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Lectures on Astronomy, Astrophysics, and Cosmology

Luis A. AnchordoquiDepartment of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA

(Dated: Spring 2007)

These lecture notes were prepared as an aid to students enrolled in (UG) Astronomy 320, in the2007 Spring semester, at UW-Milwaukee. They are for study purposes only.

I. STARS AND GALAXIES

A look at the night sky provides a strong impression ofa changeless universe. We know that clouds drift acrossthe Moon, the sky rotates around the polar star, and onlonger times, the Moon itself grows and shrinks and theMoon and planets move against the background of stars.Of course we know that these are merely local phenomenacaused by motions within our solar system. Far beyondthe planets, the stars appear motionless.

According to the ancient cosmological belief, the stars,except for a few that appeared to move (the planets),where fixed on a sphere beyond the last planet (seeFig. 1). The universe was self contained and we, here onEarth, were at its center. Our view of the universe dra-matically changed after Galileo’s first telescopic observa-tions: we no longer place ourselves at the center and weview the universe as vastly larger. The distances involvedare so large that we specify them in terms of the time ittakes the light to travel a given distance. For example, 1light second = 1 s 3×108 m/s = 3×108m = 300, 000 km,

FIG. 1: Celestial spheres of ancient cosmology.

1 light minute = 18 × 106 km, and 1 light year

1 ly = 2.998× 108 m/s 3.156× 107 s/yr

= 9.46 × 1015 m

≈ 1013 km. (1)

For specifying distances to the Sun and the Moon, weusually use meters or kilometers, but we could specifythem in terms of light. The Earth-Moon distance is384,000 km, which is 1.28 ls. The Earth-Sun distance is1.5× 1011 m or 150,000,000 km; this is equal to 8.3 lightminutes. Far out in the solar system, Pluto is about6×109 km from the Sun, or 6×10−4 ly. The nearest starto us, Proxima Centauri, is about 4.3 ly away. Therefore,the nearest star is 10,000 times farther from us that theouter reach of the solar system.

On clear moonless nights, thousands of stars with vary-ing degrees of brightness can be seen, as well as the longcloudy strip known as the Milky Way. Galileo first ob-served with his telescope that the Milky Way is comprisedof countless numbers of individual stars. A half centurylater (about 1750) Thomas Wright suggested that theMilky Way was a flat disc of stars extending to great dis-tances in a plane, which we call the Galaxy (Greek for“milky way”).

Our Galaxy has a diameter of 100,000 ly and a thick-ness of roughly 2,000 ly. It has a bulging central “nu-cleus” and spiral arms. Our Sun, which seems to be justanother star, is located half way from the Galactic centerto the edge, some 26, 000 ly from the center. The Sunorbits the Galactic center approximately once every 250million years or so, so its speed is

v =2π 26, 000× 1013 km

2.5 × 108 yr 3.156× 107 s/yr

= 200 km/s . (2)

The total mass of all the stars in the Galaxy can be esti-mated using the orbital data of the Sun about the centerof the Galaxy. To do so, assume that most of the massis concentrated near the center of the Galaxy and thatthe Sun and the solar system (of total mass m) move ina circular orbit around the center of the Galaxy (of totalmass M). Then, apply Newton’s Law, F = ma, witha = v2/r being the centripetal acceleration and F beingthe Universal Law of Gravitation:

GMm

r2= m

v2

r, (3)

2

where G = 6.67×10−11 Nm2 kg−2 is Newton’s constant.All in all,

M =r v2

G≈ 2 × 1041 kg . (4)

Assuming all the stars in the Galaxy are similar to ourSun (M⊙ = 2 × 1030 kg), we conclude that there areroughly 1011 stars in the Galaxy.

In addition to stars both within and outside the MilkyWay, we can see with a telescope many faint cloudypatches in the sky which were once all referred to as “neb-ulae” (Latin for clouds). A few of these, such as thosein the constellations of Andromeda and Orion, can actu-ally be discerned with the naked eye on a clear night. Inthe XVII and XVIII centuries, astronomers found thatthese objects were getting in the way of the search forcomets. In 1781, in order to provide a convenient list ofobjects not to look at while hunting for comets, CharlesMessier published a celebrated catalogue. Nowadays as-tronomers still refer to the 103 objects in this catalog bytheir Messier numbers, e.g., the Andromeda Nebula isM31.

Even in Messier’s time it was clear that these extendedobjects are not all the same. Some are star clusters,groups of stars which are so numerous that they ap-peared to be a cloud. Others are glowing clouds of gasor dust and it is for these that we now mainly reservethe word nebula. Most fascinating are those that be-long to a third category: they often have fairly regularelliptical shapes and seem to be a great distance beyondthe Galaxy. Immanuel Kant (about 1755) seems to havebeen the first to suggest that these latter might be cir-cular discs, but appear elliptical because we see them atan angle, and are faint because they are so distant. Atfirst it was not universally accepted that these objectswere extragalactic (i.e. outside our Galaxy). The verylarge telescopes constructed in the XX century revealedthat individual stars could be resolved within these ex-tragalactic objects and that many contain spiral arms.Edwin Hubble (1889-1953) did much of this observationalwork in the 1920’s using the 2.5 m telescope on Mt. Wil-son near Los Angeles, California. Hubble demostratedthat these objects were indeed extragalactic because oftheir great distances [1]. The distance to our nearest spi-ral galaxy, Andromeda, is over 2 million ly, a distance20 times greater than the diameter of our Galaxy. Itseemed logical that these nebulae must be galaxies sim-ilar to ours. Today it is thought that there are roughly4 × 1010 galaxies in the observable universe – that is, asmany galaxies as there are stars in the Galaxy [2].

II. DISTANCE MEASUREMENTS BY

PARALLAX

Last class we have been talking about the vast distanceof the objects in the universe. Today, we will discussdifferent methods to estimate these distances. One basic

method employs simple geometry to measure the parallaxof a star. By parallax we mean the apparent motion ofa star against the background of more distant stars, dueto Earth’s motion around the Sun. The sighting angleof a star relative to the plane of Earth’s orbit (usuallyindicated by θ) can be determined at different times ofthe year. Since we know the distance d from the Earthto the Sun, we can determine the distance D to the star.For example, if the angle θ of a given star is measuredto be 89.99994, the parallax angle is p ≡ φ = 0.00006.From trigonometry, tanφ = d/D, and since the distanceto the Sun is d = 1.5× 108 km the distance to the star is

D =d

tan φ≈ d

φ=

1.5 × 108 km

1 × 10−6= 1.5 × 1014 km , (5)

or about 15 ly.Distances to stars are often specified in terms of paral-

lax angles given in seconds of arc: 1 second (1”) is 1/60of a minute (1’) of arc, which is 1/60 of a degree, so 1”= 1/3600 of a degree. The distance is then specified inparsecs (meaning parallax angle in seconds of arc), wherethe parsec is defined as 1/φ with φ in seconds. For ex-ample, if φ = 6 × 10−5, we would say the the star is ata distance D = 4.5 pc. It is easily seen that

1 pc = 3.26 ly = 3.08 × 1016 m . (6)

Parallax can be used to determine the distance to starsas far away as about 3 kpc from Earth.1 Beyond thatdistance, parallax angles are two small to measure andmore subtle techniques must be employed.

III. LUMINOSITY AND BRIGHTNESS

A useful parameter for a star or galaxy is its luminosity(or “absolute luminosity”), L, by which we mean the to-tal power radiated in watts. Also important is the appar-ent brightness, l, defined as the power crossing unit areaat the Earth perpendicular to the path of light. Giventhat energy is conserved and ignoring any absorption inspace, the total emitted power L when it reaches a dis-tance D from the star will be spread over a sphere ofsurface area 4πD2. If D is the distance from the star toEarth, then

L = 4πD2 l. (7)

Careful analyses of nearby stars have shown that the ab-solute luminosity for most of the stars depends on the

1 The angular resolution of the Hubble Space Telescope (HST) isabout 1/20 arc sec. With HST one can measure parallaxes ofabout 2 milli arc sec (e.g., 1223 Sgr). This corresponds to adistance of about 500 pc. Besides, there are stars with radioemission for which observations from the Very Long BaselineArray (VLBA) allow accurate parallax measurements beyond500 pc. For example, parallax measurements of Sco X-1 are0.36 ± 0.04 milli arc sec which puts it at a distance of 2.8 kpc.

3

mass: the more massive the star, the greater the lumi-nosity.

IV. SURFACE TEMPERATURE

Another important parameter of a star is its surfacetemperature, which can be determined from the spec-trum of electromagnetic frequencies it emits.

The rate at which an object radiates energy has beenfound to be proportional to the fourth power of theKelvin temperature T and to the area A of the emittingobject, i.e., L ∝ AT 4. At normal temperatures (≈ 300 K)we are not aware of this electromagnetic radiation be-cause of its low intensity. At higher temperatures, thereis sufficient infrared radiation that we can feel heat ifwe are close to the object. At still higher temperatures(on the order of 1000 K), objects actually glow, such asa red-hot electric stove burner. At temperatures above2000 K, objects glow with a yellow or whitish color, suchas the filament of a lightbulb.

Planck’s law of blackbody radiation predicts the spec-tral intensity of electromagnetic radiation at all wave-lengths from a blackbody at temperature T . The spectralradiance or brightness (i.e., the energy per unit time perunit surface area per unit solid angle per unit frequencyν) is

I(ν, T ) =2hν3

c2

1

ehν/kT − 1, (8)

where h = 6.626× 10−34 J s and k = 1.38× 10−23 J K−1

are the Planck and Boltzmann constants and c is thespeed of light. The law is sometimes written in terms ofthe spectral energy density

u(ν, T ) =4π

cI(ν, T ) , (9)

which has units of energy per unit volume per unit fre-quency. Integrated over frequency, this expression yieldsthe total energy density. Using the relation λ = c/ν,the spectral energy density can also be expressed as afunction of the wavelength,

u(λ, T ) =8πh

λ3

1

ehc/λkT − 1. (10)

Stars are fairly good approximations of blackbodies.As one can see in Fig. 2, the 5500 K curve, correspond-ing to the temperature of the Sun, peaks in the visiblepart of the spectrum (400 nm < λ < 750 nm). For lowertemperatures the total radiation drops considerably andthe peak occurs at longer wavelengths. It is found exper-imentally that the wavelength at the peak of the spec-trum, λp, is related to the Kelvin temperature by

λpT = 2.9 × 10−3 mK . (11)

This is known as Wien’s law.

FIG. 2: Measured spectra of wavelengths emitted by a black-body at different temperatures.

We can now use Wien’s law and the Steffan-Boltzmannequation (power output or luminosity ∝ AT 4) to deter-mine the temperature and the relative size of a star. Sup-pose that the distance from Earth to two nearby starscan be reasonably estimated, and that their measuredapparent brightnesses suggest the two stars have aboutthe same absolute luminosity, L. The spectrum of one ofthe stars peaks at about 700 nm (so it is reddish). Thespectrum of the other peaks at about 350 nm (bluish).Using Wien’s law, the temperature of the reddish star is

Tr =2.90 × 10−3 m K

700 × 10−9 m= 4140 K . (12)

The temperature of the bluish star will be double becauseits peak wavelength is half; just to check

Tb =2.90 × 10−3 m K

350× 10−9 m= 8280 K . (13)

For a blackbody the Steffan-Boltzmann equation reads

L = σAT 4 , (14)

where σ = 5.67 × 10−8 W m−2 K−4. Thus, the powerradiated per unit of area from a star is proportional tothe fourth power of the Kelvin temperature. Now thetemperature of the bluish star is double that of the redishstar, so the bluish must radiate 16 times as much energyper unit area. But we are given that they have the sameluminosity, so the surface area of the blue star must be1/16 that of the red one. Since the surface area is 4πr2,we conclude that the radius of the redish star is 4 timeslarger than the radius of the bluish star (and its volume64 times larger) [3].

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V. HR DIAGRAM

An important astronomical discovery, made around1900, was that for most of the stars, the color is relatedto the absolute luminosity and therefore to the mass.A useful way to present this relationship is by the so-called Hertzsprung-Russell (HR) diagram [4]. On theHR diagram, the horizontal axis shows the temperatureT , whereas the vertical axis the luminosity L, each staris represented by a point on the diagram shown in Fig. 3.Most of the stars fall along the diagonal band termed themain sequence. Starting at the lowest right, we find thecoolest stars, redish in color; they are the least luminousand therefore low in mass. Further up towards the left wefind hotter and more luminous stars that are whitish likeour Sun. Still farther up we find more massive and moreluminous stars, bluish in color. Stars that fall on thisdiagonal band are called main-sequence stars. There arealso stars that fall above the main sequence. Above andto the right we find extremely large stars, with high lu-minosity but with low (redish) color temperature: theseare called red giants. At the lower left, there are a fewstars of low luminosity but with high temperature: theseare white dwarfs.

VI. DISTANCE TO A STAR USING HR

Suppose that a detailed study of a certain star suggeststhat it most likely fits on the main sequence of the HRdiagram. Its measured apparent brightness is l = 1 ×10−12 W m−2, and the peak wavelength of its spectrumis λp ≈ 600 nm. We can first find the temperature usingWien’s law and then estimate the absolute luminosityusing the HR diagram; namely,

T ≈ 2.9 × 10−3 m K

600 × 10−9 m≈ 4800 K . (15)

A star on the main sequence of the HR diagram atthis temperature has absolute luminosity of about L ≈1026 W. Then, using Eq. (7) we can estimate its distancefrom us,

D =

√

L

4πl

≈√

1026 W

4π 10−12 W m−2

≈ 3 × 1018 m , (16)

or equivalently 300 ly.

VII. STELLAR EVOLUTION

The stars appear unchanging. Night after night theheavens reveal no significant variations. Indeed, on hu-man time scales, the vast majority of stars change very

little. Consequently, we cannot follow any but the tini-est part of the life cycle of any given star since they livefor ages vastly greater than ours. Nonetheless, in today’sclass we will follow the process of stellar evolution fromthe birth to the death of a star, as we have theoreticallyreconstructed it.

There is a general consensus that stars are born whengaseous clouds (mostly hydrogen) contract due to thepull of gravity. A huge gas cloud might fragment intonumerous contracting masses, each mass centered in anarea where the density was only slightly greater thanat nearby points. Once such “globules” formed, grav-ity would cause each to contract in towards its center-of-mass. As the particles of such protostar accelerate in-ward, their kinetic energy increases. When the kinetic en-ergy is sufficiently high, the Coulomb repulsion betweenthe positive charges is not strong enough to keep hydro-gen nuclei appart, and nuclear fussion can take place. Ina star like our Sun, the “burning” of hydrogen occurswhen four protons fuse to form a helium nucleus, withthe release of γ rays, positrons and neutrinos.2

The energy output of our Sun is believed to be dueprincipally to the following sequence of fusion reactions:

11H +1

1H →21H + 2 e+ + 2 νe (0.42 MeV) , (17)

11H +2

1H →32He + γ (5.49 MeV) , (18)

and

32He +3

2He →42He +1

1H +11H (12.86 MeV) , (19)

where the energy released for each reaction (given inparentheses) equals the difference in mass (times c2) be-tween the initial and final states. Such a released energyis carried off by the outgoing particles. The net effectof this sequence, which is called the pp-cycle, is for fourprotons to combine to form one 4

2He nucleus, plus twopositrons, two neutrinos, and two gamma rays:

4 11H →4

2He + 2e+ + 2νe + 2γ . (20)

Note that it takes two of each of the first two reactions toproduce the two 3

2He for the third reaction. So the totalenergy released for the net reaction is 24.7 MeV. How-ever, each of the two e+ quickly annihilates with an elec-tron to produce 2mec

2 = 1.02 MeV; so the total energyreleased is 26.7 MeV. The first reaction, the formationof deuterium from two protons, has very low probability,and the infrequency of that reaction serves to limit therate at which the Sun produces energy. These reactions

2 The word “burn” is put in quotation marks because these high-temperature fusion reactions occur via a nuclear process, andmust not be confused with ordinary burning in air, which is achemical reaction, occurring at the atomic level (and at a muchlower temperature).

5

FIG. 3: Hertzsprung-Russell diagram. The vertical axis depicts the inherent brightness of a star, and the horizontal axis thesurface temperature increasing from right to left. The spectral class is also indicated. You can see that the Sun is a G-star.

requiere a temperature of about 107 K, corresponding toan average kinetic energy (kT ) of 1 keV.

In more massive stars, it is more likely that the energyoutput comes principally from the carbon (or CNO) cy-cle, which comprises the following sequence of reactions:

126C +1

1H →137N + γ , (21)

137N →13

6C + e+ + ν , (22)

136C +1

1H →147N + γ , (23)

147N +1

1H →158O + γ , (24)

158O →15

7N + e+ + ν , (25)

157N +1

1H →126C +4

2He . (26)

6

It is easily seen that no carbon is consumed in this cycle(see first and last equations) and that the net effect isthe same as the pp cycle. The theory of the pp cycle andthe carbon cycle as the source of energy for the Sun andthe stars was first worked out by Hans Bethe in 1939 [5].

The fusion reactions take place primarily in the coreof the star, where T is sufficiently high. (The surfacetemperature is of course much lower, on the order of afew thousand K.) The tremendous release of energy inthese fusion reactions produces an outward pressure suf-ficient to halt the inward gravitational contraction; andour protostar, now really a young star, stabilizes in themain sequence. Exactly where the star falls along themain sequence depends on its mass. The more massivethe star, the further up (and to the left) it falls in theHR diagram. To reach the main sequence requires per-haps 30 million years, if it is a star like our Sun, andit is expected to remain there 10 billion years (1010 yr).Although most of stars are billions of years old, there isevidence that stars are actually being born at this mo-ment in the Eagle Nebula.

As hydrogen fuses to form helium, the helium that isformed is denser and tends to accumulate in the centralcore where it was formed. As the core of helium grows,hydrogen continues to fuse in a shell around it. Whenmuch of the hydrogen within the core has been consumed,the production of energy decreases at the center and is nolonger sufficient to prevent the huge gravitational forcesfrom once again causing the core to contract and heat up.The hydrogen in the shell around the core then fuses evenmore fiercely because of the rise in temperature, causingthe outer envelope of the star to expand and to cool. Thesurface temperature thus reduced, produces a spectrumof light that peaks at longer wavelength (reddish). Bythis time the star has left the main sequence. It hasbecome redder, and as it has grown in size, it has becomemore luminous. Therefore, it will have moved to the rightand upward on the HR diagram. As it moves upward, itenters the red giant stage. This model then explains theorigin of red giants as a natural step in stellar evolution.Our Sun, for example, has been on the main sequencefor about four and a half billion years. It will probablyremain there another 4 or 5 billion years. When our Sunleaves the main sequence, it is expected to grow in size(as it becomes a red giant) until it occupies all the volumeout to roughly the present orbit of the planet Mercury.

If the star is like our Sun, or larger, further fusion canoccur. As the star’s outer envelope expands, its core isshrinking and heating up. When the temperature reachesabout 108 K, even helium nuclei, in spite of their greatercharge and hence greater electrical repulsion, can thenreach each other and undergo fusion. The reactions are

42He +4

2He →84Be , (27)

and

42He +8

4Be →126C , (28)

with the emission of two γ-rays. The two reactions must

occur in quick succession (because 84Be is very unstable),

and the net effect is

3 42He →12

6C (7.3 MeV) . (29)

This fusion of helium causes a change in the star whichmoves rapidly to the “horizontal branch” of the HR di-agram. Further fusion reactions are possible, with 4

2Hefusing with 12

6C to form 168O. In very massive stars, higher

Z elements like 2010Ne or 24

12Mg can be made. This processof creating heavier nuclei from lighter ones (or by absorp-tion of neutrons at higher Z) is called nucleosynthesis.

