem. Chandrasekhar, as part of his Nobel Prize in Physicsdemonstrated that white dwarfstars, if they become more massive than 1.4 times our sun can explode. They do this becauseat this point, the forces (electrons repelling electrons) which keep the star from collapsingagainst the force of gravity, lose their battle, and the white dwarf begins to collapse. As youmay recall, white dwarf stars are not made of Iron, (instead they are composed of Carbonand Oxygen) and there is still substantial amounts of nuclear energy left in their atoms. Asthe white dwarf begins to collapse against the weight of gravity, this material is ignited, andrather than collapsing further, this nuclear blast wave consumes the star in a second, creatingan explosion 10 to 100 times brighter than a Type II supernova.

The idea to measure the Universe with Supernovae is not new, it has long been contem-plated, but it is only in the past decade that it has become feasible. The first distant SN Iawas discovered in 1988 by a Danish team, but it wasn’t until 1994 that they were discoveredin large numbers. Since 1995 two teams have been discoveringthese objects: the High-Z SNSearch, and the Supernova Cosmology Project.

For redshiftsz < 0.1, curvature effects are not important and the Hubble relation cz =H0d can be used in its original version for the distance module (Fig. 2.5)

m − M = 5 log(d/pc) − 5 = 5 log(z) − 5 + 5 log(c/H0) . (2.2)

SN Ia are observable to medium redshiftsz < 1.5 giving Hubble diagrams which demonstratethat the classical Hubble relation fails at least beyound redshift 0.2 (Fig. 2.6). The deviationsfrom the straight line are the effects of the expansion of theUniverse. They can only beunderstood within a relativistic cosmological model.

The Universe is Accelerating

Recent studies of Type Ia supernovas, including measurements by the Supernova CosmologyGroup led by Saul Perlmutter at LBNL and the High Z-SN Search team lead by Brian P.Schmidt, have produced significant evidence that over cosmological distances they appeardimmer than would be expected if the universe’s rate of expansion was constant or slowingdown. This was the first direct experimental evidence for an accelerating universe potentiallydriven by a positive Cosmological Constant. However, only about 80 supernovas accumulatedover several years have been studied and other explanationshave not been completely ruledout.

A space mission is now being considered that would increase the discovery rate for suchsupernovas to about 2000 per year. Discovery of so many more supernova would help elimi-nate possible alternative explanations, give experimental measurements of several other cos-mological parameters, and put strong constraints on possible cosmological models. The satel-lite calledSNAP (Supernova / Acceleration Probe) would be a space based telescope with


Figure 2.6: The redshift of SN Ia as a function of the luminosity distance derived from the distancemodule for a set of recent data. At redshiftsz > 0.15, the Hubble relation fails. The lines correspondto luminosity distances of various cosmological models which will be discussed later on. [Data: Tonry2003]

a one square degree field of view with 1 billion pixels. Such a satellite would also comple-ment the results of proposed experiments to improve measurements of the cosmic microwavebackground.

In addition to the supernova discover program itself, LBNL’s Supernova Cosmology Grouphas unique expertise in large charge-coupled (CCD) detectors. While smaller CCD’s are nowcommon, LBNL has developed techniques to construct the large mosaics required for SNAPby stitching together several hundred of the largest ones. The group has also devised a wayto manufacture the detectors at significantly reduced cost.Technically, the CCD’s have highresistivity with excellent quantum efficiency at 1 micron, which is the same as the emissionfrom distant Type Ia supernova and where conventional CCD’shave very low sensitivity.

The SNAP baseline science objective is to obtain a high statistics calibrated dataset of TypeIa supernovae to redshifts of 1.7 with excellent control over systematic errors. The statisticalsample is to be 2 orders of magnitude greater than the currentpublished set of 42 supernovae,and is to extend much farther in distance and time. From this dataset we expect to obtaina 2% measurement of the mass density of the universe, a 5% measurement of the vacuumenergy density, a 5% measurement of the curvature, and a 5% measurement of the equationof state of the ”dark energy” driving the acceleration of theuniverse. Systematic studieswill include a measurement of the ”reddening” of spectra from ”ordinary dust” at redshiftsup to 1.7, and studying potential ”grey dust” sources. Usingtype Ia supernovae as standardcandles will require measurement of the key luminosity indicators: the light curve peak andwidth. The redshift of the host galaxy of the supernova needsto be measured, supernova type


Figure 2.7: The Supernova/Acceleration Probe (SNAP) satellite observatory is capable of mea-suring thousands of distant supernovae and mapping hundreds to thousands of square degrees ofthe sky for gravitational lensing each year. The results will include a detailed expansion historyof the universe over the last 10 billion years, determination of its spatial curvature to providea fundamental test of inflation, precise measures of the amounts of the key constituents of theuniverse,ΩM andΩΛ, and the behavior of the dark energy and its evolution over time. [SNAPHomepage]

identfied, and spectral features studied. Effects correlated with host galaxy morphology andthe position of the supernova in the host galaxy will also be studied. These properties mayindicate diferences in stellar population from which the supernova came and therefore canbe used to test whether the intrinsic brightness of the supernova changes systematically withredshift.

TheSupernovae Legacy Survey[1] is comprised of two components: an imaging surveyto detect SNe and monitor their light-curves, and a spectroscopic program to confirm thenature of the candidates and to measure the redshift. The imaging is taken as part of the deepCFHT Legacy Survey using the one sqare degree imager MegaCam. This program has beenallocated 474 nights over 5 years. Follow–up spectroscopy uses various 8–10 meter classtelescopes.

Distance measurements to 71 high redshift type Ia supernovae discovered during the firstyear of the 5-year Supernova Legacy Survey (SNLS) are presented in [1]. These events weredetected and their multi-color light-curves measured using the MegaPrime/MegaCam instru-ment at the Canada-France-Hawaii Telescope (CFHT), by repeatedly imaging four one-squaredegree fields in four bands. Follow-up spectroscopy was performed at the VLT, Gemini andKeck telescopes to confirm the nature of the supernovae and tomeasure their redshift. Withthis data set, they have built a Hubble diagram extending toz = 1 (shown in Fig. 2.8), with all


SN Redshift0.2 0.4 0.6 0.8 1

Bµ

34

36

38

40

42

44

)=(0.26,0.74)ΛΩ,mΩ(

)=(1.00,0.00)ΛΩ,mΩ(

SNLS 1st Year

SN Redshift0.2 0.4 0.6 0.8 1

) 0 H

-1 c

L (

d10

- 5

log

Bµ

-1

-0.5

0

0.5

1

Figure 2.8: Hubble diagram of SNe from the SN Legacy program [1]. The bottom plotshowsthe residuals for the best fit to a flat Lambda cosmology.

distance measurements involving at least two bands. Essentially, µB = mB − M is the dis-tance modulus for an absolute magnitude of SNIaM = −19.31± 0.03 + 5 log h70, correctedfor stretching and color. Systematic uncertainties are evaluated making use of the multi-bandphotometry obtained at CFHT. Cosmological fits to this first year SNLS Hubble diagram givethe following results:ΩM = 0.263±0.042 (stat)±0.032 (sys) for a flat LambdaCDM model; andw = −1.023±0.090 (stat)± 0.054 (sys) for a flat cosmology with constant equation of statew, when com-bined with the constraint from the recent Sloan Digital Sky Survey measurement of baryonacoustic oscillations. As a result,the Universe is definitely accelerating, and not decelerat-ing.


2.2 Cosmic Background Radiation as a Relic of the EarlyUniverse

In 1963, Arno Penzias and Robert Wilson, two scientists in Holmdale, New Jersey, were work-ing on a satellite designed to measure microwaves. When they tested the satellite’s antenna,they found mysterious microwaves coming equally from all directions. At first, they thoughtsomething was wrong with the antenna. But after checking andrechecking, they realized thatthey had discovered something real [11]. What they discovered was the radiation predictedyears earlier by Gamow [6], Alpher and Herman [1]. The radiation that Penzias and Wilsondiscovered, called the Cosmic Microwave Background Radiation (CMBR), convinced mostastronomers that the big bang theory was correct. For discovering the Cosmic MicrowaveBackground Radiation, Penzias and Wilson were awarded the 1978 Nobel Prize in Physics.

The significance of the discovery of the CMBR is that it was predicted by the basic phys-ical theory of the expanding universe, first laid down decades before, by Georges Lematre.While the cosmology wars of the 1940’s and 1950’s pitted several competing models againstone another, only the Big Bang had predcited the necessity ofthe CMBR as a consequence ofthe theory. So it’s only natural that its discovery gave support to that theory. The supportersof alternative cosmologies tried to bring such a backgroundradiation out of their models afterthe fact, but always with unsatisfying, arbitrary tricks. Only the Big Bang actually requiresthat there be a CMBR. The existence of the CMBR was one of several strong reasons thatdeveloped during the 1960’s & 1970’s, which drove Big Bang cosmology into its premiereposition amongst scientists.

Subsequent observations confirmed, as well as it could be confirmed by limited, groundbased observations, that the CMBR discovered by Penzias & Wilson indeed had the requiredthermal spectrum, and was indeed isotropic. These two factsalone are strong indicators thatthe CMBR is the predicted relic of Lematre’s primeval atom, and argues against alternativecosmologies.

After Penzias and Wilson found the Cosmic Microwave Background Radiation, astro-physicists began to study whether they could use its properties to study what the universe waslike long ago. According to Big Bang theory, the radiation contained information on howmatter was distributed over ten billion years ago, when the universe was only 500,000 yearsold. At that time, stars and galaxies had not yet formed. The Universe consisted of a hot soupof electrons and atomic nuclei. These particles constantlycollided with the photons that madeup the background radiation, which then had a temperature ofabout 3000 K.

Soon after, the Universe expanded enough, and thus the background radiation cooledenough, so that the electrons could combine with the nuclei to form atoms. Because atomswere electrically neutral, the photons of the background radiation no longer collided withthem.

When the first atoms formed, the universe had slight variations in density, which grew intothe density variations we see today - galaxies and clusters.These density variations shouldhave led to slight variations in the temperature of the background radiation, and these varia-tions should still be detectable today. Scientists realized that they had an exciting possibility:by measuring the temperature variations of the Cosmic Microwave Background Radiationover different regions of the sky, they would have a direct measurement of the density vari-ations in the early universe, over 10 billion years ago. In 1990, the a satellite called theCosmic Microwave Background Explorer (COBE) measured background radiation tempera-tures over the whole sky (Fig. 2.9). COBE found variations that amounted to only about 5

2.2 Cosmic Background Radiation as a Relic of the Early Universe 23

Figure 2.9: The spectrum measured by COBE is a perfect Planckian distribution.

Figure 2.10: The motion of the Earth generates a dipole anisotropyδT/T ≃ 0.001 in the temperaturedistribution on the Sky as observed by COBE.

parts in 100,000, but revealed the density fluctuations in the early universe (Fig. 2.10).The initial density variations would be the seeds of structure that would grow over time to

become the galaxies, clusters of galaxies, and superclusters of galaxies observed today by theSloan Digital Sky Survey. With the Sloan data, along with data from COBE, astronomers willbe able to reconstruct the evolution of structure in the universe over the last 10 to 15 billionyears. With this information, we will have a deep understanding of the history of the universe,


Figure 2.11: Temperature fluctuations on the Sky as observed by WMAP. The typical angular separationof temperature spots is about one degree, the resolution of the instrument is 13 arcminutes, in contrast tothe 7 degree resolution for the COBE instrument. [Source: Tegmark]

Figure 2.12: Harmonic analysis on the sphere: Dipole anisotropy (l = 2) and anisotropy mitl = 16 demonstrate, how a complicated fluctuation spectrum can be decomposedinto a super-position of all possiblel–values.

which will be an almost unbelievable scientific and intellectual achievement.The cosmic microwave radiation is emitted by a sphere (the socalledLast Scattering Sur-

faceat redshiftz = 1070). We can therefore consider the temperature as a scalar function onthe sphere, which can be decomposed into spherical harmonics [9]

∆T (θ, φ) = T (θ, φ) − T0 =

lmax∑

l=0

m=+l∑

m=−l

alm Ylm(θ, φ) (2.3)

Ylm are the spherical harmonics (as used e.g. in Quantum mechanics). lmax follows from theresolution on the sky, which is 7 degrees for COBE, 10 arcminutes for MAXIMA–1, and 13arcminutes for WMAP. The functionsYlm form a complete basis on the sphere, i.e.

∫

4π

Y ∗

lm(θ, φ)YLM (θ, φ) dΩ = δlL δmM . (2.4)


Figure 2.13: The power spectrum of the CMBR temperature fluctuations as a function ofmul-tipolesl, i.e. as a function of angular separation before WMAP. The angular scale is given atthe upper axis of the figure. COBE had an angular resolution of 7 degrees, ballon experimentssuch as Boomerang and Maxima could determine fluctuations uptol ≃ 800, though with con-siderable error bars due to the limited sky coverage. The solid line gives the impression offluctuations expected in a CDM model. Here, the greatest fluctuations are expected on scales ofabout one degree. On much smaller scales, fluctuations decay very rapidly.

From this orthogonality relation we obtain the representation of the coefficientsa by meansof N observed values, i.e. the temperature inN discrete directions on the sphere. The samplefunction values∆Tp can then be used to estimate the parametersalm

alm =

N−1∑

p=0

∆T (θp, φp)Y ∗

lm(θp, φp) . (2.5)


The zeroeth of the spherical harmonicsYlm divide up the sphere into invidual cells, withan extension near the equatorial plane of∆θ = π/l. This is the meaning of the multipole

Figure 2.14: The power spectrum measured by WMAP. This figure summarizes the experi-mental results as of March 2003. The existence of the resonance peakat l ≃ 200 has beenconfirmed, the secondary maximum is barely visible. [Data: Tegmark, WMAP]

of orderl. The power represented by a multipole of orderl is obtained by averaging over allm–values

Cl ≡1

2l + 1

m=+l∑

m=−l

alma∗

lm . (2.6)

The parametersCl represent indeed a complete statistical representation ofGaussian temper-ature fluctuations. The value ofl = 200 just corresponds to an angular separation of aboutone degree on the sky. COBE could only resolve scales uptol = 20, balloon measurementshad a bigger resolution, but they could not resolve structure beyondl = 750 (Fig. 2.14).

WMAP (Wilkinson Microwave Anisotropy Probe) now has confirmed these earlier mea-surements with an unprecented accuracy and a resolution of 13 arcminutes [3] (Fig. 2.14).


Figure 2.15: An image of the intensity and polarization of the cosmic microwave backgroundradiation made with the Degree Angular Scale Interferometer (DASI) telescope. The small tem-perature variations of the cosmic microwave background are shown in false color, with yellowhot and red cold. The polarization at each spot in the image is shown by a black line. Thelength of the line shows the strength of the polarization and the orientation of theline indicatesthe direction in which the radiation is polarized. The size of the white spot in the lower leftcorner approximates the angular resolution of the DASI polarization observations. [Data: DASIcollaboration]

The skymap data products derived from the WMAP observations have 45 times the sensitiv-ity and 33 time the angular resolution of the COBE DMR mission. In particular, the peakat l = 200 in the power spectrum is extremely important for the reduction of cosmologicalparameters. But in addition, the secondary peak is also visible, however with some biggeruncertainty. The data brings into high resolution the seedsthat generated the cosmic structurewe see today. These patterns are tiny temperature differences within an extraordinarily evenlydispersed microwave light bathing the Universe, which now averages a frigid 2.73 degreesabove absolute zero temperature. WMAP resolves slight temperature fluctuations, which varyby only millionths of a degree [8].

Polarisation of the CMBR: The polarization of the cosmic microwave background (Fig.2.15) was produced by the scattering of cosmic light when it last interacted with matter, nearly14 billion years ago. If no polarization had been found, astrophysicists would have to reject


all their interpretations of the remarkable data they have compiled in recent years.

Figure 2.16: Triangulation of spherical surfaces with the HealPix procedure. The surface of asphere is primarily divided up into 12 domains of equal area. These segments are then dividedup in ahierarchical procedure. top right the topography of the Earth is shown with 3,145,728pixels (corresponding to a resolution of 7 arc minutes) and left top the anisotropy of the CMBRis given for 12,582,912 pixels (resolution of 3.4 arc minutes).

Triangulation of Spherical Surfaces: Measurements of the temperatureT (~n) in a givendirection~n on the Sky means that we have to treat a scalar function on the sphere (Fig. 2.16).This requires an efficient triangulation of the 2–sphere in the sense that all pixels have aboutthe same area. A simple discretisation of the anglesφ and θ produces pixel areas which


become smaller and smaller towards the polar regions.Numerically efficient procedures should have the followingproperties:

• Equal area for all pixels.

• Pixels are distributed on lines of constant latitude. This property is essential for all har-monic analysis applications involving spherical harmonics. Due to the iso-latitude distri-bution of sampling points the speed of computation of integrals over individual sphericalharmonics scales as≃

√N with the total number of pixels, as opposed to theN–scaling

for the non-iso-latitude sampling distributions (examples of which are the Quadrilateril-ized Spherical Cube used for the NASA’s COBE data, and any distribution based on thesymmetries of the icosahedron [13]).

• The sphere is hierarchically tessellated into curvilinearquadrilaterals. The lowest resolu-tion partition is comprised of 12 base pixels. Resolution ofthe tessellation increases bydivision of each pixel into four new ones. The Figure 2.16 illustrates (clock-wise fromupper-left to bottom-left) the resolution increase by three steps from the base level, i.e.the sphere is partitioned, respectively, into 12, 48, 192, and 768 pixels.

The HealPix algorithm (Hierarchical Equal Area isoLatitude Pixelisation) satisfies these re-quirements (Fig. 2.16) [7]. Each healpix sphere contains12 × N2 pixel. Typical values areN = 256, 512, 1024. This triangulation of the spherical surface has many application in as-trophysics, in particular it is very useful for the numerical solution of the radiative transferequation.

Future CMBR Missions: Planck (formerly COBRAS/SAMBA) was officially adopted asthe third Medium-Sized Mission of ESA’s Horizon 2000 scientific program in November 1996with a scheduled launch in late 2005, now postponed to 2007. Planck currently consists of 4frequency channels (30-100 GHz) that use HEMT amplifiers running at 20K and 6 channels(100-857 GHz) that use bolometer detectors in a 0.1 K dewar. The single 1.3m x 1.5m primaryreflector produces a 0.1 degree beam at 200 GHz and above. The sensitivity of PLANCK isalmost an order of magnitude higher than WMAP.

Summary

• WMAP provided higher accuracy measurements of many cosmological parameters thanhad been available from previous instruments.

• The Hubble constant is71 ± 4 km/s/Mpc.

• The universe is composed of

1. 4% ordinary matter,

2. 23% of an unknown type of dark matter (DM),

3. and 73% of a mysterious dark energy (DE).

4. TheAge of the Universeis (13.7 ± 0.2) billion years.

• This is a confirmation of the so-calledconcordance Lambda-CDM model.

• Besides this, WMAP measured many more parameters, which willbe discussed later on.


2.3 Galaxies are not Uniformly Distributed

Matter in the universe is not distributed randomly. Galaxies, quasars, and intergalactic gasoutline a pattern that has been compared to soap bubbles - large voids surrounded by thinwalls of galaxies, with dense galactic clusters where wallsintersect. One of the primary goals

Figure 2.17: The galaxy distribution in the CfA redshift survey. Each dot representsa galaxy. Redshiftruns along the radius of the circle, we are located in the center of the circle.The radius of the circleamounts to 15,000 km/s, the Coma cluster and the Great Wall are at about8,000 km/s. The center is partof the Virgo cluster.

of modern surveys (CfA, Las Campanas, 2dF, SDSS, 6dF) is to map this structure in greatdetail, out to large distances. There are many theories about how the universe evolved, andthe theories predict different large-scale structrues forthe universe. The SDSS’s map may tellus which theories are right - or whether we will have to come upwith entirely new ideas.Galaxies are usually found near each other, in galactic clusters. The distribution of these clus-ters, and how this distribution evolves with time, are important tests of cosmological models:for instance, different cosmological models predict different numbers of galaxy clusters at dif-ferent redshifts. Additionally, not only are galaxies clustered, but the clusters themselves are

2.3 Galaxies are not Uniformly Distributed 31

Figure 2.18: The galaxy distribution in the 2dF galaxy survey for a strip of 3 degrees wide in declination.Galaxies are detected out to redshifts of 0.2, roughly corresponding toa distance of one Gigaparsec.

Figure 2.19: The redshift distribution in the 2dF galaxy redshift survey. Beyond redshift 0.15, onlythe most luminous galaxies can be detected. The solid line is the theoretical expectation based on theSchechter form for the galaxy luminosity function.

clustered! The degree to which both galaxies and clusters tend to group together is also a testof different theories. By studying the masses, distributions, and evolution of galaxy clusters,


we can learn something about the formation of mass in the universe: a fundamental goal ofcosmology.

Superclusters are simply clusters of galaxy clusters. Whereas clusters are typically foundin the filaments and walls of the universe’s ”soap bubble,” superclusters are at the intersectionsof the walls. Superclusters are the largest known structures in the universe, with some aslarge as 200,000,000 light-years! However, because these structures are very rare, only afew are known. The most famous superclusters are nearby, including the Great Wall and thePerseus-Pisces supercluster. There has been recent evidence for superclusters at redshifts ofabout 1, which places important constraints on structure formation and cosmological models.Additionally, the M/L ratios of superclusters are similar to those of clusters. This discoveryimplies that the mysterious dark matter cannot contribute more to the mass of the universethan it contributes to the mass of clusters.

Figure 2.20: A comparison between CfA and SDSS galaxy surveys. A kind of great wall is now visiblein the SDSS map, however at much larger scales.

Future Surveys: The VIRMOS-VLT Deep Survey (VVDS) is a breakthrough spectroscopicsurvey which will provide a complete picture of galaxy and structure formation over a very

2.4 Gravity is Dominated by Dark Matter 33

broad redshift range (0 < z < 5) over sixteen square degrees of the sky in four separatefields. This ambitious survey is possible thanks to the impressive multiplex gains of the VIR-MOS instruments (VIMOS and NIRMOS) built by the Franco-Italian VIRMOS consortiumfor ESO-VLT. The survey will be carried out over the consortium’s 120 nights of guaranteedtime, and will contain in total a sample of 150,000 redshifts. This unique database will enableus to trace back the evolution of galaxies, active galactic nuclei and clusters to epochs wherethe universe was a fraction (about 20 per cent) of its currentage. The VVDS will be compa-rable in size to the largest redshift surveys currently underway, but will probe to much higherredshifts. It will provide an unparalleled description of how structures and galaxy populationsevolved in the universe from high redshift to the present day.

The distribution of galaxies at higher redshifts (z ≥ 1) is the topic of deep surveys (HSTdeep fields, FORS deep fields, Subaru deep fields etc). The spatial distribution at these dis-tances is the topic of the next decade in Astronomy.

Summary

• The observations (d2F and SDSS) demonstrate for the first time that the Universe ofgalaxies is homogeneous on scales beyond 100 Mpc, but definitely not within 100 Mpc.

• Galaxies are assembled into groups, clusters and superclusters.

• The space between clusters is mainly empty of galaxies. These regions are called voids.

2.4 Gravity is Dominated by Dark Matter

The nature and identity of the dark matter in the Universe is one of the most challengingproblems facing modern cosmology. The problem has been for the first time formulated byZwicky. Given the distribution of galaxies with total luminosityL, Φ(L) dL, one can computethe mean luminosity density of galaxies

L =

∫

Φ(L) dL (2.7)

which is then determined by

L ≃ (2 ± 0.2) × 108 h0 L⊙ Mpc−3 . (2.8)

In the absence of a cosmological constant, one can define acritical mass density

ρc ≡ 3H20/8πG = 1.88 × 10−29 h2 g cm−3 . (2.9)

With this, we can define a critical mass–to–light ratio

(M/L)c = ρc/L ≃ 1390h (M⊙/L⊙) . (2.10)

For the standard valueH0 = 70 we find then(M/L)c ≃ 1000 (M⊙/L⊙). With this we obtaina value for thedensity parameter

Ωm =ρ

ρc=

(M/L)

(M/L)c. (2.11)


It turns out that mass–to–light ratios are strongly dependent on distance scale over which theyare determined: In the solar neighborhoodM/L ≃ 1, yielding values ofΩm ≃ 0.001. Inthe central parts of galaxies one findsM/L ≃ (10 − 20)h so thatΩm ≃ 0.01 for galaxies.On larger scales (i.e. for binaries and small groups of galaxies),M/L ≃ (60 − 180)h withΩm ≃ 0.1. finally, on the scale of galaxy clusters,M/L still increases and may be as large as(200−500)h giving Ωm ≃ 0.3−0.4. Thus on the scales of clusters, dark matter is inevitable.

Dark Matter in Clusters: Because galaxy clusters can be very massive (up to1014 timesthe mass of the Sun), their gravity is strong enough to hold onto extremely hot gas, withtemperatures of millions of degrees. This gas emits radiation at X-ray wavelengths, whichcan be observed by X-ray satellites like ROSAT, Chandra, andXMM. These satellites haveshown that a large fraction of clusters have structure and complicated internal motions, whichindicate that they are still evolving. Also, satellite observations have shown that the X-rayemitting gas comprises the largest fraction of the visible mass in clusters, greater than thesum of all the galaxies. This is a very interesting result - remember that galaxy clusters werediscovered as overdensities of galaxies, and now we know that galaxies are but a small partof the total mass of clusters. Some astronomers have even suggested clusters without galaxiesmay also exist – just huge clumps of gas.

Figure 2.21: The X–ray gas in the cluster Abell 2104 at redshift 0.15. The blue dots are X–ray photonsdetected by Chandra, the red dots optical galaxies in the cluster. The extension of the X–ray emittinggas is about 200 kpc. A few of the galaxies in the cluster are also emitting X–rays (these are Quasars).[Source: Chandra Homepage]


The cluster Abell 2104 is one of the lowest redshift clusters(z = 0.153) known to have agravitational lensing arc. Detailed analysis of the cluster properties such as the gravitationalpotential using the X-ray data from ROSAT (HRI) and ASCA, as well as optical imagingand spectroscopic data from the CFHT has been given. The cluster is highly luminous in theX-ray with a bolometric luminosity ofLX ≃ 3 × 1045 ergs/s and a high gas temperature ofkeV. The X-ray emission extending out to at least a radius of 1.46 Mpc, displays significantsubstructure. The total mass deduced from the X-ray data under the assumption of hydrostaticequilibrium and isothermal gas, is found to beM(r < 1.46Mpc) ≃ 8 × 1014 M⊙. The gasfraction within a radius of 1.46 Mpc is 5–10%. The cluster galaxy velocity distribution has adispersion of(1200 ± 200) km/s with no obvious evidence for substructure.

Direct evidence for dark matter in clusters follows from theobservation of the distributionof the hot baryonic gas. The hot gas in a cluster is held in the cluster primarily by the gravityof the dark matter, so the distribution of the hot gas is determined by that of the dark matter.By precisely measuring the distribution of X-rays from the hot gas, the astronomers were ableto make the best measurement yet of the distribution of dark matter in the inner region of agalaxy cluster. Under the assumption of hydrostatic equilibrium and spherical symmetry, thecluster total mass is directly related to the intracluster gas properties as

Mtot(r) = −rkBTg(r)

µmpG

(

d lnne

d ln r+

d lnTg

d ln r

)

. (2.12)

The gas density usually follows a standard law

ne(r) = n0

(

1 + (r/r0)2)−3β/2

. (2.13)

Hence the gravitational potential for a constant gas temperature is given by

Φ(r) − Φ0 =3σ2

0

2ln[1 + (r/r0)

2] (2.14)

whereσ20 = βkBTg/µmp and the total mass is given by

Mtot(r) =3σ2

0r0

G

(r/r0)3

1 + (r/r0)2. (2.15)

If we consider galaxies as test particles in the cluster potential well, then Jean’s equationfor a collisionless, steady state, non-rotating spherically symmetric system gives

Mtot(r) = −rσ2r(r)

G

(

d lnngal

d ln r+

d lnσ2r

d ln r+ 2βt

)

. (2.16)

σr is the 1D velocity dispersion of the galaxies in the cluster.By measuring the density profileand the profile of the velocity dispersion, a direct mass determination can be obtained.

Gravity can bend light, allowing huge clusters of galaxies to act as telescopes. Almost allof the bright objects in the above Hubble Space Telescope image are galaxies in the clusterknown as Abell 2218 (Fig. 2.23). The cluster is so massive andso compact that its gravitybends and focuses the light from galaxies that lie behind it.As a result, multiple images ofthese background galaxies are distorted into long faint arcs – a simple lensing effect analogousto viewing distant street lamps through a glass of wine. The cluster of galaxies Abell 2218 isitself about three billion light-years away in the northernconstellation Draco.


Figure 2.22: Mass profiles derived from X–ray observations in the Hydra cluster onthe scale from afew kpc to 200 kpc. [Source: Chandra]

Dark Matter in Galaxies: Dark matter is also required to explain the flat rotation curvesof spiral galaxies. The halo of spiral galaxies must be populated by a dark component with adensity distribution ofρH ∝ 1/r2, providing a mass distribution from the Jean’s equation

M(r) ∝ V 2r/G . (2.17)

Matter distribution in the disk cannot explain the observedrotation curves.

Dark Matter Candidates: The nature of the dark matter predicted by inflation is a profoundand unresolved puzzle. We have two choices. Either the dark matter consists of ordinary,baryonic matter, or else it consists of some more exotic formof matter. The history of theuniverse during the first few minutes provides an interesting measure of the total amount ofbaryonic matter in the universe that may help resolve the puzzle.

For a significant clue to the composition of the dark matter, we look to the abundance ofthe heavier isotope of hydrogen, weighing twice the mass, called deuterium, created duringthe big bang. There is no alternative source for the extra deuterium other than the big bang,since stars destroy deuterium rather than produce it. By now, a considerable fraction of anyprimordial deuterium present at the birth of the galaxy would have been destroyed inside stars.This is confirmed by observation: interstellar clouds contain deuterium, as do gravitationally-powered stars that have not yet developed nuclear burning cores; on the other hand, evolvedstars have no deuterium.


Figure 2.23: HST image of Abell 2218 with luminous arcs. The dark matter in the cluster operates as agravitational lense. [Data: HST]

To estimate how much deuterium was created in the big bang, one has to factor in allthe deuterium that has since been destroyed. The percentageof the isotope destroyed sincethe big bang can be calculated if one knows the its rate of destruction, which can be foundby comparing the abundance of deuterated molecules in the atmosphere of Jupiter with theabundance of deuterium in interstellar clouds. One has to choose a value for the density ofbaryons that cannot exceed about a tenth of the critical density for closure of the universe,or too little primordial deuterium would have been synthesized. Conversely, the density ofbaryons cannot be too low, below 2 or 3 percent of the criticaldensity, or else one wouldoverproduce deuterium, compared to what is observed in the solar system. If the universe isat critical density, 90 percent of the matter in the universemust be nonbaryonic.

If, in a universe at critical density, most dark matter couldnot be baryonic, what otherforms could it take? Likely relics of the early universe are species of stable, weakly interactingparticles. One example is the neutrino, if it possesses a small mass. Normally, the neutrinois assumed to be practically massless, but a finite mass is notimplausible. There are so manyneutrinos left over from the big bang that a neutrino mass of even 50 eV, or one ten-thousandththe mass of an electron, would suffice to close the universe. Laboratory experiments areunderway in several countries to determine a definitive massfor the neutrino, but at presentthese experiments are inconclusive. The current upper limit on the electron neutrino mass,which is obtained from tritium decay experiments, is about 1eV. Other species of neutrinoscould have higher masses.

If the particles are very massive, possessing more mass than, say, a proton, a special namehas been coined: the WIMP, for weakly interacting, massive particle. Exotic WIMPs suchas the photino and neutralino have been postulated to exist in sufficient quantity to close theuniverse. The problem is that there is no guarantee that these particles do exist. Disregarding


this uncertainty, the big bang theory predicts their density today, if they do exist and are stableover the age of the universe.

The existence of the photino is predicted in a theory called supersymmetry. This theorydoubles the number of known particles by postulating the existence of partner ‘-ino’ particles.These particles are almost all short-lived, and exist in large numbers only in the very earlyuniverse, when the temperature was high enough to exceed theenergy scale characteristic ofsupersymmetry, affectionately abbreviated to SUSY. As theuniverse cools, supersymmetryis broken. The relevant energy scale is not known from theory, but it must exceed 100 GeVto avoid conflict with particle experiments. In our low-energy universe today, the lightestsupersymmetric particle should still survive. It is expected to be the partner, in the sense ofhaving a complementary spin, of the photon, and is thereforeknown as a photino. Its mass isexpected to be 10 to 100 times that of the proton. The photino is uncharged and interacts veryweakly with matter.

There is strong evidence for SUSY from experiments at CERN that measure the strengthof the nuclear interactions, which increase with increasing energy. There is no guarantee,however, that the weak and strong nuclear force strengths will all converge to the same energy.That they do converge at very high energy is the thesis of grand unification of the fundamentalforces, whose breakdown in the very early universe gave riseto inflation. While this energy,some 10 to the 15 GeV, is very much higher than is directly accessible by experiment, the trendtowards convergence of the disparate forces is already apparent. Only if SUSY describes thehigh energy world do these three fundamental forces become indistinguishable at a uniqueenergy. Only therefore with SUSY could one construct a strong case for the inevitability ofgrand unification.

The most natural form for dark matter is matter that we know exists, namely baryons.The big bang explanation of the light element abundances requires the existence of baryonicdark matter. Although these same abundances imply that mostdark matter is nonbaryonic,the amount of dark baryonic matter is still most likely several times that in luminous baryonicmatter, or about 3 percent of the critical density for closing the universe. But where do welook for the baryonic dark matter? One’s first expectation might be that baryonic dark matterconsists of burnt-out stars in the galactic halo, yet other forms, such as planets and black holes,are also possible. Baryonic dark matter does exist: it is farmore uncertain whether there existsenough to solve any of the dark matter problems, that is to say, dark matter in galaxy halos,dark matter in galaxy clusters and superclusters, or dark matter in an amount suficient to closethe universe. It is most unlikely that baryonic dark matter can account for the closure density,as we will now see: for this, one must appeal to WIMPs, or some other weakly interactingparticle. However, baryonic dark matter is a serious candidate for dark matter at least in galaxyhalos, if not on larger scales. In acknowledgment of the rivalry between these two forms ofdark matter, the favored baryonic dark matter candidates have been dubbed MACHOs, formassive compact halo objects.

Among the possible astrophysical objects contained in the halo are the relics of stars, dimstars such as white dwarfs, neutron stars, or even black holes, as well as objects that havenever quite fulfilled themselves as stars because of their low mass. Because these objects areinvisible, or almost so, they are excellent candidates for dark matter. Moreover, MACHOs aremore natural candidates for the halo dark matter than WIMPs, because they are already knownto exist.

Two experiments reported in 1993 have found strong evidencefor the existence of MA-CHOs. The technique used is gravitational microlensing. Ifa MACHO in our galaxy’s halo


passes very close to the line of sight from earth to a distant star, the gravity of the otherwiseinvisible MACHO acts as a lens that bends the starlight. The star splits into multiple imagesthat are separated by a milliarc-second, far too small to observe from the ground. However,the background star temporarily brightens as the MACHO moves across the line of sight in thecourse of its orbit around the Milky Way halo. To overcome thelow probability of observinga microlensing event, the experiments were designed to monitor several million stars in theLarge Magellanic Cloud. Each star was observed hundreds of times over the course of a year.A preliminary analysis of the data, taken with both red and blue filters, revealed several eventsthat displayed the characteristic microlensing signatures. The event durations were between30 and 50 days.

