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III-3 Cosmology

(Main Ref.: Lecture notes; FK p. 721; Sec. 17-1, 17-8, 19-6, 22-4, 23-4 & 8, 25-1 through 7)

Lec. 11

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III-3a. Introduction (Ref.: Lecture notes; FK p. 721, Sec. 25-1& 2)

(i) Steady Universe vs Evolutionary Universe: Static, Steady Universe; = static, infinite expanse of universe,

with no evolution – the universe is infinitely old and lasts forever, with no major change of structure.

Evolutionary Universe; = non-static universe, with evolution – the universe changes with time leads to the big bang cosmology!

Note: Einstein’s general theory of relativity gives NO solution for the static universe – which means the universe is either expanding or it will collapse. However, in those days (when he derived the equations for general relativity first time), before Hubble’s discovery of expansion of the universe, the universe was considered to be static. To solve this problem, Einstein introduced , cosmological constant, to make his solution for the collapsing universe to become static.

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When Einstein learned the observational discovery by Hubble that the universe is indeed expanding, he said that is the greatest mistake of his life and took it out, i.e., set = 0 since he got expanding solutions. However, as we will explain later in Section III-3c we do now need , after all. Einstein’s mistake is NOT that = 0, but that it must be much larger than he predicted! See later sections.

(ii) Expanding Universe:

Hubble’s discovery, Hubble Law, established that the universe is expanding.

*Hubble Law: vc = H0 d, Eqn(III-15)

where vc is receding velocity due to cosmological redshift;

d is distance, and H0 is Hubble’s constant, and

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d = vc / H0 = z c / H0. Eqn(III-16)

where z = cosmological redshift.

Hubble Law is covered already. See Sec. II-9 (iii).

Note: Most advocates of the steady state cosmology gave it up since Hubble’s discovery, although some still fight for it, e.g., Arp, G. Burbidge. However, one of the strongest advocates, Sir Fred Hoyle, died in August 2000.

*Cosmological Principle: = Assumption that the universe is homogeneous and isotropic in large enough scale, > 100 Mpc (e.g., beyond the scale of largest clusters of galaxies) – well supported by observations. Adopted by Einstein when he developed his general theory of relativity.

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The redshifts that we see from distant galaxies are caused by this expansion, not by the motions of galaxies through space

The redshift of a distant galaxy is a measure of the scale of the universe at the time the galaxy emitted its light

Fig. III-73: Cosmological Redshift

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III-3b. Big Bang Cosmology and Microwave Background Radiation (Ref.: Lecture notes; FK Sec. 25-3 through 6)

(i) Hot Big Bang Theory: (Ref.: Lecture notes; FK Sec. 25-3 and 5)

The expanding universe emerged from a cataclysmic event called the Big Bang

• The universe began as an infinitely dense cosmic singularity which began its expansion in the event called the Big Bang, which can be described as the beginning of time

• During the first 10–43 second after the Big Bang, the universe was too dense to be described by the known laws of physics

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If the universe has been expanding, then, if we go back, the universe will get denser, until the point in time when the density gets infinite – the state called singularity, when something, e.g., like an `explosion’, may have happened

This singularity with density = , then, takes place at ~ 15 billion years ago. G. Burbidge, who still believes the steady state cosmology, coined this `singularity’ state, BIG BANG, which was adopted by the community. This singularity is more like inside of a black hole, where the distinction between space and time disappears. So, this is more like the beginning of the universe as we know it. We do not know what things were before and at the big bang, because physics as we know does not apply there!

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. *Planck Time tP: However, not only before and at the moment of the big bang, but also after the big bang but before Planck Time tP, classical mechanics breaks down. Planck Time is defined as:

tP = square root of (G h / c5 ), Eqn (III-17)

where G = gravitational constant, h = Planck constant, and c = velocity of light.

During time t < tP, classical mechanics breaks down, and so quantum mechanics must be used. Since it is near singularity (i.e., infinite density, like the center of a black hole), general relativity must be used. The combination of the two, called `Quantum Gravity’ is a very difficult field. Although some excellent ambitious physicists are attacking quantum gravity, the problems are not fully solved yet. Therefore, we do not know what went on during 0 < t < tP, although some speculate that singularity may be avoided at t = 0 if we adopt quantum gravity correctly.

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*Why Hot Big Bang? - Helium Problem: We now know that elements heavier than He (Z > 2) were produced through nucleosynthesis during the presupervona stages of massive stars. However, typical composition of ordinary stars is about 74% Hydrogen, 25% Helium, and 1% heavier elements (`metals’, i.e., elements with Z > 2 ). There is no way to produce 25% helium by nucleosynthesis within stars. Earlier we noted that high temperature of T > T☉

c (core temperature of the sun) = 1.6 x 107 K is required for hydrogen burning to convert H to He. Here comes need for Hot Big Bang!

°Idea: The universe was very hot right after the big bang. As it expands, it cools down. In the earliest stage, radiation (photons) and particles (first quarks but then protons and electrons) coexist. When T > ~ 107 K in the hot universe, H-burning converts H to He. When ~ 25% were converted to He, the universe has cooled down to T < ~ 107 K, and H-burning stops. In this way the He problem is solved.