The final fate of the star depends on its mass. Stars canlose mass as parts of their envelope drift off into space.Stars born with a mass less than about 8 solar masseseventually end up with a residual mass less than about1.4 solar masses, which is known as the Chandrasekharlimit [6]. For them, no further fusion energy can be ob-tained. The core of such low mass star (original mass. 8M⊙) contracts under gravity; the outer envelope ex-pands again and the star becomes an even larger redgiant. Eventually the outer layers escape into space, thecore shrinks, the star cools, descending downward in theHR diagram, becoming a white dwarf. A white dwarfcontracts to the point at which the electron clouds startto overlap, but collapses no further because of the Pauliexclusion principle (no two electrons can be in the samequantum state). Arriving at this point is called electrondegeneracy. A white dwarf continues to lose internal en-ergy by radiation, decreasing in temperature and becom-ing dimmer until its light goes out. It has then becomea cold dark chunk of ash.

Stars whose residual mass is greater than the Chan-drasekhar limit are thought to follow a quite differentscenario. A star with this great mass can contract un-der gravity and heat up even further. In the rangeT = 2.5 − 5 × 109 K, nuclei as heavy as 56

26Fe and 5628Ni

can be made. As massive red supergiants age, they pro-duce “onion layers” of heavier and heavier elements intheir interiors. However, the average binding energy pernucleon begins to decrease beyond the iron group of iso-topes. Thus, the formation of heavy nuclei from lighterones by fusion ends at the iron group. Further fusionwould require energy, rather than release it. As a conse-quence, a core of iron builds up in the centers of massivesupergiants.

Elements heavier than Ni are thought to form mainlyby neutron capture, particularly in supernova explosions.Large number of free neutrons, resulting from nuclearreactions, are present inside highly evolved stars and theycan readily combined with, say, a 56

26Fe nucleus to form (ifthree are captured) 59

26Fe which decays to 5927Co. The 59

27Cocan capture neutrons, also becoming neutron rich anddecaying by β− to next higher Z element and so on. Thehighest Z elements are thought to form by such neutroncapture during supernova (SN) explosions when hordesof neutrons are available.

Yet at these extremely high temperatures, well above109 K, the kinetic energy of the nuclei is so high that

7

fusion of elements heavier than iron is still possible eventhough the reactions require energy input. But the highenergy collisions can also cause the breaking apart of ironand nickel nuclei into helium nuclei and eventually intoprotons and neutrons

5626Fe → 13 4

2He + 4 n (30)

and

42He → 2 p + 2 n . (31)

These are energy-requiring reactions, but at such ex-tremely high temperature and pressure there is plentyof energy available, enough even to force electrons andprotons together to form neutrons in inverse β decay

e− + p → n + ν . (32)

Eventually, the iron core reaches the Chandrasekharmass. When something is this massive, not even electrondegeneracy pressure can hold it up. As the core collapsesunder the hughe gravitational forces, protons and elec-trons are pushed together to form neutrons and neutrinos(even though neutrinos don’t interact easily with matter,at these extremely high densities, they exert a tremen-dous outward pressure), and the star begins to contractrapidly towards forming an enormously dense neutronstar [7]. The outer layers fall inward when the iron corecollapses. When the core stops collapsing (this happenswhen the neutrons start getting packed too tightly – neu-tron degeneracy), the outer layers crash into the core andrebound, sending shock waves outward. These two effects– neutrino outburst and rebound shock wave – cause theentire star outside the core to be blow apart in a hugeexplosion: a type II supernova! It has been suggestedthat the energy released in such a catastrophic explosioncould form virtually all elements in the periodic table.3

The presence of heavy elements on Earth and in our solarsystem suggests that our solar system formed from thedebris of a supernova.

The core of a neutron star contracts to the point atwhich all neutrons are as close together as they are in anucleus. That is, the density of a neutron star is on theorder of 1014 times greater than normal solids and liquidson Earth. A neutron star that has a mass ∼ 1.5M⊙ wouldhave a diameter of only about 20 km.

If the final mass of a neutron star is less than about3 M⊙, its subsequent evolution is thought to be similarto that of a white dwarf. If the mass is greater thanthis, the star collapses under gravity, overcoming eventhe neutron exclusion principle [8]. The star eventuallycollapses to the point of zero volume and infinite density,

3 Type Ia supernovae are different. They are believed to be binarystars, one of which is a white dwarf that pulls mass from its com-panion. When the total mass reaches the Chandrasekhar limit,the star begins to collapse and then explodes as a supernova.

creating what is known as a “singularity” [9]. As thedensity increases, the paths of light rays emitted from thestar are bent and eventually wrapped irrevocably aroundthe star. Any emitted photons are trapped into an orbitby the intense gravitational field; they will never leaveit. Because no light escapes after the star reaches thisinfinite density, it is called a black hole.

Binary X-ray sources are places to find strong blackhole candidates [10]. A companion star is a perfect sourceof infalling material for a black hole. As the matter fallsor is pulled towards the black hole, it gains kinetic energy,heats up and is squeezed by tidal forces. The heating ion-izes the atoms, and when the atoms reach a few milliondegrees Kelvin, they emit X-rays. The X-rays are sentoff into space before the matter crosses the event horizon(black hole boundary) and crashes into the singularity.Thus we can see this X-ray emission. Another sign ofthe presence of a black hole is random variation of emit-ted X-rays. The infalling matter that emits X-rays doesnot fall into the black hole at a steady rate, but rathermore sporadically, which causes an observable variationin X-ray intensity. Additionally, if the X-ray source is ina binary system, the X-rays will be periodically cut off asthe source is eclipsed by the companion star. When look-ing for black hole candidates, all these things are takeninto account. Many X-ray satellites have scanned theskies for X-ray sources that might be possible black holecandidates [3].

VIII. THE OLBERS PARADOX

The XVI century finally saw what came to be a water-shed in the development of Cosmology. In 1543 NicolasCopernicus published his treatise “De Revolutionibus Or-bium Celestium” (The Revolution of Celestial Spheres)where a new view of the world is presented: the helio-centric model.

It is hard to underestimate the importance of thiswork: it challenged the age long views of the way theuniverse worked and the preponderance of the Earth and,by extension, of human beings. The realization that we,our planet, and indeed our solar system (and even ourGalaxy) are quite common in the heavens and reproducedby myriads of planetary systems, provided a sobering(though unsettling) view of the universe. All the reas-surances of the cosmology of the Middle Ages were gone,and a new view of the world, less secure and comfort-able, came into being. Despite these “problems” and themany critics the model attracted, the system was soonaccepted by the best minds of the time such as Galileo.

The simplest and most ancient of all astronomical ob-servations is that the sky grows dark when the Sun goesdown. This fact was first noted by Johannes Kepler, who,in the XVII century, used it as evidence for a finite uni-verse. In the XIX century, when the idea of an unending,unchanging space filled with stars like the Sun was wid-spread in consequence of the Copernican revolution, the

8

question of the dark night sky became a problem. Toclearly ascertain this problem, note that if absorption isneglected, the aparent luminosity of a star of absoluteluminosity L at a distance r will be l = L/4πr2. If thenumber density of such stars is a constant n, then thenumber of stars at distances r between r and r + dr isdN = 4πnr2dr, so the total radiant energy density dueto all stars is

ρs =

∫

l dN =

∫ ∞

0

(

L

4πr2

)

4π n r2dr

= Ln

∫ ∞

0

dr . (33)

The integral diverges, leading to an infinite energy den-sity of starlight!

In order to avoid this paradox, both Loys de Cheseaux(1744) [11] and Heinrich Olbers (1826) [12] postulated theexistence of an interstellar medium that absorbs the lightfrom very distant stars responsible for the divergence ofthe integral in Eq. (33). However, this resolution of theparadox is unsatisfactory, because in an eternal universethe temperature of the interstellar medium would haveto rise until the medium was in thermal equilibrium withthe starlight, in which case it would be emitting as muchenergy as it absorbs, and hence could not reduce the av-erage radiant energy density. The stars themselves areof course opaque, and totally block out the light fromsufficiently distant sources, but if this is the resolution ofthe so-called “Olbers paradox” then every line of segmentmust terminate at the surface of a star, so the whole skyshould have a temperature equal to that at the surfaceof a typical star.

IX. THE EXPANSION OF THE UNIVERSE

Even though at first glance the stars seem motion-less we are going to see that this impression is illusory.There is observational evidence that stars move at speedsranging up to a few hundred kilometers per second, soin a year a fast moving star might travel ∼ 1010 km.This is 103 times less than the distance to the clos-est star, so their apparent position in the sky changesvery slowly. For example, the relatively fast movingstar known as Barnard’s star is at a distance of about56 × 1012 km; it moves across the line of sight at about89 km/s, and in consequence its apparent position shifts(so-called “proper motion”) in one year by an angle of0.0029 degrees.4 The apparent position in the sky of themore distant stars changes so slowly that their propermotion cannot be detected with even the most patientobservation.

4 HST has measured proper motions as low as about 1 milli arc secper year. In the radio (VLBA), relative motions can be measuredto an accuracy of about 0.2 milli arc sec per year.

The observations that we will discuss in this class re-veal that the universe is in a state of violent explosion,in which the galaxies are rushing appart at speeds ap-proaching the speed of light. Moreover, we can extrapo-late this explosion backwards in time and conclude thatall the galaxies must have been much closer at the sametime in the past – so close, in fact, that neither galaxiesnor stars nor even atoms or atomic nuclei could have hada separate existence.

Our knoweledge of the expansion of the universe restsentirely on the fact that astronomers are able to measurethe motion of a luminous body in a direction directlyalong the line of sight much more accurately than theycan measure its motion at right angles to the line of sight.The technique makes use of a familiar property of anysort of wave motion, known as Doppler effect.

A. Doppler Effect

When we observe a sound or light wave from a sourceat rest, the time between the arrival wave crests at ourinstruments is the same as the time between crests asthey leave the source. However, if the source is mov-ing away from us, the time between arrivals of successivewave crests is increased over the time between their de-partures from the source, because each crest has a littlefarther to go on its journey to us than the crest before.The time between crests is just the wavelength dividedby the speed of the wave, so a wave sent out by a sourcemoving away from us will appear to have a longer wave-length than if the source were at rest. Likewise, if thesource is moving toward us, the time between arrivalsof the wave crests is decreased because each successivecrest has a shorter distance to go, and the waves appearto have a shorter wavelength. A nice analogy was put for-ward by Steven Weinberg in “The First Three Minutes.”He compared the situation with a travelling man thathas to send a letter home regularly once a week duringhis travels: while he is travelling away from home, eachsuccessive letter will have a little farther to go than theone before, so his letters will arrive a little more thana week apart; on the homeward leg of his journey, eachsuccesive letter will have a shorter distance to travel, sothey will arrive more frequently than once a week.

Suppose that wave crests leave a light source at regularintervals separated by a period T . If the source is movingat velocity V ≪ c away from the observer, then duringthe time between successive crests the source moves adistance V T . This increases the time required for thewave crest to get from the source to the observer by anamount V T/c, where c is the speed of light. Therefore,the time between arrival of successive wave crests at theobserver is

T ′ ≈ T +V T

c. (34)

9

The wavelength of the light upon emission is

λ = c T (35)

and the wavelength when the light arrives is

λ′ = c T ′ . (36)

Hence the ratio of these wavelengths is

λ′

λ=

T ′

T≈ 1 +

V

c. (37)

The same reasoning applies if the source is moving towardthe observer, except that V is replaced with −V .5 Forexample, the galaxies of the Virgo cluster are movingaway from our Galaxy at a speed of about 103 km/s.Therefore, the wavelength λ′ of any spectral line fromthe Virgo cluster is larger than its normal value λ by aratio

λ′

λ=

T ′

T≈ 1 +

103 km/s

3 × 105 km/s≈ 1.0033 . (38)

This effect was apparently first pointed out for bothlight and sound waves by Johann Christian Doppler in1842. Doppler thought that this effect might explain thedifferent colors of stars. The light from the stars thathappen to be moving away from the Earth would beshifted toward longer wavelengths, and since red lighthas a wavelength longer than the average wavelength ofvisible light, such stars might appear more red than aver-age. Similarly, light from stars that happen to be movingtoward the Earth would be shifted toward shorter wave-lengths, so that the star might appear unusually blue. Itwas soon pointed out by Buys-Ballot and others that theDoppler effect has essentially nothing to do with the colorof a star – it is true that the blue light from a recedingstar is shifted toward the red, but at the same time someof the star’s normally invisible ultraviolet light is shiftedinto the blue part of the visible spectrum, so the over-all color hardly changes. As we discussed in the previousclass, stars have different colors chiefly because they havedifferent surface temperatures.

B. Hubble Law

The Doppler effect began to be of enormous impor-tance to astronomy in 1968, when it was applied tothe study of individual spectral lines. In 1815, Joseph

5 For source and observer moving away from each other,Special Relativity’s [13] correction leads to λ′/λ =p

(1 + V/c)/(1 − V/c), where λ is the emitted wavelengthas seen in a reference frame at rest with respect to the sourceand λ′ is the wavelength measured in a frame moving withvelocity V away from the source along the line of sight; forrelative motion toward each other, V < 0 in this formula.

Frauenhofer first realized that when light from the Sunis allowed to pass through a slit and then through a glassprism, the resulting spectrum of colors is crossed withhundreds of dark lines, each one an image of the slit. Thedark lines were always found at the same colors, each cor-responding to a definite wavelength of light. The samedark spectral lines were also found in the same position inthe spectrum of the Moon and brighter stars. It was soonrealized that these dark lines are produced by the selec-tive absorption of light of certain definite wavelengths,as light passes from the hot surface of a star through itscooler outer atmosphere. Each line is due to absorptionof light by a specific chemical element, so it became pos-sible to determine that the elements on the Sun, such assodium, iron, magnesium, calcium, and chromium, arethe same as those found on Earth.

In 1868, Sir William Huggins was able to show that thedark lines in the spectra of some of the brighter stars areshifted slightly to the red or the blue from their normalposition in the spectrum of the Sun. He correctly inter-preted this as a Doppler shift, due to the motion of thestar away from or toward the Earth. For example, thewavelength of every dark line in the spectrum of the starCapella is longer than the wavelength of the correspond-ing dark line in the spectrum of the Sun by 0.01%, thisshift to the red indicates that Capella is receding fromus at 0.01% c (i.e., the radial velocity of Capella is about30 km/s).

In the late 1920’s, Hubble discovered that the spec-tral lines of galaxies were shifted towards the red by anamount proportional to their distances [14]. If the red-shift is due to the Doppler effect, this means that thegalaxies move away from each other with velocities pro-portional to their separations. The importance of thisobservation is that it is just what we should predict ac-cording to the simplest possible picture of the flow ofmatter in an expanding universe. In terms of the red-shift z ≡ (λ′ − λ)/λ, the “linear” Hubble law can bewritten as

z ≈ (H0/c) r , (39)

where c is the speed of light, H0 is the present value ofthe Hubble constant and r the distance to the galaxy.6

For small velocities (V ≪ c), from Eq. (37) the Dopplerredshift is z ≈ V/c. Therefore, V ≈ H0r, which is themost commonly used form of Hubble law. The presentday Hubble expansion rate is H0 = 100 h km s−1 Mpc−1,where h = 0.71+0.04

−0.03.

6 To avoid confusion, it should be kept in mind that λ denotes thewavelength of the light if observed near the place and time ofemission, and thus presumably take the values measured whenthe same atomic transition occurs in terrestrial laboratories,while λ′ is the wavelength of the light observed after its longjourney to us. If z > 0 then λ′ > λ and we speak of a redshift;if z < 0 then λ′ < λ , and we speak of a blueshift.

10

C. The Cosmological Principle

We would expect intuitively that at any given time theuniverse ought to look the same to observers in all typi-cal galaxies, and in whatever direction they look. (Here-after we will use the label “typical” to indicate galax-ies that do not have any large peculiar motion of theirown, but are simply carried along with the general cos-mic flow of galaxies.) This hypothesis is so natural (atleast since Copernicus) that it has been called the Cos-mological Principle by the English astrophysicist EdwardArthur Milne.

As applied to the galaxies themselves, the Cosmologi-cal Principle requires that an observer in a typical galaxyshould see all the other galaxies moving with the samepattern of velocities, whatever typical galaxy the ob-server happens to be riding in. It is a direct mathemat-ical consequence of this principle that the relative speedof any two galaxies must be proportional to the distancebetween them, just as found by Hubble. To see this con-sider three typical galaxies A, B, and C, strung out in astraight line, as shown in Fig. 4. Suppose that the dis-tance between A and B is the same as the distance be-tween B and C. Whatever the speed of B as seen from A,the Cosmological principle requires that C should havethe same speed relative to B. But note that C, which istwice away from A as is B, is also moving twice as fastrelative to A as is B. We can add more galaxies in ourchain, always with the result that the speed of recessionof any galaxy relative to any other is proportional to thedistance between them.

As often happens in science, this argument can be usedboth forward and backward. Hubble, in observing a pro-portionality between the distances of galaxies and theirspeeds of recession, was indirectly verifying the Cosmo-logical Principle. Contrariwise, we can take the Cosmo-logical Principle for granted on a priori grounds, anddeduce the relation of proportionality between distanceand velocity. In this way, through the relatively easymeasurement of Doppler shifts, we are able to judge thedistance of very remote objects from their velocities.

Before proceeding any further, two qualifications haveto be attached to the Cosmological Principle. First, it isobviously not true on small scales – we are in a Galaxywhich belongs to a small local group of other galaxies,which in turn lies near the enormous cluster of galaxiesin Virgo. In fact, of the 33 galaxies in Messier’s catalogue,almost half are in one small part of the sky, the constella-tion of Virgo. The Cosmological Principle, if at all valid,comes into play only when we view the universe on ascale at least as large as the distance between clustersof galaxies, or about 100 million light years. Second, inusing the Cosmological Principle to derive the relation ofproportionality between galactic velocities and distances,we suppose the usual rule for adding V ≪ c. This, ofcourse, was not a problem for Hubble in 1929, as none ofthe galaxies he studied then had a speed anywhere nearthe speed of light. Nevertheless, it is important to stress

FIG. 4: Homogeneity and the Hubble Law: A string of equallyspaced galaxies Z, A, B, C, . . . are shown with velocities asmeasured from A or B or C indicated by the lengths and di-rections of the attached arrows. The principle of homogeneityrequires that the velocity of C as seen by B is equal to thevelocity of B as seen by A, adding these two velocities givesthe velocity of C as seen by A, indicated by an arrow twice aslong. Proceeding in this way we can fill out the whole patternof velocities shown in the figure. As can be seen the velocitiesobey the Hubble law: the velocity of any galaxy, as seen byothers is proportional to the distance between them. This isthe only pattern of velocities consistent with the principle ofhomogeneity.

that when one thinks about really large distances char-acteristic of the universe, as a whole, one must work in atheoretical framework capable of dealing with velocitiesapproaching the speed of light.

D. The Cosmic Microwave Background

The Cosmological Principle has observational supportof another sort, apart from the measurements of theDoppler shifts. After making due allowances for the dis-tortions due to our own Galaxy and the rich nearby clus-ter of galaxies in the constellation of Virgo, the universeseems remarkably isotropic; that is, it looks the same inall directions. Now, if the universe is isotropic aroundus, it must also be isotropic about every typical galaxy.However, any point in the universe can be carried intoanother point by a series of rotations around fixed cen-ters, so if the universe is isotropic around every point,it is necessary also homogeneous. In what follows wewill discuss how the observation of the cosmic microwavebackground (CMB) provides convincing evidence for anisotropic universe.

The expansion of the universe seems to suggest thattypical objects in the universe were once much closer to-gether than they are right now. This is the idea for thebasis that the universe began about 13.7 billion yearsago as an expansion from a state of very high densityand temperature known affectionately as the Big Bang.