The duration of the microlensing event directly measures the mass of the MACHO, al-though there is some uncertainty because of the unknown transverse velocity of the MACHOacross the line of sight. The event duration is simply the time for the MACHO to cross theeffective size of the gravitational lens, known as the Einstein ring radius. The radius of theEinstein ring is approximately equal to the geometric mean of the Schwarzschild radius of theMACHO and the distance to the MACHO. For a MACHO half-way to the Large MagellanicCloud, that distance is 55 kiloparsecs. The Einstein ring radius is about equal to 1 astronomi-cal unit, or the earth-sun distance. In order to be lensed, the MACHOs must be objects that aresmaller than the lens, so they must be smaller than an astronomical unit, roughly the radius ofa red giant star. The events detected are, to within a factor of a few, as the MACHO model ofdark halo matter predicts, and the event durations suggest atypical mass of around 0.1 solarmasses; however, there is at least a factor of 3 uncertainty in either direction.

The microlensing experiments have given robust and strong limits on the baryonic contentof the halo. Much more data from the LMC and SMC will be available soon, so we expectthe statistics to improve in the near future. The LMC events,if interpreted as due to halo mi-crolensing, allow a measurement of the baryonic contribution to the halo, which is around 20%for a standard halo. In this case, the most likely Macho contribution to the Milky Way halomass is about8 × 1010 M⊙, which is roughly the same as the disk contribution to the MilkyWay mass. However, the whole story has been made more complicated (and exciting) by themuch larger than expected number of bulge microlensing events. These events imply a newcomponent of the Galaxy, and until the nature of this new component is known, unambiguousconclusions concerning the LMC events will not be possible.For example, if the Milky Waydisk is much larger than usually considered, a much smaller total halo mass will be required,and so even an all-Macho halo might be allowed. Alternatively, the new Galactic componentwhich is giving rise to the bulge events, may also be giving rise to the LMC events, and theMacho content of the halo could be zero. Fortunately, much more data is forthcoming, andmany new ideas have been proposed. Microlensing is fast becoming a new probe of Galacticstructure, and, beside the original potential to discover or limit dark matter, may well producediscoveries such as extra–solar planetary systems.

Summary

• The existence of Dark Matter (DM) is unavoidable in modern cosmology.

• Dark Matter cannot exist of dark baryons.

• There is no favorable candidate for Dark Matter.

• Dark Matter and Dark Energy are the biggest mysteries for modern cosmology.


2.5 Exercises

Hubble constant: Besides Cepheids and Supernovae, many other methods are used to esti-mate the Hubble constant. Consult the paper [5] !

CMBR Spectrum: Calculate the photon number densitynCMBR contained in the Planckspectrum and compare this number with the mean baryon numberdensitynB in the presentUniverse.

Observed galaxy redshift distributions: The overall luminosity function for galaxies canbe well approximated by means of a Schechter function (see e.g. Virgo cluster and othernearby clusters)

Φ(L) dL = Φ∗ (L∗/L)α exp(−L/L∗) d(L/L∗) . (2.18)

L∗ is a characteristic luminosity,Φ∗ the corresponding normalisation of the space density, andα ≃ 1 the power–law index of the distribution. The number of galaxies observed in a redshiftbin dz with energy fluxf in the range[f, f + df ] and in the solid angle4π∆Ω is then givenby, for z ≪ 1,

d2N

dz df= 4π∆Ω

∫ ∫ ∞

0

Φ(L/L∗) d(L/L∗) r2 dr δ(r − cz/H0) δ(L− 4πfr2) (2.19)

Show that the above integral is given by

d2N

dz df=

4π∆Ω

L∗

(

c

H0

)5

z4 Φ(κz2) , (2.20)

whereκ = 4πfc2/(H20L∗) is a dimensionless constant. In a flux–limited survey, we observe

all galaxies with fluxes above some minimum flux

dN

dz=

∫ ∞

fmin

d2N

dz dfdf = ∆Ω

(

c

H0

)3

z4

∫ ∞

κmin

Φ(κz2) dκ , (2.21)

whereκmin = 4πfminc2/(H20L∗).

Explain by simple arguments, why the observed distributionscales as∝ z4, compare withFig. 2.19.Plot the expected distributiondN/dz for the Schechter function and compare with observeddistributions for given flux limits.

Galaxy Correlation function: What is the meaning of the correlation functionξ(r) for a3D galaxy distribution ?What is the correlation lengthr0 ? (see e.g. CfA or Las Campanas surveys)How can one find estimators for this correlation function ?

Spherical Harmonics: Look up the definition of spherical harmonics as used e.g. in quan-tum mechanics. Explain the completeness of this functionalbasis.

Bibliography 41

Galaxy cluster gas: Explain the emission process behind the X–rays observed from galaxyclusters. What processes heat the gas to X–ray temperatures ?Discuss the cooling times for this cluster gas (Virgo cluster e.g.) as a function of radius fromthe cluster center.

Jean’s equation: What is the Jean’s equation for the dynamics in galaxies ?How can one extract the mass distribution in a galaxy by usingthe Jean’s equation ?

Bibliography

[1] Alpher, R.A., Herman, R.C.: 1948, Nature162, 774-775[2] P. Astier, J. Guy, N. Regnault et al.: 2005,The Supernova Legacy Survey: Mea-

surement ofΩM , ΩΛ and w from the First Year Data Set, astro–ph/0510447[3] Bennett, C.L. et al.: 2003,First Year Wilkinson Microwave Anisotropy Probe

(WMAP) Observations: Preliminary Maps and Basic Results, ApJS148, 1.[4] Freedman, W.L. et al.: 1994, Nature371, 757[5] Freedman, W.L. et al.: 2001,Final Results from the Hubble Space Telescope Key

Project to Measure the Hubble Constant, ApJ553, 47.[6] Gamow, G.: 1948, Nature162, 680-682[7] Healpix Homepage: http//www.eso.org/healpix[8] Hinshaw, G. et al.: 2003,First Year Wilkinson Microwave Anisotropy Probe

(WMAP) Observations: The Angular Power Spectrum, ApJS148, 135[9] Hu, W., Dodelson, s.: 2002,Cosmic Microwave Background Anisotropies, Ann.

Rev. Astron. Astrophys.40, 171.[10] Hubble, E.: 1929,A relation between distance and radial velocity among extra-

galactic nebulae, Proc. Nat. Acad. Sci. (USA)15, 168[11] Penzias, A.A., Wilson, R.W.: 1965, Astrophysical Journal142, 419-421 (July)[12] Tonry, J.L. et al.: 2003,Cosmological Results from High-z Supernovae, ApJ594, 1[13] Tegmark, M.: 1996, ApJ 470, 81[14] Vogeley, M.S., Park, C., Geller, M.J., Huchra, J.P.: 1992,Large–scale clustering of

galaxies in the CfA Redshift Survey, Astrophsy. J. Letters391, L5

Part II

Relativistic World Models

3 The Relativistic Cosmos

3.1 Relativity and Cosmology

General relativity is the currently accepted theory of gravitation having been introduced byEinstein in 1916, superceding the newtonian theory. It plays a major role in astrophysics insituations involving strong gravitational fields, for example the study of neutron stars, blackholes, and gravitational lensing. The theory also predictsthe existence of gravitational radi-ation, which manifests itself by the transfer of energy due to a changing gravitational field,for example that of a binary pulsar. General relativity therefore also provides the theoreticalfoundation for the subject of cosmology, in which one studies the structure and evolution ofthe Universe on the largest possible scales.

The final steps to the theory of general relativity were takenby Einstein and Hilbert atalmost the same time. Both had recognised flaws in Einstein’sOctober 1914 work and acorrespondence between the two men took place in November 1915. How much they learntfrom each other is hard to measure but the fact that they both discovered the same final formof the gravitational field equations within days of each other must indicate that their exchangeof ideas was helpful.

On the 18th November Einstein made a discovery about which hewroteFor a few days Iwas beside myself with joyous excitement. The problem involved the advance of the perihe-lion of the planet Mercury. Le Verrier, in 1859, had noted that the perihelion (the point wherethe planet is closest to the sun) advanced by 38” per century more than could be accountedfor from other causes. Many possible solutions were proposed, Venus was 10% heavier thanwas thought, there was another planet inside Mercury’s orbit, the sun was more oblate thanobserved, Mercury had a moon and, really the only one not ruled out by experiment, that New-ton’s inverse square law was incorrect. This last possibility would replace the1/d2 by 1/dp,wherep = 2+ for some very small number. By 1882 the advance was more accurately known,43” per century. From 1911 Einstein had realised the importance of astronomical observationsto his theories and he had worked with Freundlich to make measurements of Mercury’s orbitrequired to confirm the general theory of relativity. Freundlich confirmed 43” per century ina paper of 1913. Einstein applied his theory of gravitation and discovered that the advanceof 43” per century was exactly accounted for without any needto postulate invisible moonsor any other special hypothesis. Of course Einstein’s 18 November paper still does not havethe correct field equations but this did not affect the particular calculation regarding Mercury.Freundlich attempted other tests of general relativity based on gravitational redshift, but theywere inconclusive.

Also in the 18 November paper Einstein discovered that the bending of light was out bya factor of 2 in his 1911 work, giving 1.74”. In fact after manyfailed attempts (due to cloud,war, incompetence etc.) to measure the deflection, two British expeditions in 1919 were to

46 3 The Relativistic Cosmos

confirm Einstein’s prediction by obtaining 1.98”±0.30” and 1.61”±0.30”.On 25 November Einstein submitted his paperThe field equations of gravitationwhich

give the correct field equations for general relativity. Thecalculation of bending of light andthe advance of Mercury’s perihelion remained as he had calculated it one week earlier.

Five days before Einstein submitted his 25 November paper Hilbert had submitted a paperThe foundations of physicswhich also contained the correct field equations for gravitation.Hilbert’s paper contains some important contributions to relativity not found in Einstein’swork. Hilbert applied the variational principle to gravitation and attributed one of the maintheorem’s concerning identities that arise to Emmy Noetherwho was in Gottingen in 1915.No proof of the theorem is given. Hilbert’s paper contains the hope that his work will lead tothe unification of gravitation and electromagnetism.

Immediately after Einstein’s 1915 paper giving the correctfield equations, Karl Schwarzschildfound in 1916 a mathematical solution to the equations whichcorresponds to the gravitationalfield of a massive compact object. At the time this was purely theoretical work but, of course,work on neutron stars, pulsars and black holes relied entirely on Schwarzschild’s solutionsand has made this part of the most important work going on in astronomy today.

The starting point for the application of Einstein’s theoryto cosmology is what is termedcosmological principle (sometimes also called the Copernican principle):Viewed on sufficiently large distance scales, there are no preferred directions or pre-ferred places in the Universe.Stated simply, this principle means that averaged over large enough distances, one part of theUniverse looks approximately like any other part. In this sense, the Earth is not a preferredlocation in the Universe – the physical laws tested in our labs should apply to all positions inthe Universe.

In this Section we shortly describe the essential elements of Einstein’s theory of gravityand derive the most general form of isotropic world models. This leads to spaces of constantcurvature. We then discuss observational aspects for such world models: the origin of redshift,distance measurements and number counting.

Natural Scaling: From Hubble’s constantH0 we derive a characteristic scale, often calledthepresent Hubble radiusof the Universe

RH ≡ c

H0= 4286Mpc , H0 = 70

km

s Mpc. (3.1)

In comparison to the extensions of superclusters with dimensions of about 50 Mpc, this isabout a hundred times larger. Only gravitational forces canact over such large distances. TheHubble constant also means a fundamental time–scale

tH =1

H0= 13.97 × 109 h−1

0.7 yr . (3.2)

This is a measure for the age of the Universe. All distance scales are given in terms ofRH

and all time–scales in terms oftH .It is therefore clear that only a relativistic theory of gravity can deal with such huge scales

and times. Newtonian gravity with its instantaneous actionfor forces is not compatible withthe basic ideas of Special Relativity, where the speed of light is the characteristic propagationtime for all interactions between matter. The General Theory of Relativity brings togetherSpecial Relativity and gravitational forces. This theory is ageometric description of gravity.

3.2 Einstein’s Vision of Gravity 47

A central idea, the equivalence principle, suggests that motion attributed to a gravitationalforce is to interpreted as free–fall motion in a curved spacetime. Since the motion is due tocurvature, all objects are affected equivalently. This curvature may be thought as geometry’sdeviation from flat Minkowski spacetime. In the language of differential geometry, spacetimecan be described as a 4–dimensional real differentiable manifold M on which a metricgab

with Lorentzian signature is defined. This metric allows oneto meassure spacetime separa-tions in a coordinate–independent manner. This metric tensor field describes gravity since allmeasurable properties can be derived from it. In the flat space limit, all equations reduce tothose of Special Relativity. Since this theory is well tested in the Solar system and even ina few compact binary systems, we have good reasons to believethat this theory also modelsgravity in the cosmos correctly.

3.2 Einstein’s Vision of Gravity

Modern cosmology begins with Hubble’s observation that theuniverse of galaxies is expand-ing. A theoretical basis for this observation has been givenby Einstein’s theory of gravity,more than 10 years earlier. In modern terms, Einstein’s theory of gravity is a gauge theorywith the Lorentz group as the gauge group which is operating in the tangent space. Gravity istherefore modeled by means of an affine connection of a Lorentzian manifold.1 In this Sec-tion, we summarize all the elements necessary to understandthe geometry of the FriedmannUniverse and of its generalisations, such as the perturbed Friedmann Universe or Brane Cos-mology. However, it is not the purpose of this Section to introduce all concepts in sufficientdepth, for this attend a lecture on General Relativity.

3.2.1 The Concept of SpaceTime

Special relativity showed that the absolute space and time of Newtonian physics could be onlyan approximation to their true nature. However, the specialtheory of relativity is incapableof explaining gravity because SR assumes the existence of inertial frames; it does not explainhow inertial frames are to be determined. Mach’s principle,which states that the distribu-tion of matter determines space and time, suggests that matter is related to the definition ofinertial frames, but Mach never elucidated any means by which this might happen. Generalrelativity attacks this problem and in so doing, discovers that gravity is related to geometry.The equivalence principle is the fundamental basis for the general theory of relativity. Thestrict equivalence between gravity and inertial acceleration means that freefalling frames arecompletely equivalent to inertial frames. In general relativity, (GR) it is spacetime geometrythat determines freefalling (inertial, geodesic) worldlines, telling matter how to move. Mat-ter, in turn, tells spacetime how to curve. Geometry is related to matter and energy throughEinstein’s equation. The metric equation provides a general formalism for the spacetime in-terval in general geometries, not just the Minkowski (flat) spacetime of special relativity (SR).Matter and energy determine inertial frames, but within an inertial frame there is no influence

1A modern introduction into General Relativity can be found inthe textbooks by Carroll [2] and Hobson etal. [3]. The latter one does include the basic concepts for Cosmology (Friedmann models and Inflation). A moremathematically oriented treatment is given by Straumann [7]. This textbook also includes a complete overview formodern differential geometry of Riemannian manifolds (theory of tensor fields, affine connections, curvature andp–forms).


Figure 3.1: In 1916 Albert Einstein published the fundamental paperuber die relativistischeTheorie der Gravitation.

by any outside matter. Thus Mach’s principle is present morein spirit than in actuality in thegeneral theory of relativity.

The Concept of a Metric

To introduce the concept of a metric, let us consider Euclidean 2–dimensional space withCartesian coordinatesx, y. A parametrized cureve(x(t), y(t)) begins att1 and ends att2.The length of the curve is given by

s =

∫

ds =

∫

√

dx2 + dy2 =

∫ t2

t1

√

x′2 + y′2 dt . (3.3)

Here,ds =√

dx2 + dy2 is theline element. The square of the line element, also called themetric, is then given as

ds2 = dx2 + dy2 . (3.4)

For this representation, we also can use polar coordinates(r, θ) with the expression for themetric

ds2 = dr2 + r2 dθ2 . (3.5)


In a similar manner, in 3–dimensional Euclidean space, the metric is given by

ds2 = dx2 + dy2 + dz2 , (3.6)

in Cartesian coordinates, and

ds2 = dr2 + r2 (dθ2 + sin2 θ dφ2) (3.7)

in spherical coordinates.

Einstein I: Minkowski space M4 has to be generalized to a general curvedpseudo–Riemannian manifold(M, g) with metric tensor field g such that SpaceTime islocally still Minkowskian, i.e. the tangent spaceTpM = M4.

The notion of anevent is fundamental in relativity. An event is characterized by its position~x and its timet. An eventp is given by the 4 coordinates(t, x, y, z) in the 4–dimensionalSpaceTime. Already in Special Relativity, time and space appear as an entity. Two neighbor-ing events(t, x, y, z) and(t + dt, x + dx, y + dy, z + dz) have then a distanceds which isdetermined by the metric of Minkowski space,x0 = ct,

ds2 = c2 dt2 − dx2 − dy2 − dz2 = ηµν dxµ dxν . (3.8)

This distance is invariant against Lorentz transformations. We often say the spacetime of SRis flat, since the resulting curvature vanishes.

A suitable tool to picturize a spacetime is to use spacetime diagrams (Fig. 3.2). Thetime-axis is running vertically, and space is running horizontally. In Minkowski space, thelightcones have a constant openening angle of 90 degrees. Ina curved spacetime this maychange.

Gravity cannot be included into Special Relativity – despite many desperate attempts to dothis 100 years ago. Einstein postulated therefore that Minkowski space is only realized locallyin a4D pseudo–Riemannian manifold(M, g). This means strictly speaking, Minkowski spaceM4 is the tangent spaceTpM = M4 at each eventp of the spacetimeM. Einstein had the ideathat the effects of gravity are expressed in terms of a generalized Minkowski metric elementof the form

ds2 =3∑

α,β=0

gαβ(x) dxα dxβ , (3.9)

which gives the distance between neighboring events inM. The ensemble of all eventsparametrized by local coordinatest, xi is called the SpaceTime. The metric tensorg isa second–rank symmetric tensor, which in general depends onthe events.ds now measuresthe proper time of timelike curvesxµ(λ)

τ = s/c =

∫

√

gµνdxµ

dλ

dxν

dλdλ . (3.10)

Examples of Simple SpaceTimes

• TheSchwarzschild spacetimeas the expression of the gravitational field of non–rotatingstars

ds2 = exp(2Φ(r)) c2 dt2 − exp(−2Λ(r)) dr2 − r2 (dθ2 + sin2 θ dφ2) . (3.11)


Figure 3.2: The light cones in Minkowski space are flat. The time–axis runs vertically,thespatial axes horizontally. At each event we find a forward and backward light cone. Photons (andother massless particles) move along the light cone, while the trajectories ofnormal particles areconfined to the interior of the light cones. A detector can only measure photons which come infrom the backward light cone.

It has two independent functions, which only depend on the spherical radiusr.

• The gravitational field ofrapidly rotating stellar objects

ds2 = α2 c2 dt2 − R2(dφ − ω dt)2 − exp(2µr) dr2 − exp(2µθ) dθ2 (3.12)

already has 5 independent functions depending now on the radius r and onθ, but not onφ andt, α = α(r, θ) etc. This line element contains an off–diagonal termg0φ, related tothe angular momentum of the star.

• Cosmological spacetimes

ds2 = c2 dt2 − R2(t) dσ2(3) , (3.13)

wheredσ(3) is the metric of a 3–space of constant curvature. The essential degree offreedom is the expansion factorR(t) which scales all spatial lengths (see later on). Thesimplest example is the stretching of Minkowski space (called a flat Universe)

ds2 = c2 dt2 − R2(t)[

dx2 + dy2 + dz2]

, (3.14)

often written in spherical coordinates(t, r, θ, φ) as

ds2 = c2 dt2 − R2(t)[

dr2 + r2(dθ2 + sin2 θ dφ2)]

. (3.15)

All of the above spacetimes have some high degree of symmetries.


3.2.2 Gravity is an Affine Connection on SpaceTime

In order to compare tangent spaces at neighboring events, one needs aconnectionon themanifoldM. As in Riemannian geometry, this connection is required to be metric, so that thecorresponding Christoffel symbols are uniquely given by derivatives of the metric elements.This is a basic postulate of Einstein’s theory of gravity – one could construct more generaltheories of gravity which include torsion.

Physically speaking, we associateobserversea, a = 0, 1, 2, 3, i.e. an orthonormal tetrad(or Vierbein field), satisfying2

g(ea, eb) = ηab , (3.16)

whereη is the flat Minkowskian metric with signature(+−−−), or (−+ ++). An observeris a global orthonormal basis field in the tangent space of each eventp, wheree0 is timelikeandei (i = 1, 2, 3) are spacelike. One could also construct null tetrads in order to definethe geometry of the space–time. The dual elements ofea is a basis of the cotangent spaceT ∗

p , denoted byΘa, satisfyingΘa(eb) = δab. They define the metricg = ηab Θa ⊗ Θb. The

definition of these observer fields is not unique, since any observer derived by means of a localLorentz transformationΛ is also an observer

ea|x = Λba(x) eb|x , Λ(x)ηΛT (x) = η . (3.17)

These are Lorentz transformations operating in the tangentspace of each event.As an example we considerstatic observersin the Schwarzschild metric (3.11). Such an

observer is given by the following tetrad

e0 = exp(−Φ)∂t , er = exp(Λ)∂r , eθ =1

r∂θ , eφ =

1

r sin θ∂φ . (3.18)

It is then clear that they satisfyg(ea, eb) = ηab, whereηab is the Minkowski metric. The dualbasis is a basis of one–formsΘa with Θa(eb) = δa

b

Θ0 = exp(Φ) dt , Θr = exp(−Λ) dr , Θθ = r dθ , Θφ = r sin θ dφ . (3.19)

We now consider a satellite which is orbiting the central star in the equatorial plane of theSchwarzschild spacetime with 4–velocityuµ given by

u = ut(∂t + Ω∂φ) , g(u, u) = −1 . (3.20)

Ω = uφ/ut = dφ/dt is the angular velocity of the satellite (Keplerian e.g.) asmeasuredby fixed stars. The Lorentz transformation between the static observerea and the satelliteobserverea is then given by a boost transformation with 3–velocityV = r sin θΩ/α andLorentz factorγS = 1/

√1 − V 2, whereα2 = 1 − 2GM/r (c = 1). α is the redshift factor

between a static observer at radiusr and infinity. This provides us the Lorentz transformation

e0 = γS(e0 + V eφ) (3.21)

er = er (3.22)

eθ = eθ (3.23)

eφ = γS(V e0 + eφ) . (3.24)

The trajectory of this observer, with tangent vectore0 is now a helical path in spacetime.

2In the following, the convention for indices is as follows: greek indices are related to local coordinate systems,latin indicesa, b, c, ... mark observer fields, latin indicesi, k, l, ... specify spatial components.


The Concept of a Connection

A connectionis now defined as a linear mapping between the tangent space atthe eventx andthe tangent space at a neigboring event displaced bydx. It is sufficient to define this mappingfor an arbitrary basisea of the tangent space

∇eaeb = Γc

ab ec = ωcb(ea) ec , (3.25)

with the additional properties for any functionf and any vector fieldX

∇fXeb = f ∇Xeb (3.26)

∇X(feb) = f ∇Xeb + (X.f)eb . (3.27)

Thus, the one–forms defined as

ωba = Γb

ca Θc (3.28)

are calledconnection one–forms. They are identical with the Christoffel symbols, if the basisin the tangent space is the natural basis implied by the coordinate system

∇eµeλ = Γν

µλ eν . (3.29)

Remember that the first indexµ in the Christoffel symbols is a one–form index, the secondλis a matrix index.

From the duality between tangent and cotangent space we find then

∇XΘa = −ωab(X)Θb (3.30)

for any vector fieldX. From this definition we find the covariant derivative for anyvectorfield X = Xa ea

∇X = ea ⊗ (dXa + ωab Xb) , (3.31)

or in components with respect to an orthonormal basis

∇aXb = eµaXb

,µ + ωbc(ea)Xc . (3.32)

Similarly, for a one–formα = αa Θa we have

∇α = Θb ⊗ (dαb − αa ωab) , (3.33)

or in components

∇aαb = eµaαb,µ − ωc

b(ea)αc . (3.34)

When we use the coordinate basis of the chosen chart, the covariant derivatives of vector fieldsand one–forms are given in the well–known form

∇µXν = Xν,µ + Γν

µρXρ (3.35)

∇µαν = αν,µ − αρΓρµν . (3.36)


The covariant derivative for vector fields and one–forms cannow be extended to arbitrarytensor fields, in general, by requiring that the operation of∇ satisfies the Leibniz rule whenacting on tensor products

∇(S ⊗ T) = ∇S ⊗ T + S ⊗∇T . (3.37)

In this sense, the covariant derivative of a 2–tensor fieldTµν is given as follows

∇αTµν = Tµν,α + Γµ

αρTρν + Γν

αρTµρ , (3.38)

and similarly for a 2–tensor fieldAµν by means of

∇αAµν = Aµν,α − ΓραµAρν − Γρ

ανAµρ . (3.39)

Parallel Transport and Geodesics

A connection on a vector bundle (here the tangent space) specifies then the notion ofparalleltransport along curves in the manifold. Letγ be a curve on the manifold, andX a vectorfield defined onM. A vector field is calledautoparallel alongγ, if

∇γX = 0 . (3.40)

In coordinates, we have

X = Xµ∂µ , γ =dxµ

ds∂µ , (3.41)

and therefore

∇γ =

(

dXρ

ds+ Γρ

µν

dxµ

dsXν

)

∂ρ . (3.42)

The vector field is autoparallel if

dXρ

ds+ Γρ

µν

dxµ

dsXν = 0 . (3.43)

For any curveγ(s) andX0 in the tangent spaceTγ(0)M we find a unique vector fieldX(s)given alongγ(s) with the initial conditionX(0) = X0. This operation is called theparalleldisplacementof a vector field along a curveγ(s).

A curve is called ageodesic, if the tangent fieldγ is autoparallel alongγ(s). Accordingto the above analysis, this means

d2xρ

ds2+ Γρ

µν

dxµ

ds

dxν

ds= 0 . (3.44)

Geodesics are the trajectories of freely falling bodies in the gravitational field given by theaffine connection. A satellite e.g. will move on geodetic curves in the gravitational field of theEarth, planets move on the geodetic curves in the solar gravitational field.


Gravity is a Metric Connection

So far, the concept of a metric and the concept of the connection are independent of eachother. Each Riemannian manifold, however, carries a particular connection which is uniquelyassociated with the metric. For this, we say:

An affine connection is said to be a metric connection if the parallel transport along anysmooth curve in the manifold preserves the inner product.

One can then prove that this statement is equivalent to∇g = 0, which is equivalent to theRicci identity

X.g(Y,Z) = g(∇XY,Z) + g(Y,∇XZ) . (3.45)

With the condition that torsion vanishes,

∇XY −∇Y X − [X,Y ] = 0 , (3.46)

we can write the above equation as

X.g(Y,Z) = g(∇Y X,Z) + g([X,Y ], Z) + g(Y,∇XZ) . (3.47)

[X,Y ] denotes the Lie bracket for vector fields (see next Section).With cyclic permutation ofthe vector fields we obtain

Y.g(Z,X) = g(∇ZY,X) + g([Y,Z],X) + g(Y,∇XZ) (3.48)

Z.g(X,Y ) = g(∇XZ, Y ) + g([Z,X], Y ) + g(X,∇ZY ) . (3.49)

Now we add the first and third equation and subtract the secondone to get

2g(∇ZY,X) = −X.g(Y,Z) + Y.g(Z,X) + Z.g(X,Y )

− g([Z,X], Y ) − g([Y,Z],X) + g([X,Y ], Z) . (3.50)

For the fundamental vector fieldsX = ∂k, Y = ∂j andZ = ∂i the Lie bracket vanishes,[∂i, ∂j ] = 0 andg(∂i, ∂j) = gij , which means that

2g(∇∂i∂j , ∂k) = 2Γm

ijgmk = −∂kgji + ∂jgik + ∂igkj (3.51)

or

gmkΓmij =

1

2(gjk,i + gik,j − gij,k) . (3.52)

With the inverse metricgij , we now get the famous expression for the Levi–Civita connection

Γmij =

1

2gmk (gki,j + gkj,i − gij,k) . (3.53)

This affine connection is therefore uniquely determined by derivatives of the metric ten-sor and is therefore called metric connection, or Levi–Civita connection.

For any pseudo–Riemannian manifold there is then a unique affine connection∇ such that itis (i) torsion–free and (ii) metric. This particular connection is usually called theLevi–Civitaconnection, or pseudo–Riemannian connection.

Einstein II: It is now one of the fundamental postulates of Einstein’s theory of gravitythat gravity is related to the Levi–Civita connection of the Lorentzian manifold. Thismeans in particular that there is no torsion associated withgravity.


Strong Principle of Equivalence

Since the Lorentz connection transforms inhomogeneously as

ω = Λ(x)ωΛ−1(x) − (dΛ)Λ−1(x) (3.54)

for any Lorentz transformation between local observers,Θa = Λab (x)Θb, we always can find

locally an observer system such thatωp = 0, i.e. the connection can be transformed away justlocally, but not globally. This is not the case for the curvature !

The weak principle of equivalence states that effects of gravitation can be transformedaway locally by suitably accelerated frames of reference (by going to local Minkowskian co-ordinates). We can formulate, however, a much stronger requirement, the socalled

Einstein III: strong principle of equivalence, which states that any physical interaction(other than gravitation) behaves in a local inertial frame as if gravitation were absent.E.g. Maxwell’s equations will have their familiar forms as in SR.

The strong principle of equivalence allows us to extend any physical law that is expressedin a covariant way to curved SpaceTime. Ordinary derivatives are just replaced by covariantones.

3.2.3 Differential Forms on SpaceTime

Differential forms are extremely helpful concepts in direct calculation. A zero–form is a scalarfunction. The one–formsΘa defined above are the basis elements of the cotangent space, itscomponents are the components of covariant vectors. A general one–formA can alwaysbe written asA = Aµ dxµ = AaΘa. The vector potential of classical electrodynamics is thestandard example. A new operation introduced when one workswith forms is called thewedgeproduct. If x andy are coordinates, thendx anddy are one–forms, anddx ∧ dy = −dy ∧ dxis called a two–form. An example of a p–form is

A =1

p!Aµν...ρ dxµ ∧ dxν ∧ ... ∧ dxρ , (3.55)

whereAµν...ρ is a completely antisymmetric tensor withp indices. In fact, the set ofp–formsin an–dimensional manifold is a vector spaceΛp of dimensionn!/p!(n− p)! (see Table 3.1).In 4 dimensions we have one zero–form, 4 one–forms (basis in the cotangent space), 6 2–forms(the Farady tensor e.g.), 4 3–forms (currents) and only one 4–form (volume–form). Formally,the wedge product of ap–formω with a q–formα is given by the alternating operator

ω ∧ α = A(ω ⊗ α) . (3.56)

Theexterior derivative d takes ap–form into a(p + 1)–form, e.g. a one–form

dΘ = d(Θµ dxµ) = Θµ,ν dxν ∧ dxµ =1

2(Θµ,ν − Θν,µ) dxν ∧ dxµ . (3.57)

In general, for ap–formA given by

A = Aµν...ρ dxµ ∧ dxν ∧ ... ∧ dxρ (3.58)


Forms 0 1 2 3 4

dim = 3 1 3 3 1 –dim = 4 1 4 6 4 1

Table 3.1: Number of linearly independentp–forms forD = 3 andD = 4.

the exterior derivative is given by its local expression

dA = dAµν...ρ ∧dxµ ∧dxν ∧ ...∧dxρ =∂Aµν...ρ

∂xλdxλ ∧dxµ ∧dxν ∧ ...∧dxρ . (3.59)

With this explicit definition, one can show

d(A ∧ B) = dA ∧ B + (−1)p A ∧ dB (3.60)

d(dA) = 0 (3.61)

for p–formA and aq–formB.One can also define an antiderivationi which makes a(p − 1)–form out of ap–form

defined as

(iV ω)(V1, ..., Vp−1) = ω(V, V1, ..., Vp−1) , (3.62)

i.e. just by contraction with the first index. With the FaradytensorF we can e.g. build theone–formE = iV F , in componentsEν = V µFµν . This operation is called theinner productof V with ω. Applying both operations, the inner product and the exterior derivative, leavesthe degree of ap–form invariant

LX = d · iX + iX · d (3.63)

is equivalent to the Lie derivative onp–forms.TheLie derivative L is given by its action on functions

LXf = X.f = df(X) , (3.64)

its action on vector fields

LXY = [X,Y ] = (XµY α,µ − Y µXα

,µ) eα , (3.65)

and the Leibniz rule for the compatibility with higher rank tensors

LX(S ⊗ T ) = LXS ⊗ T + S ⊗ LXT . (3.66)

From the last property, we can derive for a one–formω and a vector fieldY

LX(ω ⊗ Y ) = LX(ω(Y )) = (LXω) ⊗ Y + ω ⊗ (LXY ) . (3.67)

Writing out in components, we have

Xµ(ωαXα),µ = (LXω)µY µ + ωµ(LXY )µ , (3.68)

or making use of the Lie–bracket

(LXω)µY µ = Xα(ωµ,αY µ + ωµY µ,α) − ωµ(XαY µ

,α − Y αXµ,α)

= (Xαωµ,α + ωαXα,µ)Y µ . (3.69)

Since this last equation is valid for any vector fieldY , we conclude

(LXω)µ = Xαωµ,α + ωαXα,µ . (3.70)


3.2.4 Curvature of SpaceTime

A physical theory of gravity also requires some dynamical evolution for the connection. Thisis usually formulated in terms of the curvature associated with the connection. The calculationof the Riemann tensor is therefore one of the major tasks whendealing with specific space-times. For this purpose we denote byX (M) the space of all (smooth) vector fields on themanifoldM.

Conventionally, thetorsion fields T are defined as bilinear mappingsT : X (M) ×X (M) → X (M) on the set of all vector fields on the manifold

T (X,Y ) ≡ ∇XY −∇Y X − [X,Y ] . (3.71)

Curvature is defined as a trilinear mappingR : X (M) ×X (M) ×X (M) → X (M)

R(X,Y )Z ≡ ∇X(∇Y Z) −∇Y (∇XZ) −∇[X,Y ]Z . (3.72)

The componentsRabcd of this vector field defines the Riemann tensor which has four indices.