The Big Bang was not an explosion, because an explo-sion blows pieces out into the surrounding space. Instead,the Big Bang was the start of an expansion of space itself.The volume of the observable universe was very small atthe start and has been expanding ever since. The ini-tial tiny volume of extremely dense matter is not to bethought of as a concentrated mass in the midst of a much

11

larger space around it. The initial tiny but dense volumewas the universe – the entire universe. There would nothave been anything else. When we say that the universewas once smaller than it is now, we mean that the av-erage separation between galaxies (or other objects) wasless. Therefore, it is the size of the universe itself thathas increased since the Big Bang.

In 1964, Arno Penzias and Robert Wilson were ex-periecing difficulty with what they assumed to be back-ground noise, or “static,” in their radio telescope [15].Eventually, they became convinced that it was real andthat it was coming from outside the Galaxy. They madeprecise measurements at wavelength λ = 7.35 cm, in themicrowave region of the electromagnetic spectrum. Theintensity of this radiation was found initially not to varyby day or night or time of the year, nor to depend onthe direction. It came from all directions in the universewith equal intensity, to a precision of better than 1%. Itcould only be concluded that this radiation came fromthe universe as a whole.

The intensity of this CMB as measured at λ = 7.35 cmcorresponds to a blackbody radiation at a temperatureof about 3 K. When radiation at other wavelengths wasmeasured, the intensities were found to fall on a black-body curve, corresponding to a temperature of 2.725 K.

The CMB provides strong evidence in support of theBig Bang, and gives us information about conditions inthe very early universe. In fact, in the late 1940s, GeorgeGamow calculated that the Big Bang origin of the uni-verse should have generated just such a CMB [16].

To understand why, let us look at what a Big Bangmight have been like. The temperature must have beenextremely high at the start, so high that there could nothave been any atoms in the very early stages of the uni-verse. Instead the universe would have consisted solelyof radiation (photons) and a plasma of charged electronsand other elementary particles. The universe would havebeen opaque - the photons in a sense “trapped,” trav-elling very short distances before being scattered again,primarily by electrons. Indeed, the details of the CMBprovide strong evidence that matter and radiation wereonce in thermal equilibrium at very high temperature.As the universe expanded, the energy spread out over anincreasingly larger volume and the temperature dropped.Only when the temperature had fallen to about 3,000 Kwas the universe cool enough to allow the combinationof nuclei and electrons into atoms. (In the astrophys-ical literature this is usually called “recombination,” asingularly inappropriate term, for at the time we wereconsidering the nuclei and electrons had never in the pre-vious history of the universe been combined into atoms!)The sudden disappearence of electrons broke the thermalcontact between radiation and matter, and the radiationcontinued thereafter to expand freely.

At the moment this happened, the energy in the radi-ation field at various wavelengths was governed by theconditions of the thermal equilibrium, and was there-fore given by the Planck blackbody formula, Eq. (10),

for a temperature equal to that of the matter ∼ 3, 000 K.In particular, the typical photon wavelength would havebeen about one micron, and the average distance betweenphotons would have been roughly equal to this typicalwavelength.

What has happened to the photons since then? Indi-vidual photons would not be created or destroyed, so theaverage distance between photons would simply increasein proportion to the size of the universe, i.e., in pro-portion to the average distance between typical galaxies.But we saw that the effect of the cosmological redshiftis to pull out the wavelength of any ray of light as theuniverse expands; thus the wavelength of any individualphoton would also simply increase in proportion to thesize of the universe. The photons would therefore remainabout one typical length apart, just as for blackbody ra-diation.

Before proceeding we will pursue this line of argumentquantitavively. From Eq. (10) we can obtain the Planckdistribution that gives the energy du of a blackbody radi-ation per unit volume, in a narrow range of wavelengthsfrom λ to λ + dλ,

du =8πhc

λ5dλ

1

ehc/λkT − 1. (40)

For long wavelengths, the denominator in the Planck dis-tribution may be approximated by

ehc/λkT − 1 ≃ hc/λkT , (41)

Hence, in this wavelength region,

du =8πkT

λ4dλ . (42)

This is the Rayleigh-Jeans formula. If this formula helddown to arbitrarily small wavelengths, du/dλ would be-come infinite for λ → 0, and the total energy density inthe blackbody radiation would be infinite. Fortunately,as we saw before, the Planck formula for du reaches amaximum at a wavelength λ = 0.2014052 hc/kT and thenfalls steeply off for decreasing wavelengths. The total en-ergy density in the blackbody radiation is

u =

∫ ∞

0

8πhc

λ5dλ

1

ehc/λkT − 1. (43)

Integrals of this sort can be looked up in standard ta-bles of definite integrals; the result gives the Stefan-Boltzmann law

u =8π5(kT )4

15(hc)3= 7.56464×10−15 (T/K)4erg/cm3 . (44)

(Recall that 1 J ≡ 107 erg = 6.24 × 1018 eV.)We can easily interpret the Planck distribution in

terms of quanta of light or photons. Each photon hasan energy E = hc/λ. Hence the number dn of photons

12

FIG. 5: The CMB over the entire sky, color-coded to rep-resent differences in temperature from the average 2.725 K:the color scale ranges from +300 µK (red) to −200 µK (darkblue), representing slightly hotter and colder spots (and alsovariations in density.) Results are from the WMAP satellite.The angular resolution is 0.2 [17].

per unit volume in blackbody radiation in a narrow rangeof wavelengths from λ to λ + dλ is

dn =du

hc/λ=

8π

λ4dλ

1

ehc/λkT − 1. (45)

Then the total number of photons per unit volume is

n =

∫ ∞

0

dn

= 8π

(

kT

hc

)3 ∫ ∞

0

x2 dx

ex − 1, (46)

where x = hc/(λkT ). The integral cannot be expressedin terms of elementary functions, but it can be expressedas an infinite series

∫ ∞

0

x2 dx

ex − 1= 2

∞∑

j=1

1

j3≈ 2.4 . (47)

Therefore, the number photon density is

n = 60.42198

(

kT

hc

)3

= 20.28

(

T

K

)3

photons cm−3 , (48)

and the average photon energy is

〈Eγ〉 = u/n = 3.73 × 10−16 (T/K) erg . (49)

Now, let’s consider what happens to blackbody radi-ation in an expanding universe. Suppose the size of theuniverse changes by a factor f , for example, if it doublesin size, then f = 2. As predicted by the Doppler effect,the wavelengths will change in proportion to the size ofthe universe to a new value λ′ = fλ. After the expansion,the energy density du′ in the new wavelength range λ′ toλ′ + dλ′ is less than the original energy density du in the

old wavelength range λ + dλ, for two different reasons:(i) Since the volume of the universe has increased by afactor of f3, as long as no photons have been created ordestroyed, the numbers of photons per unit volume hasdecreased by a factor of 1/f3.(ii) The energy of each photon is inversely proportionalto its wavelength, and therefore is decreased by a factorof 1/f . It follows that the energy density is decreased byan overall factor 1/f3 × 1/f = 1/f4:

du′ =1

f4du =

8πhc

λ5f4dλ

1

ehc/λkT − 1. (50)

If we rewrite Eq. (50) in terms of the new wavelengthsλ′, it becomes

du′ =8πhc

λ′5dλ′ 1

ehcf/λ′kT − 1, (51)

which is exactly the same as the old formula for du interms of λ and dλ, except that T has been replaced by anew temperature T ′ = T/f . Therefore, we conclude thatfreely expanding blackbody radiation remains describedby the Planck formula, but with a temperature that dropsin inverse proportion to the scale of expansion.

The existence of the thermal CMB gives strong sup-port to the idea that the universe was extremely hot inits early stages. As can be seen in Fig. 5, the back-ground is very nearly isotropic, supporting the isotropicand homogeneous models of the universe. Of course onewould expect some small inhomogeneities in the CMBthat would provide “seeds” around which galaxy forma-tion could have started. These tiny inhomogeneities werefirst detected by the COBE (Cosmic Background Ex-plorer) [18] and by subsequent experiments with greaterdetail, culminating with the WMAP (Wilkinson Mi-crowave Anisotropy Probe) results [19].

X. HOMOGENEOUS AND ISOTROPIC

FRIEDMANN-ROBERTSON-WALKER

UNIVERSES

In 1917 Albert Einstein presented a model of the uni-verse based on his theory of General Relativity [20]. It de-scribed a geometrically symmetric (spherical) space withfinite volume but no boundary. In accordance with theCosmological Principle, the model was homogeneous andisotropic. It was also static: the volume of the space didnot change.

In order to obtain a static model, Einstein had to in-troduce a new repulsive force in his equations [21]. Thesize of this cosmological term is given by the cosmologicalconstant Λ. Einstein presented his model before the red-shifts of the galaxies were known, and taking the universeto be static was then reasonable. When the expansion ofthe universe was discovered, this argument in favor ofa cosmological constant vanished. Einstein himself latercalled it the biggest blunder of his life. Nevertheless the

13

FIG. 6: Spherical region of galaxies with a larger radius thanthe distance between clusters of galaxies, but smaller radiusthan any distance characterizing the universe as a whole.

most recent observations seem to indicate that a non-zerocosmological constant has to be present.

The St. Petersburg physicist Alexander Friedmann [22]studied the cosmological solutions of Einstein equations.If Λ = 0, only evolving, expanding or contracting mod-els of the universe are possible. The general relativis-tic derivation of the law of expansion for the Friedmannmodels will not be given here. It is interesting that theexistence of three types of models and their law of ex-pansion can be derived from purely Newtonian consider-ations, with results in complete agreement with the rel-ativistic treatment. Moreover, the essential character ofthe motion can be obtained from a simple energy argu-ment, which we discuss next.

Consider a spherical region of galaxies of radius r. (Forthe purposes of this calculation we must take r to belarger than the distance between clusters of galaxies, butsmaller than any distance characterizing the universe asa whole, as shown in Fig. 6. We also assume Λ = 0.) Themass of this sphere is its volume times the cosmic massdensity,

M =4 π r3

3ρ . (52)

We can now consider the motion of a galaxy of mass m atthe edge of our spherical region. According to Hubble’slaw, its velocity will be V = Hr and the correspondingkinetic energy

K =1

2mV 2 . (53)

In a spherical distribution of matter, the gravitationalforce on a given spherical shell depends only on the massinside the shell. The potential energy at the edge of thesphere is

U = −GMm

r= −4πmr2ρG

3, (54)

where G = 6.67× 10−8 cm3 g−1s−2 is Newton’s constantof gravitation. Hence, the total energy is

E = K + U =1

2m V 2 − GM m

r, (55)

which has to remain constant as the universe expands.The value of ρ corresponding to E = 0 is called the crit-ical density ρc. We have,

E =1

2mH2r2 − GMm

r

=1

2mH2r2 − Gm

4π

3r2ρc

= mr2

(

1

2H2 − 4π

3Gρc

)

= 0 (56)

whence

ρc =3H2

8πG. (57)

The density parameter Ω is defined as Ω = ρ/ρc.Now consider two points at separation r, such that

their relative velocity is V . Let R(t) be a time dependentquantity representing the scale of the universe. If R in-creases with time, all distances, including those betweengalaxies, will grow. Then

r =R(t)

R(t0)r0 , (58)

and

V = r =R(t)

R(t0)r0 , (59)

where dots denote derivative with respect to t. Therefore,the Hubble constant is

H =V

r=

R(t)

R(t). (60)

From the conservation of mass it follows that ρ0R30 =

ρR3. Using Eq. (57) for the critical density one obtains

Ω =8πG

3

ρ0R30

R3H2. (61)

The deceleration expansion is described by the decelera-tion parameter q defined as

q = −RR

R2. (62)

The deceleration parameter describes the change of ex-pansion R. The additional factors have been included inorder to make it dimensionless, i.e., independent of thechoice of units of length and time.

The expansion of the universe can be compared to themotion of a mass launched vertically from the surface of a

14

FIG. 7: The two dimensional analogues of the Friedmannmodels. A spherical surface, a plane, and a pseudo-sphere.Note that the global geometry of the universe affects the sumof angles of a triangle.

celestial body. The form of the orbit depends on the ini-tial energy. In order to compute the complete orbit, themass M of the main body and the initial velocity have tobe known. In cosmology, the corresponding parametersare the mean density and the Hubble constant.

The E = 0 model corresponds to the “flat” Friedmannmodel, so-called Einstein-de Sitter model. If the densityexceeds the critial density, the expansion of any sphericalregion will turn to a contraction and it will collapse to apoint. This corresponds to the closed Friedmann model.Finally, if ρ < ρc, the ever-expanding hyperbolic modelis obtained.

These three models of the universe are called the stan-dard models. They are the simplest relativistic cosmolog-ical models for Λ = 0. Models with Λ 6= 0 are mathemat-ically more complicated, but show the same behaviour.

The simple Newtonian treatment of the expansionproblem is possible because Newtonian mechanics is ap-proximately valid in small regions of the universe. How-ever, although the resulting equations are formally sim-ilar, the interpretation of the quantities involved is notthe same as in the relativistic context. The global geom-etry of Friedmann models can only be understood withinthe general theory of relativity.

What is meant by a curved space? To answer thisquestion, recall that our normal method of viewing theworld is via Euclidean plane geometry. In Euclidean ge-ometry there are many axioms and theorems we takefor granted. Non-Euclidean geometries which involvecurved space have been independently imagined by CarlFriedrich Gauss (1777-1855), Janos Bolyai (1802 - 1860),and Nikolai Ivanovich Lobachevski (1793-1856). Let ustry to understand the idea of a curved space by using twodimensional surfaces.

Consider for example, the two-dimensional surface ofa sphere. It is clearly curved, at least to us who view itfrom outside – from our three dimensional world. Buthow do the hypothetical two-dimensional creatures de-termine whether their two-dimensional space is flat (aplane) or curved? One way would be to measure thesum of the angles of a triangle. If the surface is a plane,the sum of the angles is 180. But if the space is curved,and a sufficiently large triangle is constructed, the sum

of the angles would not be 180. To construct a triangleon a curved surface, say the sphere of Fig. 7, we mustuse the equivalent of a straight line: that is the shortestdistance between two points, which is called a geodesic.On a sphere, a geodesic is an arc of great circle (an arcin a plane passing through the center of the sphere) suchas Earth’s equator and longitude lines. Consider, for ex-ample, the triangle whose sides are two longitude linespassing from the north pole to the equator, and the thirdside is a section of the equator. The two longitude linesform 90 angles with the equator. Thus, if they make anangle with each other at the north pole of 90, the sumof the angles is 270. This is clearly not Euclidean space.Note, however, that if the triangle is small in compari-son to the radius of the sphere, the angles will add up tonearly 180, and the triangle and space will seem flat. Onthe saddlelike surface, the sum of the angles of a triangleis less than 180. Such a surface is said to have negativecurvature.

Now, what about our universe? On a large scale whatis the overall curvature of the universe? Does it havepositive curvature, negative curvature or is it flat? Bysolving Einstein equations, Howard Percy Robertson [23]and Arthur Geoffrey Walker [24], showed that the threehypersurfaces of constant curvature (the hyper-sphere,the hyper-plane, and the hyper-pseudosphere) are indeedpossible geometries for a homegeneous and isotropic uni-verse undergoing expansion.

If the universe had a positive curvature, the universewould be closed, or finite in volume. This would notmean that the stars and galaxies extended out to a cer-tain boundary, beyond which there is empty space. Thereis no boundary or edge in such a universe. If a particlewere to move in a straight line in a particular direction, itwould eventually return to the starting point – perhapseons of time later. On the other hand, if the curvatureof the space was zero or negative, the universe would beopen. It could just go on forever.

In an expanding universe, the galaxies were once muchnearer to each other. If the rate of expansion hadbeen unchanging, the inverse of the Hubble constant,tage = H−1

0 , would represent the age of the universe. InFriedmann-Robertson-Walker models, however, the ex-pansion is gradually slowing down (i.e., q < 0), and thusthe Hubble constant gives an upper limit on the age ofthe universe, tage ≈ 14 Gyr. Of course, if Λ 6= 0 thisupper limit for the age of the universe no longer holds.

In an expanding universe the wavelength of radiationis proportional to R, like all other lengths. If the wave-length at the time of emission, corresponding to the scalefactor R, is λ, then it will be λ0 when the scale factorhas increased to R0: λ0/λ = R0/R. The redshift is

z =λ0 − λ

λ=

R0

R− 1 ; (63)

i.e., the redshift of a galaxy expresses how much the scalefactor has changed since the light was emitted.

15

XI. THE DARK SIDE OF THE UNIVERSE

According to the standard Big Bang model, the uni-verse is evolving and changing. Individual stars are be-ing created, evolving and dying as white dwarfs, neutronstars, and black holes. At the same time, the universe asa whole is expanding. One important question is whetherthe universe will continue to expand forever. Just beforethe year 2000, cosmologists received a surprise. As wediscussed in the previous class, gravity was assumed tobe the predominant force on a large scale in the uni-verse, and it was thought that the expansion of the uni-verse ought to be slowing down in time because gravityacts as an attractive force between objects. But mea-surements on type Ia supernovae unexpectedly showedthat very distant (high z) supernovae were dimmer thanexpected [25]. That is, given their great distance d asdetermined from their low brightness, their speed V de-termined from the measured z was less than expectedaccording to Hubble’s law. This suggests that nearergalaxies are moving away from us relatively faster thanthose very distant ones, meaning the expansion of theuniverse in more recent epochs has sped up. This accel-eration in the expansion of the universe seems to havebegun roughly 5 billion years ago, 8 to 9 Gyr after theBig Bang. What could be causing the universe to acceler-ate in its expansion? One idea is a sort of quantum field,so-called “quintessence” [26]; another possibility suggestsan energy latent in space itself (vacuum energy) relatedto the cosmological constant. Whatever it is, it has beengiven the name of “dark energy.”

We will now re-examine the problem of the galaxy atthe edge of the massive sphere undergoing expansion toinclude the dark energy effect. The galaxy will be af-fected by a central force due to gravity and the cosmo-logical force,

mr = −4πG r3ρm

3r2+

1

3mΛr , (64)

or

r = −4πG rρ

3+

1

3Λr . (65)

In this equations, the radius r and the density ρ arechanging with time. They may be expressed in termsof the scale factor R:

r = (R/R0) r0 , (66)

where R is defined to be R0 when the radius r = r0;

ρ = (R0/R)3 ρ0 , (67)

where the density ρ = ρ0 when R = R0. SubstitutingEqs. (66) and (67) into Eq. (65), one obtains

R = − a

R2+

1

3ΛR , (68)

No Big Bang

1 2 0 1 2 3

expands forever

-1

0

1

2

3

2

3

closed

recollapses eventually

Supernovae

CMB Boomerang

Maxima

Clusters

mass density

vacu

um e

nerg

y de

nsity

(cos

mol

ogic

al c

onst

ant)

open

flat

SNAP Target Statistical Uncertainty

FIG. 8: Shown are three independent measurements of thecosmological parameters (ΩΛ, Ωm). The high-redshift super-novae [27], galaxy cluster abundance [28] and the CMB [29]converge nicely near ΩΛ = 0.7 and Ωm = 0.3. The upper-left shaded region, labeled “no Big Bang,” indicates bounc-ing cosmologies for which the universe has a turning point inits past [30]. The lower right shaded region corresponds toa universe which is younger than long-lived radioactive iso-topes [31], for any value of H0 ≥ 50 km s−1 Mpc−1. Alsoshown is the expected confidence region allowed by the futureSuperNova / Acceleration Probe (SNAP) mission [32].

where a = 4πGR30ρ0/3. If Eq. (68) is multiplied on both

sides by R, the left hand side yields

RR =1

2

d(R2)

dt, (69)

and thus Eq. (68) takes the form

d(R2) = − 2a

R2dR +

2

3Λ R dR . (70)

We now define R0 = R(t0). Integrating Eq. (70) from t0to t gives

R2 − R20 = 2a

(

1

R− 1

R0

)

+1

3Λ(R2 − R0)

2 . (71)

The constants R0 and a can be eliminated in favor of theHubble constant H0 and the density parameter Ω0 ≡ Ωm.