These quantities obviously satisfy the antisymmetry conditions

T (X,Y ) = −T (Y,X) , R(X,Y ) = −R(Y,X) , (3.73)

as well as

T (fX, gY ) = fg T (X,Y ) (3.74)

R(fX, gY )hZ = fghR(X,Y )Z (3.75)

for any functionsf , g andh.Since torsionT (X,Y ) and curvatureR(X,Y ) are antisymmetric tensors, they naturally

define corresponding two–forms

T (X,Y ) = T a(X,Y ) ea (3.76)

R(X,Y )eb = Ωab(X,Y ) ea . (3.77)

The exterior derivatives of the basic one–formsΘa and of the connection formsω satisfyCartan’s structure equations

T a = dΘa + ωab ∧ Θb (3.78)

Ωab = dωa

b + ωad ∧ ωd

b . (3.79)

The wedge operator denotes the exterior products for p–forms. The 2–formΩ is the curvature2–form which gives, when expressed locally,

Ωab =

1

2Ra

bcd Θc ∧ Θd (3.80)

the components of the Riemann tensorRabcd in orthonormal coordinates. Similarly, we have

4 torsion two–forms

T a =1

2T a

bc Θb ∧ Θc . (3.81)


For the proof of Cartan’s structure equations, we use the above definition of torsion. Writ-ten as one–forms, this means

T a(X,Y ) ea = ∇X∇Y −∇Y ∇X − [X,Y ]

= ∇X(Θb(Y )eb) −∇Y (Θb(X)eb) − Θa([X,Y ])ea

=

X.Θa(Y ) − Y.Θa(X) − Θa([X,Y ])

ea

+

Θa(Y )ωba(X) − Θa(X)ωb

a(Y )

ea

= dΘa(X,Y )ea + (ωab ∧ Θb)(X,Y )ea . (3.82)

The proof of the second structure equation is similar. Written as a two–form, this means

Ωac(X,Y ) ea = ∇X∇Y ec −∇Y ∇X ec − ωb

c([X,Y ])eb

= ∇X(ωbc(Y )eb) −∇Y (ωb

c(X)eb) − ωac([X,Y ])ea

=

X.ωac(Y ) − Y.ωa

c(X) − ωac([X,Y ])

ea

+

ωbc(Y )ωa

b(X) − ωbc(X)ωa

b(Y )

ea

= dωac(X,Y )ea + (ωa

b ∧ ωbc)(X,Y )ea . (3.83)

Local expressions: In local coordinates, a metric connection is expressed in terms of theChristoffel symbols

Γαµβ =

1

2gαρ(

gρµ,β + gρβ,µ − gµβ,ρ

)

(3.84)

such that the connection form is given in a local coordinate basis as

ωαβ = Γα

µβ dxµ , (3.85)

and therefore

dωαβ = Γα

µβ,ν dxν ∧ dxµ =1

2(Γα

µβ,ν − Γανβ,µ) dxν ∧ dxµ . (3.86)

Also,

ωαρ ∧ ωρ

β = ΓανρΓ

ρµβ dxν ∧ dxµ

=1

2(Γα

νρΓρµβ − Γα

µρΓρνβ) dxν ∧ dxµ (3.87)

Accordingly, Cartan’s second structure equation is equivalent to the conventional definition ofthe Riemann tensor in local coordinates

Rαβµν = Γα

νβ,µ − Γαµβ,ν + Γα

µρΓρνβ − Γα

νρΓρµβ . (3.88)

3.2.5 Curvature and Einstein’s Equations

Another consequence of the affine connection is an additional symmetry of the Riemann tensor

g(R(X,Y )Z,U) = −g(R(X,Y )U,Z) (3.89)

g(R(X,Y )Z,U) = g(R(Z,U)X,Y ) . (3.90)


The Riemann tensor is the fundamental entity for the construction of the field dynamics. Itsatisfies the following essential symmetries which are important for the concrete calculation

Rabcd = −Ra

bdc (3.91)

Rabcd = −Rbacd (3.92)

Rabcd = Rcdab . (3.93)

The first property results from the fact that curvature is a two–form, the second one that cur-vature is an element of the Lie algebra of the Lorentz group, and the third one gives a funda-mental relation between spacetime indices and intrinsic indices (metric condition). This lastproperty follows from thecyclic identityfor a torsion–free connection

R(X,Y )Z + R(Z,X)Y + R(Y,Z)X = 0 , (3.94)

or in components

Rabcd + Radbc + Racdb = 0 . (3.95)

Making use of the antisymmetry in the first and second pair of indices, we find

Rabcd = −Radbc − Racdb = Rdabc + Rcadb

= −(Rdcab + Rdbca) − (Rcbad + Rcdba)

= 2Rcdab + (Rbdca + Rbcad)

= 2Rcdab − Rbadc = 2Rcdab − Rabcd . (3.96)

Hence

Rabcd = Rcdab . (3.97)

In total, the Riemann tensor has 36 components, while the last symmetry reduces it to 20independent components.The Riemann tensor of apcetimes has 20 independent compo-nents. Astrophysical spacetimes have usually many symmetries such that the total number ofindependent components is drastically reduced. In comparison, the metric tensor has only 10independent components, i.e. only half of the components ofthe Riemann tensor are due tothe metric, or theRicci tensor Rab, while the other 10 components are hidden in the Weyltensor.

The Riemann tensor is now used to construct the Ricci tensor,Rbd = Rabad. For the

Schwarzschild spacetime (3.11) e.g. we get the following expressions for the Ricci tensor

R00 = Rr0r0 + R2

020 + R3030 (3.98)

Rrr = R0r0r + R2

r2r + R3r3r (3.99)

R22 = R0202 + Rr

2r2 + R3232 (3.100)

R33 = R0303 + Rr

3r3 + R2323 (3.101)

with all other components vanishing. With the Ricci scalarR = Raa as the trace of the Ricci

tensor we now can construct theEinstein tensor

Gab ≡ Rab −1

2R gab . (3.102)


The Einstein tensor is symmetric,Gab = Gba, and divergence–free,Gab;a = 0 (due to the

Bianchi identity).

Einstein IV: Einstein postulated that the tensor Gab couples to the matter content ofthe Universe

Gab =8πG

c4Tab , (3.103)

whereTab is the symmetricenergy–momentum tensor of all matterin the Universe (par-ticles, baryons, galaxies, photons, neutrinos and quantumfields). As a consequence of theabove properties, the divergence of the energy–momentum tensor vanishes identically

T ab;b = 0 . (3.104)

Einstein’s equations can be derived from the action

A =1

16πG

∫

R√−g d4 +

∫

Lmatter(Φ, ∂Φ)√−g d4x (3.105)

whereLmatter is the Lagrangian density for matter depending on some variables denotedcollectively asΦ, c = 1, since for any domainD of spacetime [7]

δ

∫

D

R√−g d4x = −

∫

D

Gµν δgµν

√−g d4x . (3.106)

The variation of this action with respect toΦ will lead to the equation of motion for matter,δLmatter/δΦ = 0, while the variation of the action with respect to the metrictensorg leads toEinstein’s equations3

Rµν − 1

2Rgµν = 16πG

δLmatter

δgµν≡ 8πGTµν . (3.107)

Here,Tµν = 2δLmatter/δgµν is the energy–momentum tensor of matter fields.

On the Cosmological Constant

Let us now consider a new matter actionL′matter = Lmatter − Λ/(8πG), whereΛ is a real

constant. The equation of motion for the matter does not change under this transformation,sinceΛ is constant. But the action now picks up an extra term proportional toΛ, which canbe written in two different ways,

A =1

16πG

∫

R√−g d4x +

∫

(

Lmatter(Φ, ∂Φ) − Λ

8πG

)√−g d4x

=1

16πG

∫

(R − 2Λ)√−g d4x +

∫

Lmatter(Φ, ∂Φ)√−g d4x (3.108)

and Einstein’s equations get modified. This simple manipulation has many backdrops in the-oretical Physics. It can be interpreted in different manners:

3see any textbook on General Relativity


• The first interpretation is based on the first line of the aboveequations, it treatsΛ as ashift in the matter Lagrangian, which in turn will lead to a shift inthe matter Hamiltonian.This could be thought of as ashift in the zero point energyof the matter system. Sucha constant shift in energy does not affect the dynamics of matter, while gravity picks upan extra contribution in the form of a new termQµν in the energy–momentum tensor

Rµν − 1

2Rδµ

ν = 8πG(Tµν + Qµ

ν ) , Qµν =

Λ

8πGδµν . (3.109)

• The second line in Eq (3.108) can be interpreted as a gravitational field, described bythe Lagrangian of the formLgrav ∝ (1/G)(R − 2Λ), interacting with matter. In thisinterpretation, gravity is described by two constants, theNewton’s constantG and thecosmological constantΛ. It is then natural to modify the left hand side of Einstein’sequations in the form of

Rµν − 1

2Rδµ

ν − δµν Λ = 8πGTµ

ν . (3.110)

In this interpretation, the spacetime is curved even in the absence of matter,Tαβ = 0,since the left hand side does not admit flat spacetimes as solutions.

• It is even possible to consider a situation where both effects can occur. If gravitationaltheories are in fact described by the Lagrangian of the form(R − 2Λ), then there isan intrinsic cosmological constant in nature, just as thereis a Newtonian constantGin nature. If the matter Lagrangian contains energy densities which change due to thedynamics, thenLmatter can pick up constant shifts during dynamical evolution. Forthiswe consider a scalar field with the Lagrangian

LΦ = (1/2)∂µΦ ∂µΦ − V (Φ) , (3.111)

which has the energy–momentum tensor

Tµν = ∂µΦ ∂νΦ − δµ

ν

(1

2∂ρΦ ∂ρΦ − V (Φ)

)

. (3.112)

For field configurations which are constant (e.g. at the minimum of the potentialV ), thiscontributes an energy–momentum tensorTµ

ν = V (Φmin) δµν , which has exactly the same

form as a cosmological constant. It is then the combination of these two effects – of verydifferent nature – which is relevant and the source will be

T effµν = [V (Φmin) + Λ/(8πG)] gµν . (3.113)

Φmin can change during the dynamical evolution, leading to a time–dependent cosmo-logical constant.

The termQµν in Einstein’s equations behaves very pecularly compared tothe energy–momentum tensor of normal matter.Qµ

ν = ρΛδµν is in the form of an energy–momentum

tensor of an ideal fluid with energy densityρΛ and pressurePΛ = −ρΛ. Obviously, either thepressure or the energy density of this fluid must be negative.

Such an equation of state,P = −ρ, also has another important implication in GR. Therelative acceleration between two geodesics,~g, satisfies in GR the following equation

∇ · ~g = −4πG(ρ + 3P ) . (3.114)


The source of this relative acceleration between geodesicsis ρ + 3P and notρ alone. Thisshows, as long asρ+3P > 0, gravity remains attractive, whileρ+3P < 0 leads to repulsiveforces. A positive cosmological constant therefore leads to repulsive gravity.

Remark: Gravity as a Gauge Theory

We now have the means to compare the formalism of connectionsand curvature in Rieman-nian geometry to that of gauge theories in particle physics.In both situations, the fields ofinterest live in vector spaces which are assigned to each point in spacetime. In Riemanniangeometry the vector spaces include the tangent space, the cotangent space, and the highertensor spaces constructed from these. In gauge theories, onthe other hand, we are concernedwith internal vector spaces. The distinction is that the tangent space and its relatives are inti-mately associated with the manifold itself, and were naturally defined once the manifold wasset up; an internal vector space can be of any dimension we like, and has to be defined as anindependent addition to the manifold. In math lingo, the union of the base manifold with theinternal vector spaces (defined at each point) is a fiber bundle, and each copy of the vectorspace is called thefiber (in perfect accord with our definition of the tangent bundle).

Non–gravitational interactions are described nowadays interms of gauge theories. In thissense, Maxwell’s theory is aU(1) gauge theory resulting from local phase transformations onquantum fields,Ψ(x) → exp(−iα(x))Ψ(x). The vector potentialsAµ(x) are the connectioncoefficients, and the Faraday tensorF = (1/2)Fµν dxµ ∧ dxν is the corresponding curvature.Strong interaction is aSU(3) gauge theory, where the internal space is defined by the colorspace – each fermion can carry a specific color. The gauge fields Ab

µa are then the localexpressions of a connection one–formω = Aµ dxµ with values in the Lie–algebra ofSU(3).In this case, we have 8 connection fieldsAλ

µ(x), λ = 1, ..., 8, corresponding to the 8 gluonfields of strong interaction.

In this sense, the gauge transformations for gravity are thelocal Lorentz transformationsoperating between different observers at the same events inspacetime. We have 6 connectionfields Aλ

µ(x), λ = 1, ..., 6, i.e. in total 24 connection coefficients. Note, however, that thedynamics proposed by Einstein is different from the Yang–Mills dynamics of non–Abeliangauge theories.

3.3 General Relativity is the Correct Theory of Gravity

Most of the tests for Einstein’s theory of gravity have been done for stellar objects, such as theSun or neutron stars. In good aprroximation, stars are spherical objects and the gravitationalfield is given in terms of spherically symmetric metric elements.4

Das einfachste metrische Feld wird von einemkugelsymmetrischen Sternerzeugt. Indiesem Falle reduziert die hohe Symmetrie (Kugelsymmetrie) die moglichen metrischen Ko-effizienten auf zwei wesentliche Funktioneng00(r) undgrr(r)

ds2 = exp(2Φ(r)) dt2 − exp(−2Λ(r)) dr2 − r2 (dθ2 + sin2 θ dφ2) . (3.115)

Die Einsteinschen Feldgleichungen, welche die Krummung der Raum–Zeit mit dem Ma-terieinhalt verknupfen, bestimmenuber Differentialgleichungen diese beiden Funktionen ein-

4A detailed analysis of all these tests is not the topic of the present lecture, see e.g. any lecture on GR or onRelativistic Astrophysics.

3.3 General Relativity is the Correct Theory of Gravity 63

deutig,r > R∗,

exp(2Φ(r)) = exp(2Λ(r)) = 1− 2GM

c2r, gθθ = −r2 , gφφ = −r2 sin2 θ . (3.116)

Diese Metrik ist alsSchwarzschild–Metrik eines Sterns bekannt. Sie enthalt nur die Gesamt-masseM des Sterns als freien Parameter. Die Große

RS =2GM

c2= 3km

M

M⊙

(3.117)

ist der der MasseM . Dies war die erste Losung der von Einstein 1915 postulierten Feldgle-ichungen. Das Gravitationsfeld der Sonne muß durch ein solches metrisches Feld beschriebenwerden, obschon die Abweichungen vom flachen Raum an der Oberflache der Sonne nurRS/R⊙ ≃ 10−6 betragen. Bei Neutronensternen sind diese Abweichungen jedoch schonbetrachtlich,RS/R∗ ≃ 1/3.

Figure 3.3: Die Lichtkegel werden in der N”ahe eines Schwarzen Lochs verzogen: sie werdenauf das Zentrum zu gerichtet. Der Zylinder wird durch die Zeitentwicklung des Horizonteserzeugt (1 Dimension unterdr”uckt).

Gravitational Redshift

Betrachten wir z.B. einen Beobachter mitr = const, θ = const undφ = const und stellenuns die Frage, wie die Zeit seiner Uhr sich zur Koordinatenzeit t verhalt (die Zeit im Un-


endlichen).dτ = ds/c mißt die Eigenzeit des Beobachters in einem lokalen Inertialsystem.Deshalb gilt

dτ =

√

1 − 2GM

c2rdt . (3.118)

Wie die Metrik zeigt, istt die im Unendlichen gemessene Zeit. Gegenuber Unendlich scheintdeshalb eine Uhr im Gravitationsfeld langsamer zu gehen. Dies konnen wir auch dadurchausdrucken, dass Photonen im Gravitationsfeld eines Sterns rotverschoben werden

νB

νA=

dτA

dτB=

√

g00(A)

g00(B)(3.119)

ergibt das Verhaltnis der Frequenzen eines Photons an verschiedenen Stellen A und B imGravitationsfeld. Beobachten wir etwa die Emission einer Linie mit Wellenlangeλ∗ auf derOberflache eines kompakten Sterns, so erhalten wir

λB

λ∗

=ν∗νB

=1

√

1 − 2GM∗

c2R∗

. (3.120)

Die Spektrallinien eines kompakten Sterns werden deshalbrotverschoben

z =λB − λ∗

λ∗

=1

√

1 − 2GM∗

c2R∗

− 1 ≃ GM

c2R∗

. (3.121)

Dies ist die sog.gravitative Rotverschiebung. Fur Weiße Zwerge hat man Rotverschiebun-genz ≃ 10−4−10−5 gemessen (Sirius B), bei Neutronensternen wurden sich Rotverschiebun-genz ≃ 0.2 ergeben und bei Schwarzen Lochernz = ∞. Das Schwarze Loch hat geradeeinen RadiusR∗ = RS (Abb. 3.3).

Post–Keplerian Effects

Apart from gravitational redshift, three other general relativistic effects are observable in thesolar system and are nowadays of principal importance for the calculation of ephemerids ofplanets:

• The perihelion precession for the Mercury orbit by43 arcsec per century (Fig. 3.4);

• Light deflection on the solar surface by

θ⊙ =4GM⊙

c2R⊙

= 1.”75 (3.122)

• The Shapiro time–delay for signals propagating in the solarsystem. This effect is due tolonger propagation of signals in the space curved by the Sun compared to a propagationfar away from the solar surface.

General Relativity predicts the bending of light by gravity, gravitational time dilation andlength contraction, gravitational redshifts and blueshifts, the precession of Mercury’s orbit,and the existence of gravitational radiation. All these effects have been measured, although

3.4 Isotropic SpaceTimes 65

Figure 3.4: Relativistische Periheldrehung im Zweik”orperproblem. F”ur die Merkurbahnbetr”agt diese post–Newtonsche Periheldrehung nur 43 Bogensekunden pro Jahrhundert (hierstark ”ubertrieben).

gravitational radiation has been observed only indirectlyvia the decay of the orbits of binarypulsars. The LIGO project is an attempt to build a giant Michelson-Morley type of interfer-ometer to detect gravitational radiation directly. Two interferometers have been built, eachone with perpendicular light-carrying vacuum pipes 5 kilometers long.

The relativistic periastron shift and Shapiro time–delay are essential effects used in astron-omy to determine the exact pulse arrival times for radio pulses emitted by pulsars in compactbinary systems (see Camenzind 2006).

3.4 Isotropic SpaceTimes

The observed high degree of isotropy of the CMBR implies strong constraints on possibleworld models. In a first part we have to regain the 3+1 aspects of a spacetime.

3.4.1 Slicing of SpaceTime

Any spacetime(M, g) can be decomposed by means of the diffeomorphism

Φ : M → Σ × I (3.123)

whereΣ is considered as a spatial part (3D space) andI as the timeflow. The entire spacetimeconsists then of a set ofΣ–slices given by some intrinsic metric. This decompositionis called


Figure 3.5: In a gravitational lense, the gravitational field of a galaxy e.g. deflects thephotonpaths so that multiple images can occur.

Figure 3.6: Photon trajectories are strongly affected by the gravitational field of a rotating BlackHole. The Black Hole is bombarded by laser photons along the equatorial plane. Photons withlow impact parameters are captured by the horizon.

the3 + 1–split of spacetime.The tangent vector field∂t can now be decomposed into a part normal to the slice and one

along the slice

∂t = α~n + ~β . (3.124)


Figure 3.7: Slicing of a spacetime into a sequence of 3–spaces with metricγ. The slicingdefines the lapse functionα and the shift vector~β.

~n is a normal vector to the sliceΣt, α is calledlapse function, and~β which is parallel toΣt

is calledshift vector. We define now a particular observer given by the vector field~n and 3orthonormal vector fieldse(3)

i in the slice

e0 = ~n =1

α

(

∂t − ~β)

, ei = e(3)i . (3.125)

Similarly, we can introduce a basis of one–forms adapted to the slice

Θ0 = dt , Θi = θi + βi dt . (3.126)

From a direct calculation, one can verify

Θa(eb) = δab , (3.127)

and the metric is expressed as

g = ηab ΘaΘb . (3.128)

With this we can calculate

g(∂t, ∂t) = −Θ0(∂t)Θ0(∂t) + δikΘi(∂t)Θ

k(∂t)

= −α2 + βiβk (3.129)

g(∂t, ei) = −Θ0(∂t)Θ0(ei) + δikΘi(∂t)Θ

k(ei) = βi . (3.130)


Space–times around stellar objects are asymptotically flat, i.e. Minkowskian far from thesurface of the star. Cosmological spaces are not asymptotically flat, but they should havetime–slices which are required to behomogeneous and isotropic. It is then a well known factin differential geometry that a Riemannian manifold of dimensionn > 2 which is required tobe isotropic about each point is necessarily of constant curvature.

In the cosmos we require the existence of a unique time–flow which is associated with theclusters of galaxies. In terms of this timet, space–time can be sliced (i, k = 1, 2, 3)

ds2 = α2 dt2 − γik(t, ~x)(dxi + βidt)(dxk + βkdt) . (3.131)

The shift vectorsβi have to vanish according to our time choice. The 3–tensorγik is then themetric of the slicesΣt for const timet. These slices are 3D Riemannian manifolds which arenow required to be isotropic around each point.

In a cosmological spacetime(M, g), there will be a family of preferred world linesrepresenting the average motion of matter at each point(these represent the histories ofclusters of galaxies, with associated fundamental observers). Their 4–velocity is

uα =dxα

dτ(3.132)

whereτ is the proper time measured along the world lines. At recent times this velocity istaken to be the 4–velocity defined by the vanishing of the CMBRdipole. There is preciselyone such 4–velocity which will set this dipole to zero. It is usually assumed that this is thesame as the average 4–velocity of matter in a suitably sized volume.

3.4.2 Isotropic Riemannian Spaces

Observations tell us that there is a special flow vector fieldu for which the Universe alwayslooks isotropic. This vector field is associated with the mass centers of galaxy clusters (Virgo,Coma etc). The flow vector field of individual galaxies alwaysincludes some particular mo-tions due to the confinement in a cluster.

Isotropy of SpaceTimes

Isotropy applies at some specific point in the space, and states that the space looks the same nomatter what direction you look in. More formally, a manifoldM is isotropic around a pointpif, for any two vectorsV andW in TpM , there is an isometry ofM such that the pushforwardof W under the isometry is parallel withV (not pushed forward).

Homogeneity is the statement that the metric is the same throughout the space. In otherwords, given any two pointsp andq in M , there is an isometry which takesp into q. Note thatthere is no necessary relationship between homogeneity andisotropy; a manifold can be ho-mogeneous, but nowhere isotropic, or it can be isotropic around a point without being homo-geneous (such as a cone, which is isotropic around its vertexbut certainly not homogeneous).On the other hand, if a space isisotropic everywherethen it is homogeneous. (Likewise if it isisotropic around one point and also homogeneous, it will be isotropic around every point.)5

The existence of this global symmetry has then essential consequences:

5This can be formulated more mathematically for the case of a spacetime: A spacetime is said to be isotropic ineach of its eventsp with respect to a given vector fieldu with g(u, u) = 1, if the setISOp of all isometries whichleave invariant the pointp is a subset of the rotational groupSO(3)(Vp). All these transformations operate in thespace orthogonal tou.


• The vector fieldu has to be orthogonal to the slicesΣt, i.e. the corresponding shift vectorfields~β have to vanish,~β = 0 for all events.

• The integral curves of the vector fieldu are geodesics,∇uu = 0. This has the conse-quence that the distance between two slicesΣt1 andΣt2 is independent of time. Thisleads to the introduction of apreferred cosmic timet such that

ds2 = dt2 − R2(t) γik dxi dxk . (3.133)

3.4.3 Spaces of Constant Curvature

If a manifoldM is isotropic, then its sectional curvatureKp(E) only depends onp and not onthe planeE in TpM , and, by Schur’s Lemma it has constant sectional curvature.This meansthat the hypersurfacesΣt are spaces of constant curvature.

As we have seen, for a given connection∇ on M curvature is given by a tensor of rank(1,3), i.e. by a linear mapping ofTpM × TpM × TpM → TpM defined by the relation (seeSect. 3.2.4)

R(u, v)w = ∇u∇vw −∇v∇uw −∇[u,v]w . (3.134)

u, v andw are vector fields onM , [u, v] = Luv is the Lie–bracket of two vector fields. Thiscan also be written in terms of a fourth vector fieldz as

R(u, v, w, z) ≡ g(R(u, v)w, z) = Rαβµν zαwβuµvν . (3.135)

This tensor is antisymmetric in both pairs of indices and satisfies the Bianchi identity.Thesectional curvaturewith respect to a planeE in TpM is given by

Kp(E) = R(e1, e2, e1, e2) , (3.136)

when(e1, e2) is a basis ofE. If the sectional curvature is constant for all planesE in TpMand for all eventsp, this defines a space of constant curvature.

Schur’s Lemma: Isotropy around each point leads to a homogeneous space. Or ex-actly: If the sectional curvature Kp(E) in a Riemannian space of dimensionn ≥ 3 isindependent ofE for each point p, thenKp(E) is also independent onp, i.e. M is a spaceof constant curvature.Mathematically, a Riemannian space that has constant curvature must have a Riemann curva-ture tensor of the following form

Rabcd = K(gacgbd − gadgbc) (3.137)

In this equationK is a constant sometimes simply called the curvature of the space. In spacesof constant curvature, the spaces are qualitatively different, depending onK > 0, K = 0 andK < 0. The Ricci tensor has then the form

Rbd = gacRabcd = K(n − 1)gbd , (3.138)

and the Ricci scalar

R = Raa = K n(n − 1) . (3.139)


The curvature 2–form assumes the simple form

Ωab = K Θa ∧ Θb . (3.140)

We now specialize to 3–space and contract this to get the Ricci–tensor

Rbd = gacRabcd = 2Kgbd . (3.141)

Hence, in the 3D–subspace, the Ricci tensor is proportionalto the metric tensor.

3.4.4 Friedmann–Robertson–Walker (FRW) SpaceTimes

Conformal Coordinates: Riemann spaces of constant curvature are conformally flat,i.e. there is a coordinate system such that6

γ =1

Ψ2

∑

i

(dxi)2 . (3.146)

In a next step we prove thatin a space of constant curvature one can always find localcoordinatesxi such that the metric is expressible as7

γ =∑

i=1,n

(dxi)2

(1 + Kρ2/4)2, ρ2 =

∑

i

(xi)2 . (3.157)

6In order to prove this we use the fact that a manifold with dimension n ≥ 3 is flat if and only if the Weyl tensorvanishes. The Riemann tensor can be decomposed into a part given by the Ricci tensor and a second part given bythe Weyl tensorCabcd

Rabcd = Cabcd −1

n − 2

(

gabRcd + gbdRac − gbcRad − gadRbc

)

+1

(n − 1)(n − 2)(gabgcd − gadgbc)R . (3.142)

This tensor has the same symmetries as the Riemann tensor, but in addition its Ricci–part vanishes

gbdCabcd = 0 . (3.143)

The deeper differential geometric meaning of the Weyl tensor is the conformal invariance, i.e. its invariance againsttransformations of the formg → exp(Φ(x)) g. Therefore, a metric is calledconformally flatif there is a transfor-mation of the form

gab → exp(Φ) δab . (3.144)

This means that the Weyl tensor vanishes and that the forms

Cab =1

2Cabcd Θc ∧ Θd (3.145)

identically vanish in a conformally flat space. One can aslo show the opposite: if the Weyl tensor of a space vanishes,then this space is conformally flat.

7For this we consider the metric of the form of Eq (3.146) and define an orthonormal frameΘi = dxi/Ψ withγ =

∑

i Θi ⊗ Θi. For the exterior derivative we obtain

dΘi =Ψ,j

Ψ2dxi ∧ dxj = Ψ,jΘ

i ∧ Θj = −ωik ∧ Θk . (3.147)

The last equality follows from the first Cartan equation. Since

dgij = ωij + ωji = 0 , (3.148)

ωij are antisymmetric. For this reason, we have the solution

ωij = −Ψ,jΘi + Ψ,iΘj (3.149)

3.5 The FRW World Models 71

For a 3–space we may express this line element in the form of

dσ2(3) =

1

(1 + Kρ2/4)2

∑

i=1,3

(dxi)2 . (3.158)

We can use the 3D forms of the line element that we have constructed to form the full 4Dcosmological line element. In 4D we have

ds2 = c2 dt2 − R2(t) dσ2(3) . (3.159)

R(t) is a general scaling function of timet. The set of all these spaces defines the Friedmann–Robertson–Walker (FRW) spacetimes. One can then always normalize the constantk =±1, 0.

3.5 The FRW World Models

In standard textbooks, the FRW model is given in a somewhat different coordinate system

ds2 = c2 dt2 − R2(t)

(

dr2

1 − kr2+ r2(dθ2 + sin2 θ dφ2)

)

. (3.160)

From the second Cartan equation we derive the curvature

Ωij = dωij + ωmi ∧ ωm

j

= Ψ(−Ψ,jmΘm ∧ Θi + Ψ,imΘm ∧ Θj) − Ψ,iΨ,mΘm ∧ Θj

+ (−Ψ,mΘi + Ψ,iΘm) ∧ (−Ψ,jΘm + Ψ,mΘj) . (3.150)

By combining different terms we obtain

Ωij = −ΨΨ,jmΘm ∧ Θi + ΨΨ,imΘm ∧ Θj − Ψ,mΨ,mΘi ∧ Θj

= KΘi ∧ Θj . (3.151)

This is only possible ifΨ,im = 0 for i 6= m. Ψ is therefore a linear combination of functionsf(xi), Ψ =∑

i f(xi).By inserting this into the curvature we get

Ωij =(

Ψ(f ′′

i + f ′′

j) −∑

m

(f ′2))

Θi ∧ Θj (3.152)

From this we find

f ′′

i + f ′′

j =(

∑

m

(f ′2) + K)

/Ψ (3.153)

Since the rhs is independent ofi and j, we must havef ′′

i = f ′′

j = const, i.e. these are quadratic functionsfi(xi) = ax2

i + bixi + ci. ThereforeΨ = αρ2/4 + 1. By inserting into the curvature, we findα = k. Therefore,we have found the solution

Ψ = 1 +K

4ρ2 (3.154)

Θi =dxi

1 + Kρ2/4(3.155)

ωij =K

4(xiΘj − xjΘi) . (3.156)

The global classification of Riemannian manifolds with constant curvature is a delicate subject.


For this we make the substitution

r =ρ

1 + Kρ2/4, dr =

r

ρdρ − K

2r2 dρ (3.161)

With this we can verify that

dr2

1 − kr2=

dρ2

(1 + Kρ2/4)2. (3.162)

Therefore the above metric can be expressed in the form

Figure 3.8: 3–spaces of constant curvature.

ds2 = dt2 − R2(t)

(1 + kρ2/4)2

(

dρ2 + ρ2(dθ2 + sin2 θ dφ2)

. (3.163)

For k = 1, we can show that ther = 1 singularity is just a coordinate singularity bychanging to the specially suited coordinates involving thenew coordinateχ defined in termsof r as

r = sin χ (3.164)

The singularity is therefore eliminated since

dr = cos χ dχ =√

1 − r2 dχ (3.165)

The time–slice is then given by

dσ2(3) = R2 [dχ2 + sin2 χ(dθ2 + sin2 θ dφ2)] . (3.166)

Simalarly, the open case,k = −1, is obtained byr = sinhχ such that

dr = cosh χ dχ =√

1 + r2 dχ . (3.167)

The corresponding time–slice has the form

dσ2(3) = R2 [dχ2 + sinh2 χ(dθ2 + sin2 θ dφ2)] , (3.168)

3.6 The Origin of the Cosmological Redshift 73

with the following ranges

0 ≤ χ < ∞ , 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π . (3.169)

Flat space,k = 0, can trivially be expressed in terms of these coordinates

dσ2(3) = R2(t)

(

dχ2 + χ2(dθ2 + sin2 θ dφ2))

. (3.170)

Summary

We have found three equivalent formulations (i.e. coordinate systems) for a Friedman–Robertson–Walker Universe

• in quasi–spherical coordinates(t, r, θ, φ)

ds2 = c2 dt2 − R2(t)[ dr2

1 − kr2+ r2(dθ2 + sin2 θ dφ2)

]

. (3.171)

• in conformal coordinates(t, ρ, θ, φ)

ds2 = c2 dt2 − R2(t)

(1 + kρ2/4)2

[

dρ2 + ρ2(dθ2 + sin2 θ dφ2)]

. (3.172)

• in hyper–spherical coordinates(t, χ, θ, φ)

ds2 = c2 dt2 − R2[

dχ2 + S2(χ)(dθ2 + sin2 θ dφ2)]

, (3.173)

whereS(χ) = sin χ, χ, sinh(χ) depending on the curvature.

The only degree of freedom is the expansion factorR(t), which has to be determined byEinstein’s equations.

3.6 The Origin of the Cosmological Redshift

The FRW model explains in a natrual way the cosmological redshift. For this we may assumethat each galaxy is given by its comoving coordinates(r, θ, φ). Let us consider a galaxy withcoordinates(r1, θ, φ) emitting photons at timet1. Photons follow null geodesics,ds2 = 0(Fig. 3.9). Thus

c dt = ± R dr√1 − kr2

. (3.174)

Sincer decreases for light propagating towards us, the minus sign is appropriate. For photonsdetected at timet0 we then have

∫ t0

t1

c dt

R(t)=

∫ r1

0

dr√1 − kr2

. (3.175)


Figure 3.9: Worl lines in an expanding universe: the middle line represents the oberver, neigh-boring world lines are associated with neighboring galaxies. The backward light cone is locallysimilar to the flat space limit, is however strongly curved in the early universe. Photons detectedby observers move along the backward light cone and are emitted at the intersections betweenthe light cone and the trajectory of the galaxy.

Let us consider wave crests, emitted att1 andt1 + ∆t1∫ t0+∆t0

t1+∆t1

c dt

R(t)=

∫ r1

0

dr√1 − kr2

. (3.176)

ForR(t) roughly constant over∆t0, we obtain by subtracting both equations

c∆t0R(t0)

− c∆t1R(t1)

= 0 . (3.177)

Therefore,

c∆t0c∆t1

=ν1

ν0=

λ0

λ1=

R(t0)

R(t1)= 1 + z , (3.178)

with z as the redshift.The wavelength of the photons get stretched by propagating throughthe universe, R(t0) > R(t1). This is the natural explanation of the redshift detected byHub-ble in 1929.

3.7 The Luminosity Distance and the Hubble–Law

Photons represent collisionless particles and must be treated in terms of phase–space distri-bution functionsf(x; ~p) with 3–momentum~p. We consider a stream of particles propagating

3.7 The Luminosity Distance and the Hubble–Law 75

rea(to)

r =

r e

to

r =

r e

dte D

r =

0

Figure 3.10: Observations in an expanding Universe. The observer’s world line is at r = 0, theemitter’s world line atr = re which is stretched toa(t0)re at present time. The emitter objecthas a physical extensionD.

freely in a FRW–spacetime. At some timet, a comoving observer findsdN particles in aproper volumedV , all having momenta in the range[~p, ~p + d3p]. The phase–space distri-bution functionf for the particles is now defined by the relationdN = f dV d3p. At alater instantt + ∆t, the proper volume of the particles would have increased by afactor[R(t + ∆t)/R(t)]3, whereas the volume in the momentum space would be redshifted bythe same factor[R(t)/R(t + ∆t)]3. The total phase volume occupied by the particles doestherefore not change during free propagation. Because the number of particlesdN is alsoconserved without interactions, thephase–space distribution functionf is conserved alongthe streamline.

Photon momenta are characterized by the energyE = hν and a direction~n on the unitsphere,~p = (hν/c)~n. The momentum space volume is therefored3p = ~p2 dp dΩ ∝ ν2 dν dΩ,so that the photon number is given by

fγ =dNγ

d3x d3p=

dNγ

[cdte dAe][ν2e dνedΩe]

=dNγ

[cdt0 dA0][ν20 dν0dΩ0]

. (3.179)

dΩ is the solid angle around around the direction of propagation ~n, andd3x ∝ cdt dA, wheredA is the area normal to the direction of propagation. The subscript e represents the processof emission and0 the process of detection. This shows that

fγ =dNγ

dt dAν2 dν dΩ(3.180)

is invariant along the streamline. In Astronomy, instead offγ the radiation field is character-


ized in terms of the specific intensityIν = Iν(x;~n) defined as

hν dNγ = Iν dt dAdν dΩ . (3.181)

This shows thatIν/ν3 is invariant along streamlines.The quantityIν will have units erg s−1 cm−2 Hz−1 Steradian−1. The specific energy

densityuν = (4π/c)Iν will have the dimensions erg cm−3 Hz−1. The energy flux in theradiation scales therefore as

Fν(1+z)(z) = Fν(0) (1 + z)3 . (3.182)

The total flux of radiation is obtained by integration over the full spectrum and varies thereforeas

F =

∫ ∞

0

Fν dν ∝ (1 + z)4 . (3.183)

The energy density in the CMBR scales therefore asu ∝ (1+z)4, which is compatible with theStephan–Boltzmann lawu = aT 4. Radiation fields of the formIν = ν3 G(ν/T ) will retainthe spectral shape under the expansion of the Universe, since the temperature is redshifted asT ∝ R−1. As a consequence, a Planckian spectrum remains a Planckianspectrum under theexpansion (see COBE results for the CMBR).

The Luminosity Distance

Cosmological observations are based on electromagnetic radiation of sources (galaxies, quasars,GRBs) at high redshift. Let an observer located atr = 0 receive at timet0 radiation of a sourcelocated atr = r1, emitted at earlier timet1 (Fig. 3.9). LetL∆t1 the total energy emittedby a galaxy per unit of time, at timet1 < t0. This energy will be detected by an observer inthe time–interval∆t0 = ∆t1 (R(t0)/R(t1)), and the energy will be redshifted by the amountR(t1)/R(t0). For isotropic emission, we can calculate the radiation fluxof a 2–sphere withradiusr1. For t = const andr = const the line–element is given by

ds2(2) = r2R2(dθ2 + sin2 θ dφ2) . (3.184)

This is the line element of an Euclidean 2–sphere with radiusrR. At time t0, the light emittedby a galaxyG is distributed on a 2–sphere with surfaceA = 4πr2

1R2(t0). The flux observed

in a detector corresponds to the emitted energy by means of the relation

F ∆t0 =L∆t1

4πr21R

2(t0)

R(t1)

R(t0), (3.185)

i.e. the observed flux is given by

F =∆t1∆t0

R(t1)

R(t0)

L

4πR2(t0)r21

=L

4πR20r

21

(

R(t1)

R(t0)

)2

. (3.186)

Here, we used relation (3.178) between the time interval in the observed system and the timeinterval of the emitter.8 This determines the expression for the bolometric luminosity

Fbol =L

4πr21R

2(t0) (1 + z)2=

L

4π d2L

(3.187)

8The stretching of time–scales has been seen in the light curves of Supernovae: the observed decay time of thelight curve is longer than the intrinsic decay time,∆t0 = (1 + z)∆SN .

3.7 The Luminosity Distance and the Hubble–Law 77

For this reason, the quantity

dL(z) ≡ r1R(t0)(1 + z) (3.188)

is called theluminosity distance of an object with redshiftz. This distance is used in theclassical distance modulus

mbol − Mbol = 5 log[dL/pc] − 5 . (3.189)

The Hubble–Law

The Hubble–law is a general consequence of the FRW model for the Universe and does notdepend on the particular expansion law. For this we make a Taylor expansion around thepresent state of the Universe at timet0

R(t) = R(t0 − [t0 − t])

= R(t0) − R(t0 − t) +1

2R(t0 − t)2 + O(∆t3)

= R0

(

1 + H0(t − t0) −1

2q0H

20 (t − t0)

2 + O(∆t3)

)

. (3.190)

Here we introduced theHubble parameter H and thedeceleration parameterq0

H0 = (R/R)t0 (3.191)

q0 = −[R/(RH2)]t0 . (3.192)

A dot always corresponds to differentiation with respect tocosmic timet. These notions arevalid at any cosmic time. We will show in the following thatH0 is indeed equivalent to theHubble constant. Since the expansion factor is equivalent to redshift,1 + z = R0/R(t), theabove formula can also be written as

1 + z =

(

1 + H0(t − t0) −1

2q0H

20 (t − t0)

2 + O(∆t3)

)−1

≃ 1 + H0(t0 − t) +1

2q0H

20 (t − t0)

2 + H20 (t − t0)

2 + O(∆t3) . (3.193)

This means for the redshift

z = H0(t0 − t) +

(

1 +1

2q0

)

H20 (t − t0)

2 + O(∆t3) . (3.194)

For not too early times, i.e.H0|t − t0| ≪ 1, or z ≪ 1, this can be inverted to give

t0 − t =1

H0

(

z − (1 + q0/2) z2

)

+ O(z3) . (3.195)

t0 − t is the time since the emission of a photon at timet (socalled look–back time). It isimportant to note thatthese relations only depend on the present–day valuesH0 and q0 ofthe Hubble and decelration parameters.


Since photons propagate along null geodesics,dΩ = 0,

ds2 = dt2 − R2(t)dr2

1 − kr2= 0 , (3.196)

we find for the relation between the propagation time and distance∫ t0

t

cdt

R(t)=

∫ r

0

dr√1 − kr2

≃ r . (3.197)

This integral gives

rR0 =

∫ t0

t

R0

R(t)dt =

1

R0

∫ t0

t

(1 + z) dt

=1

R0

∫ t0

t

[

1 + H0τ +

(

1 +1

2q0

)

H20 τ2]

dτ

= c(t0 − t)(

1 +H0

2(t0 − t) + O([H0(t − t0)]

2))

. (3.198)

Using the above relation between time and redshift, equation (3.195), this yields

rR0 =c

H0

(

z − (1 + q0/2)z2 +1

2z2 + O(z3)

)

=c

H0

(

z − 1

2(1 + q0)z

2 + O(z3))

. (3.199)

This shows that the metric distancerR0 is given by the Hubble radiusc/H0 and that curvatureeffects enter in quadratic order of redshift.

By using the general expression for the luminosity distance, dL = r1R0(1+z), we recoverHubble’s law upto second order in the redshift, which is typically valid for redshiftsz < 0.1,i.e. upto distances of about 500 Mpc (see Fig. 2.4),

dL = dL(z) =c

H0

(

z +1

2(1 − q0)z

2 + O(z3)

)

. (3.200)

Hubble’s law would be correct even in second order of redshift in a decelerating Universe withq0 = 1.

The luminosity distance for redshiftsz > 0.1 depends on the particular FRW model,i.e. on the expansion lawR(t). For a de Sitter model driven by vacuum energy,R(t) =Ri exp(H(t − ti)), one finds (q0 = −1)

ddSL (z) =

c

Hz(1 + z) . (3.201)

Luminosity distances at high–redshift behave in general quadratic with the redshift, and notlinear.

As a fundamental fact we have found that the Hubble constantH0 is just the presentexpansion velocity of the Universe

H0 =

(

R

R

)

0

. (3.202)

3.8 The Hubble Constant 79

In Astronomy, the radiation fluxf is usually expressed in terms of magnitudes

m = −2.5 log f + const , (3.203)

which can be handled by the absolute magnitude

Mbol = −2.5 logL

L⊙

− 4.75 . (3.204)

L⊙ = 3.90 × 1026 Watt is the solar luminosity. This gives rise to the distancemodulus

DM = m − M = 5.0 log[dL/10 pc] . (3.205)

In suitable units, this can be written then as,[H0] = km/s/Mpc,[cz] = km/s,

DM = 48.35 − 5 log h + 5 log(z) + 1.086(1 − q0)z . (3.206)

3.8 The Hubble Constant

As we have discussed in Sect. 2.2, the determination of the Hubble constant is one of the keytasks in observational cosmology. The various methods which lead to still somewhat differentvalues forH0 are summarized in Fig. 3.11 which is due to Tammann [8]. A key element inthis procedure is the calibration of the distance modulus for the Virgo cluster.

Method (m − M)Virgo Galaxy type

Cepheids 31.52 ± 0.20 SNovae 31.46 ± 0.40 EGlobular Clusters 31.67 ± 0.15 ETully-Fisher 31.58 ± 0.24 SDn − σ 31.85 ± 0.19 S0, E

Average 31.66 ± 0.09 → 21.5 ± 0.9 Mpc

Table 3.2: Distance modulus to the Virgo cluster according to Tammann [8].

3.9 The Apparent Angular Width of Galaxies

Another important measurable quantity is the apparent angular diameter of a galaxy . Whatis the angular extension of a galaxy at redshift one, or what is the angular width of a jet withlength of 1 Mpc at redshift 5 ? For this consideration we may assume that the corners aregiven by(θ1, φ1) and(θ1 + dθ, φ1). From this we obtain the true width of the source

ds2 = −r21R

2(t1) (∆θ1)2 = −D2 (3.207)


Figure 3.11: Distance ladder for the measurement of the Hubble constant accordingto Tam-mann. These calibrators do not include SN Ia.

with the physical diameterD of a galaxy. Therefore, the angular width is given by

∆θ0 =D

r1R(t1)=

D(1 + z)

r1R(t0). (3.208)

This can be wrtitten in terms of the luminosity distance

∆θ0 =D

dA=

D(1 + z)2

dL, dA = dL(z)/(1 + z)2 . (3.209)

This demonstrates that the classical resultD/dL is wrong for higher redshifts. SincedL ∝c/H0, the overall angular width of cosmological objects is always of the order ofθ ≃ D/RH .

3.10 Number Counts in the Expanding Universe 81

For a galaxy cluster e.g. this meansθcl ≃ 10−3. In a de Sitter model, the apparent angularwidth of an object will be constant at high redshifts

∆θdS0 =

D

dA(z)=

D

RH

(1 + z)2

z(1 + z)→ D

RH. (3.210)

This is in fact the minimal angular width, since in a realistic model the angular width willslightly increase with redshift beyond redshift one. A typical disk galaxy with extension of 30kpc will subtend an angle of7×10−6 rad, which amounts to 1.4 arcseconds. A galaxy clusterwith an extension of 3 Mpc will subtend a minimal angle of about 2 arcminutes.

3.10 Number Counts in the Expanding Universe

The number of cosmological sources in a volume section at a point P down the null cone is

dN = d2A dΩ0 [n(−kaua)]P dy . (3.211)

n is the number density of radiating sources (galaxies, quasars, GRBs) per unit proper volumein a section of bundle of light rays converging towards the observer and subtending a solidangledΩ at the observer’s position,dA is the area distance of this section from the observer’spoint of view (also called angular distance),ua is the observer’s 4–velocity,ka the tangentvector of the null rays andy the affine parameter distance down the light rays (Fig. 3.12). nis measured in the rest frame of the counted sources. This expression is metric independentand therefore valid for all cosmological models (first derived by Ellis 1973).

The above form is written in unobservable variables. Since the redshift is given by

1 + z =(uaka)P

(uaka)Observer(3.212)

the number density can be expressed in terms of redshift

dN = d2A dΩ0 [n]P (1 + z)(−kaua)]O dy . (3.213)

The minus sign is chosen because we are interested in incoming light rays.k is a null vectorsatisfying

kaka = 0 , kbka;b = 0 . (3.214)

Thereforekaua = −1 and

dN = d2A dΩ0(1 + z)n(y) dy . (3.215)

n anddA are functions ofz, as well asy, and therefore we refer to the quantitydN/dz insteadof dN/dy.

FRW Model: The distance along the line of sight corresponding to a redshift interval dz isgiven by

dl = c dt = cdt

dR

dR

dzdz = − dz

1 + z

cR

R= − dz

1 + zdH(z) , (3.216)


Figure 3.12: Number counts of objects detected in an expanding universe.

wheredH(z) = c/H(z) is the Hubble–radius at redshiftz. In a given cosmological modelwe can determinedl/dz = dH/(1 + z). From this we obtain the proper volume element atredshiftz

dV =dH(z) dz

1 + zd2

A dΩ =R2

0r2em(z)dH(z)

(1 + z)3dz dΩ . (3.217)

Thus the number count of galaxies per unit solid angle and perredshift interval should begiven by

dN

dz dΩ= n(z)

dV

dz dΩ= n(z)

R20r

2em(z)dH(z)

(1 + z)3= n(z) dH(z)

d2L(z)

(1 + z)5. (3.218)

To compute this quantity explicitly, we need the luminositydistancedL(z) as well as theHubble radius as functions of redshift, which will be determined by means of the Friedmannequation.

3.11 Slightly Inhomogeneous World Models 83

3.11 Slightly Inhomogeneous World Models

The CMBR demonstrates that the present Universe is highly isotropic. In lowest order ofapproximation, the CMBR temperature does not depend on directions on the Sky. The sameholds for the Hubble constant. One could, however, imagine of world models where theexpansion slightly depends on directions on the Sky, but thematter would still be uniformlydistributed. Such Universes are called homogeneous. Thesemodels played some role e.g. ininvestigations of singularities.

The metric element can still be split in the 3+1 manner

ds2 = dt2 − γik dxi dxk . (3.219)

The manifoldsM are then locallyM = R × G the product of the time–axis with a groupspace. Completely inhomogeneous world models are difficultto construct. They are treatedonly in the limit of small perturbations (ripples of spacetime including gravitational waves).An approach to this will be discussed in Chapter 8.

3.12 Summary

• A consistent theory for the description of gravity in the Universe has to based on a curvedspacetime with a metric tensorgµν which embodies the gravitational potentials.

• Gravitational forces are related to the metric connectionΓ of this pseudo–Riemannianmanifold. Torsion is absent in Einstein’s theory of gravity.

• Curvature of this manifold is generated via Einstein’s equations by the matter distributionin form of the energy–momentum tensor for all kind of particles and fields.

• The Copernican principle requires the cosmic spacetime to be homogeneous and isotropic.The Friedman-Robertson-Walker Universe (FRW) is the most general model, its spatialsections are either flat, a 3–sphere or a 3–hyperboloid. The corresponding line–elementis very simple

ds2 = c2 dt2 − R2(t)[ dr2

1 − kr2+ r2(dθ2 + sin2 θ dφ2)

]

. (3.220)

(r, θ, φ) are adapted to Sky coordinates and are called comoving coordinates,k = 0,±1.

• In FRW, cosmological redshift of galaxies and quasars is a mere result of the stretchingof the 3–space.

• In FRW, the Hubble-law is a mere consequence of the expansionof the Universe, inde-pendent of the particular form of the expansion.

• The Hubble-law does not extend to redshifts beyondz ≃ 0.1. On scales withz >0.1, luminosity distance, apparent angular widths and number counts will depend on thespecific world model.

• The Hubble parameter is the relative expansion velocity andis therefore redshift–dependent.


3.13 Exercises

Distances in a de Sitter Model: Let us consider a particular FRW–world model given bythe expansion lawR(t) = Ri exp(H(t− ti)), H = const, k = 0 (de Sitter model). Calculate

• the deceleration parameterq;

• the luminosity distancedL(z);

• the angular width of an object with intrinsic diameterD as a function of redshift;

• the observed number densitydN/dz for a constant comoving density of objects.

Proper Volume: For the total cosmos, the number of galaxies in a spherical shell at radiusr andr + dr is then given by

dN =4πr2 dr√1 − kr2

n(t) . (3.221)

The timet depends onr over the null cone condition

∫ t0

t

cdt′

R(t′)=

∫ r

0

dr′√1 − kr2

. (3.222)

The total number of galaxies upto the radiusr1 is then

N(r1) =

∫ r1

0

4πr2n(t) dr√1 − kr2

. (3.223)

On the other hand, the volumeV over a distancer follows for k = 1

V+(r) =4π

3R3r3

[3

2

arcsinr

r3− 3

2

√1 − r2

r2

]

, (3.224)

and fork = −1 correspondingly

V−(r) =4π

3R3r3

[3

2

√1 + r2

r2− 3

2

arsinhr

r3

]

. (3.225)

For samll distancesr ≪ 1 we get the Euclidean result with a small correction from curvature

Vk(r) =4π

3R3r3

[

1 +3

10kr2 + O(r4)

]

. (3.226)

Bibliography

[1] Camenzind, M.: 2007,Compact Objects in Astrophysics – White Dwarfs, Neu-tron Stars and Black Holes, Springer–Verlag (Heidelberg)

[2] Carroll, S.M.: 2004,Spacetime and Geometry, Addison Wesley (San Francisco)(this is a modern introduction into the basics of General Relativity)

Bibliography 85

[3] Hobson, M.P., Efstathiou, G.P., Lasenby, A.N.: 2006,General Relativity – AnIntroduction for Physicists, Cambridge University Press

[4] Einstein, A.: 1917, Kosmologische Betrachtungen zur allgemeinen Rela-tivit atstheorie, Preuss. Akad. Wiss. Berlin, Sitzber. 142

[5] Hartle, J.B.: 2002,Gravity: An Introduction to Einstein’s General Relativity,Addison–Wesley

[6] Robertson, H.P.: 1935,Kinematics and world structure, ApJ 82, 248[7] Straummann, N.: 2004,General Relativity. With Application to Astrophysics,

Springer–Verlag, Heidelberg (this is a more mathematically oriented approach toGeneral Relativity, with a complete introduction into modern differential geometry)

[8] Tammann, G., Parodi, B.R, Reindl, R.: 1999, inProc. IAU Coll. 167, ASP Conf.Series

[9] Walker, A.G.: 1936,On Milne’s theory of world–structure, Proc. Lond. Math. Soc.42, 90

4 The Universe with Matter and Dark Energy

The FRW world models essentially only depend on the expansion factor R(t). The timeevolution for the expansion factor results from Einstein’sequations, which couple curvaturewith the matter content of the Universe. As a first input we need a model for the descriptionof matter in the Universe, which includes all types of particles and fluids.

4.1 Description of Matter in a Relativistic Cosmos

The present Universe contains various forms of matter: photons, neutrinos, baryons, galaxies,dark matter and dark energy.

4.1.1 Fluid Approach

A great many astrophysical systems may be approximately regarded asperfect fluids. A per-fect fluid is defined as having at each point a velocity~v, such that an observer moving withthis velocity sees the fluid around him as isotropic. This will be the case if the mean free pathbetween collisions is small compared with the scales used bythe observer.

First we suppose that we are in a frame of reference in which the fluid is at rest at some par-ticular position and time. The perfect fluid hypothesis tells us then that the energy–momentumtensor takes the form

T 00 = ρ (4.1)

T 0i = 0 (4.2)

T ik = Pδik . (4.3)

The coefficientsρ andP are proper energy density and pressure, respectively. Now we goto a reference system at rest in the lab, and suppose that the fluid in this frame appears to bemoving with velocity~v. The connection between the two systems is given by a Lorentzboost

Λαβ(~v) with Lorentz factorγ = 1/

√

1 − ~β2, where~β = ~v/c. SinceT is a symmetric tensorof second rank, we have

Tαβ = Λαγ(~v)Λβ

δ(~v) T γδ . (4.4)

This gives explicitly

T 00 = γ2 (ρ + P~v2) (4.5)

T i0 = γ2 (ρ + P )vi (4.6)

T ik = Pδik + γ2(ρ + P )vivk . (4.7)

88 4 The Universe with Matter and Dark Energy

This can be integrated into a single equation for the energy–momentum tensor of a perfectfluid

Tαβ = (ρ + P )UαUβ − P gαβ , (4.8)

whereUα is the velocity 4–vector

U0 =dt

dτ= γ , ~U =

d~x

dτ= γ~v . (4.9)

Apart from energy and momentum, a fluid will aslo carry a conserved number current.If n is the particle number density in a Lorentz frame that moves with the fluid, then in thisframe the particle current is given by

N0 = n , N i = 0 . (4.10)

In any other Lorentz frame the expression follows from a boost transformation

N0 = Λ0β(~v)Nβ = γn (4.11)

N i = Λiβ(~v)Nβ = γ nvi , (4.12)

or more concisely

Nα = nUα . (4.13)

This current is conserved,∂αNα = 0, and also the energy–momentum is conserved,Tαβ,α = 0.

This latter equations correspond to Euler equations

∂t~v + (~v · ∇)~v = −1 − ~v2

ρ + P(∇P + ~v ∂tP ) (4.14)

and the entropy conservation

∂ts + (~v · ∇)s = 0 (4.15)

for perfect fluids.s denotes the specific entropy of the fluid.According to the covariance principle, the form of the energy–momentum tensor is main-

tained when going to curved spaces with the same interpretation for ρ andP , keeping the formof Eq (4.8), and the equations of motion follow from the covariant divergence

Tαβ;α = 0 , (4.16)

where the semicolon refers to the covariant derivative on the space–time. Similarly, particlenumber conservation follows fromNα

;α = 0.

4.1.2 Kinetic Description

Galaxies (pressureless fluids,P = 0) and baryons can be modeled in terms of the fluid ap-proach. But photons, dark matter and neutrinos are collisionless species and require a kineticapproximation. To describe a non–uniform system of particles in Minkowski spacetime oneintroduces a local densityn(t, ~x). This quantity is defined in such a way thatn(t, ~x)∆3x

4.1 Description of Matter in a Relativistic Cosmos 89

gives the average number of particles in the spatial volume element∆3x at the position~x andtime t. Similarly one defines a particle flow~j(t, ~x). Together they form a four–vector fieldNα = (cn(t, ~x),~j(t, ~x)). Let us consider a simple system of relativistic particles of massmwith momenta~p and energycp0. If the number of particles is large, we introduce a functionf(x, ~p) which gives the distribution of 4–momentapα at each event. The definition is suchthatf(x, ~p)∆3x∆3p gives the average number of particles which at timet are located in thevolume element∆3x at the point~x with momenta lying in the range[~p, ~p + ∆~p].

Particle Current

With the help of this distribution function we define the particle density as

n(t, ~x) =

∫

d3p f(x, ~p) . (4.17)

In the same way we introduce the particle flow as

~j(t, ~x) =

∫

d3p~v f(x, ~p) = n < ~v > , (4.18)

where~v = c~p/p0 is the velocity of the relativistic particle with momentump. The particleflow–vector has therefore the covariant form

Nα(x) = c

∫

d3p

p0pα f(x, ~p) . (4.19)

Since the volumed3p/p0 on the mass–shell is invariant under Lorentz transformations, thephase space distributionf(x, p) must be a scalar object,d3p/p0 is the Lorentz–invariant vol-ume on the mass–shell.

Energy–Momentum Tensor

Since the energy per particle iscp0, the average energy density can be written as

T 00 = c

∫

d3p p0 f(x, ~p) , (4.20)

whereT 00 denotes the energy density, as indicated by the above expression for perfect fluids.In a similar manner, we may define the energy flowcT 0i with

T 0i =

∫

d3p p0 vi f(x, ~p) . (4.21)

Finally, we introduce the momentum flow (or pressure tensor)which is the flow in directionkof the momentum in directioni

T ik =

∫

d3p pi vk f(x, ~p) . (4.22)

In fact, this object is a two–tensor which can be written in compact and covariant form

Tαβ(x) = c

∫

d3p

p0pαpβ f(x, ~p) . (4.23)


Hence, in the relativistic kinetic theory,the energy–momentum tensor is defined as the secondmoment of the distribution function, and thus a symmetric tensor. The particle current is thefirst moment.

In this form, the energy–momentum tensor takes only the restenergy and the kinetic en-ergy of the particles into account. This is the case for dilute systems in the sense that theinteraction energy of the particles is small as compared to their kinetic energies. Otherwise,the energy–momentum tensor should include a potential energy contribution.

Entropy Flow

TheH−function, introduced by Boltzmann, implies that the local entropy density for a systemoutside equilibrium may be defined as

S0(x)/c = −kB

∫

d3p f(x, ~p)[

log[h3f(x, ~p)] − 1]

(4.24)

with a constanth such thath3f(x, p) is dimensionless. Conveniently,h is taken as the Planckconstant. The entropy flow has then the form

~S(x) = −kB

∫

d3p~vf(x, ~p) [log[h3f(x, ~p)] − 1] . (4.25)

This can be combined into a covariant form

Sα(x) = −kBc

∫

d3p

p0pαf(x, ~p)

[

log h3f(x, ~p) − 1]

. (4.26)

Equilibrium Distributions

In the early universe, the plasma is extremely hot, completely ionized and so dense that typ-ically interactions occur on time–scales much shorter thanthe expansion time–scale. Denseplasmas are described in terms of a phase–space distribution feq(t, ~p), also called occupationnumber, which is isotropic in phase–space. As shown above, the number densityn, the energydensityρ and the pressureP follow then from integrals over phase–space, given in the restsystem of the plasma,

n =g

(2π~)3

∫

feq(t, ~p) d3p (4.27)

ρc2 =g

(2π~)3

∫

E(~p) feq(t, ~p) d3p (4.28)

P =g

(2π~)3

∫ |~p|23E

feq(t, ~p) d3p . (4.29)

E is the relativistic energy of a particle corresponding to momentum~p, andg is the spin–degeneracy factor for a particle species. For particles in kinetic equilibrium, the distributionfunction is either the Fermi– or Bose–distribution, which only depend on energy and temper-ature, but not on angles in phase–space

feq(|~p|) =1

exp[(E − µ)/kT ] ± 1(4.30)

4.1 Description of Matter in a Relativistic Cosmos 91

with chemical potentialµ.In thermal equilibrium, these plasma parameters are then given by

n =g

2π2~3c3

∫ ∞

m

√E2 − m2c4

exp[(E − µ)/kT ] ± 1E dE (4.31)

ρc2 =g

2π2~3c3

∫ ∞

m

√E2 − m2c4

exp[(E − µ)/kT ] ± 1E2 dE (4.32)

P =g

6π2~3c3

∫ ∞

m

(E2 − m2c4)3/2

exp[(E − µ)/kT ] ± 1dE . (4.33)

These integrals can be evaluated for relativistic bosons (kT ≫ mc2, kT ≫ µ)1

nB =ζ(3)

π2~3c3g (kT )3 (4.35)

ρBc2 =π2

30 ~3c3g (kT )4 (4.36)

PB =1

3ρBc2 , (4.37)

and for relativistic fermions

nF =3

4

ζ(3)

π2~3c3g (kT )3 =

3

4nB (4.38)

ρF c2 =7

8

π2

30 ~3c3g (kT )4 =

7

8ρBc2 (4.39)

PF =1

3ρF c2 . (4.40)

With respect to bosons (photons e.g.), the number density offermions has a weight factor of3/4 and the energy density a weight factor of 7/8.

Relativistic particles always satisfy the EOSP = ρc2/3.

4.1.3 EOS for Vacuum Energy

When long–range fields are present in the Universe, they also contribute to the energy–momentumtensor. As an example we consider a scalar fieldΦ(t, ~x) given by its Lagrangian density

LΦ =1

2∂µΦ ∂µΦ − V (Φ) (4.41)

with potential energyV (Φ). This provides us the energy–momentum tensor

Tµν = ∂µΦ ∂νΦ − δµ

ν

(1

2∂ρΦ ∂ρΦ − V (Φ)

)

. (4.42)

1The Riemannζ–function is defined as

ζ(x) =1

Γ(x)

∫

∞

0

ux−1

exp(u) − 1du (4.34)

with its valuesζ(3) = 1.202... andζ(4) = π4/90.


The corresponding density and pressure in a FRW–model are given by

ρΦ =1

2Φ2 +

1

2R−2(t)(∇Φ)2 + V (Φ) (4.43)

PΦ =1

2Φ2 − 1

6R−2(t)(∇Φ)2 − V (Φ) . (4.44)

If there is a non–vanishing vacuum expectation value< Φ >= Φ0 6= 0, the energy densityρΦ = V (Φ0) is constant and the corresponding pressurePΦ = −ρΦ is negative for positiveV (Φ0).Any form of matter with an EoS of the form P < −ρ/3 is called Dark Energy.

4.2 Einstein’s Equations for FRW Models

According to Einstein’s equations, the total energy–momentum tensorT including all sorts ofmatter is the source of gravity in the Universe

Rab −1

2gabR− Λgab = κTab . (4.45)

κ is the corresponding coupling constant determined by its Newtonian limit

κ =8πG

c4. (4.46)

R is the Ricci scalar.

4.2.1 Derivation of Friedmann’s Equations

We work in the conformal 3–metric (i, k = 1, 2, 3)

ds2 = dt2 − R2(t)

(1 + kρ2/4)2ηik dxi dxk (4.47)

with the corresponding natural one–forms

Θ0 = dt (4.48)

Θi =R(t)

1 + kρ2/4dxi , ρ2 =

3∑

i=1

(xi)2 . (4.49)

In these forms, the metric is expressed asg = Θa ⊗ Θb ηab. It is easy to work out theChristoffel symbols and from there the Riemann tensor in a coordinate basis. Here, I showhow the method with the Cartan equations can be used (Sect. 3.2.4). In this method, we havefirst to calculate the exterior derivatives of the observer frames (see Sect. 3.2.3)

dΘ0 = 0 (4.50)

dΘi =R

RΘ0 ∧ Θi − k

2Rxj Θi ∧ Θj . (4.51)

With Cartan’s first equation

dΘa = −ωab ∧ Θb (4.52)

4.2 Einstein’s Equations for FRW Models 93

we can then solve for the connection forms

ωi0 = −ω0i =R

RΘi (4.53)

ωij = −ωji =k

2R[xiΘj − xjΘi] , (4.54)

since the connection form has to be antisymmetric

dgαβ = ωαβ + ωβα = 0 . (4.55)

ωij is the connection of the slicest = const. With this we get the exterior derivatives of the

connection one–forms,

dωi0 =R

RΘ0 ∧ Θi − kR

2R2xjΘi ∧ Θj (4.56)

dωij =k

R2

(

1 +kρ2

4

)

Θi ∧ Θj

− k2

4R2[xixmΘj ∧ Θm − xjxmΘi ∧ Θm] . (4.57)

This can now be used in the second structure equation

Ωab = dωa

b + ωac ∧ ωc

b , (4.58)

or for the individual components

Ω0i = dω0

i + ω0m ∧ ωm

i (4.59)

and

Ωij = dωi

j + ωi0 ∧ ω0

j + ωim ∧ ωm

j . (4.60)

Inserting the above expressions, we can read off from these equations the curvature 2–forms

Ω0i = − R

RΘ0 ∧ Θi (4.61)

Ωij = −k + R2

R2Θi ∧ Θj . (4.62)

This shows explicitly that the Riemann tensor is isotropic

R0i0k =

R

Rδik , R0

ijm = 0 (4.63)

Rikik =

k + R2

R2, Ri

k0m = 0 . (4.64)

The FRW–model is a very simple spacetime where only two components of the Riemanntensor are independent ! These expressions also demonstrate that the Riemann tensor does notdepend on the chosen coordinate system, since it is constanton 3–surfaces.


The Ricci tensors can be derived from the above expressions

R00 = R1010 + R2

020 + R3030 = −3

R

R(4.65)

R11 = R0101 + R2

121 + R3131 =

(

R

R+ 2

k + R2

R2

)

(4.66)

R22 = R11 (4.67)

R33 = R11 . (4.68)

This can be written in compressed form, now including the speed of ligth2

R00 = − 3

c2

R

R(4.69)

R0i = 0 (4.70)

Rik =1

c2

(

R

R+ 2

kc2 + R2

R2

)

δik , (4.71)

as well as for the Ricci–scalar

R = − 6

c2

(

R

R+

kc2 + R2

R2

)

. (4.72)

With this, we obtain the Einstein tensor

G00 =3

c2

kc2 + R2

R2(4.73)

G0i = 0 (4.74)

Gik = − 1

c2

(

2R

R+

kc2 + R2

R2

)

δik . (4.75)

The Einstein tensor together with the energy–momentum tensor determines the Friedmannequations

R2 + kc2

R2− c2Λ

3=

8πG

3c2T 0

0 (4.76)

2R

R+

R2 + kc2

R2− c2Λ =

8πG

c2T 1

1 . (4.77)

Here, we use the concrete expressions for the energy–momentum tensor,

T 00 = ρc2 , (4.78)

T 11 = T 2

2 = T 33 = −P , (4.79)

resulting then in theFriedmann equations[6]

R2 + kc2

R2=

8πGρ

3+

c2Λ

3(4.80)

2Remember that these components are expressed in terms of orthonormal systems.

4.2 Einstein’s Equations for FRW Models 95

and

2R

R+

R2 + kc2

R2= −8πGP

c2+ c2Λ . (4.81)

By means of the first equation, we may simplify the second one

R

R= −4πG(ρc2 + 3P )

3c2+

c2Λ

3. (4.82)

The second Friedmann equation, which is known as the Raychaudhuri equation for thetidal forceR0

i0i = R/R, implies the essential conditionR < 0, providedΛ = 0, andR > 0,whenever vacuum energy is dominant.

Models including a dominant non–vanishing vacuum energy arealways accelerated atlate times.

4.2.2 Energy Conservation

The expansion factorR(t) satisfies the following identity

d

dt[R(R2 + kc2)] = R [2RR + R2 + kc2] (4.83)

and therefore as a consequence of the Friedmann equation,

d

dR(ρc2R3) + 3P R2 = 0 . (4.84)

This is known as thecosmic energy conservationin the form of the first law of thermody-namics for the energyE = ρc2R3

dE + P dV = 0 , (4.85)

sinceV = R3 is a measure for the comoving volume. For pressureless matter (galaxies ordark matter), this is nothing else than the number conservation

ρ(t) = ρ0

(

R0

R(t)

)3

. (4.86)

For relativistic matter, such as photons, neutrinos etc., with EOSP = ρc2/3 we find

ρ(t) = ρ0

(

R0

R(t)

)4

. (4.87)

For a general EOS of the formP = wρc2 with w = const the energy conservation leadsto the density evolution

ρ(t) = ρ0

(

R0

R(t)

)3(1+w)

. (4.88)

This includes the above special cases. In particular forw = −1 the density must stay constant.


4.2.3 Density Parameters for Friedman Models

With the Hubble constant we can define a characteristic orcritical density

ρc ≡ 3H20

8πG= 1.88 × 10−29 h2 g cm−3 = 2.77 × 1011 h2 M⊙ Mpc−3 . (4.89)

In modern Cosmology, it is now a tradition to parametrize thesource terms in the Friedmannequation by means of fourΩ–parameters

H2(z) = H20

[

ΩR(1 + z)4 + ΩM (1 + z)3 + Ωk(1 + z)2 + ΩΛ

]

. (4.90)

The various parameters follow from the definition of the critical density

• Density parameter for non–relativistic matter:

ΩM ≡ 8πG

3H20

ρM,0 =ρM,0

ρc(4.91)

• Vacuum energy parameter:

ΩΛ =Λc2

3H20

(4.92)

• Curvature parameter:

Ωk = − kc2

R20H

20

= −kR2H

R20

. (4.93)

Curvature is small in a Universe, whenever the Hubble radiusRH is much smaller thanthe scaling radiusR0. This is the basic idea behind inflationary models. The observableUniverse is then practically flat, which does not mean that itis globally flat.

• Radiation density parameter:

ΩR ≡ 8πG

3H20

ρRad,0 =ρRad,0

ρc(4.94)

The radiation density consists of the CMB and the neutrino background, if neutrinosare massless. The present contribution from CMB is negligibly small, Ωγ = (2.471 ±0.004) × 10−5/h2 = (4.9 ± 0.5) × 10−5.