16

Because ρc = 3H20/8πG,

2a = 8πGR30 ρ0/3 = H2

0R30 ρ0/ρc = H2

0R30Ω0 , (72)

where Ω0 = ρ0/ρc. Using Eq.(72) and R0 = H0R0 inEq. (71), and defining the vacuum energy density ΩΛ =Λ/(3H2

0 ), one obtains

R2

H20R2

0

= Ω0R0

R+ ΩΛ

(

R

R0

)2

+ 1 − Ω0 − ΩΛ (73)

as the basic differential equation governing R(t). Asshown in Fig. 8, the data from the WMAP survey andother recent experiments agree well with theories whenthey input dark energy as providing 70% of the energy inthe universe, and when the total energy density ρ equalsthe critical density ρc. Interestingly, this ρ cannot be onlybaryonic matter (atoms are 99.9% baryons – protons andneutrons – by weight). WMAP data indicate that theamount of normal baryonic matter in the universe is only4% of the critical density. What is the other 96%?

There is a strong evidence for a significant amount ofnonluminous matter in the universe referred to as darkmatter. For example, observations of the rotation ofgalaxies suggest that they rotate as they had consider-ably more mass than we can see. From Eq. (3) it is easilyseen that the speed of stars revolving around the Galacticcenter,

v =√

GM/r , (74)

depends on the galactic mass M and their distance fromthe center r. Observations show that stars farther fromthe Galactic center revolve much faster than expectedfrom visible matter, suggesting a great deal of invisiblematter. Similarly, observations of the motions of galaxieswithin clusters also suggest they have considerably moremass than can be seen. What might this nonluminousmatter in the universe be? We do not know yet. It cannotbe made of ordinary (baryonic) matter, so it must consistof some other sort of elementary particle. The evolutionof the scale factor for an accelerating universe as derivedfrom Eq. (73) assuming 70% of dark energy, 26% of darkmatter, and 4% of baryons, is shown in Fig. 9.

XII. GRAVITATIONAL REDSHIFT

We have seen that the global geometry of spacetimemay (in principle) be non-Euclidean; but how does theuniverse look locally (i.e., on a small scale)? Accordingto Einstein’s theory, spacetime is curved near massivebodies. We might think of space as being like a thinrubber sheet: if a heavy weight is hung from it, it curves.The weight corresponds to a huge mass that causes space(space itself!) to curve. Thus, in Einstein’s theory wedo not speak of the “force” of gravity acting on bodies.Instead we say that bodies and light rays move as they dobecause spacetime is curved. A body at rest or moving

FIG. 9: The three possibilities for expansion in Friedmann-Robertson-Walker models, plus a forth possibility that in-cludes dark energy.

slowly near a great mass would follow a geodesic towardsthat body.

The extreme curvature of spacetime could be producedby a black hole. A black hole, as we previously learned,is so dense that even light cannot escape from it. Tobecome a black hole, a body of mass M must undergogravitational collapse, contracting by gravitational selfattraction to within a radius called the Schwarzschild ra-dius [33],

rs =2GM

c2, (75)

where G is the gravitational constant and c the speed oflight. The Schwarzschild radius also represents the eventhorizon of the black hole. By event horizon we mean thesurface beyond which no signals can ever reach us, andthus inform us of events that happen. As a star collapsestoward a black hole, the light it emits is pulled harderand harder by gravity, but we can still see it. Once thematter passes within the event horizon the emitted lightcannot escape, but is pulled back in by gravity. Thus,a light ray propagating in a curved spacetime would beredshifted by the gravitational field. For example, theredshift of radiation from the surface of a star of radiusR and mass M is

zg =1

√

1 − rs/R− 1 . (76)

The gravitational redshift of the radiation from the galax-ies is normally insignificant [34].

XIII. ENERGY DENSITY OF STARLIGHT

To see how modern cosmological models avoid the Ol-bers paradox, we discuss next how the apparent lumi-nosity of a star l and the number of sources per unitvolume become corrected due to the expansion of theuniverse. First, note that each photon emitted with en-ergy hν1 will be red-shifted to energy hν1R(t1)/R(t0) andphotons emitted at time intervals δt1 will arrive at time

17

intervals δt1R(t0)/R(t1), where t1 is the time the lightleaves the source, and t0 is the time the light arrives atEarth. Second, it is easily seen that the total emittedpower by a source at a distance r1 at the time of emis-sion will spread over a sphere of surface area 4πR2(t0)r

21 .

Thus, the apparent luminosity is

l = L

[

R(t1)

R(t0)

]21

4π R2(t0) r21

. (77)

Now, we assume that at time t1 there are n(t1, L) sourcesper unit volume with absolute luminosity between Land L + dL. The proper volume element of space isdV = R3(t1) dr1 sin θ1 dθ1 dφ1, so the number of sourcesbetween r1 and r1+dr1 with absolute luminosity betweenL and L + dL is

dN = 4π R3(t1) r21 n(t1, L) dr1 dL . (78)

The coordinates r1 and t1 are related via r1 = r(t1),where r(t) is defined by

∫ t0

t

dt′

R(t′)≡

∫ r(t)

0

dr . (79)

Differentiation of Eq. (79) leads to dr1 = −dt1/R(t1) andso Eq. (78) can be re-written as

dN = 4π R2(t1) n(t1, L) r21 |dt1| dL . (80)

The total energy density of starlight is therefore

ρs =

∫

l dN =

∫ t0

−∞

L(t1) [R(t1)/R(t0)]4 dt1 , (81)

where L is the proper luminosity density

L(t1) ≡∫

n(t1, L) dL . (82)

In the hot Big Bang model there is obviously no paradox,because the integral in Eq. (81) is effectively cut off at alower limit t1 = 0, and the integrand vanished at t1 = 0,roughly like R(t1). In other words, the modern explana-tion of the paradox is that the stars have only existed fora finite time, so the light from very distant stars has notyet reached us. Rather than providing the world to befinite in space, as was suggested by Kepler in 1610, theOlbers paradox has shown it to be of a finite age.

XIV. LOOKBACK TIME

When we look out from the Earth, we look back intime. Any other observer in the universe would see thesame view. The farther an object is from us, the earlierin time the light we see left it. No matter which directionwe look, our view of the very early universe is blocked bythe “early plasma” - we can only see as far as its surface,

called “the surface of last scattering,” but not into it.Wavelengths from there are red-shifted by z ≈ 1000.

We now discuss how the parameters of the universechange when looking back toward the Big Bang. As wehave seen, in the matter-dominated era, the total masswithin a co-moving sphere of radius R(t) is proportionalto the number of nuclear particles within the sphere, andit must remain constant (i.e., 4π ρ(t) R3(t)/3 = con-stant). Consequently, ρ(t) ∝ R−3(t). Nevertheless, if themass density is dominated by the mass equivalent to theenergy of radiation (the radiation-dominated era), thenρ(t) is proportional to the fourth power of the temper-ature; and because the temperature varies like R−1(t),ρ(t) ∝ R−4(t).

Let us assume that at a time t a typical galaxy ofmass m is at a distance R(t) from some arbitrarily chosengalaxy, say the Milky Way. From Eq. (55) we see that

E = mR2(t)

[

1

2H2(t) − 4

3πρ(t)G

]

. (83)

Now, ρ(t) increases as R(t) → 0 at least as fast as R−3(t),and thus ρ(t)R2(t) grows at least as fast as R−1(t) forR(t) going to zero. Therefore, in order to keep the energyE constant, the two terms in brackets must nearly canceleach other, so that for R(t) → 0, H2(t) → 8πρ(t)G/3 .The characteristic expansion time texp is the reciprocalof the Hubble constant, hence

texp(t) ≡1

H(t)=

√

3

8πρ(t)G. (84)

However, the Hubble constant is proportional to√

ρ, and

therefore we conclude that H ∝ R−n/2(t). The velocityof a typical galaxy is then

V (t) = H(t)R(t) = R ∝ R1−n/2(t) . (85)

Integration of Eq. (85) leads to the relation

t1 − t2 =2

n

[

R(t1)

V (t1)− R(t2)

V (t2)

]

, (86)

or equivalently,

t1 − t2 =2

n

[

1

H(t1)− 1

H(t2)

]

. (87)

We can express H(t) in terms of ρ(t) and find that

t1 − t2 =2

n

√

3

8πG

[

1√

ρ(t1)− 1

√

ρ(t2)

]

. (88)

Hence, whatever the value of n, the time elapsed is pro-portional to the change in the inverse square root of thedensity. This general result can also be expressed moresimply by saying that the time required for the densityto drop to a value ρ from some value very much greaterthan ρ is

t =2

n

√

3

8πGρ=

1/2 texp radiation − dominated2/3 texp matter − dominated

,

18

where we have neglected the second term in Eq. (88)because ρ(t2) ≫ ρ(t1).

In summary, we have shown that the time required forthe density of the universe to drop to a value ρ from muchhigher earlier values is proportional to 1/

√ρ, while the

density ρ ∝ R−n. The time is therefore proportional toRn/2, or equivalently

R ∝ t2/n =

t1/2 radiation− dominated erat2/3 matter − dominated era

. (89)

This remains valid until the kinetic and potential energieshave both decreased so much that they are beginning tobe comparable to their sum, the total energy.

The universe then has a sort of horizon, which shrinksrapidly as we look back toward the beginning. No signalcan travel faster than the speed of light, so at any time wecan only be affected by events occurring close enough sothat a ray of light would have had time to reach us sincethe beginning of the universe. Any event that occurredbeyond this distance could as yet have no effect on us – itis beyond the horizon. Note that as we look back towardsthe beginning, the distance to the horizon shrinks fasterthan the size of the universe. As can be read in Eq. (89)the size of the universe is proportional to the one-half ortwo-thirds power of the time, whereas the distance to thehorizon is simply proportional to the time. Therefore, forearlier and earlier times, the horizon encloses a smallerand smaller portion of the universe. This implies thatat a sufficiently early time any given “typical” particle isbeyond the horizon.

Though we can “see” only as far as the surface of lastscattering, in recent decades a convincing theory of theorigin and evolution of the “early universe” has been de-veloped. Most of this theory is based on recent theo-retical and experimental advances in elementary particlephysics. Hence, before continuing our look back throughtime, we make a detour to discuss the current state ofthe art in High Energy Physics.

XV. ELEMENTARY PARTICLES

A. The Four Forces in Nature

Since the years after World War II, particle acceler-ators have been a principal means of investigating thestructure of nuclei. The accelerated particles are pro-jectiles that probe the interior of the nuclei they strikeand their constituents. An important factor is that fastermoving projectiles can reveal more detail about the nu-clei. The wavelength of the incoming particles is given byde Broglie’s wavelength formula λ = h/p, showing thatthe greater the momentum p of the bombarding particle,the shorter the wavelength and the finer the detail thatcan be obtained. (Here h is Planck’s constant.)

The accepted model for elementary particle physics to-day views quarks and leptons as the basic constituents of

ordinary matter. To understand our present-day viewwe need to begin with the ideas up to its formulation.While the classic discoveries of Thomson [35] (the elec-tron) and Rutherford [36] (the proton) opened successivedoors to subatomic and nuclear physics, particle physicsmay be said to have started with the discovery of thepositron in cosmic rays by Carl Anderson at Pasadenain 1932 [37], verifying Paul Dirac’s almost simultaneousprediction of its existence [38]. By the mid 1930s, it wasrecognized that all atoms can be considered to be madeup of neutrons, protons and electrons. The structure ofmatter seemed fairly simple in 1935 (with a total of sixelementary particles: the proton, the neutron, the elec-tron, the positron, the neutrino and the photon), but inthe decades that followed, hundreds of other elementaryparticles were discovered. The properties and interac-tions of these particles, and which ones should be consid-ered as basic or elementary, is the essence of elementaryparticle physics.

In 1935 Hideki Yukawa predicted the existence of anew particle that would in some way mediate the strongnuclear force [39]. In analogy to photon exchange thatmediates the electromagnetic force, Yukawa argued thatthere ought to be a particle that mediates the strong nu-clear force – the force that holds nucleons together in thenucleus. Just as the photon is called the quantum of theelectromagnetic force, the Yukawa particle would repre-sent the quantum of the strong nuclear force. Yukawapredicted that this new particle would have a mass in-termediate between that of the electron and the proton.Hence it was called a meson (meaning “in the middle”).We can make a rough estimate of the mass of the mesonas follows. For a nucleon at rest to emit a meson wouldrequire energy (to make the mass) that would have tocome from nowhere, and so such a process would violateconservation of energy. But the uncertainty principle al-lows non-conservation of energy of an amount ∆E if itoccurs only for a time ∆t given by

∆E ∆t ≥ ℏ = h/2π . (90)

We set ∆E = mc2, the energy needed to create the massm of the meson. Now conservation of energy is violatedonly as long as the meson exists, which is the time ∆trequired for the meson to pass from one nucleon to theother. If we assume that the meson travels at relativisticspeeds, close to the speed of light c, then ∆t would beat most about ∆t = d/c, where d ≈ 1.5 × 10−15 cmis the maximum distance that can separate interactingnucleons. Thus, replacing in Eq. (90) we have

mc2 ≈ 2.2 × 10−11 J = 130 MeV . (91)

The mass of the predicted meson is thus very roughly130 MeV/c2 or about 250 times the electron mass, me ≈0.51 MeV/c2. (Note, incidentally, that the electromag-netic force that has an infinite range, d = ∞, requires amassless mediator, which is indeed the case of the pho-ton.)

19

TABLE I: Relative strength for two protons in a nucleus ofthe four forces in nature.

Type Relative Strength Field Particle

Strong Nuclear 1 gluons

Electromagnetic 10−2 photon

Weak Nuclear 10−6 W± Z0

Gravitational 10−38 graviton (?)

Just as photons can be observed as free particles, aswell as acting in an exchange, so it was expected thatmesons might be observed directly. Such a meson wassearched on the cosmic radiation that enters the Earth’satmosphere. In 1937 a new particle was discovered whosemass is 106 MeV (207 times the electron mass) [40]. Thisis quite close to the mass predicted, but this new parti-cle called the muon, did not interact strongly with mat-ter. Then it cannot mediate the strong nuclear force ifit does not interact by means of the strong nuclear force.Thus, the muon, which can have either a + or − chargeand seems to be nothing more than a very massive elec-tron, is not the Yukawa particle. The particle predictedby Yukawa was finally found in 1947 [41]. It is calledthe pion (π). It comes in three charged states, +, −,or 0. The π+ and π− have a mass of 140 MeV and theπ0 a mass of 135 MeV. All three interact strongly withmatter. Soon after their discovery in cosmic rays, pionswere produced in the laboratory using a particle accel-erator [42]. Reactions observed included p p → p p π0

and pp → pnπ+. The incident proton from the accelera-tor must have sufficient energy to produce the additionalmass of the pion.

Yukawa’s theory of pion exchange as the carrier of thestrong force is now out of date, and has been replaced byquantum chromodynamics in which the basic entities arequarks, and the carriers of the strong force are gluons.However, the basic idea of Yukawa’s theory, i.e., thatforces can be understood as the exchange of particlesremains valid.

There are four known types of force – or interaction– in nature. The electromagnetic force is carried by thephoton, the strong force by gluons. What about the othertwo: the weak nuclear force and gravity? These two arealso mediated by particles. The particles that transmitthe weak force are referred to as the W+, W−, and Z0,and were detected in 1983 [43]. The quantum (or carrier)of the gravitational force is called the graviton, and hasnot yet been observed.

A comparison of the four forces is given in Table I,where they are listed according to their (approximate)relative strengths. Note that although gravity may bethe most obvious force in daily life (because of the hugemass of the Earth), on a nuclear scale it is the weakestof the four forces, and its effect at the particle level cannearly always be ignored.

B. Particle’s Zoo

In the decades following the discovery of the pion, agreat many other subnuclear particles were discovered.One way of arranging the particles in categories is ac-cording to their interactions, since not all particles inter-act by means of all four of the forces known in nature(though all interact via gravity).

The gauge bosons include the gluons, the photon andthe W and Z particles; these are the particles that medi-ate the strong, electromagnetic, and weak interactions re-spectively. The leptons are particles that do not interactvia the strong force but do interact via the weak nuclearforce. The leptons include the electron e−, the electronneutrino νe, the muon µ−, the muon neutrino νµ, thetau τ−, the tau neutrino ντ and their corresponding an-tiparticles (e+, νe, µ+, νµ, τ+, ντ ). The third categoryof particles is the hadron. Hadrons are those particlesthat interact via the strong nuclear force. Therefore theyare said to be strongly interacting particles. They alsointeract via the other forces, but the strong force pre-dominates at short distances. The hadrons include theproton, the neutron, the pion, the kaon K, the Λ, and alarge number of other particles. Hadrons can be dividedinto two subgroups: baryons, which are those particlesthat have baryon number B = +1 (or B = −1 in case oftheir antiparticles), and mesons, which have B = 0.

Conservation laws are indispensable in ordering thissubnuclear world. The laws of conservation of energy,of momentum, of angular momentum, of electric charge,and baryon number are found to hold precisely in all par-ticle interactions. Also useful are the conservation lawsfor the three lepton numbers (for a given lepton l, l−

and νl are assigned Ll = +1 and l+ and νl are assignedLl = −1, whereas all other particles have Ll = 0), asso-ciated with weak interactions including decays. Never-theless, an important result has come to the fore in ouryoung XXI century: neutrinos can occasionally changeinto one another in certain circumstances, a phenomenoncalled neutrino flavor oscillation [44]. This result suggeststhat neutrinos are not massless particles and that the lep-ton numbers Le, Lµ, and Lτ are not perfectly conserved.The sum Le +Lµ +Lτ , however, is believed to be alwaysconserved.

The lifetime of an unstable particle depends on whichforce is more active in causing the decay. When astronger force influences the decay, that decay occursmore quickly. Decays caused by the weak force typicallyhave lifetimes of 10−13 s or longer (W and Z are excep-tions). Decays via the electromagnetic force have muchshorter lifetimes, typically about 10−16 to 10−19 s andnormally involve a photon. Hundreds of particles havebeen found with very short lifetimes, typically 10−23 s,that decay via the strong interaction. Their lifetimes areso short that they do not travel far enough to be detectedbefore decaying. The existence of such short-lived parti-cles is inferred from their decay products. In 1952, usinga beam of π+ with varying amounts of energy directed

20

through a hydrogen target (protons), Enrico Fermi [45]found that the number of interactions (π+ scattered)when plotted versus the pion kinetic energy had a promi-nent peak around 200 MeV. This led Fermi to concludethat the π+ and proton combined momentarily to forma short-lived particle, before coming apart again, or atleast that they resonated together for a short time. This“new particle” – so called the ∆++ – is referred to as aresonance. Since then many other resonances have beenfound. The width of these excited states is an interestingapplication of the uncertainty principle. If the particleonly lives 10−23 s, then its mass will be uncertain by anamount ∆E ≈ ℏ/∆t ≈ 10−11 J ≈ 100 MeV, which iswhat is observed. Indeed the lifetimes of ∼ 10−23 s forsuch resonances are inferred by the reverse process, i.e.,from the measured width being ∼ 100 MeV.

In the early 1950s, the newly found particles K0-mesonand Λ0-baryon were found to behave strangely in twoways. (a) They were always produced in pairs; for exam-ple the reaction π−p → K0Λ0 occurred with high prob-ability, but the similar reaction π−p → K0n was neverobserved to occur. (b) These strange particles, as theycame to be called, were produced via the strong interac-tion (i.e., at high rate), but did not decay at a fast ratecharacteristic of the strong interaction (even though theydecay into strongly interacting particles).