These parameters satisfy, as a consequence of the Friedmannequation, the following relation

ΩM + ΩΛ + Ωk = 1 . (4.95)

Friedmann models of the cosmos are given in the cubus(ΩM ,ΩΛ,Ωk) by a plane, also calledthe fundamental plane of cosmology. Baryons contribute a small fraction to the cosmicmatter

ΩB = (0.0224 ± 0.0009)/h2 = 0.044 ± 0.004 . (4.96)

For this we can always write

ΩM = ΩB/fGas (4.97)

with 0.1 ≤ fGas ≤ 0.2 from observations in galaxy clusters. This gives a present value of

ΩM = (0.135 ± 0.08)/h2 = 0.27 ± 0.04 . (4.98)

4.3 FRW Models without Vacuum Energy 97

4.3 FRW Models without Vacuum Energy

At present time, the Universe is dominated by cold matter, i.e. matter in galaxies and darkmatter. Also, the energy density in the CMBR is many orders ofmagnitude below the criticaldensity and has not to be included in the source terms for gravity. Hence, neglecting vacuumenergy, the above equations simplify

R2 + kc2

R2=

8πGρ

3(4.99)

2R

R+

R2 + kc2

R2= 0 . (4.100)

The first equation now determines the Hubble constant,t = t0,

H20 +

kc2

R20

=8πGρ0

3. (4.101)

If the present matter densityρ0 were known exactly, we could decide whether the Universeis presently closed or open, sinceΩM + Ωk = 1. Compared to present densities, the criticaldensity seems to dominate all forms of matter. The contribution for galaxies to thedensityparameter ΩM is small

ΩG ≃ 0.005 . (4.102)

The Universe could however be dominated by some form of dark matter (invisible to tele-scopes), such thatΩM ≃ 1. The density parameterΩM is therefore the second parameterwhich determines the dynamical state of the present Universe. Models withΩΛ = 0 andΩM = 1 were calledStandard Cold Dark Matter models, orSCDM. These models werethe standard model for Cosmology in the 90s.

As we have seen, it is also common to describe the dynamical state in terms of a deceler-ation parameter

q0H20 = −

(

R

R

)

0

, (4.103)

which is related to the curvature of the expansion factora(t). For models with vanishingcosmological constant we find the important relation

q0 =1

2ΩM , ΩΛ = 0 . (4.104)

According to this, the Universe is flat, ifΩM = 1 (q0 = 0.5), closed forΩM > 1 and openfor ΩM < 1.

The energy conservation provides us the evolution of the density as a function of thepresent densityρ0

ρ(t) = ρ0

(

R0

R(t)

)3

= ΩMρc

(

R0

R(t)

)3

. (4.105)

With this expression we can determine the solution of the Friedmann equation for all valuesof k.


4.3.1 The Euclidean Universe, k=0

Fork = 0 we find the equation

R2 =8πGρ0

3

R30

R= H2

0

R30

R, (4.106)

which has the simple solution, called Einstein–de Sitter model,

R(t) = R0

(

t

t0

)2/3

. (4.107)

The Hubble constant gives then the age of the Universe

t0 =2

3H0. (4.108)

This also yields time as a function of redshift

t(z) = t0 (1 + z)−3/2 . (4.109)

4.3.2 The Closed Universe, k=1

Fork = 1 the Friedman equation yields

2R

R+

R2 + c2

R2= 0 (4.110)

R2 + c2

R2− 8πGρ0R

30

3R3= 0 . (4.111)

The second equation is the essential equation which gives

(

R

R0

)2

= H20

[

1 − 2q0 + 2q0R0

R

]

. (4.112)

Its solution is given ast = t(R)

t =1

H0

∫ R/R0

0

[

1 − 2q0 +2q0

x

]−1/2

dx (4.113)

with the present age defined as

t0 =1

H0

∫ 1

0

[

1 − 2q0 +2q0

x

]−1/2

dx <1

H0. (4.114)

In this case one obtains the solution of the Friedman equation over the transformation

1 − cos Θ =2q0 − 1

q0

R(t)

R0. (4.115)


The relation betweenΘ and time follows from the equation (4.112)

H0t =q0

(2q0 − 1)3/2(Θ − sinΘ) . (4.116)

This represents now an implicit equation for the parameterΘ as a function of time. Allto-gether, this represents the equation for a cycloid with its maximum atΘm = π, i.e. at thetime

tm =πq0

H0(2q0 − 1)3/2, Rm =

2q0R0

2q0 − 1. (4.117)

For the present time we find thereforeΘ0 = Θ(t0), or

cos Θ0 =1 − q0

q0, sin Θ0 =

√2q0 − 1

q0. (4.118)

With this information we are able to calculate the age of the closed Universe

t0 =Rm

2c(Θ0 − sin Θ0) (4.119)

=1

H0

q0

(2q0 − 1)3/2

(

cos−1(1 − q0

q0

)

−√

2q0 − 1

q0

)

. (4.120)

As example, one obtains forq0 = 1

t0 =1

H0

(π

2− 1)

. (4.121)

In comparison to a flat Universe, the closed Universe is younger.The solution for the expansion factor describes a cycloid (see Fig. 4.1). The radius of the

Universe reaches its maximum atΘ = π with the value

Rmax = Rm =c

H0

2q0

(2q0 − 1)3/2. (4.122)

A closed Universe cycles therefore between expansion and contraction and reaches its mini-mumR = 0 after a total time

tL =πRm

c=

1

H0

2πq0

(2q0 − 1)3/2. (4.123)

For q0 = 1, the life–cycle of the Universe istL = 2π/H0.

4.3.3 The Open Universe, k=–1

Finally, we consider the casek = −1 with the Friedman equation

2R

R+

R2 − c2

R2= 0 (4.124)

R2 − c2

R2− 8πGρ0R

30

3R3= 0 . (4.125)


The essential dynamics is hidden in the equation

R2 = c2

(

1 +Rm

R

)

(4.126)

with

Rm =2q0

(1 − 2q0)3/2

c

H0. (4.127)

Similar to the closed Universe, one can obtain the solution in terms of the transformation

R(t) =Rm

2(cosh Ψ(t) − 1) , ct =

Rm

2(sinh Ψ − Ψ) . (4.128)

This is valid in the range0 ≤ q0 < 1/2, or 0 ≤ Ω < 1. The present value ofΨ is given by

cosh Ψ0 =1 − q0

q0, sinhΨ0 =

√1 − 2q0

q0. (4.129)

This gives the age of the Universe in terms of the Hubble agetH = 1/H0

t0 =Rm

2c(sinh Ψ0 − Ψ0) (4.130)

=1

H0

q0

(1 − 2q0)3/2

(√1 − 2q0

q0− ln

(1 − q0 +√

1 − 2q0

q0

)

)

. (4.131)

4.3.4 The Singularity at t=0: Big–Bang

All three Friedman models have the genuine property thatR = 0 is reached after a finite time.NearR = 0, the Hubble parameter explodes and becomes finally unbounded. At the sametime, we observe that all components of the Riemann tensor become singular at this point.This point is therefore calledBig–Bang.

This singularity is an inherent property of the Einstein equations. The curvature of the thesolutionR(t) is always negativeR < 0, fallsρc2 +3P > 0. This energy condition is certainlysatisfied for all type of classical matter, even in the early Universe at high densities. If thisenergy condition is never violated, the Universe has to passthrough a singular state att = 0.One can even show that the singularity survives in even less symmetric spaces. At such smallscales, however, quantum effects are expect to play an essential role (space will be quantised).

4.3.5 The Mattig Formula for the Luminosity Distance

As an example of the general considerations in Sect. 3.7 we want to calculate the luminositydistance for these classical Friedman models,dL = remR(t0)(1 + z).


Figure 4.1: The expansion factor of classical Friedman models.

Flat Universe

In the flat case we simply get

r1 =

∫ t0

t1

c dt

R(t)=

c

R0

∫ t0

t1

t2/30 t−2/3 dt

= 3c

R0t2/30 (t

1/30 − t

1/31 )

=3c

R0t0

[

1 −(

t1t0

)1/3]

. (4.132)

Together with redshift this gives

r1 =3ct0R0

[1 − (1 + z)−1/2] =2c

R0H0[1 − (1 + z)−1/2] . (4.133)

The luminosity distance follows therefore from

dL = r1R0(1 + z) =2c

H0[(1 + z) −

√1 + z] . (4.134)

This correspond to the classical Hubble–law forz ≪ 1, dL ≃ (c/H0)z.


Closed Universe

In the closed Universe we find

∫ r1

0

dr√1 − r2

=

∫ t0

t1

c dt

R(t). (4.135)

The left integral is simple,sin−1 r1, and the right hand side follows from

∫ t0

t1

c dt

R(t)=

∫ t0

t1

dR√

R(Rm − R)=

∫ Θ0

Θ1

dΘ = Θ0 − Θ1 . (4.136)

Therefore

r1 = sin(Θ0 − Θ1) , (4.137)

and together with redshift we have

1 + z =R(t0)

R(t1)=

sin2(Θ0/2)

sin2(Θ1/2)(4.138)

the relation

sin Θ1 =2

1 + zsin

Θ0

2

√

z + cos2Θ0

2(4.139)

cos Θ1 =z + cos Θ0

1 + z. (4.140)

On the other hand, the parameterΘ0 satisfies the Friedman equation

sinΘ0

2=

√

2q0 − 1

2q0(4.141)

cosΘ0

2=

√

1

2q0. (4.142)

With this we obtain the expression forr1

r1 =

√2q0 − 1

q20(1 + z)

(

q0z + (1 − q0)[1 −√

1 + 2q0z])

. (4.143)

This provides the well–known relation for the distance as a function of redshift, the socalledMattig formula

dL = r1R0(1 + z) =c

H0

1

q20

(

q0z + (q0 − 1)[√

1 + 2q0z − 1])

. (4.144)

Be aware of the fact that forz ≪ 1 the parameterq0 disappears, and we find once again theclassical Hubble–law. As already discussed in Sect. 3.7, the Hubble–law is independent ofthe curvature of the space.

4.4 The Present Universe with Dark Energy 103

Open Universe

For open Universe, the calculation is similar

r1 =

√1 − 2q0

q20(1 + z)

(

q0z + (1 − q0)[1 −√

1 + 2q0z])

, (4.145)

and therefore for the distance

dL =c

H0

1

q20

(

q0z + (q0 − 1)[√

1 + 2q0z − 1])

. (4.146)

This expression is exactly the expression we found for the closed Universe, and in the limitDieser Ausdruck unterscheidet sich nicht von dem im geschlossenen Modell, undq0 → 1/2we also get the luminosity distance of the flat Universe Therefore, equation (4.144) is validfor all three types of models, i.e. for all values ofq0 ≥ 0.

4.4 The Present Universe with Dark Energy

Results from WMAP and the analysis of the luminosity distancefor high redshift Supernovaeindicate that the present state of the Universe cannot be satisfactorily described in terms ofclassical Friedmann models not including vacuum energy. Wewill discuss in the followingsoltuions of the Friedmann equation including some form of vacuum energy given by an EOSof the form (see the discussion in 3.2)

PV = wV ρV . (4.147)

As we have seen in 3.2, a cosmological constant is equivalentto a vacuum energy densitydefined as

ρΛ =c2

8πGΛ . (4.148)

The total energy density of the present Universe has then three contributions

ρ(t) = ρM (t) + ρRad(t) + ρV , (4.149)

and similarly for the pressure

P (t) = PM (t) + PRad(t) + PV , PV = wρV c2 (4.150)

with wV = −1.The energy densityρV of the vacuum always results from ground state energy in field

theories. Both termsρV andρΛ occur as sum in Einstein’s equations auf(

R

R

)2

=8πG

3(ρM + ρRad + ρV + ρΛ) − kc2

R2, (4.151)

R

R= −4πG

3(ρM + 6PRad − 2ρV c2 − 2ρΛc2) . (4.152)


The sum of all energy densities cannot excede the present critical density, i.e.Λ ≤ 4× 10−56

cm−2. Natural values for the vacuum energy density are either given by QCD or the Planckepoch

ρQCDV c2 = (0.3GeV )4 = 1.6 × 1036 erg/cm3 (4.153)

ρPlanckV c2 = (1018 GeV )4 = 2 × 10110 erg/cm3 . (4.154)

It is one of the big mysteries, why this vacuum energy is so small, ρV ≃ ρcrit.

4.4.1 Cosmological Parameters

It is usual to normalize the radius to its present value,a(t) = R(t)/R0. Then the Friedmannequation simplifies to

a2 = H20 (

ΩM

a+ ΩV a2) − kc2

R20

. (4.155)

Time is measured in units ofH−10 . ΩV is the density parameter of the vacuum.

For t = t0 this is equivalent to

1 = ΩM + ΩV − kc2

R20H

20

, (4.156)

or to

R0 =c

H0

√

k

ΩM + ΩV − 1. (4.157)

From here, we get the differential equation

a2 = H20

(

ΩM

a+ ΩV a2 + 1 − ΩM − ΩV

)

. (4.158)

This corresponds to the equation of motion of a particle in a potential−ΩM/x − ΩV x2 withenergy1 − ΩM − ΩV . For ΩV > 0 the potential has a maximum and it is negative for allvaluesx, i.e. the solutions withk = 0,−1 expand away from a singularity. Fork = +1 wefind a critical valueΩV , so thatR2 = 0.

As in classical Friedmann models, the deceleration parameter q0 is quite often used toparametrize the present state of the Universe

q0 = − RR

R2= − R

RH2. (4.159)

Together with the second Friedmann–equation,R/R = −4πGR(ρ + 3P/c2)/3 + c2Λ/3 weobtain the relation

q0 =1

2ΩM − ΩΛ . (4.160)

q0 is therefore not an independent parameter.The existence of the vacuum energy determines therefore thegeometry of the Uni-

verse (Fig. 4.3). A nonvanishing vacuum energy leads to a flatUniverse, even if thematter density is smaller than the critical density. A positive vacuum energy densityaccelerates the expansion of the Universe.In the follwoing, we only discuss models withw = −1 (Fig. 4.3).


Figure 4.2: Expansion of the Universe dominated by vacuum energy. The transitionfrom decelerationto acceleration occurs around redshift one.

4.4.2 Solutions for Inflationary Universes

The solutions of the Friedmann equation including vacuum energy are not trivial, but mustbe determined by means of numerical techniques. There is however a special case, which isanalytically solvable. In a flat Universe, the Friedmann equation is simply

(

a

a

)2

= H20

[

ΩM

a3+ 1 − ΩM

]

. (4.161)

One can easily show that in the flat case,k = 0, the following ansatz satisfies the aboveFriedmann equation

a(t) =

[

√

ΩM

1 − ΩMsinh

(

3√

1 − ΩM H0t

2

)

]2/3

(4.162)

As expected, for late times we find an exponential expansion,a(t) ∝ exp(√

1 − ΩM H0t),while the Taylor expansion for small time–scales,sinh(x) ≃ x provides the classical expan-sion law of a flat Friedmann Universe with vanishing vacuum energy,a(t) ∝ t2/3 (Fig. 4.4).

4.4.3 Age of the Universe

We write the Friedmann equation for the dimensionless expansion factora(t) ≡ R(t)/R0 =1/(1 + z) with a(t0) = 1

da

dt= H0

√

ΩM

a+ ΩΛ a2 + 1 − ΩM − ΩΛ (4.163)


Figure 4.3: Transition from a decelerating expansion to the accelerating Universe dominated by darkenergy (top). The lower panel shows the future evolution of the Universe dominated by vacuum energy.

From this expression we can derive the aget = t(z) as function of redshiftz, parametrizedby H0, ΩM andΩΛ. Since

dt =dt

dada = − dt

da

dz

(1 + z)2

= − 1

H0

dz

(1 + z)2√

ΩM (1 + z) + ΩΛ/(1 + z)2 + 1 − ΩM − ΩΛ

= − 1

H0

dz

(1 + z)√

ΩM (1 + z)3 + (1 − ΩM − ΩΛ)(1 + z)2 + ΩΛ

, (4.164)


Figure 4.4: Expansion of the Universe for vanishing curvature (inflationary cosmos). The scalefactor is normalized to the present radius, time is measured in billions of years from today.Curves in the blue region represent models with accelerated expansion,with a vacuum energyof 95% to 40% of the critical density.

we find for the age of the Universe as a function of redshift

t(z) =

∫ t

0

dt =1

H0

∫ ∞

z

dz′

(1 + z′)E(z′)(4.165)

with E(z) = H(z)/H0. This expression can easily be integrated numerically for any Fried-man equationH = H(z) (see Fig. 4.6).

The special case of the flatinflationary Universe Eq (4.162),Ωk = 0, can be simplifiedto

t(z) =

∫ t

0

dt =1

H0

∫ ∞

z

dz′

(1 + z′)√

ΩM (1 + z′)3 + ΩΛ

. (4.166)

With the transformation

tan θ(z) =

√

ΩM

ΩΛ(1 + z)3/2 (4.167)


Figure 4.5: Age of the Universe as a function of redshiftz. The upper curve represents a flatΛCDM–model withΩM = 0.3, ΩΛ = 0.7, the lower curve (dashed line) a classical CDM–model (SCDM) withΩM = 1. Hubble constantH0 = 65 km/s/Mpc.

the age can be solved analytically (Fig. 4.5)

t(z) =2

3H0

√ΩΛ

ln

[

1 + cos θ(z)

sin θ(z)

]

. (4.168)

The Empirical Age of the Universe

There are at least 3 ways that the age of the Universe can be estimated. The most importantones are:

• The age of the oldest star clusters (globular clusters).

• The age of the chemical elements.

• The age of the oldest white dwarf stars.

As we have seen, the age of the Universe can also be estimated from a cosmological modelbased on the Hubble constant and the densities of matter and dark energy. This model-basedage is currently13.7 ± 0.2 Gyr. The actual age measurements are in fact consistent withthemodel–based age. This increases our confidence in the Big Bang model.

Age of Globular Clusters: The life cycle of a star depends upon its mass. High massstars are much brighter than low mass stars, thus they rapidly burn through their supply ofhydrogen fuel. A star like the Sun has enough fuel in its core to burn at its current brightnessfor approximately 11 billion years. A star that is twice as massive as the Sun will burn through


Figure 4.6: Age of the Universe as a function ofΩm andΩΛ. The numbers of the contour linesgive the age of the Universe in terms of the Hubble agetH = 1/H0.


its fuel supply in only 800 million years. A 10 solar mass star, a star that is 10 times moremassive than the Sun, burns nearly a thousand times brighterand has only a 20 million yearfuel supply. Conversely, a star that is half as massive as theSun burns slowly enough for itsfuel to last more than 20 billion years.

All of the stars in a globular cluster formed at roughly the same time, thus they can serve ascosmic clocks. If a globular cluster is more than 20 million years old, thenall of its hydrogenburning stars will be less massive than 10 solar masses. Thisimplies that no individual hydro-gen burning star will be more than 1000 times brighter than the Sun. If a globular cluster ismore than 2 billion years old, then there will be no hydrogen-burning star more massive than2 solar masses.

When stars are burning hydrogen to helium in their cores, theyfall on a single curve inthe luminosity-temperature plot known as the HR diagram after its inventors, Hertzsprung andRussell. This track is known as the main sequence, since moststars are found there. Sincethe luminosity of a star varies likeM3 or M4, the lifetime of a star on the main sequencevaries liket =constM/L = k/L0.7. Thus if you measure the luminosity of the most lu-minous star on the main sequence, you get an upper limit for the age of the cluster: Age< k/L(MSmax)0.7.

This is an upper limit because the absence of stars brighter than the observedL(MSmax)could be due to no stars being formed in the appropriate mass range. But for clusters withthousands of members, such a gap in the mass function is very unlikely, the age is equal tok/L(MSmax)0.7. Chaboyer, Demarque, Kernan and Krauss (1996, Science271, 957) applythis technique to globular clusters and find that the age of the Universe is greater than 12.07Gyr with 95% confidence. They say the age is proportional to one over the luminosity of theRR Lyra stars which are used to determine the distances to globular clusters. Chaboyer (1997)gives a best estimate of14.6 ± 1.7 Gyr for the age of the globular clusters. But Hipparcosresults have shown that the globular clusters are further away than previously thought, so theirstars are more luminous. Gratton et al. give ages between 8.5and 13.3 Gyr with 12.1 beingmost likely, while Reid gives ages between 11 and 13 Gyr, and Chaboyer et al. give11.5±1.3Gyr for the mean age of the oldest globular clusters.

The Age of the Elements: The age of the chemical elements can be estimated using radioac-tive decay to determine how old a given mixture of atoms is. The most definite ages that canbe determined this way are ages since the solidification of rock samples. When a rock solidi-fies, the chemical elements often get separated into different crystalline grains in the rock. Forexample, sodium and calcium are both common elements, but their chemical behaviours arequite different, so one usually finds sodium and calcium in different grains in a differentiatedrock. Rubidium and strontium are heavier elements that behave chemically much like sodiumand calcium. Thus rubidium and strontium are usually found in different grains in a rock. ButRb-87 decays into Sr-87 with a half-life of 47 billion years.And there is another isotope ofstrontium, Sr-86, which is not produced by any rubidium decay. The isotope Sr-87 is calledradiogenic, because it can be produced by radioactive decay, while Sr-86 is non-radiogenic.The Sr-86 is used to determine what fraction of the Sr-87 was produced by radioactive decay.This is done by plotting the Sr-87/Sr-86 ratio versus the Rb-87/Sr-86 ratio. When a rock isfirst formed, the different grains have a wide range of Rb-87/Sr-86 ratios, but the Sr-87/Sr-86ratio is the same in all grains because the chemical processes leading to differentiated grainsdo not separate isotopes. After the rock has been solid for several billion years, a fraction ofthe Rb-87 will have decayed into Sr-87. Then the Sr-87/Sr-86ratio will be larger in grains


with a large Rb-87/Sr-86 ratio.

The Age of Oldest White Dwarfs: A white dwarf star is an object that is about as heavy asthe Sun but only the radius of the Earth. The average density of a white dwarf is a million timesdenser than water. White dwarf stars form in the centers of redgiant stars, but are not visibleuntil the envelope of the red giant is ejected into space. Whenthis happens the ultravioletradiation from the very hot stellar core ionizes the gas and produces a planetary nebula. Theenvelope of the star continues to move away from the central core, and eventually the planetarynebula fades to invisibility, leaving just the very hot corewhich is now a white dwarf. Whitedwarf stars glow just from residual heat. The oldest white dwarfs will be the coldest and thusthe faintest. By searching for faint white dwarfs, one can estimate the length of time the oldestwhite dwarfs have been cooling. Oswalt, Smith, Wood and Hintzen (1996, Nature, 382, 692)have done this and get an age of9.5 + 1.1 − 0.8 Gyr for the disk of the Milky Way. Theyestimate an age of the Universe which is at least 2 Gyr older than the disk, sot0 > 11.5 Gyr.Hansen et al. have used the HST to measure the ages of white dwarfs in the globular clusterM4, obtaining12.7 ± 0.7 Gyr. In 2004 Hansen et al. updated their analysis to give an age forM4 of 12.1±0.9 Gyr, which is very consistent with the age of globular clusters from the mainsequence turnoff. Allowing for the time between the Big Bangand the formation of globularclusters (and its uncertainty) implies an age for the Universe of12.8 ± 1.1 Gyr.

4.4.4 The Event Horizon

We consider an event(t1, r1) which we wish to observe at our locationr = 0

∫ r1

0

dr√1 − kr2

=

∫ t

t1

c dt′

R(t′). (4.169)

An observer atr = 0 will be able to receive signals from any event (after a suitable longwait), provided the integral on the rhs diverges (t → ∞). FroR(t) ∝ tp, this impliesp < 0,or a decelerating Universe. In an accelerating Universe, the integral converges, signaling thepresence of anevent horizon. For this we need

∫ r1

0

dr√1 − kr2

≤∫ ∞

0

c dt′

R(t′). (4.170)

Observers beyond a distance

RH = R0

∫ ∞

t0

c dt

R(t)(4.171)

are not able to communicate with the rest of the world. For a deSitter UniverseR(t) =R1 exp(H(t − t1)), this limit is RH = c/H.

4.4.5 The Particle Horizon

Alternatively, we may ask the question: Can we nowadays see the entire cosmos ? Or is therea limit rH for r1, providedz → ∞ ? This would imply the existence ofRH satisfying

RH = R0

∫ t0

te→0

c dt

R= R0

∫ rH

0

dr√1 − kr2

. (4.172)


In an Einstein–de Sitter cosmos withR(t) ∝ t2/3, t → 0, this limit will be achieved

RH =3c

H0, k = 0 . (4.173)

For a radiation–dominated Universe,R(t) ∝√

t we findRH = 2c/H. This is calledparticlehorizon: particles withr1 > RH are hidden for us.

As an application we consider the particle horizon at recombination, wheretR ≃ 106

years. The particle horizon at that time wasRH = 2c/H(tR). At that time, the domain ofinfluence was much smaller than today, and despite this fact,we observe a large degree ofisotropy in the background radiation. This fact is called the causality problem of the standardUniverse (Fig. 4.7).

Figure 4.7: Causality problem at recombination: at redshiftz = 1100 (blue line) we observeradiation from patches of spaces which were not in causal contact. Thetime trajectories aregiven here in conformal time,dη = dt/R(t).

4.4.6 Conformal Maps of the Present Universe

Cartographers mapping the Earth’s surface were faced with the challenge of mapping a curvedsurface onto a plane. No such projection can be perfect, but it can capture important features.Perhaps the most famous map projection is the Mercator projection (presented by GerhardusMercator in 1569). This is a conformal projection which preserves shapes locally. Linesof latitude are shown as straight horizontal lines, while meridians of longitude are shown asstraight vertical lines.

The Hammer–Aitoff projection shows the Earth as a horizontal ellipse with 2:1 axis ratio.The equator is shown as a straight horizontal line marking the long axis of the ellipse. It isproduced in the following way: map the entire sphere onto itswestern hemisphere by simplycompressing each longitude by a factor of 2. Now map this western hemisphere onto a planeby the Lambert equal area azimuthal projection. This map is acircular disk. This is thenstretched by a factor of 2 (undoing the previous compressionby a factor of 2) in the equatorialdirection to make an ellipse with a 2:1 axis ratio. Thus, the Hammer projection preservesareas.


In galaxy surveys we useslice maps of the universeto make flat maps. The CfA mapssurveyed a slice of sky,117 degrees long and6 degrees wide, of constant declination. In 3Dthis slice had the geometry of a cone, and they flattened this onto a plane. (A cone has zeroGaussian curvature and can therefore be constructed from a piece of paper. A cone cut alonga line and flattened onto a plane looks like a pizza with a slicemissing .) If the cone is atdeclinationδ, the map in the plane will bex = r cos(λ cos(δ)), y = r sin(λ cos(δ)), whereλis the right ascension (in radians), andr is the co-moving distance (as indicated by the redshiftof the object). This will preserve shapes. Many times a360 degree slice is shown as a circlewith the Earth in the center, wherex = r cos(λ), y = r sin(λ). If r is measured in co-movingdistance, this will preserve shapes only if the universe is flat (k = 0), and the slice is in theequatorial plane (δ = 0), (if δ 6= 0, structures (such as voids) will appear lengthened in thedirection tangential to the line of sight by a factor of1/ cos(δ)). Therefore, it is importantto investigate map projections which willpreserve shapes locally. If one has the correctcosmological model, and uses such a conformal map projection, isotropic features in the largescale structure will appear isotropic on the map.

Our objective here is to produce aconformal map of the universewhich will show thewide range of scales encountered while still showing shapesthat are locally correct. Considerthe general Friedmann metrics in the spherical coordinates

ds2 = −dt2 + a2(t)(dχ2 + sin2 χ(dθ2 + sin2 θdφ2)), k = +1 (4.174)

ds2 = −dt2 + a2(t)(dχ2 + χ2(dθ2 + sin2 θdφ2)), k = 0 (4.175)

ds2 = −dt2 + a2(t)(dχ2 + sinh2 χ(dθ2 + sin2 θdφ2)), k = −1 (4.176)

wheret is the cosmic time since the Big Bang,a(t) is the expansion parameter, and individualgalaxies participating in the cosmic expansion follow geodesics with constant values ofχ, θ,andφ. These three are calledco-moving coordinates. Neglecting peculiar velocities, galaxiesremain at constant positions in co-moving coordinates as the universe expands. The expansionfactora(t) obeys Friedmann’s equations.

We can define a conformal timeη by the relationdη = dt/a, so that

η(t) =

∫ t

0

dt

a(4.177)

Light travels on radial geodesics withdη = ±dχ so a galaxy at a co-moving distanceχ fromus emitted the light we see today at a conformal timeη(t) = η(t0)−χ. Thus, we can calculatethe timet and redshiftz = a(t0)/a(t) − 1 at which that light was emitted. Conversely, if weknow the redshift, given a cosmological model we can calculate the co-moving radial distanceof the galaxy from us from its redshift, again ignoring peculiar velocities.

The WMAP data implies thatw ≈ −1 for dark energy (ie.pvac = wρvac ≈ −ρvac),suggesting that a cosmological constant is an excellent model for the dark energy, so we aresimply adopting that. The current Hubble radiusRH0

= c/H0 = 4220Mpc. The cosmicmicrowave background is at a redshiftz = 1089. Substituting, using geometrized units inwhich c = 1, and integrating the first Friedmann equation we find the conformal time may becalculated

η(t) =

∫ t

0

dt

a=

∫ a(t)

0

da

a2 H(a)=

1

H0

∫ a(t)

0

[

Ωr + Ωma + Ωka2 + ΩΛa4]−1/2

da

(4.178)


whereρm ∝ a−3 andρr ∝ a−4. This formula will accurately track the value ofη(t), provid-ing that this is interpreted as the value of the conformal time since the end of the inflationaryperiod at the beginning of the universe. (During the inflationary period at the beginning ofthe universe, the cosmological constant assumed a large value, different from that observedtoday, and the formula would have to be changed accordingly.So we simply start the clockat the end of the inflationary period where the energy densityin the false vacuum [large cos-mological constant] is dumped in the form of matter and radiation. Thus, when we trace backto the big bang, we are really tracing back to the end of the inflationary period. After that,the model does behave just like a standard hot-Friedmann bigbang model. This standardmodel might be properly referred to as an inflationary-big bang model, with the inflationaryepoch producing the Big Bang explosion at the start.) Now,a(t) is the radius of curvatureof the universe for thek = +1 andk = −1 cases, but for theΩk = 0 case, which we willbe investigating first and primarily, there is no scale and sowe arefree to normalize, set-ting a(t0) = RH0

= c/H0 = 4220Mpc. Then,χ measures co-moving distances at thepresent epoch in units of the current Hubble radiusRH0

. Thus, for theΩk = 0 case, usinggeometrized units, we have

η(a) = η(a(t)) =

∫ a

0

[ a

a0Ωm + Ωr + (

a

a0)4ΩΛ

]−1/2 da

a0(4.179)

whereΩm, ΩΛ, Ωr are the values at the current epoch. Given the values adoptedfrom WMAPwe find

η(a0) = 3.38 (4.180)

That means that when we look out now att = t0 (whena = a0) we can see out to a distanceof

χ = 3.38 (4.181)

or aco-moving distanceof

χRH0= 3.38RH0

= 14, 300Mpc . (4.182)

This is theeffective particle horizon, where we are seeing particles at the moment of theBig Bang. This is a larger radius than 13.7 billion light years – the age of the universe (thelookback time) times the speed of light – because it shows theco-moving distance the mostdistant particles we can observe now will have from us when they are as old as we are now, i.e.measured at the current cosmological epoch. We may calculate the value ofη as a function ofa, or equivalently as a function of observed redshiftz = (a0/a) − 1. Recombination occursatzrec = 1089, which is the redshift of the cosmic microwave background seen by WMAP.

η(zrec) = 0.0671 (4.183)

So, theco-moving radius of the cosmic microwave backgroundis

χRH0= (η0 − ηrec)RH0

= 14, 000Mpc (4.184)

That is the radius at the current epoch, so at recombination the WMAP sphere has a physicalradius that is 1090 times smaller or about13 Mpc.


Redshift z r(z) (Mpc) Remark

∞ 14,283 Big Bang (end of inflationary period)3233 14,165 Equal matter and radiation density epoch1089 14,000 Recombination

6 8,4225 7,9334 7,3053 6,4612 5,2451 3,317

0.5 1,8820.2 8090.1 413

Table 4.1: Co-moving radii for different redshifts

We may computeco-moving radii r = χRH0for different redshifts, as shown in table 4.1.

We can also calculate the value ofη(t = ∞) = 4.50 which shows how far a photon can travelin co-moving coordinates from the inflationary Big Bang to the infinite future. Thus, if wewait until the infinite future we will eventually be able to see out to a co-moving distance of

rt=∞ = 4.50RH0= 19, 000Mpc . (4.185)

This is theco-moving future visibility limit, or future horizon . No matter how long wewait, we will not be able to see further than this. This is surprisingly close. The number ofgalaxies we will eventually ever be able to see is only largerthan number observable today bya factor of(rt=∞/rt0)

3 = 2.36.If we send out a light signal now, byt = ∞ it will reach a radiusχ = η(t = ∞)−η(t0) =

4.50 − 3.38 = 1.12, or

r = 4, 740Mpc (4.186)

to which we refer to as the “outward limit of reachability”. We cannot reach (with light signalsor rockets) any galaxies that are further away than this. Whatredshift does this correspondto? Galaxies we observe today with a redshift ofz = 1.69 are at this co-moving distance.Galaxies with redshifts larger than1.69 today are unreachable. This is a surprisingly smallredshift.

We can see many galaxies at redshifts larger than1.69 that we will never be able to visit orsignal. In the accelerating universe, these galaxies are accelerating away from us so fast thatwe can never catch them. The total number of stars that our radio signals will ever pass is oforder2 × 1021.


A Map Projection for the Universe

We will choose a conformal map that will cover the wide range of scales from the Earth’sneighborhood to the cosmic microwave background. First we will consider the flat case (Ωk =0) which the WMAP data tells us is the appropriate cosmologicalmodel. Our map will be twodimensional so that it can be shown on a wall chart. CfA surveyshowed with their slice of theUniverse, just how successful a slice of the Universe can be in illustrating large scale structure.The Sloan Digital Survey includes spectra and accurate positions for about 1 million galaxiesand quasars in a 3D sample. We only use an equatorial slice4 degrees wide (−2o < δ < 2o)centered on the celestial equator covering both northern and southern galactic hemispheres.This shows many interesting features including many prominent voids and a great wall longerthan the great wall found in CfA.

Since the observed slice is already in a flat plane (k = 0 model, along the celestial equa-tor) we may project this slice directly onto a flat sheet of paper using polar coordinates withr = χRH0

being the co-moving distance, andθ being the right ascension. We wish to showlarge scale structure and the extent of the observable universe out to the cosmic microwavebackground radiation including all the SDSS galaxies and quasars in the equatorial slice. Itsco-moving radius is14.0 Gpc. (Since the size of the universe at the epoch of recombination issmaller that that a present by a factor of1 + z = 1090, the true radius of this circle is about12.84 Mpc.) Slightly beyond the cosmic microwave background in co-moving coordinates isthe Big Bang at a co-moving distance of14.3 Gpc.

(Imagine a point on the cosmic microwave background circle.Draw a radius around thatpoint that is tangent to the outer circle labeled Big Bang, asshown in the figure, in other words,a circle that has a radius equal to the difference in radius between the cosmic microwavebackground circle and the Big Bang circle. That circle has a co-moving radius of283 Mpc.That is the co-moving horizon radius at recombination. If the Big Bang model – withoutinflation – were correct we would expect a point on the cosmic microwave background circleto be causally influenced only by things inside that horizon radius at recombination. Theangular radius of this small circle as seen from the Earth is(283Mpc/14, 000Mpc) radiansor 1.16 degrees. If the Big Bang model without inflation were correctwe would expect thecosmic microwave background to be correlated on scales of atmost1.16 degrees. Inflation, byhaving a short period of accelerated expansion during the first10−34 seconds of the universe,puts distant regions in causal contact because of the slightadditional time allowed when theuniverse was very small. So, with inflation, we can understand why the cosmic microwavebackground is uniform to one part in 100,000 all over the sky.Furthermore, random quantumfluctuations predicted by inflation add a series of adiabaticfluctuations which are expected tohave a peak in the power spectrum at an angular scale about thesize of the horizon radius atrecombination calculated above,∼ 0.86 degrees.)