To explain these observations, a new quantum num-ber, strangeness, and a new conservation law, conserva-tion of strangeness were introduced. By assigning thestrangeness numbers S = +1 for the kaon and S = −1for the Λ, the production of strange particles in pairswas explained. Antiparticles were assigned oppositestrangeness from their particles. Note that for the reac-tion π−p → K0Λ0, both the initial and final states haveS = 0 and hence strangeness is conserved. However, forπ−p → K0n, the initial state has S = 0 and the finalstate has S = 1, so strangeness would not be conserved;and this reaction is not observed.

To explain the decay of strange particles, it is postu-lated that strangeness is conserved in the strong interac-tion, but is not conserved in the weak interaction. There-fore, strange particles are forbidden by strangeness con-servation to decay to non-strange particles of lower massvia the strong interaction, but could decay by means ofthe weak interaction at the observed longer lifetimes of10−10 to 10−8 s. The conservation of strangeness was thefirst example of a partially conserved quantity.

C. The “Standard Model”

All particles, except the gauge bosons, are either lep-tons or hadrons. The principal difference between thesetwo groups is that hadrons interact via the strong inter-action, whereas the leptons do not.

The six leptons are considered to be truly elementaryparticles because they do not show any internal structure,and have no measurable size. (Attempts to determine

TABLE II: Quark quantum numbers: charge Q, baryon num-ber B, strangeness S, charm c, “beauty” or bottomness b, and“truth” or topness t

name symbol Q B S c b t

up u 2

3e 1

30 0 0 0

down d −1

3e 1

30 0 0 0

strange s − 1

3e 1

3−1 0 0 0

charm c 2

3e 1

30 1 0 0

bottom b − 1

3e 1

30 0 −1 0

top t −1

3e 1

30 0 0 1

the size of the leptons have put an upper limit of about10−18 m.)

On the other hand, there are hundreds of hadronsand experiments indicate they do have internal struc-ture. In the early 1960s, Murray Gell-Mann [46] andGeorge Zweig [47] proposed that none of the hadrons aretruly elementary, but instead are made up of combina-tions of three pointlike entities called, somewhat whim-sically, quarks. Today quarks are considered the trulyelementary particles like leptons. The three quarks origi-nally proposed were labeled, u, d, s and have the namesup, down, and strange; respectively. At present the the-ory has six quarks, just as there are six leptons – basedon a presumed symmetry in nature. The other threequarks are called charmed, bottom, and top (labeled c,b, and t). The names apply also to new properties ofeach that distinguish the new quarks from the old quarksand which (like strangeness) are conserved in strong, butnot in weak interactions. The properties of the quarksare given in Table II. Antiquarks have opposite signs ofelectric charge, baryon number, strangeness, charm, bot-tomness, and topness.

All hadrons are considered to be made up of com-binations of quarks, and their properties are describedby looking at their quark content. Mesons consist of aquark-antiquark pair. For example, a π+ meson is a udcombination: note that for ud pair Q = (2/3+1/3)e = e,B = 1/3 − 1/3 = 0, S = 0 + 0, as they must for the π+;and a K+ = us, with Q = 1, B = 0, S = 1. Baryons onthe other hand, consist of three quarks. For example, aneutron is n = ddu, whereas an antiproton is p = uud.

The truly elementary particles are the leptons, thequarks, and the gauge bosons. Matter is made up mainlyof fermions which obey the Pauli exclusion principle, butthe carriers of the forces are all bosons (which do not).

Quarks are fermions with spin 1/2 and therefore shouldobey the exclusion principle. Yet for three particularbaryons (∆++ = uuu, ∆− = ddd, and Ω− = sss), allthree quarks would have the same quantum numbers, andat least two quarks have their spin in the same directionbecause there are only two choices, spin up (ms = +1/2)or spin down (ms = −1/2). This would seem to violatethe exclusion principle!

Not long after the quark theory was proposed, it was

21

suggested that quarks have another property called color,or “color charge” (analogous to the electric charge). Thedistinction between the six quarks (u, d, s, c, b, t) was re-ferred to as flavor. Each of the flavors of quark can havethree colors usually designated red, green, and blue. Theantiquarks are colored antired, antigreen, and antiblue.Baryons are made up of three quarks, one with each color.Mesons consist of a quark-antiquark pair of a particularcolor and its anticolor. Both baryons and mesons arethus colorless or white. Because the color is different foreach quark, it serves to distinguish them and allows theexclusion principle to hold. Even though quark color wasoriginally an ad hoc idea, it soon became the central fea-ture of the theory determining the force binding quarkstogether in a hadron.

Each quark is assumed to carry a color charge, analo-gous to the electric charge, and the strong force betweenquarks is referred to as the color force. This theory of thestrong force is called quantum chromodynamics (QCD).The particles that transmit the color force are called thegluons. There are eight gluons, all massless and all havecolor charge. Thus gluons have replaced mesons as par-ticles responsible for the strong (color) force.

One may wonder what would happen if we try to see asingle quark with color by reaching deep inside a hadron.Quarks are so tightly bound to other quarks that extract-ing one would require a tremendous amount of energy, somuch that it would be sufficient to create more quarks.Indeed, such experiments are done at modern particlecolliders and all we get is not an isolated quark, but morehadrons (quark-antiquark pairs or triplets). This prop-erty of quarks, that they are always bound in groups thatare colorless, is called confinement. Moreover, the colorforce has the interesting property that, as two quarks ap-proach each other very closely (or equivalently have highenergy), the force between them becomes small. Thisaspect is referred to asymptotic freedom [48].

The weak force, as we have seen, is thought to be me-diated by the W+, W−, and Z0 particles. It acts be-tween the “weak charges” that each particle has. Eachelementary particle can thus have electric charge, weakcharge, color charge, and gravitational mass, althoughone or more of these could be zero.

The fundamental particles can be classified into spin-1/2 fermions (6 leptons and 6 quarks), and spin-1 gaugebosons (γ, W±, Z0, and g). The leptons have 18 degreesof freedom: each of the 3 charged leptons has 2 pos-sible chiralities and its associated anti-particle, whereasthe 3 neutrinos and antineutrinos have only one chirality(neutrinos are Left-handed and antineutrinos are Right-handed).7 The quarks have 72 degrees of freedom: each

7 A phenomenon is said to be chiral if it is not identical to itsmirror image. The spin of a particle may be used to define ahandedness for that particle. The chirality of a particle is Right-handed if the direction of its spin is the same as the direction ofits motion. It is Left-handed if the directions of spin and motion

of the 6 quarks, has the associated antiparticle, three dif-ferent color states, and 2 chiralities. The gauge bosonshave 27 degrees of freedom: a photon has two possiblepolarization states, each massive gauge boson has 3, andeach of the eight independent types of gluon in QCD has2.

One important aspect of on-going research is the at-tempt to find a unified basis for the different forces innature. A so-called gauge theory that unifies the weakand electromagnetic interactions was put forward in the1960s by Sheldon Lee Glashow [49], Abdus Salam [50],and Steven Weinberg [51]. In this electroweak theory, theweak and electromagnetic forces are seen as two differ-ent manifestations of a single, more fundamental elec-troweak interaction. The electroweak theory has hadmany successes, including the prediction of the W±

particles as carriers of the weak force, with masses of81 ± 2 GeV/c2 in excellent agreement with the mea-sured values of 80.482 ± 0.091 GeV/c2 [52] (and simi-lar accuracy for the Z0), and the prediction of the massof the top quark mt = 174.1+9.7

−7.6 GeV/c2 in remarkableagreement with the top mass measured at the Tevatron,mt = 178.0 ± 4.3 GeV/c2 [53]. The combination of theelectroweak theory plus QCD for the strong interactionis referred today as the Standard Model (SM).

Another intriguing aspect of particle physics is to ex-plain why the W and Z have large masses rather thanbeing massless like the photon. Electroweak theory sug-gests an explanation by means of the Higgs boson, whichinteracts with the Z and the W to “slow them down.”In being forced to go slower than the speed of light, theymust acquire mass. The search for the Higgs boson hasbeen a priority in particle physics. So far, searches haveexcluded a Higgs lighter than 115 GeV/c2 [55]. Yet it isexpected to have a mass no larger than 200 GeV/c2. Weare narrowing in on it.

Summing up, the SM is conceptually simple and con-tains a description of the elementary particles and forces.The SM particles are 12 spin-1/2 fermions (6 quarks and6 leptons), 4 spin-1 gauge bosons and a spin-0 Higgs bo-son. Seven of the 16 particles (charm, bottom, top, tauneutrino, W, Z, gluon) were predicted by the SM beforethey were observed experimentally! There is only oneparticle predicted by the SM which has not yet been ob-served [54].

D. Beyond the Standard Model

With the success of the unified electroweak theory, at-tempts are being made to incorporate it and QCD for thestrong (color) force into a so-called grand unified theory(GUT). One type of GUT has been worked out, in which

are opposite. By convention for rotation, a standard clock, tossedwith its face directed forwards, has Left-handed chirality.

22

FIG. 10: Symmetry around a table.

leptons and quarks belong to the same family (and areable to change freely from one type to the other) andthe three forces are different aspects of the underlyingforce [56]. The unity is predicted to occur, however, onlyon a scale less than about 10−32 m, corresponding to anextremely high energy of about 1016 GeV. If two elemen-tary particles (leptons or quarks) approach each other towhithin this unification scale, the apparently fundamen-tal distinction between them would not exist at this level,and a quark could readily change to a lepton, or viceversa. Baryon and lepton numbers would not be con-served. The weak, electromagnetic, and strong (color)force would blend to a force of a single strength.

The unification of the three gauge coupling constantsof the SM is sensitive to the particle content of the the-ory. Indeed if the minimal supersymmetric extension ofthe SM [57] is used, the match of the gauge couplingconstants becomes much more accurate [58]. It is com-monly believed that this matching is unlikely to be acoincidence. Supersymmetry (SUSY) predicts that in-teractions exist that would change fermions into bosonsand vice versa [59], and that all known fermions have asupersymmetric boson partner. Thus, for each quark weknow (a fermion), there would be a squark (a boson) orSUSY quark. For every lepton there would be a slep-ton. Likewise for every gauge boson (photons and gluonsfor example), there would be a SUSY fermion. But whyhaven’t all these particles been detected. The best guessis that SUSY particles may be heavier than their con-ventional counterparts, perhaps too heavy to have beenproduced in today’s accelerators. Until a supersymmet-ric particle is found, and this may be possible in the nowcoming-on-line Large Hadron Collider (LHC), SUSY isjust an elegant guess.

What happens between the unification distance of10−32 m and more normal (larger) distances is referred toas symmetry breaking. As an analogy, consider a tablethat has four identical place settings as shown in Fig. 10.

The table has several kinds of symmetries. For example,it is symmetric to rotations of 90 degrees; that is, thetable will look the same if everyone moved one chair tothe left or to the right. It is also north-south symmetricand east-west symmetric, so that swaps across the tabledo not affect the way the table looks. It also does notmatter whether any person picks up the spoon to the leftof the plate or the spoon to the right. However, once thefirst person picks up either spoon, the choice is set forall the rest of the table as well. The symmetry has beenbroken. The underlying symmetry is still there – the blueglasses could still be chosen either way – but some choicemust be made, and at that moment, the symmetry of thediners is broken.

Since unification occurs at such tiny distances and hugeenergies, the theory is difficult to test experimentally.But it is not completely impossible. One testable pre-diction is the idea that the proton might decay (via, forexample p → π0e+) and violate conservation of baryonnumber. This could happen if two quarks approached towithin 10−31 m of each other. This is very unlikely at nor-mal temperature and energy; consequently the decay ofa proton can only be an unlikely process. In the simplestform of GUT, the theoretical estimate of the proton life-time for the decay mode p → π0e+ is about 1031 yr, andthis has just come within the realm of testability. Protondecays have still not been seen, and experiments put alower limit on the proton lifetime for the above mode tobe about 1033 yr [60], somewhat greater than the theo-retical prediction. This may seem a disappointment, buton the other hand, it presents a challenge. Indeed morecomplex GUTs are not affected by this result.

Even more ambitious than GUTs are the attempts toalso incorporate gravity, and thus unify all forces of na-ture into a single theory. Superstring theory [61], inwhich elementary particles are imagined not as points,but as one-dimensional strings (perhaps 10−35 m long),is at present the best hope for unification of all forces.8

8 A point worth noting at this juncture: Very recently a newframework with a diametrically opposite viewpoint was put for-ward. The new premise suggests that the weakness of gravitymay be evidence from large extra spatial compactified dimen-sions [62]. This is possible because SM particles are confined toa 4-dimensional world (corresponding to our apparent universe)and only gravity spills into the higher dimensional spacetime.Hence, if this picture is correct, gravity is not intrinsically weak,but of course appears weak at relatively large distances of com-mon experience because its effects are diluted by propagationin the extra dimensions. The distance at which the gravita-tional and electromagnetic forces might have equal strength isunknown, but a particularly interesting possibility is that it isaround 10−19 m, the distance at which electromagnetic and weakforces are known to unify to form the electroweak force. Thiswould imply a fundamental Planck mass, M∗ ∼ MW ∼ 1 TeV,at the reach of experiment [63].

23

XVI. THE EARLY UNIVERSE

In today’s class, we will go back to the earliest of times– as close as possible to the Big Bang – and follow theevolution of the universe. It may be helpful to consultFig. 11 as we go along.

We begin at a time only a minuscule fraction of asecond after the Big Bang: it is thought that prior to10−35 s, perhaps as early as 10−44 s, the four forces of na-ture were unified – the realm Superstring theory. This isan unimaginably short time, and predictions can be onlyspeculative.9 The temperature would have been about1032 K, corresponding to “particles” moving about everywhich way with an average kinetic energy of

K ≈ kT ≈ 1.4 × 10−23 J/K 1023 K

1.6 × 10−10 J/GeV≈ 1019 GeV , (92)

where we have ignored the factor 2/3 in our order of mag-nitude calculation. At t = 10−44 s, a kind of “phase tran-sition” is believed to have occured during which the grav-itational force, in effect, “condensed out” as a separateforce. The symmetry of the four forces was broken, butthe strong, weak, and electromagnetic forces were stillunified, and the universe entered the grand unified era.There were no distinctions between quarks and leptons;baryon and lepton numbers were not conserved. Veryshortly thereafter, as the universe expanded considerablyand the temperature had dropped to about 1027 K, therewas another phase transition and the strong force con-densed out at about 10−35 s after the Big Bang. Now theuniverse was filled with a “soup” of leptons and quarks.The quarks were initially free, but soon began to “con-dense” into mesons and baryons. With this confinementof quarks, the universe entered the hadron era.

About this time, the universe underwent an incredi-ble exponential expansion, increasing in size by a factorof 1040 or 1050 in a tiny fraction of a second, perhaps10−32 s. The usefulness of this inflationary scenario isthat it solved major problems with earlier Big Bang mod-els, such as explaining why the universe is flat, as wellas the thermal equilibrium to provide the nearly uniformCMB [65]. Inflation is now a generally accepted aspectof the Big Bang theory.

After the very brief inflationary period, the universewould have settled back into its more regular expansion.The universe was now a “soup” of leptons and hadrons.We can think of this soup as a grand mixture of parti-cles and antiparticles, as well as photons – all in roughlyequal numbers – colliding with one another frequentlyand exchanging energy.

9 Though for the moment this is a matter of conjecture, it is ap-pealing that Superstring theory predicts a concrete candidatefor the quintessence field that can drive the acceleration of theuniverse in the present epoch [64].

By the time the universe was only about a microsec-ond (10−6 s) old, it had cooled to about 1013 K, corre-sponding to an average kinetic energy of 1 GeV, and thevast majority of hadrons disappeared. To see why, let usfocus on the most familiar hadrons: nucleons and theirantiparticles. When the average kinetic energy of parti-cles was somewhat higher than 1 GeV, protons, neutrons,and their antiparticles were continually being created outof the energies of collisions involving photons and otherparticles. But just as quickly, particle and antiparticleswould annihilate. Hence the process of creation and anni-hilation of nucleons was in equilibrium. The numbers ofnucleons and antinucleons were high – roughly as many asthere were electrons, positrons, or photons. But as theuniverse expanded and cooled, and the average kineticenergy of particles dropped below about 1 GeV, whichis the minimum energy needed in a typical collision tocreate nucleons and antinucleons (940 MeV each), theprocess of nucleon creation could not continue. However,the process of annihilation could continue with antinu-cleons annihilating nucleons, until there were almost nonucleons left; but not quite zero! To explain our presentworld, which consists mainly of matter with very littleantimatter in sight, we must suppose that earlier in theuniverse, perhaps around 10−35 s after the Big Bang, aslight excess of quarks over antiquarks was formed. Thiswould have resulted in a slight excess of nucleons overantinucleons, and it is these “leftover” nucleons that weare made of today. The excess of nucleons over antin-ucleons was about one part in 109. Earlier, during thehadron era there should have been about as many nucle-ons as photons. After it ended, the “leftover” nucleonsthus numbered only about one nucleon per 109 photons,and this ratio has persisted to this day. Protons, neu-trons, and all other heavier particles were thus tremen-dously reduced in number by about 10−6 s after the BigBang. The lightest hadrons, the pions, were the last onesto go, about 10−4 s after the Big Bang. Lighter particles(those whose threshold temperature are below 1011 K),including electrons, positrons, and neutrinos, were thedominant forms of matter, and the universe entered thelepton era. The universe was so dense that even theweakly interacting (anti)neutrinos (that can travel foryears through lead bricks without being scattered) werekept in thermal equilibrium with the electrons, positronsand photons by rapid collision with them and with eachother.

When the temperature of the universe is 3 × 1010 K,0.11 s have elapsed. Nothing has changed qualitatively– the contents of the universe are still dominated byelectrons, positrons, neutrinos, antineutrinos, and pho-tons, all in thermal equilibrium, and all high above theirthreshold temperatures. By the time the first second haspast (certainly the most eventful second in history!), theuniverse has cooled to about 1010 K. The average ki-netic energy is 1 MeV. This is still sufficient energy tocreate electrons and positrons and balance their annihi-lations reactions, since their masses correspond to about

24

FIG. 11: Compressed graphical representation of events as the universe expanded and cooled after the Big Bang. The LaserInterferometer Space Antenna (LISA) mission, whose launch is envisaged for 2013, is also shown. LISA will be a trio ofspacecrafts orbiting the Sun trying to observe gravitational waves by using laser interferometry over astronomical distances.

0.5 MeV. However, about this time the decreasing den-sity and temperature have increased the mean free pathof neutrinos and antineutrinos so much that they are be-ginning to behave like free particles, no longer in ther-mal equilibrium with the electrons, positrons, or photons.From now on neutrinos will cease to play an active rolein our discussion, except that their energy will continueto provide part of the source of the gravitational field ofthe universe.

After 13.8 s, the temperature ∼ 3×109 K had droppedsufficiently so that e+e− could no longer be formed. An-nihilation (e+e− → photons) continued. Electrons andpositrons have now also disappeared from the universe,except for a slight excess of electrons over positrons (laterto join with nuclei to form atoms). Thus, about 14 sec-onds after the Big Bang, the universe entered the radia-tion era. Its major constituents were photons and neu-trinos, but neutrinos partaking only in the weak force,rarely interacted.

As long as thermal equilibrium was preserved, the totalentropy remained fixed. The entropy per unit volume is

found to be: S ∝ NTT 3, where NT is the effective numberof species (total number of degrees of freedom) of parti-cles in thermal equilibrium whose threshold temperaturesare below T. In order to keep the total entropy constant,S must be proportional to the inverse cube of the sizeof the universe. That is, if R is the separation betweenany pair of typical particles, then SR3 ∝ NTT 3R3 =constant.

Just before the annihilation of electrons and positrons(at about 5×109 K) the neutrinos and antineutrinos hadalready gone out of thermal equilibrium with the restof the universe, so the only abundant particles in equi-librium were the electrons, positrons and photons. Theeffective total number of particle species before annhila-tion were Nbefore = 6. On the other hand, after the an-nihilation of electrons and positrons, the only remainingabundant particles in equilibrium were photons. Hencethe effective number of particle species was Nafter = 2. Itfollows from the conservation of entropy that

6 (TR)3∣

∣

before= 2 (TR)3

∣

∣

after. (93)

25

That is, the heat produced by the annihilation of elec-trons and positrons increases the quantity TR by a factor

(TR)|after(TR)|before

= 31/3 ≃ 1.4 . (94)

Before the annihilation of electrons and positrons, theneutrino temperature Tν was the same as the photontemperature T . But from then on, Tν simply droppedlike R−1, so for all subsequent times, TνR equals thevalue before annihilation,

(TνR)|after = (TνR)|before = (TR)|before . (95)

We conclude therefore that after the annihilation pro-cess is over, the photon temperature is higher than theneutrino temperature by a factor

(

T

Tν

)∣

∣

∣

∣

after

=(TR)|after(TνR)|after

≃ 1.4 . (96)

Therefore, even though out of thermal equilibrium, theneutrinos and antineutrinos make an important contri-bution to the energy density. The effective number ofneutrinos and antineutrinos is 6, or 3 times the effectivenumber of species of photons. On the other hand, thefourth power of the neutrinos temperature is less than

the fourth power of the photon temperature by a factorof 3−4/3. Thus the ratio of the energy density of neutri-nos and antineutrinos to that of photons is

uν/uγ = 3−4/3 3 = 0.7 . (97)

From Eq. (44) we obtain the photon energy density atphoton temperature T ; hence the total energy densityafter electron positron annihilation is

uν + uγ = 1.7uγ ≃ 1.3 × 10−14(T/K)4 erg/cm3

. (98)

We can convert this to an equivalent mass density,

ρ = u/c2 ≃ 1.22 × 10−35 (T/K)4 g/cm3

. (99)

We have seen that the baryonic matter is about 4%of the total energy in the universe ρb ≈ 0.04 ρc =0.04 × 10−26 kg/m

3. Using the proton as typical bary-

onic matter (mp = 1.67 × 10−27 kg), this implies thatthe number density of baryons is nb = 0.24 nucleons/m3.From Eqs. (48) and (97) we see that the neutrino num-ber density is about 109 that of nucleons, i.e., nν =2.4× 108 neutrinos/m3. By assuming that neutrinos sat-urate the dark matter density we can set an upper boundon the neutrino mass

mν <0.26 ρc

nν≈ 2.6 × 10−27 kg/m

3

2.4 × 108 neutrino/m3 ≈ 10−35 kg

neutrino× 9.315× 108eV/c2

1.67 × 10−27 kg∼ 6 eV/c2 . (100)

Meanwhile, during the next few minutes, crucial eventswere taking place. Beginning about 2 or 3 minutes afterthe Big Bang, nuclear fusion began to occur. The tem-perature had dropped to about 109 K, corresponding toan average kinetic energy 〈K〉 ≈ 100 keV, where nucleonscan strike each other and be able to fuse, but now coolenough so that newly formed nuclei would not be brokenapart by subsequent collisions. Deuterium, helium, andvery tiny amounts of litium nuclei were probably made.Because the universe was cooling too quickly, larger nu-clei were not made. After only a few minutes, probablynot a quarter of an hour after the Big Bang, the temper-ature dropped far enough that nucleosynthesis stopped,not to start again for millions of years (in stars). Thus,after the first hour or so of the universe, matter consistedmainly of bare nuclei of hydrogen (about 75%) and he-lium (about 25%) and electrons. Nevertheless radiationcontinues to dominate.

The evolution of the early universe is almost com-plete. The next important event is presumed to haveoccurred 380,000 years later. The universe had expandedto about 1/1000 of its present size, and the temperaturehad dropped to about 3000 K. The average kinetic en-

ergy of nuclei, electrons, and photons was less than 1 eV.Since ionization energies of atoms are O(eV), as the tem-perature dropped below this point, electrons could orbitthe bare nuclei and remain there (without being ejectedby collisions), thus forming atoms. With the birth ofatoms, the photons – which had been continually scat-tering from free electrons – now became free to spreadunhindered throughout the universe, i.e., the photons be-came decoupled from matter. The total energy containedin radiation had been decreasing (lengthening in wave-length as the universe expanded), and the total energycontained in matter became dominant. The universe wassaid to have become matter-dominated. As the universecontinued to expand the photons cooled further, to 2.7 Ktoday, forming the CMB we detect from everywhere inthe universe.

After the birth of atoms, stars and galaxies begin toform – presumably by self-gravitation around mass con-centrations (inhomogeneities). Stars began to form about200 million years after the Big Bang, galaxies after almost109 yr. The universe continued to evolve until today,some 13.7 billion years later [66].

26

FIG. 12: Simulations of structure formation with varyingamounts of matter in the neutrino component, i.e., vary-ing neutrino mass: (top left) massless neutrinos, (top right)mν = 1 eV, (bottom left) mν = 7 eV, (bottom right)mν = 4 eV.

XVII. WIMPs

At present, the only dark matter particle which isknown to exist from experiment is the neutrino. As welearned in the last class, neutrinos decoupled from ther-mal equilibrium while still relativistic, constituting a hotdark matter candidate. Their contribution to the uni-verse’s matter balance is similar to that of light, but neu-trinos play a secondary role. The role is however identifi-able: because of their large mean free path, they preventthe smaller structures in the cold dark matter from fullydeveloping. This is visible in the observed distributionof galaxies shown in Fig. 12. Simulations of structureformation with varying amounts of matter in the neu-trino component can match to a variety of observationsof today’s sky, including measurement of galaxy-galaxycorrelations and temperature fluctuations on the surfaceof last scattering. These analyses suggest a more restric-tive limit for neutrino masses than the one derived inEq. (100), namely mν . 1 eV.

The simplest model for cold dark matter consists ofWIMPs - weakly interacting massive particles. GenericWIMPs were once in thermal equilibrium, but decou-pled while strongly non-relativistic. The most promisingWIMP candidate is probably the lightest SUSY particle(hereafter denoted by χ) [67]. The relic abundance of χ’sis determined by its thermally average annihilation crosssection at freezeout. At high temperatures the numberdensity of χ’s is roughly the same as the number densityof photons, but as the temperature drops below the massMχ, the number density of WIMPs drops exponentially.

This continues until the total annihilation cross sectionis no longer large enough to mantain equilibrium and theWIMP number density “freezes out.” The largest an-nihilation cross section in the early universe is expectedto be roughly ∼ M−2

χ . This implies that very massiveWIMPs have such a small annihilation cross section thattheir present abundance would be too large. Indeed, toaccount for the observed non-baryonic matter the ther-mally average annihilation cross section at freezeout hasto be larger than about 1 pb, yielding Mχ < 10 TeV.

Many approaches have been developed to attempt todetect dark matter. Such endeavors include direct detec-tion experiments which hope to observe the scatteringof dark matter particles with the target material of thedetector and indirect detection experiments which aredesigned to search for the products of WIMP annihila-tion into gamma-rays, anti-matter and neutrinos. Thedetection sensitivity of current experiments has been im-proving at a steady rate, and new dark matter huntershave been proposed for the coming years [68].

XVIII. MULTI-MESSENGER ASTRONOMY

A. The Photon Window

Conventional astronomy expands 18 decades in photonwavelengths, from 104 cm radio waves to 10−16 cm γ raysof TeV energy. The images of the Galactic Plane shownin Fig. 13 summarize the different wave-bands. Ourrudimentary understanding of the GeV γ-ray sky wasgreatly advanced in 1991 with the launch of the EnergeticGamma Ray Experiment Telescope (EGRET) on boardof the Compton Gamma Ray Observatory (CGRO). Thescience returns from EGRET observations exceed pre-launch expectations. In particular, the number of pre-viously known GeV γ ray sources increased from 1–2dozen to the 271 listed in the 3rd EGRET Catalog [69].However, of this multitude of sources, only about 100have been definitively associated with known astrophys-ical objects. Therefore, most of the γ ray sky, as wecurrently understand it, consists of unidentified objects.One of the reasons that such a small fraction of thesources were identified is the size of the typical γ-ray er-ror box of EGRET, that was about 1× 1, an area thatcontains several candidate sources preventing straight-forward identification. This leaves intringuing puzzlesfor the next generation of GeV γ ray instruments to un-cover [70]. What happens at higher energies?

Above a few 100 GeV the universe becomes opaque tothe propagation of γ rays, because of e+e− production onthe radiation fields permeating the universe (see Fig. 14).The pairs synchrotron radiate on the extragalactic mag-netic field before annihilation and so the photon flux issignificantly depleted. Moreover, the charged particles

also suffer deflections on the ~B-field camouflaging theexact location of the sources. In other words, the injec-tion photon spectrum is significantly modified en route

27

FIG. 13: The Milky Way as it appears at (a) radio, (b) infrared, (c) visible, (d) X-ray, and (e) γ-ray wavelenghts. Each frameis a panoramic, view covering the entire sky in galactic coordinates. The center of the Galaxy, which lies in the direction of theconstellation Sagittarius is at the center of each map. (NRAO; NASA; Lund Observatory; K. Dennerl and W. Voges; NASA.)

to Earth. This modification becomes dramatic at around106 GeV where interaction with the CMB dominates andthe photon mean free path is smaller than the Galacticradius.

Therefore, to study the high energy behavior of dis-tance sources we need new messengers. Nowadays thebest candidates to probe the high energy universe are cos-mic rays, neutrinos and gravitational waves. Of coursein doing multi-messenger astronomy one has to face newchallenges that we discuss next.

B. Cosmic Rays

In 1912 Victor Hess carried out a series of pioneer-ing balloon flights during which he measured the levelsof ionizing radiation as high as 5 km above the Earth’ssurface [72]. His discovery of increased radiation at highaltitude revealed that we are bombarded by ionizing par-ticles from above. These cosmic ray (CR) particles arenow known to consist primarily of protons, helium, car-bon, nitrogen and other heavy ions up to iron.

Below 1014 eV the flux of particles is sufficiently largethat individual nuclei can be studied by detectors carriedaloft in balloons or satellites. From such direct experi-ments we know the relative abundances and the energyspectra of a variety of atomic nuclei, protons, electrons

10−5 10−4 10−3 10−2 10−1 100 101 102 103

redshift z

7

8

9

10

11

12

13

14

15

16

ener

gy l

og10

[E/e

V]

10−5 10−4 10−3 10−2 10−1 100 101 102 103

7

8

9

10

11

12

13

14

15

16

Eew

Gal

actic

Cen

ter

Mrk

501

max

imum

of s

tar

form

atio

n

first

obj

ects

form

γe γe

γp e+e−p

γγ e+e−

3K

IR

VIS

UV

FIG. 14: Mean interaction length for photons on the ultravi-olet (UV), visible (VIS), infrared (IR), and microwave back-grounds. The electroweak scale is indicated by a dashed line.The redshifts of the star formation epoch and the famous γ-ray source Markarian 501 are also indicated [71].

and positrons as well as the intensity, energy and spa-tial distribution of X-rays and γ-rays. Measurementsof energy and isotropy showed conclusively that one ob-

28

vious source, the Sun, is not the main source. Onlybelow 100 MeV kinetic energy or so, where the solarwind shields protons coming from outside the solar sys-tem, does the Sun dominate the observed proton flux.Spacecraft missions far out into the solar system, wellaway from the confusing effects of the Earth’s atmosphereand magnetosphere, confirm that the abundances around1 GeV are strikingly similar to those found in the ordi-nary material of the solar system. Exceptions are theoverabundance of elements like lithium, beryllium, andboron, originating from the spallation of heavier nucleiin the interstellar medium.

Above 1014 eV, the flux becomes so low that onlyground-based experiments with large apertures and longexposure times can hope to acquire a significant numberof events. Such experiments exploit the atmosphere asa giant calorimeter. The incident cosmic radiation inter-acts with the atomic nuclei of air molecules and producesextensive air showers which spread out over large areas.Already in 1938, Pierre Auger concluded from the sizeof extensive air showers that the spectrum extends up toand perhaps beyond 1015 eV [73]. Nowadays substantialprogress has been made in measuring the extraordinarilylow flux (∼ 1 event km−2 yr−1) above 1019 eV. Contin-uously running experiments using both arrays of parti-cle detectors on the ground and fluorescence detectorswhich track the cascade through the atmosphere, havedetected events with primary particle energies somewhatabove 1020 eV [74].

The mechanism(s) responsible for imparting an energyof more than one Joule to a single elementary parti-cle continues to present a major enigma to high energyphysics. It is reasonable to assume that, in order to ac-celerate a proton to energy E in a magnetic field B, thesize R of the accelerator must encompass the gyro radiusof the particle: R > Rgyro = E/B, i.e. the accelerat-ing magnetic field must contain the particle’s orbit. Bydimensional analysis, this condition yields a maximumenergy E = ΓBR. The Γ-factor has been included toallow for the possibility that we may not be at rest inthe frame of the cosmic accelerator, resulting in the ob-servation of boosted particle energies. Opportunity forparticle acceleration to the highest energies is limited todense regions where exceptional gravitational forces cre-ate relativistic particle flows. All speculations involvecollapsed objects and we can therefore replace R by theSchwarzschild radius R ∼ GM/c2 to obtain E < ΓBM.

At this point a reality check is in order. Such a di-mensional analysis applies to the Fermilab accelerator:10 kilogauss fields over several kilometers (covered with arepetition rate of 105 revolutions per second) yield 1 TeV.The argument holds because, with optimized design andperfect alignment of magnets, the accelerator reaches ef-ficiencies matching the dimensional limit. It is highlyquestionable that nature can achieve this feat. Theoristscan imagine acceleration in shocks with an efficiency ofperhaps 1 − 10%.

Given the microgauss magnetic field of our galaxy, no

FIG. 15: Compilation of measurements of the differential en-ergy spectrum of cosmic rays. The dotted line shows an E−3

power-law for comparison. Approximate integral fluxes (persteradian) are also shown.

structures are large or massive enough to reach the en-ergies of the highest energy cosmic rays. Dimensionalanalysis therefore limits their sources to extragalacticobjects. A common speculation is that there may berelatively nearby active galactic nuclei powered by a bil-lion solar mass black holes. With kilo-Gauss fields wereach 1011 GeV. The jets (blazars) emitted by the cen-tral black hole could reach similar energies in acceleratingsub-structures boosted in our direction by a Γ-factor of10, possibly higher.

In contrast to the irregular shape of the isotropicelectromagnetic background spectrum from, say, 108 −1020 Hz, the CR energy spectrum above 109 eV canbe described by a series of power laws, with the fluxfalling about 3 orders of magnitude for each decade in-crease in energy (see Fig. 15). In the decade centered at∼ 1015.5 eV (the knee) the spectrum steepens from E−2.7

to E−3.0. This feature, discovered around 40 years ago,is still not consistently explained. The spectrum steepensfurther to E−3.3 above ∼ 1017.7 eV (the dip) and thenflattens to E−2.7 at ∼ 1018.5 eV (the ankle). Within thestatistical uncertainty of the data, which is large around1020 eV, the tail of the spectrum is consistent with asimple extrapolation at that slope to the highest ener-

29

gies, possibly with a hint of a slight accumulation around1019.5 eV. A very widely held interpretation of the ankleis that above 1018.5 eV a new population of CRs withextragalactic origin begins to dominate the more steeplyfalling Galactic population. The extragalactic compo-nent seems to be dominated by protons [75].

The main reason why this impressive set of data failsto reveal the origin of the particles is undoubtedly thattheir directions have been scrambled by the microgaussgalactic magnetic fields. However, above 1019 eV pro-ton astronomy could still be possible because the ar-rival directions of electrically charged cosmic rays are nolonger scrambled by the ambient magnetic field of ourown Galaxy. Protons point back to their sources withan accuracy determined by their gyroradius in the inter-galactic magnetic field B,

θ ≃ d

Rgyro=

dB

E, (101)

where d is the distance to the source. Scaled to unitsrelevant to the problem,

θ

0.1≃ (d/Mpc) (B/nG)

E/1020.5 eV. (102)

Speculations on the strength for the inter-galactic mag-netic field range from 10−7 to 10−9 G. For the dis-tance to a nearby galaxy at 100 Mpc, the resolutionmay therefore be anywhere from sub-degree to nonex-istent. Moreover, neutrons with energy & 1018 eV havea boosted cτn sufficiently large to serve as Galactic mes-sengers.10 The decay mean free path of a neutron isc Γn τn = 9.15 (En/109 GeV) kpc, the lifetime beingboosted from its rest-frame value, τn = 886 s, to its labvalue by Γn = En/mn. It is therefore reasonable to ex-pect that the arrival directions of the very highest energycosmic rays may provide information on the location oftheir sources.

At very high energies, however, the universe becomesopaque to the propagation of cosmic rays. Shortly af-ter the discovery of the CMB, Greisen, Zatsepin, andKuzmin (GZK) [77] pointed out that this photonic mo-lasses makes the universe opaque to protons of sufficientlyhigh energy, i.e., protons with energies beyond the pho-topion production threshold,

EthpγCMB

=mπ (mp + mπ/2)

EγCMB

≈ 1020

(

EγCMB

10−3 eV

)−1

eV , (103)

where mp (mπ) denotes the proton (pion) mass andEγCMB

∼ 10−3 eV is a typical CMB photon energy. Af-ter pion production, the proton (or perhaps, instead, a

10 Neutron astronomy from nearby extragalactic sources may alsobe possible [76].

neutron) emerges with at least 50% of the incoming en-ergy. This implies that the nucleon energy changes by ane-folding after a propagation distance . (σpγ nγ y)−1 ∼15 Mpc. Here, nγ ≈ 400 cm−3 is the number density ofthe CMB photons, σpγ > 0.1 mb is the photopion produc-tion cross section, and y is the average energy fraction (inthe laboratory system) lost by a nucleon per interaction.Therefore, if ultra-high energy cosmic rays originate atcosmological distances, the net effect of their interactionswould yield a pile-up of particles around 1019.6 eV withthe spectrum dropping sharply thereafter. This so-calledGZK cutoff has been recently observed [78], suggestingthat cosmic ray astronomy (if possible) would be limitedto 1019 . E/eV . 1020 [79].

Cosmic accelerators are also cosmic beam dumps pro-ducing secondary photons and neutrino beams. Particlesaccelerated near black holes pass through intense radia-tion fields or dense clouds of gas leading to productionof secondary photons and neutrinos that accompany theprimary cosmic ray beam. The target material, whethera gas or photons, is likely to be sufficiently tenuous sothat the primary beam and the photon beam are onlypartially attenuated. Neutrinos propagate in straightlines and they can reach the Earth from far distancesources, but as we will discuss in the next class, thesemessengers need large detectors with sensitivity to weakinteractions.

C. Cosmic Neutrinos

For a deep, sharply focused examination of the uni-verse a telescope is needed which can observe a particlethat is not much affected by the gas, dust, and swirlingmagnetic fields it passes on its journey. The neutrino isthe best candidate. As we have seen, neutrinos consti-tute much of the total number of elementary particles inthe universe, and these neutral, weakly-interacting par-ticles come to us almost without any disruption straightfrom their sources, traveling at very close to the speed oflight. A (low energy) neutrino in flight would not noticea barrier of lead fifty light years thick. When we are ableto see outwards in neutrino light we will no doubt receivea wondrous new view of the universe.

Neutrinos were made in staggering numbers at the timeof the Big Bang. Like the CMB photons, the relic neu-trinos now posess little kinetic energy due to expansionof the universe. There are expected to be at least 114neutrinos per cubic centimeter, averaged over all space.There could be many more on Earth because of conden-sation of neutrinos, now moving slowly under the grav-itational pull of our galaxy. As of now, we only have alower limit on the total mass in this free-floating, ghostlygas of neutrinos, but even so it is roughly equivalent tothe total mass of all the visible stars in the universe.