Beyond the Big Bang circle is the circle showing the future co-moving visibility limit. Ifwe wait until the infinite future, we will be able to see out to this circle. (In other words, in theinfinite future, we will be able to see particles at the futureco-moving visibility limit as theyappeared at the Big Bang.)

The SDSS quasars extend out about halfway out to the cosmic microwave backgroundradiation. The distribution of quasars shows several features. The radial distribution showsseveral shelves due to selection effects as different spectral features used to identify quasarscome into view in the visible. Several radial spokes appear due to incompleteness in somenarrow right ascension intervals. Two large fan shaped regions are empty and not surveyedbecause they cover the zone of avoidance close to the galactic plane which is not included


Figure 4.8: Galaxies and quasars in the equatorial slice (−2 < δ < 2 degrees) of the SloanDigital Sky Survey displayed in co-moving coordinates out to the horizon.The co-movingdistances to galaxies are calculated from measured redshift, assuming Hubble flow and WMAPcosmological parameters. This is a conformal map – it preserves shapes. While this map canconformally show the complete Sloan survey, the majority of interesting large scale structure iscrammed into a blob in the center. The dashed circle marks the outer limit of figure 4.9. Thecircle labeled ’Unreachable’ marks the distance beyond which we cannot reach (i.e. we cannotreach with light signals any object that is further away). This radius corresponds to a redshiftof z = 1.69. As ’Future comoving visibility limit’ we label the co-moving distance to which aphoton would travel from the inflationary Big Bang to the infinite future. This isthe maximumradius out to which observations will ever be possible. At4.50RH0

, it is suprisingly close. [Gottet al. 2003]


in the Sloan survey. These excluded regions run from approximately 3.7 h . α . 8.7 hand approximately16.7 h . α . 20.7 h. The quasars do not show noticeable clustering orlarge scale structure. This is because the quasars are so widely spaced that the mean distancebetween quasars is larger than the correlation length at that epoch.

The circle of reachability is also shown. Quasars beyond this circle are unreachable. Radiosignals emitted by us now will only reach out as far as this circle, even in the infinite future.

Figure 4.9: Zoom in of the region marked by the dashed circle in figure 4.8, out to 0.06 rhorizon

(= 858 Mpc). The points shown are galaxies from the main and bright red galaxysamples of theSDSS. Compared to figure 4.8, we can now see a lot of interesting structure. The Sloan GreatWall can be seen stretching from8.7h to 14h in R.A. at a median distance of about310 Mpc.Although the large scale structure is easier to see, a ”zoom in” like this fails to capture anddisplay, in one map, the sizes of modern redshift surveys.


The SDSS galaxies appear as a black blob in the center. There is much interesting largescale structure here but the field is too crowded and small to show it. This illustrates theproblem of scale in depicting the universe. If we want a map ofthe entire observable universeon one page, at a nice scale, the galaxies are crammed into a blob in the center. Let us enlargethe central circle of radius0.06 times the distance to the Big Bang circle by a factor of 16.6and plot it again in figure 4.9. This now shows a circle of co-moving radius858 Mpc. Almostall of these points are galaxies from the galaxy and bright red galaxy samples of the SDSS.Now we can see a lot of interesting structure. The most prominent feature is a Sloan GreatWall at a median distance of about310 Mpc stretching from8.7h to 14h in R.A. There arenumerous voids. A particularly interesting one is close in at a co-moving distance of 125 Mpcat 1.5h R.A. At the far end of this void are a couple of prominent clusters of galaxies whichare recognizable as ”fingers of God” pointing at the Earth. Redshift in this map is taken asthe co-moving distance indicator assuming participation in the Hubble flow, but galaxies alsohave peculiar velocities and in a dense cluster with a high velocity dispersion this causes thedistance errors due to these peculiar velocities to spread the galaxy positions out in the radialdirection producing the ”finger of God” pointing at the Earth. Numerous other clusters canbe similarly identified. This is a conformal map, that preserves shapes – excluding the smalleffects of peculiar velocities. The original CfA survey in which Geller and Huchra discoveredthe Great Wall had a co-moving radius of only 211 Mpc, which isless than a quarter of theradius shown in figure 4.9. Figure 4.9 is a quite impressive picture, but it does not captureall of the Sloan Survey. If we displayed figure 4.8 at a scale enlarged by a factor of 16.6 thecentral portion of the map would be as you see displayed at thescale shown in figure 2 whichis adequate, but the Big Bang circle would have a diameter of 6.75 feet. You could put thison your wall, but if we were to print it in the journal for you toassemble it would require thenext 256 pages. This points out the problem of scale for even showing the Sloan Survey allon one page. Small scales are also not represented well. The distance to the Virgo Cluster infigure 4.9 is only about 2 mm and the distance from the Milky Wayto M31 is only1/13th ofa millimeter and therefore invisible on this Map. Figure 4.9, dramatic as it is, fails to capturea picture of all the external galaxies and quasars. The nearby galaxies are too close to see andthe quasars are beyond the limits of the page.

We may try plotting the Universe inlookback time rather than co-moving coordinates.The result is in figure 4.10. The outer circle is the cosmic microwave background. It isindistinguishable from the Big Bang as the two are separatedby only 380, 000 years outof 13.7 billion years. The SDSS quasars now extend back nearly to thecosmic microwavebackground radiation (since it is true that we are seeing back to within a billion years of theBig Bang). Lookback time is easier to explain to a lay audience than co-moving coordinatesand it makes the SDSS data look more impressive, but it is a misleading portrayal as faras shapes and the geometry of space are concerned. It misleads us as to how far out we areseeing in space. For that, co-moving coordinates are appropriate. figure 4.10 does not preserveshapes – it compresses the large area between the SDSS quasars and the cosmic microwavebackground into a thin rim. This is not a conformal map. The SDSS galaxies now occupya larger space in the center, but they are still so crowded together that one can not see thelarge scale structure clearly. Figure 4.11 shows the central 0.2 radius circle (shown as a dottedcircle in figure 4.10) enlarged by a factor of 5. Thus if we wereto make a wall map of theobservable universe using lookback time at the scale of figure 4.11 it would only need tobe 2 feet across and would only require the next 25 pages in thejournal to plot. This is anadvantage of the lookback time map. It makes the interestinglarge scale structure that we see


Figure 4.10: Galaxies and quasars in the equatorial slice of the SDSS, displayed in lookbacktime coordinates. The radial distance in the figure corresponds to lookback time. While theGalaxies at the center occupy a larger area, this map is a misleading portrayal as far as shapesand the geometry of space are concerned. It is not conformal – it compresses the area close tothe horizon (this compression is more explicitly shown in figure 4.12). Also, the galaxies arestill too crowded in the center of the map to show all of the intricate details of theirclustering.Figure 4.11 shows a zoom in of the region inside the dashed circle. [Gott et al. 2003]

locally (figure 4.11) a factor of slightly over 3 larger in size relative to the cosmic microwavebackground circle than if we had used co-moving coordinates. Figure 4.11 looks quite similarto figure 4.9. At co-moving radii less than 858 Mpc, the lookback time and co-moving radiusare rather similar. Still, figure 4.11 is not perfectly conformal. Near the outer edges there is


a slight radial compression that is beginning to occur in thelookback time map as one goestoward the Big Bang.

Figure 4.11: Zoom in of the region marked by the dotted circle in figure 4.10, showing SDSSgalaxies out to 0.2thorizon. The details of galaxy clustering are now displayed much better.However, like figure 4.9, it still fails to capture the whole survey in one, reasonably sized, map.[Gott et al. 2003]

The effects of radial compression are illustrated in figure 4.12, where we have plotteda square grid in co-moving coordinates in terms of lookback time as would be depicted infigure 4.10. Each grid square would contain an equal number ofgalaxies in a flat slice ofconstant vertical thickness. This shows the distortion of space that is produced by using thelookback time. The squares become more and more distorted inshape as one approaches theedge.


Figure 4.12: Square comoving grid shown in lookback time coordinates. Grid spacing is0.1RH0

= 422.24 Mpc. Each grid square would contain an equal number of galaxies in aflat slice of constant vertical thickness. The distortion of space that is produced by using thelookback time is obvious as the squares become more and more distorted inshape as one ap-proaches the horizon. [Gott et al. 2003]

4.5 Measuring the Acceleration of the Present Universe

The three primary methods to measure curvature are luminosity, scale length and number.Luminosity requires an observer to find some standard candle, such as the brightest quasars orSupernovae, and follow them out to high redshifts. Scale length requires that some standardsize be used, such as the size of the largest galaxies. Lastly, number counts are used whereone counts the number of galaxies in a box as a function of distance. Several groups aremeasuring distant supernovae with the goal of determining whether the Universe is open orclosed by measuring the curvature in the Hubble diagram. Thefigure 4.14 shows a binnedversion of the latest dataset (Kowalski et al. (2008)).

4.5 Measuring the Acceleration of the Present Universe 123

4.5.1 Luminosity Distance and Hubble–Diagrams forΛCDM

With standard candles, one can measure the cosmos over the distance modulus

m(z) − M = 5 log dL(z; ΩM ,ΩΛ,H0) + 25 , (4.187)

which connects apparent magnitudem(z) und absolute magnitudeM with the distancedL

in units of Mpc. Hereby,dL is the luminosity distancedL = r1R0(1 + z), wherer1 is thecoordinate distance of the object at redshiftz

R0

∫ r1

0

dr√1 − kr2

= R0

∫ t0

t1

c dt

R(t)

=c

H0

∫ t0

t1

da

a√

ΩM/a + ΩΛa2 + 1 − ΩM − ΩΛ

. (4.188)

The second equality follows from the Friedman equation. Since da = −dz/(1 + z)2, we

d_L [Gpc]1 10

z

10-1

1 Vacuum

Luminosity Distance Inflationary Universe

Figure 4.13: Luminosity distance as a function of redshift in the flat inflationary Universe. Thelinear relation corresponds to the classical Hubble law (or a closed Universe withΩM = 2,ΩΛ = 0), the boxes to a matter–dominated modelΩM = 1 (SCDM), the dotted curve to anopen modelΩM = 0.3, ΩΛ = 0, the star symbols refer to aΛCDM–Modell ΩM = 0.3,ΩΛ = 0.7 (ΛCDM), the lowest curve to a vacuum dominated de Sitter Universe withΩΛ = 1

(H0 = 70 km/s/Mpc).

have the equation

R0dr√1 − kr2

= − c

H0

dz

E(z)(4.189)


with

E(z) ≡√

Ωk(1 + z)2 + ΩM (1 + z)3 + ΩΛ (4.190)

andE(0) = 1. This defines thecomoving distancedC(z) = r1R0, wherer1 has to be thesolution of

R0

∫ r1

0

dr√1 − kr2

=1√κ0

R0 arcsin(h)(√

κ0r1) =c

H0

∫ z

0

dz′

E(z′). (4.191)

For our analysis we use the luminosity distancedL(z) = (1 + z)dC(z) in the general form of

dL(z) =c

H0

1 + z√κ0

S(

√κ0

∫ z

0

dz′√∑

i Ωi(1 + z′)3+3wi − κ0(1 + z′)2

)

. (4.192)

S(x) = sin(x), x, or sinh(x) for closed, flat, and open models respectively, and the curvatureparameterκ0, is defined asκ0 = −Ωk =

∑

i Ωi − 1. This formula includes the special caseof a constant vacuum contribution withwV = −1.

The Hubble parameterH(z), appearing in these equations is provided by the Einstein fieldequations. If the different sources which populate the universe do not interact with each otherand each of them is represented by an equation of statewi ≡ Pi/ρi (which can be a functionof time in general), energy conservation

dρ

dz= 3ρ

1 + w(z)

1 + z(4.193)

leads to the solution for the density as a function of redshift

ρ(z) = ρ0 exp

3

∫ z

0

1 + wi(z′)

1 + z′dz′

. (4.194)

The Friedman equations then yield the solution for the Hubble parameter

H2(z) = H20

[

∑

i

Ωi exp

3

∫ z

0

1 + wi(z′)

1 + z′dz′

− Ωk (1 + z)2]

, (4.195)

and for the curvature

q(z) =H2

0

H2(z)

∑

i

Ωi

21 + 3wi(z) exp

3

∫ z

0

1 + wi(z′)

1 + z′dz′

, (4.196)

whereΩi are, as usual, the present day energy densities of the different source components inunits of the critical density3H2

0/8πG andΩk ≡ k/R20H

20 (i denoting non-relativistic matter,

radiation, cosmological constant, quintessence etc.). The present value of the scale factorR0,which measures the curvature of spacetime, can now be calculated from

R0 = H−10

√

k

(∑

i Ωi − 1). (4.197)


Figure 4.14: The curves show a closed Universe (Ω = 2, classical Hubble law) in red, thecritical density Universe (Ω = 1, SCDM) in black, the empty Universe (Ω = 0) in green, thesteady state model in blue, and the WMAP based concordance model withΩm = 0.27 andΩDE = 0.73 in purple. This model givesH0 = 71 km/sec/Mpc which has been used to scalethe luminosity distances in the plot. The data show an accelerating Universe at low to moderateredshifts but a decelerating Universe at higher redshifts, consistentwith a model having both acosmological constant and a significant amount of dark matter. The dashed black curve showsan Einstein-de Sitter model with a constant co-moving dust density which can be ruled out. Thedashed purple curve shows a closed LCDM model which is a good fit to thedata. The dashedblue curve shows an evolving supernova model which is also a good fit. [Gold Sample: NedWright 2008]

We note that the coordinate distancer1, and hencedL, are sensitive toΩi for the distant SNeonly. For the nearby SNe (in the low-redshift limit), equation (4.187) then reduces to

m(z) = M + 5 log z , (4.198)

which can be used to measureM by using low-redshift supernovae-measurements that are farenough into the Hubble flow so that their peculiar velocitiesdo not contribute significantly totheir redshifts.


Special Cases

For dark matter (i = M , wM = 0) and a vacuum energy (i = V , wV = −1), the integral canbe reduced to the form

dL(z; ΩM ,ΩΛ,H0) =c(1 + z)

H0

√

|Ωk|S(x[z]) , (4.199)

where

x[z] =√

|Ωk|∫ z

0

dz′√

(1 + z′)2(1 + ΩMz′) − z′(2 + z′)ΩΛ

. (4.200)

The curvature parameterΩk follows from the condition (4.95). This integral is easily com-puted numerically. From this we get the apparent magnitude for given absolute magnitudeMand therefore areduced distance moduluswhich does not depend on the Hubble constant

DM ≡ m(z) −M = 5 log DL(z; ΩM ,ΩΛ) (4.201)

with dL ≡ (c/H0)DL and the definition of

M ≡ M − 5 log

(

H0 Mpc

c

)

+ 25 = M − 5 log h + 42.38 , (4.202)

which is constant within an ensemble of standard candles. From fitting low–redshift SNe,mB(z) = MB + 5 log z, one can derive a value forM. Perlmutter et al. have pointed out in1997 that the apparent magnitude of SNIa at redshiftsz > 0.1 only depends on the functionDL(z; ΩM ,Ωλ), which does not contain the Hubble constantH0 (Fig. 4.16). This idea is thebasic driver behind all high redshift supernovae projects.

For a flat Universe,Ωk = 0, the above relation is simplified to

dL(z; ΩM ,ΩΛ,H0) =c(1 + z)

H0

∫ z

0

dz′√

ΩM (1 + z′)3 + 1 − ΩM

. (4.203)

The special case withΩM = 1, i.e. ΩΛ = 0, reduces to the well known luminosity distancefor flat Universe without vacuum energy

dL(z;H0) =2c

H0

[

1 + z −√

1 + z]

. (4.204)

In general, the objects appear dimmer in a Universe including vacuum energy when comparedto classical CDM models. Please note that this is also the case for Quasars. At high redshifts,this difference can amount upto two magnitudes (Fig. 4.15).

The K–Correction

When we measure magnitudes of cosmological objects, we have to take into account that thespectrum is shifted with respect to the measured wavelengths due to cosmic expansion. Let usconsider a quasar with redshiftz, which has emitted its light at timet1. Let Eλ(t) denote the


Redshift0 1 2 3 4 5

bri

gh

t

DM

f

ain

t

-8

-6

-4

-2

0

2

4

6

8

Vacuum

Closed Universe

Reduced Distance Modulus

Figure 4.15: Reduced distance modulusDM = m − M as function of redshift for variouscosmological models. For high redshift Supernovae we see differences upto two magnitudes.The starry symbols represent a model withΩM = 0.3 andΩΛ = 0.7. The vacuum model hasΩΛ = 1, the closed Universe corresponds toΩM = 2 andΩΛ = 0 (global linear Hubble–law).The dotted curve is an open model withΩM = 0.3 andΩΛ = 0, and the boxes relate to theclassical SCDM. In this redshift–range, an open model is not very much different from LCDM.

monochromatic luminosity for wavelengthλ and timet, measured in the rest system of thequasar. LetLλ0

be the measured luminosity in a spectral band centered atλ0 (in erg/s/cm2)

Lλ0=

∫ ∞

0

Fλ Sλ dλ . (4.205)

Sλ denotes the transmission function of the instrument. We findthen3

Lλ0=

1

4πd2(1 + z)

∫ ∞

0

Eλ/(1+z)(t1)Sλ dλ

=

∫∞

0Eλ(t0)Sλ dλ

4πd2(1 + z)×∫∞

0Eλ/(1+z)(t0)Sλ dλ∫∞

0Eλ(t0)Sλ dλ

×∫∞

0Eλ/(1+z)(t1)Sλ dλ

∫∞


. (4.206)

3Tinsley, B.M. 1970, Ap&SS 6, 344


Figure 4.16: Hubble Diagram for Supernovae Type Ia. Top panel: Hubble Diagram for SCPlow–extinction subsample; Bottom panel: Residuals relative to an empty Universe. Data fromKnop et al. 2003 [9].

with d as the luminosity distance of the quasar. From this expression we obtain the apparentmagnitude

mλ0= Mλ0

(t0) + 5 log d + const

+

[

2.5 log(1 + z) + 2.5 log

∫∞

0Eλ(t0)Sλ dλ

∫∞


]

+ 2.5 log

[

∫∞


∫∞


]

(4.207)


The expression in the brackets denotes the K–correction, the last term corresponds to evolu-tionary effects. The observed magnitude consists therefore of 5 different contributions D

• the absolut luminosity in the rest system of the object;

• a factor depending on the luminosity distance;

• a constant factor from normalisation;

• the K–correction – the difference in magnitudes between twoobjects of the same spec-trum, however shifted with respect to each other (the difference between observed mag-nitude at wavelengthλ1 = λ0/(1 + z) and the magnitude of the same quasar at timet0with wavelengthλ0);

• evolutionary effects (ageing of quasars and galaxies).

4.5.2 Measuring Cosmology with Supernovae

Understanding the global history of the Universe is a fundamental goal of cosmology. Oneof the conceptually simplest tests in the repertoire of the cosmologist is observing how astandard candle dims as a function of redshift. The nearby Universe provides the current rateof expansion, and with more distant objects it is possible tostart seeing the varied effectsof cosmic curvature and the Universe’s expansion history (usually expressed as the rate ofacceleration/deceleration). Over the past several decades a paradigm for understanding theglobal properties of the Universe has emerged based on General Relativity with the assumptionof a homogeneous and isotropic Universe. The relevant constants in this model are the Hubbleconstant (or current rate of cosmic expansion), the relative fractions of species of matter thatcontribute to the energy density of the Universe, and these species’ equation of state.

Early luminosity distance investigations used the brightest objects available for measuringdistance - bright galaxies (Fig. 4.17), but these efforts were hampered by the imprecisenessof the distance indicators and the changing properties of the distance indicators as a functionof look–back time. Although many other methods for measuring the global curvature andcosmic deceleration exist, supernovae (SNe) have emerged as one of the preeminent distancemethods due to their significant intrinsic brightness (which allows them to be observable inthe distant Universe), ubiquity (they are visible in both the nearby and distant Universe), andtheir precision (type Ia SNe provide distances that have a precision of approximately 8%).

Over the past decade, supernovae have emerged as some of the most powerful tools formeasuring extragalactic distances. A well developed physical understanding of type II super-novae allow them to be used to measure distances independentof the extragalactic distancescale. Type Ia supernovae are empirical tools whose precision and intrinsic brightness makethem sensitive probes of the cosmological expansion. Both types of supernovae are consistentwith a Hubble Constant within≃ 10% of H0 = 70 km/s/Mpc. Two teams have used type Iasupernovae to trace the expansion of the Universe to a look-back time more than 60% of theage of the Universe. These observations show an accelerating Universe which is currently bestexplained by a cosmological constant or other form of dark energy with an equation of statenearw = P/ρ = −1. While there are many possible remaining systematic effects, none ap-pears large enough to challenge these current results. Future experiments are planned to bettercharacterize the equation of state of the dark energy leading to the observed acceleration byobserving hundreds or even thousands of objects. These experiments will need to carefully


Figure 4.17: Hubble diagram for the brightest galaxies in clusters (elliptical galaxies). Magni-tudes are corrected for various effects. [Source: Sandage].

control systematic errors to ensure future conclusions arenot dominated by effects unrelatedto cosmology.

Type Ia Supernovae as Standardized Candles

SNIa have been used as extragalactic distance indicators since Kowal first published his Hub-ble diagram (σ = 0.6 mag) for type I SNe. We now recognize that the old type I SNe spec-troscopic class is comprised of two distinct physical entities: SNIb/c which are massive starsthat undergo core collapse (or in some rare cases might undergo a thermonuclear detonationin their cores) after losing their hydrogen atmospheres, and SNIa which are most likely ther-monuclear explosions of white dwarfs. In the mid-1980s it was recognized that studies of thetype I SN sample had been confused by these similar appearingSNe, which were henceforthclassified as type Ib and type Ic. By the late 1980s/early 1990s, a strong case was being madethat the vast majority of the true type Ia SNe had strikingly similar light curve shapes, spectraltime series, and absolute magnitudes. A 1992 review by Branch and Tammann [2] of a varietyof studies in the literature concluded that the intrinsic dispersion inB andV maximum fortype Ia SNe must be< 0.25 mag, making themthe best standard candles known so far.

In fact, the Branch and Tammann review indicated that the magnitude dispersion wasprobably even smaller, but the measurement uncertainties in the available datasets were too


large to tell. The Calan/Tololo Supernova Search (CTSS), a program begun by Hamuy et al.in 1990, took the field a dramatic step forward by obtaining a crucial set of high quality SNlight curves and spectra. By targeting a magnitude range that would discover type Ia SNe inthe redshift rangez = 0.01 − 0.1, the CTSS was able to compare the peak magnitudes ofSNe whose relative distance could be deduced from their Hubble velocities. As the CTSSdata began to become available, several methods were presented that could select for the moststandard subset of the type Ia standard candles, a subset which remained the dominant majorityof the ever-growing sample. Phillips [?] found a tight correlation between the rate at whicha type Ia SN’s luminosity declines and its absolute magnitude, a relation which apparentlyapplied not only to the Branch Normal type Ia SNe, but also to the peculiar type Ia SNe.Phillips plotted the absolute magnitude of the existing setof nearby SNIa, which had densephotoelectric or CCD coverage, versus the parameterm15(B), the amount the SN decreasedin brightness in the B-band over the 15 days following maximum light. The sample showeda strong correlation which, if removed, dramatically improved the predictive power of SNIa.Hamuy et al. [7] used this empirical relation to reduce the scatter in the Hubble diagram to< 0.2 mag inV for a sample of nearly 30 SNIa from the CTSS search.

Impressed by the success of them15(B) parameter, Riess et al. [17] developed the multi-color light curve shape method (MLCS), which parameterizedthe shape of SN light curves asa function of their absolute magnitude at maximum. This method also included a sophisticatederror model and fitted observations in all colors simultaneously, allowing a color excess to beincluded. This color excess, which we attribute to intervening dust, enabled the extinction tobe measured. Another method that has been used widely in cosmological measurements withSNIa is the ”stretch” method described in Perlmutter et al. [14]. This method is based onthe observation that the entire range of SNIa light curves, at least in theB andV –bands, canbe represented with a simple time stretching (or shrinking)of a canonical light curve. Thecoupled stretchedB andV light curves serve as a parameterized set of light curve shapes,providing many of the benefits of the MLCS method but as a much simpler (and constrained)set. This method, as well as recent implementations ofm15(B), also allows extinction to bedirectly incorporated into the SNIa distance measurements. Other methods that correct forintrinsic luminosity differences or limit the input sampleby various criteria have also beenproposed to increase the precision of type Ia SNe as distanceindicators. While these lattertechniques are not as developed as them15(B), MLCS, and stretch methods, they all providedistances that are comparable in precision, roughlyσ = 0.18 mag about the inverse square law,equating to a fundamental precision of SNIa distances of≃ 6% (0.12 mag), once photometricuncertainties and peculiar velocities are removed. Finally, a poor man’s distance indicator, thesnapshot method, combines information contained in one or more SN spectra with as little asone night’s multi-color photometry. This method’s accuracy depends critically on how muchinformation is available.

To illustrate the effect of cosmological parameters on the luminosity distance, in Fig. 4.19we plot a series of models. In the top panel, the various models show the same linear behaviorat z < 0.1 with models having the sameH0 being indistinguishable to a few percent. Byz = 0.5 the models with significant are clearly separated, with luminosity distances that aresignificantly further than the zero- universes. Unfortunately, two perfectly reasonable uni-verses, given our knowledge of the local matter density of the Universe (M ≃ 0.2), one witha large cosmological constant,ΩΛ = 0.7, ΩM = 0.3 and one with no cosmological constant,ΩM = 0.2, show differences of less than 25%, even to redshifts ofz > 5. Interestingly, themaximum difference between the two models is atz ≃ 0.8, not at largez. Fig. 4.20 illus-


Figure 4.18: The relationship between the light curve width and the luminosity of SNIa. Thetop panel shows various light curves from the Calan/Tololo SN survey. The bottom graph showshow stretch can be used to describe the light curve with just one parameter.


Figure 4.19:dL expressed as distance modulus(m−M) for four relevant cosmological models;ΩM = 0, ΩΛ = 0 (empty Universe, solid line);ΩM = 0.3, ΩΛ = 0 (short dashed line);ΩM = 0.3, ΩΛ = 0.7 (hatched line); andΩM = 1.0, ΩΛ = 0 (long dashed line). In the bottompanel the empty Universe has been subtracted from the other models to highlight the differences.

trates the effect of changing the equation of state of the non-matter, dark energy component,assuming a flat universe, totΩ = 1. If we are to discern a dark energy component that is not acosmological constant, measurements better than 5% are clearly required, especially since thedifferences in this diagram include the assumption of flatness and also fix the value ofΩM . Infact, to discriminate among the full range of dark energy models with time varying equationsof state will require much better accuracy than even this challenging goal.

The intrinsic brightness of SNIa allow them to be discoveredto z > 1.5 with currentinstrumentation (while a comparably deep search for type IISNe would only reach redshiftsof z ≃ 0.5). In the 1980s, however, finding, identifying, and studyingeven the impressivelyluminous type Ia SNe was a daunting challenge, even towards the lower end of the redshiftrange shown in Fig. 4.19. At these redshifts, beyondz ≃ 0.25, Fig. 4.19 shows that relevantcosmological models could be distinguished by differencesof order 0.2 mag in their predictedluminosity distances. For SNIa with a dispersion of 0.2 mag,10 well observed objects shouldprovide a 3 separation between the various cosmological models. It should be noted that theuncertainty described above in measuringH0 is not important in measuring the parameters fordifferent cosmological models. Only the relative brightness of objects near and far is beingexploited and the absolute value ofH0 scales out.


Figure 4.20: dL for a variety of cosmological models containingΩM = 0.3 andΩx = 0.7 witha constant (not time-varying) equation of statewx. Thewx = −1 model has been subtractedoff to highlight the differences between the various models.

The first distant SN search was started by the Danish team of Norgaard–Nielsen et al.With significant effort and large amounts of telescope time spread over more than two years,they discovered a single SNIa in az = 0.3 cluster of galaxies (and one SNII atz = 0.2).The SNIa was discovered well after maximum light on an observing night that could not havebeen predicted, and was only marginally useful for cosmology. However, it showed that suchhigh redshift SNe did exist and could be found, but that they would be very difficult to use ascosmological tools.

Just before this first discovery in 1988, a search for high redshift type Ia SNe using athen novel wide field camera on a much larger (4m) telescope was begun at the LawrenceBerkeley National Laboratory (LBNL) and the Center for Particle Astrophysics, at Berkeley.This search, now known as the Supernova Cosmological Project (SCP), was inspired by theimpressive studies of the late 1980s indicating that extremely similar type Ia SN events couldbe recognized by their spectra and light curves, and by the success of the LBNL fully robotic


low-redshift SN search in finding 20 SNe with automatic imageanalysis.The SCP targeted a much higher redshift range,z > 0.3, in order to measure the (pre-

sumed) deceleration of the Universe, so it faced a differentchallenge than the CTSS search.The high redshift SNe required discovery, spectroscopic confirmation, and photometric followup on much larger telescopes. This precious telescope time could neither be borrowed fromother visiting observers and staff nor applied for in sufficient quantities spread throughout theyear to cover all SNe discovered in a given search field, and with observations early enoughto establish their peak brightness. Moreover, since the observing time to confirm high redshiftSNe was significant on the largest telescopes, there was a clear ”chicken and egg” problem:telescope time assignment committees would not award follow-up time for a SN discoverythat might, or might not, happen on a given run (and might, or might not, be well past max-imum) and, without the follow-up time, it was impossible to demonstrate that high redshiftSNe were being discovered by the SCP.

By 1994, the SCP had solved this problem, first by providing convincing evidence thatSNe, such as SN1992bi, could be discovered near maximum (andK-corrected) out toz =0.45, and then by developing and successfully demonstrating a new observing strategy thatcould effectively guarantee SN discoveries on a predetermined date, all before or near maxi-mum light. Instead of discovering a single SN at a time on average (with some runs not findingone at all), the new approach aimed to discover an entire ”batch” of half-a-dozen or more typeIa SNe at a time by observing a much larger number of galaxies in a single two or three dayperiod a few nights before new Moon. By comparing these observations with the same obser-vations taken towards the end of dark time almost three weeksearlier, it was possible to selectjust those SNe that were still on the rise or near maximum. Thechicken and egg problem wassolved, and now the follow-up spectroscopy and photometry could be applied for and sched-uled on a pre-specified set of nights. The new strategy worked- the SCP discovered batchesof high redshift SNe, and no one would ever again have to hunt for high-redshift SNe withoutthe crucial follow-up scheduled in advance.

The High-Z SN Search (HZSNS) was conceived at the end of 1994,when this group ofastronomers became convinced that it was both possible to discover SNIa in large numbers atz > 0.3 by the efforts of Perlmutter et al., and also use them as precision distance indicators asdemonstrated by the CTSS group. Since 1995, the SCP and HZSNShave both worked avidlyto obtain a significant set of high redshift SNIa.

The two high redshift teams both used this pre-scheduled discovery and follow-up batchstrategy. They each aimed to use the observing resources they had available to best scientificadvantage, choosing, for example, somewhat different exposure times or filters. Quantita-tively, type Ia SNe are rare events on an astronomer’s time scale - they occur in a galaxy likethe Milky Way a few times per millennium and the chapter by Cappellaro in this volume).With modern instruments on 4 meter-class telescopes, whichobserve 1/3 of a square degreeto R = 24 mag in less than 10 minutes, it is possible to search a milliongalaxies toz < 0.5for SNIa in a single night.

Since SNIa take approximately 20 days to rise from undetectable to maximum light, thethree-week separation between observing periods (which equates to 14 rest frame days atz = 0.5) is a good filter to catch the SNe on the rise. The SNe are not always easily identifiedas new stars on the bright background of their host galaxies,so a relatively sophisticatedprocess must be used to identify them. The process, which involves 20 Gigabytes of imagingdata per night, consists of aligning a previous epoch, matching the image star profiles (throughconvolution), and scaling the two epochs to make the two images as identical as possible. The


difference between these two images is then searched for newobjects which stand out againstthe static sources that have been largely removed in the differencing process. The dramaticincrease in computing power in the 1980s was an important element in the development ofthis search technique, as was the construction of wide-fieldcameras with ever larger CCDdetectors or mosaics of such detectors.

This technique is very efficient at producing large numbers of objects that are, on average,at or near maximum light, and does not require unrealistic amounts of large telescope time.It does, however, place the burden of work on follow-up observations, usually with differentinstruments on different telescopes. With the large numberof objects discovered (50 in twonights being typical), a new strategy is being adopted by both the SCP and HZSNS teams,as well as additional teams like the Canada France Hawaii Telescope (CFHT) legacy survey,where the same fields are repeatedly scanned several times per month, in multiple colors, forseveral consecutive months. This type of observing programprovides both discovery of newobjects and their follow up, all integrated into one efficient program. It does require a largeblock of time on a single telescope - a requirement which was not politically feasible in yearspast, but is now possible.

Obstacles to Measuring Luminosity Distances at High-Z

As shown above, the distances measured to SNIa are well characterized atz < 0.1, but com-paring these objects to their more distant counterparts requires great care. Selection effectscan introduce systematic errors as a function of redshift, as can uncertain K-corrections and apossible evolution of the SNIa progenitor population as a function of look-back time. Theseeffects, if they are large and not constrained or corrected,will limit our ability to accuratelymeasure relative luminosity distances, and have the potential to reduce the efficacy of high-ztype Ia SNe for measuring cosmology

• K–correction: As SNe are observed at larger and larger redshifts, their light is shiftedto longer wavelengths. Since astronomical observations are normally made in fixed bandpasses on Earth, corrections need to be applied to account for the differences causedby the spectrum shifting within these band passes. These corrections take the form ofintegrating the spectrum of an SN over the relevant band passes, shifting the SN spectrumto the correct redshift, and re-integrating.

• Extinction: In the nearby Universe we see SNIa in a variety of environments, and about10% have significant extinction. Since we can correct for extinction by observing two ormore wavelengths, it is possible to remove any first order effects caused by a changingaverage extinction of SNIa as a function of z. However, second order effects, such aspossible evolution of the average properties of intervening dust, could still introducesystematic errors. This problem can also be addressed by observing distant SNIa over adecade or so of wavelength in order to measure the extinctionlaw to individual objects.Unfortunately, this is observationally very expensive. Current observations limit the totalsystematic effect to< 0.06 mag, as most of our current data is based on two colorobservations.

An additional problem is the existence of a thin veil of dust around the Milky Way.Measurements from the Cosmic Background Explorer (COBE) satellite accurately deter-mined the relative amount of dust around the Galaxy, but there is an uncertainty in the


absolute amount of extinction of about 2 - 3%. This uncertainty is not normally a prob-lem, since it affects everything in the sky more or less equally. However, as we observeSNe at higher and higher redshifts, the light from the objects is shifted to the red andis less affected by the Galactic dust. Our present knowledgeindicates that a systematicerror as large as 0.06 mag is attributable to this uncertainty.