These relic neutrinos would be wonderful to observe,and much thought has gone into seeking ways to doso [80]. The problem is that the probability of neutrinos

30

interacting within a detector decreases with the square ofthe neutrino’s energy, for low energies. And even in therare case when the neutrino does react in the detectorthe resulting signal is frustratingly miniscule. Nobodyhas been able to detect these lowest-energy neutrinos asof yet. Prospects are not good for a low-energy neutrinotelescope, at least in the near future.

Next best are neutrinos from the nuclear burning ofstars. Here we are more fortunate, as we have the Sunclose-by, producing a huge flux of neutrinos, which hasbeen detected by several experiments [81]. A forty yearmystery persists in the deficit of about one half of thenumbers of neutrinos detected compared to expectations,the so-called “Solar Neutrino Problem” [82]. This deficitis now thought probably to be due to neutrino oscilla-tions.

A marvelous event occurred at 07:35:41 GMT on23 February 1987, when two detectors in deep minesin the US (the IMB experiment [83]) and Japan (theKamiokande experiment [84]) recorded a total of 19 neu-trino interactions over a span of 13 seconds. Two anda half hours later (but reported sooner) astronomers inthe Southern Hemisphere saw the first Supernova to bevisible with the unaided eye since the time of Kepler,250 years ago, and this occurred in the Large Magel-lanic Clouds at a distance of some 50 kiloparsecs (roughly150,000 light years). As we have seen, supernovae ofthe gravitational collapse type, occur when elderly starsrun out of nuclear fusion energy and can no longer re-sist the force of gravity. The neutrinos wind up carry-ing off most of the in-fall energy, some 10% of the totalmass-energy of the inner part of star of about 1.4M⊙

masses. Approximately 3 × 1053 ergs are released withabout 1058 neutrinos over a few seconds. This is a stag-gering thousand times the solar energy released over itswhole lifetime! The awesome visible fireworks consist ofa mere one thousandth of the energy release in neutrinos.From this spectacular beginning to neutrino astronomyfollowed many deductions about the nature of neutrinos,such as limits on mass, charge, gravitational attraction,and magnetic moment.

Moving up in energy, neutrinos would also be in-evitably produced in many of the most luminous and en-ergetic objects in the universe. Whatever the source, themachinery which accelerates cosmic rays will inevitablyalso produce neutrinos, guaranteeing that high energyneutrinos surely arrive to us from the cosmos. The burn-ing question for would-be neutrino astronomers is, how-ever, are there enough neutrinos to detect?

Neutrino detectors must be generally placed deep un-derground, or in water, in order to escape the back-grounds caused by the inescapable rain of cosmic raysupon the atmosphere. These cosmic rays produce manymuons which penetrate deeply into the earth, in even thedeepest mines, but of course with ever-decreasing num-bers with depth. Hence the first attempts at high energyneutrino astronomy have been initiated underwater andunder ice. The lead project, called DUMAND was can-

celed in 1995 on account of slow progress and budgetdifficulties, but managed to make great headway in pio-neering techniques, studying backgrounds, and exploringdetector designs.

High energy neutrinos are detected by observing theCherenkov radiation from secondary particles producedby neutrinos interacting inside large volumes of highlytransparent ice or water, instrumented with a lattice ofphotomultiplier tubes. To visualize this technique, con-sider an instrumented cubic volume of side L. Assume,for simplicity, that the neutrino direction is perpendic-ular to a side of the cube (for a realistic detector theexact geometry has to be taken into account as well asthe arrival directions of the neutrinos). To a first ap-proximation, a neutrino incident on a side of area L2

will be detected provided it interacts within the detectorvolume, i.e. within the instrumented distance L. Thatprobability is

P = 1 − exp(−L/λν) ≃ L/λν , (104)

with λν = (ρNAσν)−1. Here ρ is the density of the iceor water, NA Avogadro’s number and σν the neutrino-nucleon cross section. A neutrino flux F (neutrinos percm2 per second) crossing a detector with cross sectionalarea A (≃ L2) facing the incident beam, will produce

N = ATPF (105)

events after a time T. In practice, the quantities A, Pand F depend on the neutrino energy and N is obtainedby a convolution over neutrino energy above the detectorthreshold.11

The Antartic Muon And Neutrino Detector Array(AMANDA), using a 1-mile-deep Antartic ice area as aCherenkov detector, has operated for more than 5 years.The AMANDA group has reported detection of upcom-ing atmospheric neutrinos produced in cosmic ray show-ers, a demonstration of feasibility [85]. IceCube, the suc-cesor experiment to AMANDA, would reach the sensitiv-ity close to the neutrino flux anticipated to accompanythe highest energy cosmic rays, dubbed the Waxman-Bahcall bound [86]. This telescope, which is currentlybeing deployed near the Amundsen-Scott station, com-prises a cubic-kilometer of ultra-clear ice about a milebelow the South Pole surface, instrumented with longstrings of sensitive photon detectors which record lightproduced when neutrinos interact in the ice [87]. Com-panion experiments in the deep Mediterranean are mov-ing into construction phase.

11 The ”effective” telescope area A is not strictly equal to the ge-ometric cross section of the instrumented volume facing the in-coming neutrino because even neutrinos interacting outside theinstrumented volume may produce a sufficient amount of lightinside the detector to be detected. In practice, A is thereforedetermined as a function of the incident neutrino direction bysimulation of the full detector, including the trigger.

31

x

y

FIG. 16: Initial configuration of test particles on a circle ofradius L before a gravitational wave hits them.

y

x

FIG. 17: The effect of a plus-polarized gravitational wave on aring of particles. The amplitude shown in the figure is roughlyh = 0.5. Gravitational waves passing through the Earth aremany billion billion times weaker than this.

With the Sun and supernova neutrino observations asproofs of concepts, next generation neutrino experimentswill also scrutinize their data for new particle physics,from the signatures of dark matter to the evidence forsuperstring theory.

Summing up, it seems likely that real high energy neu-trino astronomy with kilometer scale projects is only afew years away. Meanwhile, underground detectors waitpatiently for the next galactic supernova. In the verylong run, as has been the case with every venture intonew parts of the electromagnetic spectrum, one can besure that neutrino astronomy will teach us many new andunexpected wonders as we open a new window upon theuniverse [88].

D. Gravitational Waves

Ever since Isaac Newton in the XVII century, we havelearned that gravity is a force that acts immediately on

x

y

FIG. 18: The effect of cross-polarized gravitational waves ona ring of particles.

y

x

y

x

h h+ +

FIG. 19: Two linearly independent polarizations of a grav-itational wave are illustrated by displaying their effect on aring of free particles arrayed in a plane perpendicular to thedirection of the wave. The figure shows the distortions in theoriginal circle that the wave produces if it carries the plus-polarization or the cross-polarization. In general relativitythere are only 2 independent polarizations. The ones shownhere are orthogonal to each other and the polarizations aretransverse to the direction of the wave.

an object. In Einstein theory of General Relativity, how-ever, gravity is not a ”force” at all, but a curvature inspace [20]. In other words, the presence of a very mas-sive body does not affect probed objects directly; it warpsthe space around it first and then the objects move in thecurved space. Inherit from such a redefinition of gravityis the concept of gravitational waves: as massive bodiesmove around, disturbances in the curvature of spacetimecan spread outward, much like a pebble tossed into apond will cause waves to ripple outward from the source.Propagating at (or near) the speed of light, these distur-bances do not travel “through” spacetime as such – thefabric of spacetime itself is oscillating!

The simplest example of a strong source of gravita-tional waves is a spinning neutron star with a smallmountain on its surface. The mountain’s mass will causecurvature of the spacetime. Its movement will “stir up”spacetime, much like a paddle stirring up water. Thewaves will spread out through the universe at the speed

32

of light, never stopping or slowing down.As these waves pass a distant observer, that observer

will find spacetime distorted in a very particular way: dis-tances between objects will increase and decrease rhyth-mically as the wave passes. To visualize this effect, con-sider a perfectly flat region of spacetime with a groupof motionless test particles lying in a plane, as shown inFig. 16. When a weak gravitational wave arrives, pass-ing through the particles along a line perpendicular tothe ring of radius L, the test particles will oscillate ina ”cruciform” manner, as indicated in Figs. 17 and 18.The area enclosed by the test particles does not change,and there is no motion along the direction of propaga-tion. The principal axes of the ellipse become L + ∆Land L−∆L. The amplitude of the wave, which measuresthe fraction of stretching or squeezing, is h = ∆L/L. Ofcourse the size of this effect will go down the farther theobserver is from the source. Namely, h ∝ d−1, where dis the source distance. Any gravitational waves expectedto be seen on Earth will be quite small, h ∼ 10−20.

The frequency, wavelength, and speed of a gravita-tional wave are related through λ = cν. The polariza-

tion of a gravitational wave is just like polarization ofa light wave, except that the polarizations of a gravita-tional wave are at 45, as opposed to 90. In other words,the effect of a “cross”-polarized gravitational wave (h×)on test particles would be basically the same as a wavewith plus-polarization (h+), but rotated by 45. The dif-ferent polarizations are summarized in Fig. 19.

In general terms, gravitational waves are radiated byvery massive objects whose motion involves acceleration,provided that the motion is not perfectly spherically sym-metric (like a spinning, expanding or contracting sphere)or cylindrically symmetric (like a spinning disk). Forexample, two objects orbiting each other in a quasi-Keplerian planar orbit will radiate. The power given offby a binary system of masses M1 and M2 separated adistance R is

P = −32

π

G4

c5

(M1 M2)2(M1 + M2)

R5. (106)

For the Earth-Sun system R is very large and M1 andM2 are relatively very small, yielding

P = −32

π

(6.7 × 10−11 m3

kg s2 )4

(3 × 108 m/s)5(6 × 1024 kg 2 × 1030 kg)2(6 × 1024 kg + 2 × 1030 kg)

(1.5 × 1011m)5= 313 W . (107)

Thus, the total power radiated by the Earth-Sun systemin the form of gravitational waves is truly tiny comparedto the total electromagnetic radiation given off by theSun, which is about 3.86 × 1026 W. The energy of thegravitational waves comes out of the kinetic energy ofthe Earth’s orbit. This slow radiation from the Earth-Sun system could, in principle, steal enough energy todrop the Earth into the Sun. Note however that thekinetic energy of the Earth orbiting the Sun is about2.7 × 1033 J. As the gravitational radiation is given off,it takes about 300 J/s away from the orbit. At this rate,it would take many billion times more than the currentage of the Universe for the Earth to fall into the Sun.

Although the power radiated by the Earth-Sun systemis minuscule, we can point to other sources for which theradiation should be substantial. One important exampleis the pair of stars (one of which is a pulsar) discovered byRussell Hulse and Joe Taylor [89]. The characteristics ofthe orbit of this binary system can be deduced from theDoppler shifting of radio signals given off by the pulsar.Each of the stars has a mass about 1.4 M⊙. Also, theirorbit is about 75 times smaller than the distance betweenthe Earth and Sun, which means the distance betweenthe two stars is just a few times larger than the diameterof our own Sun. This combination of greater masses andsmaller separation means that the energy given off by theHulse-Taylor binary will be far greater than the energy

given off by the Earth-Sun system, roughly 1022 times asmuch.

The information about the orbit can be used to predictjust how much energy (and angular momentum) shouldbe given off in the form of gravitational waves. As theenergy is carried off, the orbit will change; the stars willdraw closer to each other. This effect of drawing closeris called an inspiral, and it can be observed in the pul-sar’s signals. The measurements on this system were car-ried out over several decades, and it was shown that thechanges predicted by gravitational radiation in GeneralRelativity matched the observations very well, providingthe first experimental evidence for gravitational waves.

Inspirals are very important sources of gravitationalwaves. Any time two compact objects (white dwarfs,neutron stars, or black holes) come close to each other,they send out intense gravitational waves. As the ob-jects come closer and closer to each other (that is, as Rbecomes smaller and smaller), the gravitational waves be-come more and more intense. At some point these wavesshould become so intense that they can be directly de-tected by their effect on objects on the Earth. This directdetection is the goal of several large experiments aroundthe world.

The great challenge of this type of detection, though,is the extraordinarily small effect the waves would pro-duce on a detector. The amplitude of any wave will

33

fall off as the inverse of the distance from the source.Thus, even waves from extreme systems like merging bi-nary black holes die out to very small amplitude by thetime they reach the Earth. For example, the amplitudeof waves given off by the Hulse-Taylor binary as seenon Earth would be roughly h ≈ 10−26. However, somegravitational waves passing the Earth could have some-what larger amplitudes, h ≈ 10−20. For an object 1 m inlength, this means that its ends would move by 10−20 mrelative to each other. This distance is about a billionthof the width of a typical atom.

A simple device to detect this motion is the laser inter-ferometer, with separate masses placed many hundreds ofmeters to several kilometers apart acting as two ends ofa bar. Ground-based interferometers are now operating,and taking data. The most sensitive is the Laser Inter-ferometer Gravitational Wave Observatory (LIGO) [90].This is actually a set of three devices: one in Livingston,Louisiana; the other two (essentially on top of each other)in Hanford, Washington. Each consists of two light stor-age arms which are 2 to 4 km in length. These are at 90

angles to each other, and consist of large vacuum tubesrunning the entire 4 kilometers. A passing gravitationalwave will then slightly stretch one arm as it shortens theother. This is precisely the motion to which an interfer-ometer is most sensitive.

Even with such long arms, a gravitational wave willonly change the distance between the ends of the arms byabout 10−17 m at most. This is still only a fraction of thewidth of a proton. Nonetheless, LIGO’s interferometersare now running routinely at an even better sensitivitylevel. As of November 2005, sensitivity had reached theprimary design specification of a detectable strain of onepart in 1021 over a 100 Hz bandwidth.

All detectors are limited at high frequencies by shotnoise, which occurs because the laser beams are made upof photons. If there are not enough photons arriving ina given time interval (that is, if the laser is not intenseenough), it will be impossible to tell whether a measure-ment is due to real data, or just random fluctuations inthe number of photons.

All ground-based detectors are also limited at low fre-quencies by seismic noise, and must be very well iso-lated from seismic disturbances. Passing cars and trains,falling logs, and even waves crashing on the shore hun-dreds of miles away are all very significant sources of noisein real interferometers.

Space-based interferometers, such as LISA, are also be-ing developed. LISA’s design calls for test masses to beplaced five million kilometers apart, in separate space-crafts with lasers running between them, as shown inFig. 11. The mission will use the three spacecrafts ar-ranged in an equilateral triangle to form the arms of agiant Michelson interferometer with arms about 5 millionkilometers long. As gravitational waves pass through thearray, they slowly squeeze and stretch the space betweenthe spacecrafts. Although LISA will not be affected byseismic noise, it will be affected by other noise sources,

including noise from cosmic rays and solar wind and (ofcourse) shot noise.

As we have seen in our look back through time, afterthe GUT era, the universe underwent a period of fan-tastic growth. This inflationary phase concluded with aviolent conversion of energy into hot matter and radia-tion. This reheating process also resulted in a flood ofgravitational waves. Because the universe is transparentto the propagation of gravitational waves, LISA will beable to search for these relic ripples probing energy scalesfar beyond the eV range, associated to the surface of lastscattering [91].

XIX. QUANTUM BLACK HOLES

As we have seen, black holes are the evolutionary end-points of massive stars that undergo a supernova explo-sion leaving behind a fairly massive burned out stellarremnant. With no outward forces to oppose gravitationalforces, the remnant will collapse in on itself.

The density to which the matter must be squeezedscales as the inverse square of the mass. For example,the Sun would have to be compressed to a radius of3 km (about four millionths its present size) to becomea black hole. For the Earth to meet the same fate, onewould need to squeeze it into a radius of 9 mm, abouta billionth its present size. Actually, the density of asolar mass black hole (∼ 1019 kg/m3) is about the high-est that can be created through gravitational collapse.A body lighter than the Sun resists collapse because itbecomes stabilized by repulsive quantum forces betweensubatomic particles.

However, stellar collapse is not the only way to formblack holes. The known laws of physics allow matter den-sities up to the so-called Planck value 1097 kg/m3, thedensity at which the force of gravity becomes so strongthat quantum mechanical fluctuations can break downthe fabric of spacetime, creating a black hole with a ra-dius ∼ 10−35 m and a mass of 10−8 kg. This is thelightest black hole that can be produced according to theconventional description of gravity. It is more massivebut much smaller in size than a proton.

The high densities of the early universe were a pre-requisite for the formation of primordial black holes butdid not guarantee it. For a region to stop expandingand collapse to a black hole, it must have been denserthan average, so the density fluctuations were also neces-sary. As we have seen, such fluctuations existed, at leaston large scales, or else structures such as galaxies andclusters of galaxies would never have coalesced. For pri-mordial black holes to form, these fluctuations must havebeen stronger on smaller scales than on large ones, whichis possible though not inevitable. Even in the absenceof fluctuations, holes might have formed spontaneouslyat various cosmological phase transitions – for example,when the universe ended its early period of acceleratedexpansion, known as inflation, or at the nuclear density

34

epoch, when particles such as protons condensed out ofthe soup of their constituent quarks.

The realization that black holes could be so smallprompted Stephen Hawking to consider quantum effects,and in 1974 his studies lead to the famous conclusion thatblack holes not only swallow particles but also spit themout [92]. The strong gravitational fields around the blackhole induce spontaneous creation of pairs near the eventhorizon. While the particle with positive energy can es-cape to infinity, the one with negative energy has to tun-nel through the horizon into the black hole where thereare particle states with negative energy with respect toinfinity.12 As the black holes radiate, they lose mass andso will eventually evaporate completely and disappear.The evaporation is generally regarded as being thermalin character,13 with a temperature inversely proportionalto its mass MBH,

TBH =1

8πGMBH=

1

4 π rs, (108)

and an entropy S = 2 π MBH rs, where rs is theSchwarzschild radius and we have set c = 1. Note thatfor a solar mass black hole, the temperature is around10−6 K, which is completely negligible in today’s uni-verse. But for black holes of 1012 kg the temperature isabout 1012 K hot enough to emit both massless particles,such as γ-rays, and massive ones, such as electrons andpositrons.

The black hole, however, produces an effective po-tential barrier in the neighborhood of the horizon thatbackscatters part of the outgoing radiation, modifing theblackbody spectrum. The black hole absorption crosssection, σs (a.k.a. the greybody factor), depends uponthe spin of the emitted particles s, their energy Q, andthe mass of the black hole [95]. At high frequencies(Qrs ≫ 1) the greybody factor for each kind of particlemust approach the geometrical optics limit. The inte-grated power emission is reasonably well approximatedtaking such a high energy limit. Thus, for illustrativesimplicity, in what follows we adopt the geometric opticsapproximation, where the black hole acts as a perfectabsorber of a slightly larger radius, with emitting areagiven by [95]

A = 27πr2s . (109)

Within this framework, we can conveniently write thegreybody factor as a dimensionless constant normalizedto the black hole surface area seen by the SM fields Γs =σs/A4, such that Γs=0 = 1, Γs=1/2 ≈ 2/3, and Γs=1 ≈1/4.

12 One can alternatively think of the emitted particles as comingfrom the singularity inside the black hole, tunneling out throughthe event horizon to infinity [93].

13 Indeed both the average number [92] and the probability dis-tribution of the number [94] of outgoing particles in each modeobey a thermal spectrum.