• Selection Effects:As we discover SNe, we are subject to a variety of selection effects,both in our nearby and distant searches. The most significanteffect is the Malmquist Bias- a selection effect which leads magnitude limited searchesto find brighter than averageobjects near their distance limit since brighter objects can be seen in a larger volume thantheir fainter counterparts. Malmquist Bias errors are proportional to the square of theintrinsic dispersion of the distance method, and because SNIa are such accurate distanceindicators these errors are quite small,≃ 0.04 mag. Monte Carlo simulations can beused to estimate such selection effects, and to remove them from our data sets. The totaluncertainty from selection effects is≃ 0.01 mag and, interestingly, may be worse forlower redshift objects because they are, at present, more poorly quantified.

• Gravitational Lensing: Several authors have pointed out that the radiation from anyobject, as it traverses the large scale structure between where it was emitted and whereit is detected, will be weakly lensed as it encounters fluctuations in the gravitationalpotential. On average, most of the light travel paths go through under-dense regions andobjects appear de-magnified. Occasionally, the light path encounters dense regions andthe object becomes magnified. The distribution of observed fluxes for sources is skewedby this process such that the vast majority of objects appearslightly fainter than thecanonical luminosity distance, with the few highly magnified events making the mean ofall light paths unbiased. Unfortunately, since we do not observe enough objects to capturethe entire distribution, unless we know and include the skewed shape of the lensing a biaswill occur. At z = 0.5, this lensing is not a significant problem: If the Universe isflat innormal matter, the large scale structure can induce a shift of the mode of the distributionby only a few percent. However, the effect scales roughly as z2, and byz = 1.5 the effectcan be as large as 25%. While corrections can be derived by measuring the distortion ofbackground galaxies near the line of sight to each SN, atz > 1, this problem may beone which ultimately limits the accuracy of luminosity distance measurements, unlessa large enough sample of SNe at each redshift can be used to characterize the lensingdistribution and average out the effect. For thez ≃ 0.5 sample, the error is< 0.02 mag,but it is much more significant atz > 1, especially if the sample size is small.

• Evolution: SNIa are seen to evolve in the nearby Universe. Hamuy et al. [29] plotted theshape of the SN light curves against the type of host galaxy. SNe in early hosts (galaxieswithout recent star formation) consistently show light curves which rise and fade morequickly than SNe in late-type hosts (galaxies with on-goingstar formation). However,once corrected for light curve shape the luminosity shows nobias as a function of hosttype. This empirical investigation provides reassurance for using SNIa as distance indica-tors over a variety of stellar population ages. It is possible, of course, to devise scenarioswhere some of the more distant SNe do not have nearby analogues, so as supernovaeare studied at increasingly higher redshifts it can become important to obtain detailedspectroscopic and photometric observations of every distant SN to recognize and rejectexamples that have no nearby analogues.


Figure 4.21: Upper panel: The Hubble diagram for high redshift SNIa from both the HZSNSand the SCP teams. Lower panel: The residual of the distances relative toa ΩM = 0.3, ωΛ =

0.7 Universe. Thez < 0.15 objects for both teams are drawn from CTSS sample, so manyof these objects are in common between the analyses of the two teams. Lowest panel: In thelast few years distant supernovae with redshifts up to 1.755 have beenobserved by the HubbleSpace Telescope. These objects show that the trend toward fainter supernovae seen at moderateredshifts has reversed. The green curve is theΩ = 0 Universe. The solid magenta curve showsthe best fit flat accelerating vacuum-dominated model. The dashed magenta curve is the bestclosed dark energy dominated fit to the supernova data alone.


High Redshift SNIa Observations

The SCP [?] in 1997 presented their first results with 7 objects at a redshift aroundz = 0.4.

Figure 4.22: The supernova data as of April 2008 published by Kowalski et al. (2008) providethe best fit, 1, 2 and 3 standard deviation contours shown as the green, blue, red and black ellipsesin the figure at left. The CMB data using WMAP five year results provide the cloud of dots froma Monte Carlo Markov chain sampling of the likelihood function. The CMB degeneracy trackdoes not follow the flat Universe line, but crosses the flat line at a pointreasonably consistentwith the supernova fit. Each CMB model has an implied Hubble constant which provides thecolor code for the dots. A model that fits both the supernova data and the CMB data has a Hubbleconstant that agrees reasonably well with the Hubble Space Telescope Key Project value of theHubble constant. The addition of high redshift supernovae has had two effects on the supernovaerror ellipse. The long axis of the ellipse has gotten shorter, and the slope of the ellipse hasgotten higher. The best fit model has gotten closer to the CMB degeneracy track in absoluteterms, and it has also gotten closer in terms of standard deviations in the Kowalski et al. (2008)dataset. [Plot: Ned Wright 2008]

These objects hinted at a decelerating Universe with a measurement ofΩM = 0.88+0.69−0.60,

but were not definitive. Soon after, the SCP published a further result, with az ≃ 0.84 SNIaobserved with the KECK I and HST added to the sample, and the HZSNS presented the resultsfrom their first four objects. The results from both teams nowruled out aΩM = 1 Universewith greater than 95% significance. These findings were againsuperceded dramatically whenboth teams announced results including more SNe (10 more HZSNS SNe, and 34 more SCP


SNe) that showed not only were the SN observations incompatible with aΩM = 1 Universe,they were also incompatible with a Universe containing onlynormal matter. Fig. 4.21 showsthe Hubble diagram for both teams. Both samples show that SNeare, on average, fainter thanwould be expected, even for an empty Universe, indicating that the Universe is accelerating.The agreement between the experimental results of the two teams is spectacular, especiallyconsidering the two programs have worked in almost completeisolation from each other.

The easiest solution to explain the observed acceleration is to include an additional com-ponent of matter with an equation of state parameter more negative thanw < −1/3; the mostfamiliar being the cosmological constant (w = −1). Fig. 4.23 shows the joint confidence con-tours for values ofΩM andΩΛ from both experiments. If we assume the Universe is composedonly of normal matter and a cosmological constant, then withgreater than 99.9% confidencethe Universe has a non-zero cosmological constant or some other form of dark energy. Of

Figure 4.23: Left panel: Contours ofΩM versuswx from current observational data. RightPanel: Contours ofΩM versuswx from current observational data, where the current value ofΩM is obtained from the 2dF redshift survey. For both panelsΩM + Ωx = 1 is taken as a prior.

course, we do not know the form of dark energy which is leadingto the acceleration, and itis worthwhile investigating what other forms of energy are possible additional components.Fig. 4.23 shows the joint confidence contours for the HZSNS+SCP observations forΩM andwx (the equation of state of the unknown component causing the acceleration). Because thisintroduces an extra parameter, we apply the additional constraint thatΩM + Ωx = 1, as in-dicated by the CMB experiments. The cosmological constant is preferred, but anything witha w < −0.5 is acceptable. Additionally, we can add information about the value ofΩM , assupplied by recent 2dF redshift survey results, as shown in the 2nd panel, where the constraintstrengthens tow < −0.6 at 95% confidence [16].


The m − z Fit Procedure

Now for givenM andΩi, these equations can provide the predicted value ofm(z) at any givenz. We compare this value with the corresponding observed magnitudemobs and computeχ2

from

χ2 =

N∑

j=1

[

m(zj ; M, Ωi) − mobs,j

σmobs,j

]2

, (4.208)

where the quantityσmobs,jis the uncertainty in the observed magnitudemobs,j of thej-th SN.

It may be noted that some data sets are given in terms of the distance modulusµ = m(z)−M ,instead ofm. However, the zero-point absolute magnitude is set arbitrarily. In this case alsowe can use equation (4.208) for fitting the data by usingµobs in place ofmobs, the constantM(which plays the role of the normalization constant in this case) simply gets modified suitably.Sometimes in the data we have also been provided with independent uncertainties on someother variable. In this case the equation forχ2 gets modified as

χ2 =

N∑

j=1

[

m(zj ; M, Ωi) − mobs,j2

σ2mobs,j

+ σ2int + σ2

v,j

]

, (4.209)

whereσint is the uncertainty due to the intrinsic dispersion of SNe absolute magnitude, andσv,j due to peculiar velocity.

The key point about the SNe Ia data sets is that the absolute luminositiesM of all the SNe,distant or nearby, are regarded same (standard candle-hypothesis). Hence so is the constantM, as it has only one extra parameterH0 which certainly does not differ from SN to SN.Thus there are two ways of the actual data fitting: (i) estimateM by using low-redshift SNe,and use this value in equation (4.208) to estimateΩi from the high-redshift data alone; (ii) uselow-, as well as, high-redshift data simultaneously to evaluate all the parameters from equation(4.208) by keepingM as a free parameter. Obviously the second method gives a better fitting,as shown in Table 4.2.

It is obvious from equation (4.208) that if the model represents the data correctly, then thedifference between the predicted magnitude and the observed one at each data point should beroughly the same size as the measurement uncertainties and each data point will contribute toχ2 roughly one, giving the sum roughly equal to the number of data pointsN (more correctlyN−number of fitted parameters≡ number of degrees of freedom ‘dof’). Ifχ2 is large, thenthe fit is bad. However we must quantify our judgment and decision about thegoodness-of-fit, in the absence of which, the estimated parameters of the model (and their estimateduncertainties) have no meaning at all. An independent assessment of the goodness-of-fit ofthe data to the model is given in terms of theχ2-probability: if the fitted model provides atypical value ofχ2 asx atn dof, this probability is given by

P (x, n) =1

Γ(n/2)

∫ ∞

x/2

e−uun/2−1du. (4.210)

P (x, n) gives the probability that a model which does fit the data atn dof, would give a valueof χ2 as large or larger thanx. If P is very small, the model is ruled out. For example, if weget aχ2 = 20 at 5 dof for some model, then the hypothesis thatthe model describes the datagenuinelyis unlikely, as the probabilityP (20, 5) = 0.0012 is very small.


Fitting with Various SNe Data

In a recent paper, Vishwakarma [24] has discussed the most recent data on SNe projects. Theystart the analysis with the data from Perlmutter et al. (1999), which is one of the important datasets of the first generation of SN Ia cosmology programs (Perlmutter et al. 1998; Garnavichet al. 1998; Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999). They particularlyfocus on the sample of 54 SNe from their ‘primary fit’ C. Results from this fitting procedureare shown in Table 4.2. TheΛCDM, as well as the models with the constantw have a goodfit. For example the concordance model (flatΛCDM) has aχ2 = 57.7 at 52 dof with theprobabilityP = 27.3%, representing a good fit. However, we also note that thedata favour aspherical universe. The Einstein de Sitter (EdS) model (Ωm = 1, Λ = 0) has a bad fit withP = 0.06%.

Table 4.2: Fits of different cosmologies to available data sets [24].

Models Constraint Ωm ΩΛ or ΩX w χ2 dof P

54 SNe from Perlmutter et al. (1999)ΛCDM Ωm + ΩΛ = 1 0.28 ± 0.08 1 − Ωm -1 57.7 52 0.273

ΛCDM none 0.79 ± 0.47 1.40 ± 0.65 -1 56.9 51 0.266

constw Ωm + ΩX = 1 0.48 ± 0.15 1 − Ωm −2.10 ± 1.83 57.2 51 0.257

EdS Ωm = 1, Λ = 0 1 1 − Ωm 92.9 53 0.0006

Gold sample of 157 SNe from Riess et al. (2004)ΛCDM Ωm + ΩΛ = 1 0.31 ± 0.04 1 − Ωm -1 177.1 155 0.108

ΛCDM none 0.46 ± 0.10 0.98 ± 0.19 -1 175.0 154 0.118

constw Ωm + ΩX = 1 0.49 ± 0.06 1 − Ωm −2.33 ± 1.07 173.7 154 0.132

EdS Ωm = 1, Λ = 0 1 1 − Ωm 324.7 156 10−13

164 SNe from Gold+ ESSENCE (Krisciunas et al. 2005)ΛCDM Ωm + ΩΛ = 1 0.30 ± 0.04 1 − Ωm -1 190.3 162 0.064

ΛCDM none 0.48 ± 0.10 1.04 ± 0.18 -1 187.2 161 0.077

constw Ωm + ΩX = 1 0.50 ± 0.06 1 − Ωm −2.69 ± 1.30 185.5 161 0.091

EdS Ωm = 1, Λ = 0 1 1 − Ωm 348.2 163 10−15

168 SNe from Gold+ ESSENCE+ 4 SNe from Clocchiatti et al. (2005)ΛCDM Ωm + ΩΛ = 1 0.29 ± 0.04 1 − Ωm -1 200.79 166 0.034

ΛCDM none 0.51 ± 0.09 1.12 ± 0.16 -1 195.8 165 0.051

constw Ωm + ΩX = 1 0.50 ± 0.04 1 − Ωm −3.18 ± 1.46 193.4 165 0.065

EdS Ωm = 1, Λ = 0 1 1 − Ωm 367.5 167 10−17

The existing data points coming from a wide range of different observations were compiledby Tonry et al. [23]. With many new important additions towards higher redshifts, a refinedsample of 194 SNe was presented by Barris et al. (2004). However, the data we are goingto consider next, is theGold sample of Riess et al. [18], which is a more reliable set ofdata with reduced calibration errors arising from the systematics. It contains 143 points fromthe previously published data, plus 14 new points withz > 1 discovered with the HubbleSpace Telescope (HST). While compiling this sample, variouspoints from the previously


published data were discarded where the classification of the supernova was not certain orthe photometry was incomplete. The fit-results of this data show that although the fits totheΛCDM and the constantw-models are reasonable, they are deteriorated considerably; theprobabilitiesP , which represent the goodness-of-fit, have reduced to less than half of thecorresponding probabilities obtained in the case of the Perlmutter et al. data.

We now consider the first results of the ESSENCE project (Krisciunas et al. 2005, madepublic in August 2005) under which 9 SNe with redshift in the range 0.5 - 0.8 were discoveredjointly with HST and Cerro Tololo 4-m telescope. In order to minimize the systematic errors,all the ground-based photometry was obtained with the same telescope and instrument. Weconsider 7 SNe of this project which have unambiguous redshift and definite classification,and add them to the ‘gold sample’ resulting in a reliable sample of 164 SNe. The Table 1shows that the fits to different cosmologies have further worsened considerably and do notrepresent good fit, in any case. Increasing the number of fitted parameters (for example, in themodels with a constantωφ) improves the fit marginally only.

Next we consider the recent discovery of 5 SNe at redshiftz ≈ 0.5 by the High-z Super-nova Search Team (Clocchiatti et al. 2005, results made public in October 2005). We consider4 SNe from this sample for which distances estimated from theMLCS2k2 (Multi-colour LightCurve Shape) method are available, so that we can include them in the previous sample of‘gold + ESSENCE’ which also use the MLCS2k2 method to determine the distance moduli.The fit-results of the resulting sample of 168 SNe are very disappointing. The quality of thefits to different models has deteriorated to such extent thatthe concordance model can be re-jected at 96.6 % confidence level! This is an alarming situation. Other models have marginallysimilar fit and increasing the number of fitted parameters does not help significantly. Modelswith variableωφ(t) do not help either. For example, if we considerw(z) = w0 +w1 z/(1+z)with w0, w1 as constants, we obtainΩm = 0.42, ΩX = 0.42, w0 = −4.95 andw1 = 2.83 asthe best-fitting solution withχ2 = 193.07 at 163 dof andP = 5.4%.

Finally, the author also considers the recently published (made public in October 2005)first year-data of the planned five-year SuperNova Legacy Survey (SNLS) (Astier et al. [1]).In this survey the authors claim to have achieved high precision from improved statistics andbetter control of systematics by using the multi-band rolling search technique and a singleimaging instrument to observe the same fields. Their data setincludes 71 high redshift SNeIa in the redshift range 0.2 - 1 from the SNLS, together with 44low redshift SNe Ia compiledfrom the literature but analyzed in the same manner as the high-z sample. As the SNLS datahave been analyzed differently, the fitting procedure followed by the authors is also different.In order to calculateχ2, they use

χ2 =

N∑

j=1

[

µ(zj ; M, Ωi) − µobs,j2

σ2µobs,j

+ σ2int

]

, (4.211)

whereσint, is the (unknown) intrinsic dispersion of the SN absolute magnitude which, unlikethe other data sets, is not included in theσµobs; rather it has been used as an adjustable freeparameter to obtainχ2/dof = 1. We shall return to this issue later for our comments.Firstwe want to verify if we can reproduce the results of Astier et al. from equation (4.211) byusing their calculatedµobs (given in columns 7 of their Tables 8 and 9) instead of using their‘stretch’ and ‘color’ parameters, which do not seem necessary once we haveµobs. We findthat by fixingσint = 0.13, we getΩm = 1 − ΩΛ = 0.26, M = 43.16 with χ2/dof = 1.00;andΩm = 0.31, ΩΛ = 0.81, M = 43.15 with χ2/dof = 1.01. This is exactly what Astier et alhave obtained.


However, the introduction ofσint in equation (4.211) is justified only when we use in-dependent measurement uncertaintiesσint,j on the parameter, instead of using it as a freeparameter. The latter case is just equivalent to increasingthe error bars suitably in order tohave a desired fit. In this way one can fit any model to the data. For example, the EdS modelcan also have an excellent fit by consideringσint = 0.258: M = 43.46 with χ2/dof = 1.00. Itshould also be noted that this approach (which is equivalentto assuming that the data have agood fit to the model) prohibits an independent assessment ofthe goodness-of-fit-probabilityP , in the absence of which the estimated parameters do not havemuch significance. However,one can still compare the goodness-of-fit of different models. For example, withσint = 0.13,the EdS model has a worse fit than theΛCDM model, givingχ2/dof = 2.7. As the SNLS datahave been analyzed in a different way than the other available data, it does not make sense toadd this data to the earlier samples for a joint analysis.

The High–z Supernovae Search team has recently discovered 8new SNe in the redshiftintervalz = 0.3− 1.2 [23]. From these observations they derive a SN Ia rate of(1.4± 0.5)×10−4 h3 Mpc−1 yr−1 at a mean redshift of 0.5. Including the constraint of a flat Universe,they findΩM = 0.28 ± 0.05.

Nesseris et al. [13] have performed a similar comparative analysis with three SNe datasets:full Gold sample [18], a truncated Gold sample, and the SNLS dataset consiting of 115 SNe[1].

Gamma–Ray Burster as Standard Candles

One of the few ways to measure the properties of Dark Energy isto extend the Hubble daigram(HD) to higher redshifts with Gamma-Ray Bursts (GRBs). GRBshave at least five properties(their spectral lag, variability, spectral peak photon energy, time of the jet break, and theminimum rise time) which have correlations to the luminosity of varying quality. In Schaefer2006, they construct a GRB HD with 69 GRBs over a redshift range of 0.17 to> 6, with halfthe bursts having a redshift larger than 1.7. This paper usesover 3.6 times as many GRBs and12.7 times as many luminosity indicators as any previous GRBHD work. The GRB HD iswell-behaved and nicely delineates the shape of the HD. The reduced chi-square for the fit tothe concordance model is 1.05 and the RMS scatter about the concordance model is 0.65 mag.

The plot 4.24 shows the difference in distance modulus between the empty model and thesupernova and the GRB binned data. It looks a bit inconsistent at redshifts near 0.5 but theresiduals from the fits are not much bigger than the stated errors. When fitting to both the SNeand GRB datasets, there should be two free parameters for Hubble constant changes, one foreach dataset. These free parameters can be thought of as adjustments to the overall luminositycalibration of SNe and GRBs respectively.

The Future

How far can we push the SN measurements ? Finding more and moreSNe allows us to beatdown statistical errors to arbitrarily small levels but, ultimately, systematic effects will limitthe precision to which SNIa magnitudes can be applied to measure distances (Fig. 4.25).Our best estimate is that it will be possible to control systematic effects from ground–basedexperiments to a level of≃ 0.03 mag. Carefully controlled ground-based experiments on 200SNe will reach this statistical uncertainty inz = 0.1 redshift bins in the rangez = 0.3 − 0.7,and is achievable within five years. A comparable quality lowredshift sample, with 300 SNein z = 0.02 − 0.08, will also be achievable in that time frame.


Figure 4.24: The difference in distance modulus between the empty model and the supernovaand the GRB binned data. The GRB data are from Schaefer (2006). [Plot: Ned Wright 2008]

Figure 4.25: SNAP constraints on dark energy models. The current set of 18 + 42 SNe issupplemented by a set of SNAP SNe (each SNAP point represents data of 50 supernovae).

The SuperNova/Acceleration Probe (SNAP) collaboration has proposed to launch a dedi-


cated cosmology satellite - the ultimate SNIa experiment. This satellite will, if funded, scanmany square degrees of sky, discovering well over a thousandSNIa per year and obtain theirspectra and light curves out toz = 1.7. Besides the large numbers of objects and their ex-tended redshift range, space-based observations will alsoprovide the opportunity to controlmany systematic effects better than from the ground. Fig. 4.26 shows the expected precisionin the SNAP and ground–based experiments for measuring w, assuming a flat Universe. Per-haps the most important advance will be the first studies of the time variation of the equationof state w (see the right panel of Fig. 4.26).

Figure 4.26: Future expected constraints on dark energy: Left panel: Estimated 68%confidenceregions for a constant equation of state parameter for the dark energy, w, versus mass density,for a ground-based study with 200 SNe betweenz = 0.3 − 0.7 (open contours), and for thesatellite-based SNAP experiment with 2,000 SNe betweenz = 0.3−1.7 (filled contours). Bothexperiments are assumed to also use 300 SNe betweenz = 0.02 − 0.08. A flat cosmologyis assumed (based on Cosmic Microwave Background (CMB) constraints) and the inner (solidline) contours for each experiment include tight constraints (from largescale structure surveys)on ΩM , at the±0.03 level. Such ground-based studies will test the hypothesis that the darkenergy is in the form of a cosmological constant, for whichw = −1 at all times. Middlepanel: The same confidence regions for the same experiments not assuming the equation ofstate parameter, w, to be constant, but instead marginalizing over w’, where w(z) = w0 + w’z.(Weller and Albrecht [25] recommend this parameterization of w(z) over the others that havebeen proposed to characterize well the current range of dark energy models.) Note that theseplanned ground-based studies will yield impressive constraints on the value of w today, w0,even without assuming constant w. In fact, these constraints are comparable to the currentmeasurements of w assuming it is constant. Right panel: Estimated 68% confidence regionsof the first derivative of the equation of state,w′, versus its value today,w0, for the sameexperiments.

With rapidly improving CMB data from interferometers, the satellites Microwave AnisotropyProbe (MAP) and Planck, and balloon-based instrumentationplanned for the next several

4.6 Angular Width in FRW Models 147

years, CMB measurements promise dramatic improvements in precision on many of the cos-mological parameters. However, the CMB measurements are relatively insensitive to the darkenergy and the epoch of cosmic acceleration. SNIa are currently the only way to directlystudy this acceleration epoch with sufficient precision (and control on systematic uncertain-ties) that we can investigate the properties of the dark energy, and any time dependence inthese properties. This ambitious goal will require complementary and supporting measure-ments of, for example,ΩM from CMB, weak lensing, and large scale structure. The SN mea-surements will also provide a test of the cosmological results independent from these othertechniques, which have their own systematic errors. Movingforward simultaneously on theseexperimental fronts offers the plausible and exciting possibility of achieving a comprehensivemeasurement of the fundamental properties of our Universe.

4.6 Angular Width in FRW Models

We already have derived a general expression for the angularwidth ∆θ of an object in anexpanding Universe, based on the angular distance which is related to the luminosity distance

∆θ =D(1 + z)2

dL. (4.212)

The redshift factor in the nominator cancels to a large extent the increase in the luminositydistance with redshift. For a Universe dominated by vacuum energy, the apparent angularwidth reaches a constant value which is just given byD/RH (Fig. 4.27). For all othermodels, the angular width attains a minimum at some redshiftaroundz ≃ 1. For matter–dominated models withΩΛ = 0 (SCDM),q0 = 1/2 the minimum can easily be determined

∆θ =DH0

2c

(1 + z)3/2

√1 + z − 1

. (4.213)

We attain a minimum for

zmin = 1.25 (4.214)

and the corresponding minimal angular width is given by4

∆θmin = 3.375DH0

c= 0.12 (2h)mas

(

D

pc

)

. (4.215)

In the SCDM model, a galaxy with dimension of 30 kpc would havean angular diameterof 3.6 arcsec at redshiftz = 1.2. It would be nicely resolvable with present–day telescopes.At higher redshifts, the angular diameter would even increase. In fact, on deep images of theSky (such as Hubble–Deep field or the FORS–Deep field) galaxies appear resolved and havetypical extensions of about one to a few arcseconds. This focussing property of the expandingUniverse is essential for observations of high–redshift objects. The jets of quasars remainmore or less of constant angular extension beyond redshift one. It is however very difficultto find a cosmic standard rod which could be used to measure thecosmological parameters(extension of galaxy clusters or the length of compact jets in quasars).

41 mas = 1 milli–arcsecond


z10-1

1 10

D/R

Hu

bb

le

1

10

Angular Width Inflationary Universe

Figure 4.27: Apparent angular width as a function of redshift in the flat inflationary Universe.The decay with1/z corresponds to the classical redshift behaviour without expansion, the bluesolid line to a matter–dominated flat model withΩM = 1 (SCDM), the dotted line to a matter–dominated open model forΩM = 0.3 (OCDM), the star symbols refer to aΛCDM–model withΩM = 0.3, ΩΛ = 0.7, the lowest curve to a vacuum dominated de Sitter Universe withΩΛ = 1

(in all models:H0 = 70 km/s/Mpc).

Angular Width of Structure at Recombination

For the interpretation of the anisotropies in the CMBR it is important to know what the intrin-sic scale is e.g. for an angular width of one degree. As we see from Fig. 4.28, the SCDMmodel provides an upper limit for the angular width

∆Θ(rec) <DH0

c

zrec

2. (4.216)

A reasonable value is∆Θ(rec) ≃ 300DH0/c. A scale of one degree corresponds thereforeto a structureD ≃ 0.2 Mpc at recombination, i.e. a structure of 200 Mpc today. Typicaltemperature fluctuations have been measured to occur on angular scales of about half a degree(WMAP, Fig. 2.9), corresponding to a scale of roughly 100 Mpc in the present Universe. Thisis in fact the mean distance between large clusters which form the corners of huge filaments(Fig. 4.29). There is no structure beyond this scale, i.e. the homogeneity is indeed satisfied onthose scales. This is in agreement with the findings that the CMBR spectrum is flat on scaleslarger than about one degree.

TheMillennium Simulation (Fig. 4.29) employed more than 10 billion particles of mat-ter to trace the evolution of the matter distribution in a cubic region of the Universe over 2billion light-years on a side.5 It kept the principal supercomputer at the Max Planck Society’s

5The Virgo Consortium is an international grouping of scientists carrying out supercomputer simulations of the

4.7 Redshift Distribution of Cosmological Objects 149

10-1

1 10 102

103

1

10

102

Angular Width Recombination

Vacuum

z

SCDM

OCDM

Figure 4.28: Apparent angular width in units ofD/RH as a function of redshift at the recombi-nation era. The highest angular widths are obtained for classical SCDM models, while vacuumenergy reduces the angular width (starry line:ΛCDM with ΩM = 0.3). In aΛCDM model wefind for the angular width at recombination∆Θ(Rec) ≃ 300 D/RH .

Supercomputing Centre in Garching, Germany occupied for more than a month. By applyingsophisticated modelling techniques to the 25 Terabytes of stored output, Virgo scientists areable to recreate evolutionary histories for the approximately 20 million galaxies which popu-late this enormous volume and for the supermassive black holes occasionally seen as quasarsat their hearts.

4.7 Redshift Distribution of Cosmological Objects

We assume that there aren(t1;L) dL galaxies at timet1 having absolute luminosity betweenL andL + dL. The number of such objects is then given by

dN =R(t1) dr1√

1 − kr21

R2(t1)r21 n(t1;L) dLdΩ . (4.217)

Since

dr1 = −√

1 − kr21 c dt1/R(t1) (4.218)

formation of galaxies, galaxy clusters, large-scale structure, and of the evolution of the intergalactic medium. Al-though most of the members are based in Britain and at the MPA in Germany, there are important nodes in Japan,Canada and the United States. The primary platforms used by theconsortium are the IBM supercomputer at theComputing Centre of the Max Planck Society in Garching and theSun Microsystems ”Cosmology Machine” at theInstitute for Computational Cosmology of Durham University.


Figure 4.29: The distribution of dark matter in the Universe on different scales. The backgroundpicture shows a cut through the Millennium Simulation with a total extension of more than 9billion lightyears on a side. On such huge scales, the universe appears nearly homogeneous, butthe series of enlargements overlayed show a complex cosmic web of dark matter up to scales oforder≃ 100 Mpc. This large-scale structure consists of filaments which surround large voidsand cross in massive halos of matter. The largest of these halos are rich clusters of galaxies,containing more than one thousand galaxies which are still resolved as halosubstructure in thesimulation. [22]

4.7 Redshift Distribution of Cosmological Objects 151

we find

dN = R2(t1)r21 n(t1;L) c|dt1| dLdΩ . (4.219)

From the redshift condition

dt = − dz

(1 + z)E(z)(4.220)

the number is given by

dN = RH R2(t0)r21 n(t1;L)

dz dLdΩ

(1 + z)3 E(z). (4.221)

This may be expressed in terms of the luminosity distance

dN = RHd2

L(z)

(1 + z)5 E(z)n(t1;L) dz dLdΩ . (4.222)

For a classical Friedmann Universe,ΩΛ = 0, we recover the old expression

dN = RHd2

L(z)

(1 + z)6√

1 + ΩMzn(t1;L) dz dLdΩ . (4.223)

This is equivalent to the expression derived in Sect. 3.6 forthe number counts of galaxies perunit solid angle and per redshift interval

Redshift10-1

1 10

(1/V

_H)

dV

/ d

z d

O

10-3

10-2

OCDM

SCDM

LCDM

Comoving Volume Inflationary Universe

Figure 4.30: The comoving cosmic volume per redshift interval and per steradian asa function ofredshift for various FRW models (in units of the Hubble volumeVH = R3

H ). The highest volume isachieved for models including dark energy, while in an inflationary Universe with vanishing dark energy,the lowest volume is obtained.

dN

dz dΩ= n(z) dH(z)

d2L(z)

(1 + z)5(4.224)


dH(z) = c/H(z) is the Hubble–radius at redshiftz anddL(z) the luminosity distance. Forobjects which are conserved per proper volume,n remains constant as a function of redshift,or

n(z;L) = n(0;L) (1 + z)3 , (4.225)

we obtain for the total distribution

dN(z) = RH

∫ ∞

0

dLd2

L(z)

(1 + z)2 E(z)n(0;L) dz dΩ . (4.226)

For a class of objects with time–independent luminosity (Supernovae e.g.) we would have adistribution of the kind

dN(z) = RHd2

L(z)

(1 + z)2 E(z)n(0) dz dΩ . (4.227)

In distinction to the above proper volume, thecomoving volumedefined as

dVC = dH(z) d2C(z) dz dΩ = dH(z)

d2L

(1 + z)2dz dΩ (4.228)

is often used. Here,dC(z) = R0rem is the comoving distance. When compared to a non–expanding Universe, the proper volume attains a maximum around redshift one (Fig. 4.30).The location of the maximum depends on the particular FRW–model.

In Fig. 4.31 we show the behaviour of the differential numbercounts according to relation(4.227). In all FRW–models, the number counts reach a maximum around redshift two andthen slowly decline towards higher redshifts. In flux–limited samples, this decline wouldoccur much more rapidly, since at high redshifts only the brightest sources can be observed.If we were observing galaxies with sufficient sensitivity, we should see this maximum aroundredshift two. This behaviour is typically seen in the redshift distribution of Quasars. It is,however, quite interesting that the redshift distributionof Quasars attains its maximum aroundredshift two.

How much of the Sky is covered by galaxies ? Another interesting question is the proba-bility of a given line of sight to intersect a galaxy with redshift in the interval[z, z + dz]. Thisis given by

dPG = πr2G ndl (4.229)

resulting in the expression

dPG

dz= πr2

Gn(z)dH(z)

1 + z. (4.230)

If galaxies are neither created nor destroyed,n(z) = n0(1 + z)3, showing that

dPG

dz= πr2

GRH n0

(

(dH(z)/RH) (1 + z)2)

. (4.231)

4.8 The Cosmological Fundamental Plane 153

Redshift0 1 2 3 4 5

(1/n

V_H

) d

N /

dz

dO

0

0.1

0.2

0.3

0.4

0.5

OCDM

SCDM

LCDM

Number Counts Inflationary Universe

Figure 4.31: The number counts of galaxies per redshift interval and per steradianas a function ofredshift for various FRW models (in units of the Hubble volume numbernVH ). In all models, theobserved number counts reach a maximum around redshift two and then decline towards higher redshifts.

The total optical depth for intersection of objects upto some redshiftz is then given by theintegral

τG(z) =

∫ z

0

dPG = πr2GRH n0

∫ z

0

dx (dH(z)/RH)(1 + x)2 . (4.232)

This integral can be done e.g. in a classical inflationary FRW–model withΩM = 1, wheredH(z) = RH/(1 + z)3/2,

τG(z) =2

3n0 πr2

GRH (1 + z)3/2 . (4.233)

For bright galaxies,n0 ≃ 0.02h3 Mpc−3 andrG ≃ 10/h kpc, we getτG(z) ≃ 0.01 (1+z)3/2.At redshift one, about 4% of the Sky is covered by galaxies. The optical depth would reachunity at a redshift ofz ≃ 20, but galaxies are formed at redshifts lower than this value.

4.8 The Cosmological Fundamental Plane

Due to the restrictionΩk+ΩM+ΩΛ = 1, given by the Friedmann equation, the present state ofthe Universe is determined by a point in the(ΩM ,ΩΛ)–plane. This plane is nowadays calledFundamental Plane of Cosmology(Fig. 4.32). In particular, the transition from acceleratedto decelerated models is given by the lineq0 = 0. Three types of data are available to constrainthe location of the present Universe

• Supernovae data, as discussed in the previous section;


• CMBR anisotropies (WMAP and other experiments, see Sect. 8);

• mass determinations in clusters of galaxies (Chandra and XMM–Newton, Sect. 2.4).

All these data are compatible with the assumption of a non–vanishing vacuum energy. Thebest–buy modelof the year 2005 (calledconcordance model) is given by the following pa-rameters

• H0 = (71 ± 5) km/s/Mpc

• Ωk = 0.02 ± 0.02 (the Universe is flat)

• ΩM = (0.135 ± 0.08)/h2 = 0.27 ± 0.04

• ΩΛ = 0.73 ± 0.04

• ΩB = (0.0224 ± 0.0009)/h2 = 0.044 ± 0.004.

• Dark Energy is compatible with vacuum energy,w0 = −1.

It is the first time since 1929 that realistic error bars are given for these parameters. TheUniverse is completely dominated by Dark Energy and Dark Matter.

4.9 On the Origin of the Dark Energy

Not much is known about dark matter, and even less about dark energy. Cosmologists havetaken to discussing the enigmatic properties of the dark energy with the use of a new pa-rameter,w, which is the ratio of its average pressure to energy density. The degree of thisrunaway expansion impulse is expressed byw. What is the nature of dark energy and howdoes it overcome the attractive pull of gravitation in orderto speed up the cosmic expansion,and what is the proper value ofw? In the best known model, the cosmological constant inEinstein’s famous equations of general relativity corresponds to energy and pressure of theuniversalquantum vacuum, and is constant in space and time. Here the value ofw is -1. In asecond popular model, thequintessence model, the dark energy is associated with a universalquantum field relaxing towards some final state. Here the energy density and pressure of thedark energy are slowly decreasing with time, and the value ofw is somewhere between -1/3and -1 (w must be smaller than -1/3 in order for cosmic acceleration tooccur).