All in all, a black hole emits particles with initial totalenergy between (Q, Q + dQ) at a rate

dNi

dQ=

σs

8 π2Q2

[

exp

(

Q

TBH

)

− (−1)2s

]−1

(110)

per degree of particle freedom i. The change of variablesu = Q/T, brings Eq. (110) into a more familar form,

Ni =27 Γs TBH

128 π3

∫

u2

eu − (−1)2sdu. (111)

This expression can be easily integrated using

∫ ∞

0

zn−1

ez − 1dz = Γ(n) ζ(n) (112)

and∫ ∞

0

zn−1

ez + 1dz =

1

2n(2n − 2) Γ(n) ζ(n) , (113)

yielding

Ni = f27 Γs

128 π3Γ(3) ζ(3)TBH , (114)

where Γ(x) (ζ(x)) is the Gamma (Riemann zeta) functionand f = 1 (f = 3/4) for bosons (fermions).14 Therefore,the black hole emission rate is found to be

Ni ≈ 7.8 × 1020

(

TBH

GeV

)

s−1 , (115)

Ni ≈ 3.8 × 1020

(

TBH

GeV

)

s−1 , (116)

Ni ≈ 1.9 × 1020

(

TBH

GeV

)

s−1 , (117)

for particles with s = 0, 1/2, 1, respectively.At any given time, the rate of decrease in the black

hole mass is just the total power radiated

MBH

dQ= −

∑

i

ciσs

8π2

Q3

eQ/TBH − (−1)2s, (118)

where ci is the number of internal degrees of freedom ofparticle species i. A straightforward calculation yields

MBH = −∑

i

ci f27 Γs

128 π3Γ(4) ζ(4) T 2

BH, (119)

14 The Gamma function is an extension of the factorial function fornon-integer and complex numbers. If s is a positive integer, thenΓ(s) = (s − 1)!. The Riemann zeta function of a real variables, defined by the infinite series ζ(s) =

P

∞

n=11/ns, converges

∀s > 1. Using these two definitions one can now verify Eq. (47).

35

where f = 1 (f = 7/8) for bosons (fermions). Assum-ing that the effective high energy theory contains ap-proximately the same number of modes as the SM (i.e.,cs=1/2 = 90, and cs=1 = 27), we find

dMBH

dt= 8.3 × 1073 GeV4 1

M2BH

. (120)

Ignoring thresholds, i.e., assuming that the mass of theblack hole evolves according to Eq. (120) during the en-tire process of evaporation, we can obtain an estimate forthe lifetime of the black hole,

τBH = 1.2 × 10−74 GeV−4

∫

M2BH dMBH . (121)

Using ℏ = 6.58 × 10−25 GeV s, Eq. (121) can then bere-written as

τBH ≃ 2.6 × 10−99 (MBH/GeV)3 s

≃ 1.6 × 10−26 (MBH/kg)3 yr . (122)

This implies that for a solar mass black hole, the lifetimeis unobservably long 1064 yr, but for a 1012 kg one, it is∼ 1.5 × 1010 yr, about the present age of the universe.Therefore, any primordial black hole of this mass wouldbe completing its evaporation and exploding right now.

The questions raised by primordial black holes moti-vate an empirical search for them. Most of the mass ofthese black holes would go into gamma rays (quarks andgluons would hadronize mostly into pions which in turnwould decay to γ-rays and neutrinos), with an energyspectrum that peaks around 100 MeV. In 1976, Hawkingand Don Page realized that γ-ray background observa-tions place strong upper limits on the number of suchblack holes [96]. Specifically, by looking at the observedγ-ray spectrum, they set an upper limit of 104/pc3 on thedensity of these black holes with masses near 5×1011 kg.Even if primordial black holes never actually formed,thinking about them has led to remarkable physical in-sights because they linked three previously disparate ar-eas of physics: general relativity, quantum theory, andthermodynamics [97].

XX. HOMEWORKS

1. Using the definitions of the parsec and light year,show that 1 pc = 3.26 ly.

2. About 1350 J of energy strikes the atmosphere ofthe Earth from the Sun per second per square meter ofarea at right angle to the Sun’s rays. What is (a) theapparent brightness l of the Sun, and (b) the absoluteluminosity L of the Sun.

3. Estimate the angular width that our Galaxy wouldsubtend if observed from Andromeda. Compare to theangular width of the Moon from the Earth.

4. (a) In a forest there are n trees per hectare, evenlyspaced. The thickness of each trunk is D. What is the

distance of the wood not seen for the trees? (b) How isthis related to the Olbers paradox?

5. According to special relativity, the Doppler shift isgiven by

λ′ = λ√

(1 + V/c)/(1 − V/c) , (123)

where λ is the emitted wavelength as seen in a referenceframe at rest with respect to the source, and λ′ is thewavelength measured in a frame moving with velocity Vaway from the source along the line of sight. Show thatthe Doppler shift in wavelength is z ≈ V/c for V ≪ c.

6. Through some coincidence, the Balmer lines fromsingle ionized helium in a distant star happen to overlapwith the Balmer lines from hydrogen in the Sun. Howfast is that star receding from us? [Hint: the wavelengthsfrom single-electron energy level transitions are inverselyproportional to the square of the atomic number of thenucleus.]

7. Astronomers have recently measured the rotationof gas around what might be a supermassive black holeof about 3.6 million solar masses at the center of theGalaxy. If the radius of the Galactic center to the gascloud is 60 light-years, what Doppler shift ∆λ/λ do youestimate they saw?

8. Find the photon density of the 2.7 K backgroundradiation.

9. What is the maximum sum-of-the-angles for a tri-angle on a sphere?

10. Estimate the age of the Universe using the Eisntein-de Sitter approximation.

11. To explore the distribution of charge within nuclei,very-high-energy electrons are used. Electrons are usedrather than protons because they do not partake in thestrong nuclear force, so only the electric charge is inves-tigated. Experiments at the Stanford linear acceleratorwere using electrons with 1.3 GeV of kinetic energy toobtain the charge distribution of the bismuth nucleus.What is the wavelength, and hence the expected resolu-tion for such a beam of electrons.

12. In the rare decay π+ → e+ +νe, what is the kineticenergy of the positron? Assume that the π+ decays fromrest and that neutrinos are massless.

13. Show, by conserving momentum and energy, thatit is impossible for an isolated electron to radiate only asingle photon.

14. Are any of the following reactions/decays possible?For those forbidden, explain what conservation law is vio-lated. (a) µ+ → e++νµ, (b) π−p → K0n (c) π+p → nπ0,(d) p → ne+νe, (e) π+p → pe+, (f) p → e+νe, and(g) π−p → K0Λ0.

15. (a) Symmetry breaking occurs in the electroweaktheory at about 10−18 m. Show that this corresponds toan energy that is on the order of the mass of the W±. (b)Show that the so-called unification distance of 10−32 min a grand unified theory is equivalent to an energy ofabout 1016 GeV.

16. An experiment uses 3300 tons of water waiting tosee a proton decay of the type p → π0e+. If the experi-

36

ment is run for 4 yr without detecting decay, estimate alower limit on the proton lifetime.

17. What is the temperature that corresponds to1.8 TeV collisions at the Fermilab collider? To what erain cosmological history does this correspond?

18. Estimate the minimum energy energy of a cosmicray proton to excite the ∆ resonance scattering off CMBphotons.

19. Evaluate the prospects for neutrino detection atthe km3 IceCube facility. Estimate the total number ofultra-high energy events (108 GeV < Eν < 1011 GeV)expected to be detected at IceCube during its lifetime of15 years, if the cosmic neutrino flux is [86]

Fν(Eν) ≃ 6.0×10−8(Eν/GeV)−2 GeV−1 cm−2 s−1 sr−1 ,(124)

and the neutrino-nucleon cross section rises with energyas [98]

σ(Eν) ≃ 6.04 (Eν/GeV)0.358 pb . (125)

Assume that for this energy range the Earth becomescompletely opaque to the propagation of neutrinos.[Hint: 1b = 10−28 m2].

20. (a) Estimate the power radiated in gravitationalwaves by a neutron star of M⋆ = 1.4M⊙ orbiting a blackhole of MBH = 20M⊙, assuming the orbital radius isR = 6GMBH/c2. (b) If the kinetic energy of the neutronstar orbiting the black hole is about 7×1047 J, how muchtime will it take the neutron star to fall into the blackhole?

21. Hawking has calculated quantum mechanicallythat a black hole will emit particles as if it were a hotbody with temperature T proportional to its surfacegravity. Since the surface gravity is inversely propor-tional to the black hole mass M , and the emitting area isproportional to M2, the luminosity or total power emit-ted is proportional to AT 4, or equivalently L ∝ M−2.Show that as M decreases at this rate, the black holelifetime will be proportional to M3. Estimate the life-time of a black hole with a temperature T = 1012 Kcorresponding to M = 1012 kg, i.e., about the mass of amountain.

22. The spectrum of black holes scales like 1/E−3,just as the ultra high energy cosmic ray background isobserved to do above the so-called “knee.” Using the up-per bound on the primordial black hole density, estimatean upper limit on the black hole contribution to the highenergy cosmic ray spectrum. Can primordial black holesbe the sources of the observed cosmic rays?

Solutions:

1. The parsec is the distance D when the angle φis 1 second of arc. Hence φ = 4.848 × 10−6, and sincetan φ = d/D, we obtain D = 3.086 × 1016 m = 3.26 ly.(d = 1.5 × 108 km is the distance to the Sun.)

2.(a) The apparent brightness is l = 1.3 × 103 W/m2.

2.(b) The absolute luminosity is L = 3.7 × 1026 W.

3. The angular width is the inverse tangent of thediameter of our Galaxy divided by the distance to An-dromeda,

φ = arctan

[

Galaxy diameter

Distance to Andromeda

]

≈ 2.4 . (126)

For the Moon we obtain,

φ = arctan

[

Moon diameter

Distance to Moon

]

≈ 0.52 . (127)

Therefore, the galaxy width is about 4.5 times the Moonwidth.

4.(a) Imagine a circle with radius x around the ob-server. A fraction s(x), 0 ≤ s(x) ≤ 1, is covered bytrees. Then we’ll move a distance dx outward, and drawanother circle. There are 2πnxdx trees growing in theannulus limited by these two circles. They hide a dis-tance 2πxnDdx, or a fraction nDdx of the perimeter ofthe circle. Since a fraction s(x) was already hidden, thecontribution is only [1 − s(x)]nDdx. We get

s(x + dx) = s(x) + [1 − s(x)] n D dx , (128)

which gives a differential equation for s:

ds(x)

dx= [1 − s(x)] n D . (129)

This is a separable equation which can be integrated:

∫ s

0

ds

1 − s=

∫ x

0

n D dx . (130)

This yields the solution

s(x) = 1 − e−nDx . (131)

This is the probability that in a random direction wecan see at most to a distance x. This function x is acumulative probability distribution. The correspondingprobability density is its derivative ds/dx. The mean freepath λ is the expectation of this distribution

λ =

∫ ∞

0

xds(x)

dxdx =

1

nD. (132)

For example, if there are 2000 trees per hectare, and eachtrunk is 10 cm thick, we can see to a distance of 50 m,on average.

4.(b) The result can be easily generalized into 3 dimen-sions. Assume there are n stars per unit volume, and eachhas a diameter D and a surface A = πD2 perpendicularto the line of sight. Then we have

s(x) = 1 − e−nAx , (133)

37

where λ = (nA)−1. For example, if there were onesun per cubic parsec, the mean free path would be1.6× 104 pc. If the universe were infinite old and infinitein size, the line of sight would eventually meet a stellarsurface in any direction, although we could see very farindeed.

5. To show that the Doppler shift in wavelength is∆λ/λ ≈ V/c for v ≪ c we use the binomial expansion:

λ′ = λ (1 + V/c)1/2 (1 − V/c)−1/2

≈ λ

[

1 +V

2c+ O

(

V 2

c2

)] [

1 −(

− V

2c

)

+ O(

V 2

c2

)]

≈ λ[1 + V/c + O(V 2/c2)] . (134)

6. The wavelengths from single electron energy leveltransitions are inversely proportional to the square of theatomic number of the nucleus. Therefore, the lines fromsingly-ionized helium are usually one fourth the wave-length of the corresponding hydrogen lines. Because oftheir redshift, the lines have 4 times their usual wave-length (i.e., λ′ = 4λ) and so

4λ = λ√

(1 + V/c)/(1 − V/c) ⇒ V = 0.88 c . (135)

7. We assume that gravity causes a centripetal accel-eration on the gas. Using Newton’s law,

Fgravity = Gmgas MBH

r2=

mgasV2gas

r, (136)

we solve for the speed of the rotating gas,Vgas = 2.9 × 104 m/s. The Doppler shift as com-pared to the light coming from the center of the Galaxyis then ∆λ/λ ≈ Vgas/c ≈ 9.6 × 10−5.

8. Substituting the CMB temperature in Eq. (48) weobtain n ≈ 400 photons cm−3.

9. The limiting value for the angles in a triangle on asphere is 540. Imagine drawing an equilateral trianglenear the north pole, enclosing the north pole. If thattriangle were small, the surface would be approximateflat, and each angle on the triangle would be 60. Thenimagine “stretching” each side of the triangle downtowards the equator, while keeping sure that the northpole stayed inside the triangle. The angle at each vertexof the triangle would expand, with a limiting value of180. The three 180 angles in the triangle would addup to 540.

10. Setting Ω0 = 1 and ΩΛ = 0 in Eq. (73) we obtaina differential equation,

R2 = H20R3

0/R , (137)

that can be easily integrated

R = (3H0t/2)2/3 R0 . (138)

Then, for R = R0 we obtain

t0 =2

3

1

H0∼ 10 Gyr . (139)

11. The momentum of the electron is given by therelativistic identity E2 = p2c2 + m2c4. Each electron hasan energy of 1300 MeV, which is about 2500 times themass of the electron me ≈ 0.51 MeV/c2, thus we canignore the second term and solve for p = E/c. Thereforethe de Broglie wavelength is λ = h/p = hc/E,

λ =6.63 × 10−34 Js 3 × 108 m/s

1.3 × 109 eV 1.6 × 10−19 J/eV

≃ 0.96 × 10−15 m . (140)

This resolution, of approximately 1 fm, is on the orderof the size of a nucleus.

12. We use conservation of energy,

mπ+c2 = Ee+ + Eν , (141)

and momentum. Since the pion decays from rest, themomentum before the decay is zero. Thus, the momen-tum after the decay is also zero, and so the magnitudeof momenta of the positron and the neutrino are equal,pe+ = pν . Therefore, p2

e+ c2 = p2ν c2. The positron and

neutrino momenta are obtained from the relativistic iden-tity, yielding E2

e+ − m2e+ c4 = E2

ν . (Recall neutrinos aretaken as massless particles). From Eq. (141) we first find

E2e+ − m2

e+ c4 = (mπ+ c2 − Ee+)2 (142)

and then solve for

Ee+ =1

2mπ+ c2 +

m2e+ c2

2 mπ+

. (143)

Finally the total kinetic energy of the positron is

Ke+ =mπ+ c2

2− me+c2 +

m2e+ c2

2 mπ+

≃ 139.6 MeV

2− 0.511 MeV +

0.511 MeV

2 × 139.6 MeV≃ 69.3 MeV . (144)

13. We work in the rest frame of the isolated electron,so that it is initially at rest. Energy conservation leadsto

mec2 = Ke + mec

2 + Eγ . (145)

38

Hence, Ke = −Eγ (i.e., Ke = Eγ = 0). Because thephoton has no energy, it does not exist. Consequently,it has not been emitted.

14.(a) The reaction is forbidden because lepton numberis not conserved.

14.(b) The reaction is forbidden because strangeness isnot conserved.

14.(c) The reaction is forbidden because charge is notconserved.

14.(d) The reaction is forbidden because energy is notconserved.

14.(e) The reaction is forbidden because lepton numberis not conserved.

14.(f) The reaction is forbidden because baryon num-ber is not conserved.

14.(g) The reaction is possible via the strong interac-tion.

15.(a) We use Eq. (90) to estimate the mass of theparticle based on the given distance,

mc2 ≈ hc

2πd= 1.98 × 1011 eV ≈ 200 GeV . (146)

This value is of the same order of magnitude as the massof the W±.

15.(b) We take the de Broglie wavelength, λ = h/p,as the unification distance. The energy is so high thatwe assume E = pc, and thus EGUT = 3.1× 1015 GeV. Asimilar result (EGUT = 2 × 1016 GeV) can be obtainedusing Eq. (90). Therefore, this energy is the amountthat could be violated in conservation of energy if theuniverse were the size of the unification distance.

16. As with radioactive decay, the number of decaysis proportional to the number of parent species (N), thetime interval ∆t, and the decay constant λ,

∆N = −λN∆t, (147)

where the minus sign means that N is decreasing. Therate of decay is then ∆N/∆t = −λN, or equivalently,

dN

dt= −λN ⇒ ln

(

N

N0

)

= e−λt, (148)

where N0 is the number of protons present at time t = 0and N is the number remaining after a time t. Therefore,N = N0e

−λt. The half-lifetime is defined as the time forwhich 1/2 of the original amount of protons in a givensample have decayed,

T1/2 =ln 2

λ⇒ λ =

ln 2

T1/2. (149)

Hence,

∆N = − ln 2

T1/2N ∆t; (150)

thus, for ∆N < 1 over the four-year trial,

T1/2 > N ∆t ln 2 . (151)

To determine N, we note that each molecule of waterH2O contains 10 protons. So one mole of water (18 g,6 × 1023 molecules) contains 6 × 1024 protons in 18 gof water, or about 3 × 1026 protons per kg. One ton is103 kg, so the chamber contains

N = 3.3 × 106 kg 3 × 1026 protons/kg

≈ 1033 protons . (152)

Then a very rough estimate for the lower limit on theproton half-life is T1/2 > 1033 4 yr 0.7 ≈ 3 × 1033 yr.It should be stressed that this naıve calculation doesnot take into account either the required study ofbackground events or a statistical analysis of smallsignals [99].

17. We can approximate the temperature from thekinetic energy relationship given in Eq. (92),

T =2

3

K

k

=2

3

1.8 × 1012 eV 1.6 × 10−19 J/eV

1.38 × 10−23 J/K

≈ 1.4 × 1016 K . (153)

From Sec. XVI, we see that this temperature correspondsto the hadron era.

18. The average square center-of-mass energy in apγCMB collision can be expressed in the Lorentz invari-ant form (note that the cosine of the angle between theparticles average away)

s = m2p + 2 Ep EγCMB

, (154)

where EγCMB∼ 10−3 eV is the typical CMB photon en-

ergy and Ep is the energy of the incoming proton. The∆+ decays according to

s = m2p + m2

π + 2mπmp . (155)

Thus, from Eqs. (154) and (155), we obtainEth

pγCMB≈ 1020 eV.

19. The total number of target nucleons present in onekm3 of ice is: NT = ρice NA V ∼ 6 × 1038, where ρice ∼1 g/cm

3is the density of ice, NA = 6.02 × 1023 mol−1

is Avogadro’s number, and V = 1015 cm3 is the totalvolume. Assuming the Earth is completely opaque tothe propagtion of neutrinos (i.e., an effective aperturefor detection at IceCube of 2π sr) the total number ofevents expected in T = 15 yr is

N = 2π NT T

∫ 1011 GeV

108 GeV

F(Eν)σ(Eν ) dEν

≈ 7 . (156)

39

20. Substituting in Eq. (106) we obtainP = 2.4 × 1047 W. At this emission rate the neu-tron star will fall into the black hole in t ≈ 2.9 s.

21. Substituting in Eq. (122) we obtainτBH ∼ 1.5 × 1010 yr, about the present age of theuniverse.

22. The upper bound on the primordial black hole den-sity yields an upper limit on the black hole contributionto the high energy cosmic ray background as observed onEarth given by

dJ/dE ≃ [1016 eV2/m2 s sr]/(E eV)3 . (157)

The details of this tedious calculation have been pub-

lished in [100]. Comparing to the observed backgroundshown in Fig. 15,

dJ/dE ≃ [1024.8 eV2/m2 s sr]/(E eV)3 , (158)

we find that the black signal is down by about 9 ordersof magnitude.

Acknowledgments

I’m thankful to Walter Lewin for a very thorough read-ing of the notes and insightful comments. I’d also like tothank Heinz Andernach for helpful remarks.

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