A new cosmic doomsday scenario takes the present acceleration of the expansion of theuniverse to new extremes. Dartmouth physicist Robert Caldwell and his colleagues MarcKamionkowski and Nevin Weinberg at Caltech have determinedthat if the supposed dark en-ergy responsible for the acceleration is potent enough not only will the space between galaxiescontinue to increase but that the galaxies themselves will fly apart as will, at successive timesstars, planets, and even atoms and nuclei.

In Caldwell’s phantom energymodel, there is no stable vacuum quantum state and theenergy density and the expansionary pressure exerted on theuniverse seems to increase evenas the spacetime itself expands (with ordinary gases, pressure falls with expansion). In thisscenariow is less than -1. The implications of this new type of cosmology are that boundsystems should in the course of time be ripped up. For example, at aw value of -1.5 theuniverse would last for 35 billion years before being rippedapart. About 60 million yearsbefore the end, the Milky Way would be torn apart. About 3 months before the end the so-lar system would become undone. About 30 minutes before thatthe Earth would explode.

4.9 On the Origin of the Dark Energy 155

Figure 4.32: The Fundamental Plane of Cosmology with the restrictions from SNe, CMBR–anisotropies and dark matter in clusters. All these data are compatible with a flat Universedominated by dark energy. In the future, the question whether the Universe is really flat can beattacked (SNAP e.g.).

And about10−19 seconds before the ultimate moment of doom, atoms would be pulled apart.Caldwell suggests that deciding between this model and the others might be possible in com-ing years with much better data coming from microwave background, supernovae, and galaxymeasurements [3].


4.9.1 The Vacuum is not Empty

According to quantum mechanics, the vacuum is not empty, butteeming with virtual particlesthat constantly wink in and out of existence. One strange consequence of this sea of activity istheCasimir effect: Two flat metal surfaces automatically attract one another if they get closeenough. The Casimir force is so weak that it has rarely been detected at all, but now a teamreports in the 23 November PRL that they have made the most precise measurement ever ofthe phenomenon. They claim that their technique, using an atomic force microscope, has thecapacity to test the strangest aspects of the Casimir effect, ones that have never before beentested.

The simplest explanation of the Casimir effect is that the two metal plates attract becausetheir reflective surfaces exclude virtual photons of wavelengths longer than the separationdistance. This reduces the energy density between the plates compared with that outside, and–like external air pressure tending to collapse a slightly evacuated vessel–the Casimir forcepulls the plates toward one another. But the most puzzling aspect of the theory is that theforce depends on geometry: If the plates are replaced by hemispherical shells, the force isrepulsive. Spherical surfaces somehow enhanced the numberof virtual photons. There is nosimple or intuitive way to tell which way the force will go before carrying out the complicatedcalculations.

Since the discovery of the theory by Casimir 50 years ago, there have been only twoprevious documented detections of the effect. One was in 1958 and had 100% uncertainty,and the second was in 1997, when the theory was verified to within 5%. Umar Mohideenand Anushree Roy, of the University of California at Riverside, claim their results verify thetheory to within 1% [Phys. Rev. Lett. 81, 4549 (1998)].

Technically, quantum fields have an infinite number of possible energy states, all of whichshould contribute virtual particles to the vacuum. Theorists normally assume that the num-ber of states is actually finite because they can’t exceed theso-called Planck energy, whichcorresponds to the smallest distance quantum mechanics allows for.

4.9.2 Quintessence

Today’s cosmologists find Lambda to be just as objectionable, but for a different reason. Allquantum fields possess a finite amount of ”zero-point” vacuumenergy as a result of the uncer-tainty principle. A naive estimate of the zero-point energypredicts a vacuum energy densitythat is 120 orders of magnitude greater than the energy density of all the other matter in theuniverse. If the vacuum energy density really is so enormous, it would cause an exponentiallyrapid expansion of the universe that would rip apart all the electrostatic and nuclear bonds thathold atoms and molecules together. There would be no galaxies, stars or life. Since we cannotignore quantum mechanics, some other mechanism must nullify this vacuum energy. One ofthe major goals of unified theories of gravity has been to explain why the vacuum energy iszero.

Einstein’s blunder has been resurrected as a possible solution to the dark-energy problem.Maybe there is a miraculous cancellation mechanism, but perhaps it is slightly imperfect.Instead of making Lambda exactly zero, the mechanism only cancels to 120 decimal places.Then, vacuum energy would comprise the missing two-thirds of the critical density. Therequirements seem bizarre, though. Some constant that is naturally enormous must be cutdown by 120 orders of magnitude, but with such precision thattoday it has just the right valueto account for the missing energy.


Extrapolating back in time to the early universe, the story seems even more bizarre. Whenthe volume of the universe was 100 orders of magnitude smaller, say, the mass density was100 orders of magnitude greater, but the vacuum energy density had to have the same value astoday. In other words, the vacuum energy density remained constant as the universe expanded,but the total vacuum energy increased as the volume of space increased. This extra energycame from the gravitational potential energy of the universe. Whatever physical processescreated the initial energy in the universe had to arrange foran exponentially large differencebetween the two forms of energy, but somehow this differencehad to have exactly the rightvalue for the vacuum energy to become important 15 billion years later.

It would seem more natural for the dark energy to start with anenergy density similarto the density of matter and radiation in the early universe (Fig. 4.33). The dark energy andmatter density could both then decrease at similar rates as the universe expanded, with the darkenergy density overtaking the matter density only after structure has formed in the universe.However, if the dark energy density has been changing, it cannot consist of vacuum energy.Therefore, the concept of quintessence was introduced to overcome this problem in 1998.Quintessence is a dynamic, time-evolving and spatially dependent form of energy withnegative pressure sufficient to drive the accelerating expansion. Whereas the cosmologicalconstant is a very specific form of energy - vacuum energy - quintessence encompasses a wideclass of possibilities.

The simplest model proposes that the quintessence is a quantum field with a very longwavelength, approximately the size of the observable universe. Some examples had beenexplored almost a decade earlier by Bharat Ratra and James Peebles at Princeton University,and by Wetterich (Heidelberg). A particle is usually thought of as a bundle of oscillations ina quantum field, but since this bundle is much larger than any conventional length scale, theparticle description is impractical.

The energy is composed of kinetic energy, which depends on the rate of oscillations in thefield strength, and potential energy, which depends on the interaction of the field with itself andmatter. The pressure is determined by the difference between the kinetic and potential energy,with kinetic energy contributing positively to the pressure. However, since the oscillation hasan extremely long wavelength and period - essentially the size and age of the universe - itskinetic energy is negligible. The behaviour of the quintessence field is therefore dominated byhow it interacts with itself. Much like a stretched spring, this self-interaction potential leadsto negative pressure.

Within certain models that seek to unify the four fundamental forces of nature there existfields, called ”tracker fields”, that can make quintessence behave in this way (see Zlatev et al.in further reading). First, the dark energy density in the early universe can be comparable withthe matter density. The model is insensitive to the precise initial value because the dynamicalequations that determine the time evolution of the tracker field have solutions that cause theenergy to follow the same evolution, independent of initialconditions (similar to the classicalattractor solutions found in conventional nonlinear dynamics). In particular, the energy densitytracks the radiation and matter density (see Figure 4.33)).For most of the history of theuniverse, the quintessence occupies a very small fraction of the critical density, but the fractiongrows slowly until it catches up with and ultimately overtakes the matter density.

It seems natural to ask if there are any direct gravitationalinteractions between ordinarymatter and dark energy. If the dark energy is vacuum energy, then the two do not interactbecause vacuum energy is inert and unchanging. But if the dark energy is quintessence, theycan interact under certain conditions. Ordinary particlesare physically very small compared


Figure 4.33: This figure illustrates the different scaling of energy densities of radiation(dottedcurve), matter (dashed curve), a cosmological constant and two quintessence models. The solidgreen line is an early dark energy model where the amount of dark energy is non-negligible atearly times. The dashed green line depicts a phantom dark energy model.a is the scale factor,normalized such thata = 1 today. The energy scale at the right side of the figure is convenientlychosen as Gev per cubic meter: one hydrogen atom corresponds to one Gev and hence the meandensity of our Universe today corresponds to a few hydrogen atoms per cubic meter.

with the Compton wavelength,λc = ~/mc, of the quintessence particles, so individual par-ticles have a completely negligible effect on quintessence(and vice versa). However, verylarge clumps of ordinary matter - spread out over a region comparable with the Comptonwavelength - can interact gravitationally with quintessence and create inhomogeneities in itsdistribution that may produce detectable signals in the cosmic microwave background.

What cosmologists find most difficult to explain is why the acceleration should begin atthis particular moment in cosmic history. Is it a coincidence that, just when thinking beingshave evolved, the universe suddenly shifts into overdrive?The situation is peculiar becausethe energy associated with the cosmological constant or quintessence is very tiny, less than amillielectron-volt. If new ultra-low-energy physics is responsible, it should have already beenobserved in other experiments.

Perhaps a more satisfying possibility is that the acceleration is triggered by natural eventsin the recent history of the universe. According to the big-bang model, the energy densityin the universe was predominantly in the form of hot, relativistic particles until the universewas a few tens of thousands of years old. At that time, the universe had cooled enough thatthe mass energy of non-relativistic particles became more important than both their kineticenergy and the energy of radiation, resulting in a change in the cosmic expansion rate. Thismarked the beginning of the ”matter-dominated epoch”. Onlythen could gravity begin to


clump matter together to form stars, galaxies and large-scale structure. Is it possible that thistransition triggered the onset of quintessence?

Quintessence leaves its mark on the universe in several ways, so experimenters have anumber of methods they can use to test for this exotic form of energy. The acceleration effectof a dark-energy component depends on the ratio of its pressure to its energy density. Morenegative values of this ratio, w, lead to greater acceleration. Quintessence and vacuum energyhave different values of w, so more precise measurements of supernovae over a longer span ofdistances may be able to separate these two possibilities.

This challenge is the motivation for two proposals - the Earth-based Large-Aperture Syn-optic Survey Telescope (LSST) and the space-based Supernova Acceleration Project (SNAP)- that will monitor the sky for supernovae and other time-varying astrophysical phenomena.The proposers of these projects are currently seeking funding.

Cosmic acceleration also affects the number of galaxies to be found as one explores deeperand deeper into space. With appropriate corrections for evolution and other effects, the av-erage density of galaxies is uniform throughout space. Consequently, for a fixed range ofdistances, one should find the same number of galaxies nearbyand far away. But cosmolo-gists measure the redshift of distant galaxies, not their distance. The conversion from redshiftto distance follows a simple linear relation (the Hubble law) if the distances are small, but anonlinear relation depending on the acceleration of the universe if the distances are large. Thenonlinear relation will cause the number of galaxies found for a fixed range of redshifts tochange systematically as one probes deeper into space. The Deep Extragalactic EvolutionaryProbe (DEEP), an advanced spectrograph on the Keck II telescope in Hawaii, is poised to testthis prediction with an accuracy that may be sufficient to distinguish between quintessenceand a cosmological constant.

Quintessence should also have an effect on the cosmic microwave background becausedifferences in the acceleration rate will produce small differences in the angular size of hotand cold spots. Moreover, unlike a cosmological constant, quintessence is not spatially ho-mogeneous. Small variations in the amount of quintessence across the sky should be seen asripples in the microwave background temperature. Measurements by the WMAP and Plancksatellites may be able to detect these effects, which will beat the level of a few per cent,although it will be difficult to separate them from other effects.

In many cases, quintessence interacts with matter in a way that affects the forces betweenparticles. Then, if the quintessence field is varying temporally or spatially, it will cause thestrengths of the forces between particles to change as well.Hence, ongoing tests for changesin the values of the fundamental physical constants with time could be another source ofevidence for quintessence. It might be possible to search for such effects with astrophysicalobservations (e.g. a variation of hyperfine splitting with redshift) or in ultrahigh-precisionlaser–spectroscopy experiments.

4.9.3 Braneworld Models – Higher Dimensions

Pretend you lived on your computer screen and could only moveon that two dimensionalsurface. The computer exists in three space dimensions but you can only move on a twodimensional subspace made by the screen, so the spacetime that you experience would looklike three dimensions (two space plus time) rather than four. That’s sort of the idea in abraneworld higher dimensional theory. Our observed four dimensional spacetime is like thecomputer screen, a subspace of some bigger space that we cannot see because all matter and


forces are constrained to move (mainly) on our subspace, or brane (as in membrane). The totalspace is called the bulk and the subspace or brane on which we would live is called the brane.

Gravity is the force that determines the shape of spacetime.Therefore, at least in principle,gravity should propagate in all dimensions equally. That means we should be able to detectlarge extra dimensions by looking for suspicious behavior in the gravitational force. But incertain braneworld models, gravity can actually be confinedor bound to our brane so that itdoes not propagate very far in the bulk. That makes the extra dimensions harder to detectusing gravity.

Braneworld models have been extensively applied to cosmology, where they demonstratequalitatively new and very interesting properties (see [12, 19] for recent reviews). Theorieswith the simplest generic action involving scalar-curvature terms both in the bulk and on thebrane are not only good in modeling cosmological dark energybut, in doing so, they alsoexhibit some interesting specific features, for example, the possibility of superacceleration(supernegative effective equation of state of dark energyweff ≤ −1) and the possibility ofcosmological loitering even in a spatially flat universe.

Figure 4.34: The Universe is considered to be a Brane in a 5–dimensional spacetime,called bulk. Onlygravity can expand into the bulk, matter is confined to the Brane.

It was noted some time ago that the cosmological evolution inbraneworld theory, fromthe viewpoint of the Friedmann universe, proceeds with a time-dependent gravitational con-stant. It turns out that, for a broad range of parameter values, the braneworld model behavesexactly as a LCDM (Λ + Cold Dark Matter) universewith different values of the effectivecosmological density parameterΩm at different epochs. This allows the model to share manyof the attractive features of LCDM, but with the important new property that the cosmologicaldensity parameter inferred from the observations of the large-sale structure and cosmic mi-crowave background (CMB) and that determined from neoclassical cosmological tests suchas observations of supernovae (SN) can potentially have different values.

An important feature of this model is that, although it is very similar to LCDM at thepresent epoch, its departure from “concordance cosmology”can be significant at intermediateredshifts, leading to new possibilities for the universe atthe end of the “dark ages” which maybe worth exploring.


A Simple Model

We consider the simplest generic braneworld model with an action of the form

S = M3

[∫

bulk

(R− 2Λb) − 2

∫

brane

K

]

+

∫

brane

(

m2R − 2σ)

+

∫

brane

L (hab, φ) .

(4.234)

Here,R is the scalar curvature of the metricgAB (A,B = 0, ..., 4) in the five-dimensionalbulk, andR is the scalar curvature of the induced metrichAB = gAB − nAnB on the brane,wherena is the vector field of the inner unit normal to the brane, whichis assumed to be aboundary of the bulk space. The quantityK = hABKAB is the trace of the symmetric tensorof extrinsic curvatureKAB = hC

A∇CnB of the brane. The symbolL(hab, φ) denotes theLagrangian density of the four-dimensional matter fieldsφ whose dynamics is restricted tothe brane so that they interact only with the induced metrichab. All integrations over the bulkand brane are taken with the corresponding natural volume elements. The symbolsM andm =

√

~c/G denote the five-dimensional and four-dimensional Planck masses, respectively,Λb is the bulk cosmological constant, andσ is the brane tension. In general,M ≪ m.

Action (4.234) leads to the Einstein equation with cosmological constant in the bulk:

GAB + ΛbgAB = 0 , (4.235)

while the field equation on the brane is

m2Gab + σhab = τab + M3 (Kab − habK) , (4.236)

whereτab is the stress–energy tensor on the brane stemming from the last term in action(4.234).

The cosmological evolution on the brane that follows from (4.235) and (4.236) is describedby the main equation [21, 4, 12]

H2 +κ

a2=

ρ + σ

3m2+

2

ℓ2

[

1 ±√

1 + ℓ2(

ρ + σ

3m2− Λb

6− C

a4

)

]

, (4.237)

whereC is the integration constant describing the so-called “darkradiation” and correspond-ing to the black-hole mass of the Schwarzschild–(anti)-de Sitter solution in the bulk,H ≡ a/ais the Hubble parameter on the brane, and the termκ/a2 corresponds to the spatial curvatureon the brane. The length scaleℓ is defined as

ℓ =2m2

M3= 2

(m

M

)2

ΛP (M) , ΛP (M) = ~/Mc . (4.238)

This length scale can be comparable to the Hubble radiusc/H0, m = 1.3 × 1019 GeV/c2

andM > 1 GeV/c2. On short scale–lengths,r ≪ ℓ, one recovers Einstein’s gravity, on largescale–lengths,r ≫ ℓ, brane specific effects become important, leading to acceleration of theUniverse.

The “±” signs in (4.237) correspond to two different branches of the braneworld solutions.Models with the lower (“−”) sign were called Brane 1, and models with the upper (“+”) signwere called Brane 2.


In what follows, we consider a spatially flat universe (κ = 0) without dark radiation(C = 0). It is convenient to introduce the dimensionless cosmological parameters

Ωm =ρ0

3m2H20

, Ωσ =σ

3m2H20

, Ωℓ =1

ℓ2H20

, ΩΛb= − Λb

6H20

, (4.239)

where the subscript “0” refers to the current values of cosmological quantities. The cosmo-logical equation with the energy densityρ dominated by dust-like matter can now be writtenin a transparent form:

H2(z)

H20

= Ωm(1+z)3 +Ωσ +2Ωℓ ± 2√

Ωℓ

√

Ωm(1+z)3 + Ωσ + Ωℓ + ΩΛb. (4.240)

The model satisfies the constraint equation

Ωm + Ωσ ± 2√

Ωℓ

√

1 + ΩΛb= 1 (4.241)

reducing the number of independentΩ parameters. The sign choices in Eqs. (4.240) and(4.241) always correspond to each other if1+ΩΛb

> Ωℓ . In the opposite case,1+ΩΛb< Ωℓ ,

both signs in (4.241) correspond to the lower sign in (4.240), and the option with both uppersigns in Eqs. (4.240) and (4.241) does not exist. The signs in(4.241) correspond to the twopossible ways of bounding the Schwarzschild–(anti)-de Sitter bulk space by the brane [4].

In the “normal” case1 + ΩΛb> Ωℓ , substitutingΩσ from (4.241) into (4.240), we get

H2(z)

H20

= Ωm(1+z)3 + 1 − Ωm + 2Ωℓ ∓ 2√

Ωℓ

√

1 + ΩΛb

± 2√

Ωℓ

√

Ωm(1+z)3 − Ωm +(

√

1 + ΩΛb∓√

Ωℓ

)2

. (4.242)

In the “special” case1 + ΩΛb< Ωℓ , the Brane 1 model [the lower sign in (4.240)] corre-

sponds to both signs of (4.241), and the cosmological equation reads as follows:

H2(z)

H20

= Ωm(1+z)3 + 1 − Ωm + 2Ωℓ ∓ 2√

Ωℓ

√

1 + ΩΛb

− 2√

Ωℓ

√

Ωm(1+z)3 − Ωm +(

√

Ωℓ ∓√

1 + ΩΛb

)2

. (4.243)

For sufficiently high redshifts, the first term on the right-hand side of (4.242) or (4.243)dominates, and the model reproduces the matter-dominated Friedmann universe with the den-sity parameterΩm. Now we note that, for the values ofz and parametersΩΛb

andΩℓ whichsatisfy

Ωm(1+z)3 ≪(

√

1 + ΩΛb∓√

Ωℓ

)2

, (4.244)

both Eqs. (4.242) and (4.243) can be well approximated as

H2(z)

H20

≃ Ωm(1+z)3 + 1 − Ωm −√

Ωℓ√Ωℓ ∓

√

1 + ΩΛb

[

Ωm(1+z)3 − Ωm

]

. (4.245)


We introduce the positive parameterα by the equation

α =

√

1 + ΩΛb√Ωℓ

. (4.246)

Then, defining a new density parameter by the relation

ΩLCDMm =

α

α ∓ 1Ωm , (4.247)

we get

H2(z)

H20

≃ ΩLCDMm (1+z)3 + 1 − ΩLCDM

m , (4.248)

which is precisely the Hubble parameter describing a LCDM universe. [Note that the braneworldparametersΩℓ andΩΛb

have been effectively absorbed into a “renormalization” ofthe mat-ter densityΩm → ΩLCDM

m , defined by (4.247). Since the value of the parameterΩLCDMm

should be positive by virtue of the cosmological considerations, in the case of the upper signin (4.245) and (4.247), we restrict ourselves to the parameter regionα > 1, i.e., we excludethe special case of Brane 1 model with the upper sign in (4.241) from consideration.]

Thus, our braneworld displays the following remarkable behaviour which we refer to as“cosmic mimicry”:

• A Brane 1 model, which at high redshifts expands with densityparameterΩm, at lowerredshiftsmasquerades as a LCDM universewith a smaller valueof the density parame-ter. In other words, at low redshifts, the Brane 1 universe expands as the LCDM model(4.248) withΩLCDM

m < Ωm [whereΩLCDMm is determined by (4.247) with the lower

(“+”) sign].

• A Brane 2 model at low redshifts also masquerades as LCDM but with a larger valueofthe density parameter. In this case,ΩLCDM

m > Ωm with ΩLCDMm being determined by

(4.247) with the upper (“−”) sign.

The range of redshifts over which this cosmic mimicry occursis given by0 ≤ z ≪ zm,with zm determined by (4.244). Specifically,

zm =

(√

1 + ΩΛb∓

√Ωℓ

)2/3

Ω1/3m

− 1 , (4.249)

which can also be written as

(1 + zm)3 =Ωm (1 + ΩΛb

)

(ΩLCDMm )

2 (4.250)

for both braneworld models.In view of relation (4.250), it is interesting to note that wecan use the equations de-

rived above to relate the three free parameters in the braneworld model: Ωℓ ,ΩΛb,Ωm to


Figure 4.35: An illustration of cosmic mimicry for the Brane 2 model. The Hubble parameter in threelow-density Brane 2 models withΩm = 0.04 is shown. Also shown is the Hubble parameter in a LCDMmodel (dotted line) which mimics this braneworld but has a higher mass density ΩLCDM

m = 0.2 (Ωℓ =0.8). The brane matter-density parameter (Ωm) and the corresponding parameter of the masquerading

LCDM model are related asΩm = ΩLCDMm ×

[

1 −√

Ωℓ/ (1 + ΩΛb)]

, so thatΩm ≤ ΩLCDMm . The

redshift interval during which the braneworld masquerades as LCDM,so thatHBRANE2 = HLCDM forz ≪ zm, is zm = 8, 32, 130 (left to right) for the three braneworld models.

Ωm, zm,ΩLCDMm

. These relations (which turn out to be the same for Brane 1 andBrane 2models) are

1 + ΩΛb

ΩLCDMm

=ΩLCDM

m

Ωm(1 + zm)3 , (4.251)

Ωℓ

ΩLCDMm

=

√

ΩLCDMm

Ωm−√

Ωm

ΩLCDMm

2

(1 + zm)3 . (4.252)

Furthermore, if we assume that the value ofΩLCDMm is known (say, from the analysis of SN

data), then the two braneworld parametersΩℓ andΩΛbcan be related to the two parameters

Ωm andzm, so it might be more convenient to analyze the model in terms of Ωm andzm

(instead ofΩℓ andΩΛb). We also note that, under condition (4.244), the brane tension σ,

determined by (4.241), is positive for Brane 1 model, and negative for Brane 2 model.

Since the Hubble parameter in braneworld models departs from that in LCDM atinterme-diate redshifts (z > zm), this could leave behind an important cosmological signature espe-cially since several key cosmological observables depend upon the Hubble parameter eitherdifferentially or integrally. Examples include:


Figure 4.36: The age of the universe in the Brane 2 model is shown with respect to the LCDM value.The mimicry redshift iszm = 4 so thatHbrane(z) ≃ HLCDM(z) at z ≪ 4. The braneworld modelshaveΩm = 0.2, 0.1, 0.04 (bottom to top) whereasΩLCDM

m = 0.3. Note that the braneworld models areconsiderably older than LCDM.

• the luminosity distancedL(z):

dL(z)

1 + z= c

∫ z

0

dz′

H(z′), (4.253)

• the angular-size distance

dA(z) =c

1 + z

∫ z

0

dz′

H(z′), (4.254)

• the productd2A(z)H−1(z), which is used in the volume-redshift test

• the deceleration parameter:

q(z) =H ′(z)

H(z)(1 + z) − 1 , (4.255)

• the effective equation of state of dark energy:

w(z) =2q(z) − 1

3 [1 − Ωm(z)], Ωm(z) = Ωm

[

H0

H(z)

]2

(1 + z)3 , (4.256)

• the age of the universe:

t(z) =

∫ ∞

z

dz′

(1 + z′)H(z′), (4.257)


• the electron-scattering optical depth to a redshiftzreion

τ(zreion) = c

∫ zreion

0

ne(z)σT dz

(1 + z)H(z), (4.258)

wherene is the electron density andσT is the Thomson cross-section describing scatter-ing between electrons and CMB photons.

4.9.4 Effects from Inhomogeneous Universe

Recently, there have been several papers discussing the possibility that the apparent accel-erated expansion of the universe is not caused by this mysterious dark energy, but rather byinhomogeneities in the distribution of matter [10, 26]. Thebasic idea behind this line of ex-planation is that we live in an underdense region of the universe, and the evolution of thisunderdensity is what we percieve as an accelerated expansion. An analysis of early supernovadata by Zehavi et al. gave the first indications that there might indeed exist such an underdensebubble centered near us.

Specific models that give rise to such underdensities have been studied previously in theform of a local homogeneous void. In these works both the underdensity and the region out-side it are assumed to be perfectly homogeneous Friedmann-Robertson-Walker (FRW) modelswith a singular mass shell separating the two regions. The inhomogeneity manifests itself asa discontinuous jump at the location of the mass shell.

Figure 4.37: Backreaction in the inhomogeneous Universe.

In a homogeneous universe, it is possible to infer the time evolution of the cosmic expan-sion from observations along the past light cone, since the expansion rate is a function of timeonly. In the inhomogeneous case, however, the expansion rate varies both with time and space.

4.10 Summary 167

Therefore, if the expansion rates inferred from observations of supernovae are larger for lowredshifts than higher redshifts, this must be attributed tocosmic acceleration in a homoge-neous universe, whereas in our case it can simply be the consequence of a spatial variation,with the expansion rate being larger closer to us.

However, the supernova observations are not the only data that support the claim of anaccelerating expansion. As mentioned above, CMB observations also seem to lend support tothis claim. Therefore, in order for a model to be considered realistic, it should also be able toexplain the observed CMB temperature power spectrum. It is possible to obtain a very goodmatch to both the supernova data and the location of the first acoustic peak simultaneously. Infact, the match to the supernova data is better than for theΛCDM model.

The observed isotropy of the CMB radiation implies that we must be located close to thecenter of the inhomogeneity. According to this picture, we are positioned at a rather specialplace in the Universe. On the other hand, this model has the attractive feature that there isno need for dark energy. Also the model is sufficiently simplethat it can be solved exactly.It is therefore a good toy model for testing the ideas of inhomogeneities as a solution to themystery of the dark energy.

4.10 Summary

• The time evolution of a Friedman Universe follows from the Friedman equations forH = R/R

H2 =8πG

3ρ − kc2

R2+

c2Λ

3(4.259)

and

R

R= −4πG(ρc2 + 3P )

3+

c2Λ

3. (4.260)

• The present state of the FRW Universe is determined by 5 parameters:

1. H0 = (R/R)0: the Hubble constant

2. Ωm = ρm/ρcrit: non–relativistic matter density

3. Ωk = −kc2/R20: curvature parameter

4. ΩΛ = ρDE/ρcrit: Dark Energy parameter

5. w0 = PDE/ρDE ≃ −1: EoS for Dark Energy.

• Only two of the density parameters are independent

Ωm + Ωk + ΩΛ = 1 . (4.261)

• The presentconcordance modelhas only two free parametersH0 andΩm, sinceΩk = 0andΩΛ = 1 − Ωm,

a(t) =

[

√

Ωm

1 − Ωmsinh

(

3√

1 − Ωm H0t

2

)

]2/3

(4.262)


• Models can be tested with the following relations followingfrom the Friedman equationH = H(z)

1. the luminosity distancedL(z):

dL(z)

1 + z= c

∫ z

0

dz′

H(z′), (4.263)

2. the angular-size distance

dA(z) =c

1 + z

∫ z

0

dz′

H(z′), (4.264)

3. the productd2A(z)/H(z), which is used in the volume-redshift test

4. the deceleration parameter:

q(z) =H ′(z)

H(z)(1 + z) − 1 , (4.265)

5. the effective equation of state of dark energy:

w(z) =2q(z) − 1

3 [1 − Ωm(z)], Ωm(z) = Ωm

[

H0

H(z)

]2

(1 + z)3 , (4.266)

6. the age of the universe:

t(z) =

∫ ∞

z

dz′

(1 + z′)H(z′), (4.267)

7. the electron-scattering optical depth to a redshiftzreion

τ(zreion) = c

∫ zreion

0

ne(z)σT dz

(1 + z)H(z), (4.268)

wherene is the electron density andσT is the Thomson cross-section describingscattering between electrons and CMB photons.

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[20] Schmidt, B. et al.: 1998, Astrophys. J. 507, 46[21] Shtanov Yu V,On brane-world cosmology, 2000Preprinthep-th/0005193[22] Springel, V., White, S.D.M. et al.: 2005,Simulations of the formation, evolution

and clustering of galaxies and quasars, Nature, 2 June 2005[23] Tonry, J.L. et al.: 2003,Cosmological Results from High-z Supernovae, ApJ594, 1[24] Vishwakarma, R.G.: 2005,Recent Supernovae Ia observations tend to rule out all

the cosmologies, astro–ph/0511628[25] Weller, Albrecht, : 2002, ApJ[26] Wiltshire, D.L.: 2005, gr–qc/0503099

List of Figures

1.1 The Aristotelian Universe . . . . . . . . . . . . . . . . . . . . . . . . .. . . 4

2.1 Universe scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Superclusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Cepheid luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162.4 Hubble expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Hubble expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 SNIa Hubble diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 SNAP Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 SNIa Hubble diagram from SNLegacy program . . . . . . . . . . . .. . . . 212.9 CMB Planck distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .. 232.10 COBE dipole anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . .. 232.11 WMAP temperature ripples . . . . . . . . . . . . . . . . . . . . . . . . . .. 242.12 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 242.13 Power spectrum of temperature fluctuations before WMAP .. . . . . . . . . 252.14 WMAP power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.15 DASI polarisation maps . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 272.16 Triangulation of spheres . . . . . . . . . . . . . . . . . . . . . . . . .. . . 282.17 Galaxy distribution in CfA survey . . . . . . . . . . . . . . . . . .. . . . . 302.18 Galaxy distribution in 2dF galaxy survey . . . . . . . . . . . .. . . . . . . . 312.19 Redshift distribution in the 2dF survey . . . . . . . . . . . . .. . . . . . . . 312.20 SDSS vs CfA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.21 Abell 2104 at redshift 0.15 . . . . . . . . . . . . . . . . . . . . . . . .. . . 342.22 Hydra cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.23 Abell 2218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Einstein’s Theorie der Relativit”at von 1916 . . . . . . . . .. . . . . . . . . 483.2 Light cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Light cone structure near a Black Hole . . . . . . . . . . . . . . . .. . . . . 633.4 Periheldrehung im Zweik”orperproblem . . . . . . . . . . . . . .. . . . . . 653.5 Gravitational lense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 663.6 Photon trajectories around a Black Hole . . . . . . . . . . . . . .. . . . . . 663.7 Slicing of SpaceTime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .673.8 Spaces of constant curvature . . . . . . . . . . . . . . . . . . . . . . .. . . 723.9 World lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.10 Observations in FRW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .753.11 Distance ladder for the Hubble constant . . . . . . . . . . . . .. . . . . . . 80

172 List of Figures

3.12 Light bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1 Classical Friedman models . . . . . . . . . . . . . . . . . . . . . . . . .. . 1014.2 Accelerated Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1054.3 Expansion of the Universe with dark energy . . . . . . . . . . . .. . . . . . 1064.4 Flat solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.5 Age of the Universe as a function of redshift . . . . . . . . . . .. . . . . . . 1084.6 Age of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.7 Causality problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1124.8 Equatorial slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1174.9 Zoom in of the region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.10 Galaxies and quasars in the equatorial slice, displayed in lookback time coor-

dinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.11 Zoom in of the region marked . . . . . . . . . . . . . . . . . . . . . . . .. 1214.12 Square comoving grid shown in lookback time coordinates . . . . . . . . . . 1224.13 Luminosity distance as a function of redshift . . . . . . . .. . . . . . . . . . 1234.14 Luminosity distance as a function of redshift . . . . . . . .. . . . . . . . . . 1254.15 Reduced distance modulus . . . . . . . . . . . . . . . . . . . . . . . . .. . 1274.16 Hubble diagram for SN Ia . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1284.17 Hubble diagram for the brightest ellipticals . . . . . . . .. . . . . . . . . . 1304.18 Light curve stretching of SNIa . . . . . . . . . . . . . . . . . . . . .. . . . 1324.19 Distance modulus for SNIa . . . . . . . . . . . . . . . . . . . . . . . . .. . 1334.20 Distance modulus for SNIa . . . . . . . . . . . . . . . . . . . . . . . . .. . 1344.21 Hubble diagram for cosmological Supernovae . . . . . . . . .. . . . . . . . 1384.22 Fundamental plane for cosmological Supernovae . . . . . .. . . . . . . . . 1394.23 Fundamental plane for cosmological Supernovae . . . . . .. . . . . . . . . 1404.24 The difference in distance modulus between the empty model and the super-

nova and the GRB binned data . . . . . . . . . . . . . . . . . . . . . . . . . 1454.25 Expected SNAP accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1454.26 Future constraints on SN . . . . . . . . . . . . . . . . . . . . . . . . . .. . 1464.27 Apparent angular width . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 1484.28 Apparent angular width . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 1494.29 Clustering of Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1504.30 Comoving cosmic volume . . . . . . . . . . . . . . . . . . . . . . . . . . .1514.31 Number counts in the expanding Universe . . . . . . . . . . . . .. . . . . . 1534.32 Empirical Fundamental Plane of Cosmology . . . . . . . . . . .. . . . . . . 1554.33 Matter densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1584.34 Bulk and Brane in Braneworld . . . . . . . . . . . . . . . . . . . . . . .. . 1604.35 Cosmic mimicry for the Brane2 models . . . . . . . . . . . . . . . .. . . . 1644.36 Braneworld age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.37 Inhomogeneous Universe . . . . . . . . . . . . . . . . . . . . . . . . . .. . 166

List of Tables

3.1 Number of linearly independentp–forms forD = 3 andD = 4. . . . . . . . 563.2 Distance modulus Virgo cluster . . . . . . . . . . . . . . . . . . . . .. . . . 79

4.1 Co-moving radii for different redshifts . . . . . . . . . . . . .. . . . . . . . 1154.2 Fits of different cosmologies to available data sets . . .. . . . . . . . . . . . 142

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