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• The observable universe extends about 14 billion light-years in every direction from the Earth

We cannot see objects beyond this distance because light from these objects has not had enough time to reach us

Fig. III-74: The Observable Universe

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*Age of the Universe To:

We can roughly estimate age of the universe by assuming the universe has been expanding uniformly at the constant rate (ie., Hubble Constant Ho = constant with time). Then, To = d / v (where d = current separation between two galaxies and v = velocity with which they are separating). Then, from Hubble Law, Eqn(III-15), we get:

T0 = 1 / H0. Eqn (III-18)

See class notes for derivation

EX 62: If Ho = 70 km/sec-Mpc, what is To? Ans. To = 14 billion years.

See class notes for details.

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Note: The oldest observed stars are in globular clusters, which are ~ 12 – 16 billion years old. The numbers roughly agree, and so o.k. (Remember that the time after the big bang before first stars are created must be very short compared with the age of the current universe.)

Note: Later, we will learn that Ho has not been constant with time (see Section III-3c), but this rough estimate is still o.k. – tells roughly the age of the universe – the uncertainty involved (mostly due to the uncertainty in distances which causes uncertainty in the Ho value) is larger.

(ii) Evolution of the Universe: (Ref.: Lecture notes; FK Sec. 25-5)

After the hot big bang, the universe expands and cools down. What will Happen as time goes on?

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Radiation Dominated Era and Matter Dominated Era:*The universe is made of matter

and energy. Through Einstein’s

E = m c2 , Eqn (III-19)energy E and mass m are similar –

They can be converted from one

to the other.

*Radiation density R:

Define radiation energy ER and

the equivalent mass for radiation

mR, for convenience. Then, mass

density for radiation

R = mR/volume will become:

Fig. III-44: Evolution of Density

Fig. III-75:The Evolution of Density

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R = 4 T / c3 , Eqn (III-20a) (Blackbody)

where = Stefan Boltzman constant, c = velocity of light, and T is the temperature of microwave background radiation today, which is

T = 2.725 K (see next Section (iii)). (Derivation of Eqn(III-20a) skipped.)

Then, Eqn(III-20a) gives:

R = 4.6 X 10–31 kg/m3, Eqn (III-20b), now.

*Total matter density m: = total mass of all matter in the

universe/volume today, including all matter emitting radiation (e.g., stars, galaxies, nebulae, etc.), and `invisible’ dark matter. Then, observation gives:

m = (2 – 4) X 10–27 kg/m3, Eqn (III-21), now.

That means, today the universe is matter dominated. However, that is not so earlier, because radiation density declines more steeply with time, than matter density does (see Fig. III-75).

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Then, we find m = R, at time tM, where: tM = 2500 years (after the big bang); when zM = 25,000; M = 40 nm; and TM = 75000 K (comes in UV), where zM = redshift, M = wavelength of em radiation (photons), and TM = temperature, at time tM. The period BEFORE tM; t < tM; is called Radiation Dominated Era.The period AFTER tM; t > tM; is called Matter Dominated Era. See class notes for further details.

Recombination Era:

After tM, as the universe keeps cooling down, another critical time tR is reached when temperature TR is so low that protons and electrons combine to make Hydrogen atom. Here, tR = 3 x 105

years, with redshift zR = 1000, and TR = 3000 K. The period AFTER tR is called Recombination Era.

Note: blackbody radiation photons at 3000K is visible red light.

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Note: Before tR, both protons and electrons

are free – matter is called ionized `plasma’

(charged particles). Then, photons hit the

particles and scatter – random walk (like inside

the sun). That means the matter is NOT

transparent, but opaque.

This hot opaque matter with photons is

sometimes called `primodial fire ball’.

After tR, however, when electrons and protons

are combined and become atoms, photons do

NOT interact with H atoms – they travel freely, meaning the matter becomes transparent to photons. Since photons and matter do not interact, they no longer have the same temperature. Until tR, photons and matter have the same temperature. (See Fig. III-76.)

(See class notes for further details.)

Fig. III-76: Evolution of Radiation Temperature

17Fig. III-77a:The Era of Recombination (before)

18Fig. III-77b:The Era of Recombination (after)

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Formation of Stars and Galaxies:

Right after tR, stars and galaxies are formed out of density fluctuations at tR (see Section (iii) below). Which are formed first is still controversial.

See class notes for the details.

(iii) Cosmic Microwave Background Radiation - CBR:

(Ref.: Lecture notes; FK Sec. 25-4)

The microwave radiation that fills all space is evidence of a hot Big Bang.

The background radiation was hotter and more intense in the past•The cosmic microwave background radiation, corresponding to radiation from a blackbody at a temperature of nearly 3 K, is the greatly redshifted remnant of the hot universe as it existed about 380,000 years after the Big Bang

20Fig. III-78a: The Spectrum of CBR

•During the first 380,000 years of the universe, radiation and matter formed an opaque plasma called the primordial fireball

•We can never see photons from

t < tR because matter is opaque,

but we should be able to see

photons emitted at tR because

matter is transparent for t > tR.

These photons should have

reached us here today. However,

due to the expansion of the universe,

the wavelength should have

stretched to the microwave region.

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Fig. III-78b:

(i) The Bell Labs Horn Antenna

(ii) COBE

(ii)

(i)

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★Peebles and Dicke of Princeton predicted that if the hot big bang theory is correct, we should detect this microwave background radiation. Indeed, Penzias and Wilson of Bell Lab discovered this CBR, for which they received Nobel Prize. From Big Bang Theory we can calculate the wavelength we should detect today, and then from Wien’s Law for blackbody radiation we can calculate the temperature of CBR. The results are:

= 10 m – 1cm; T = 2.725 K. Eqn(III-22)

Note: Fig. III-78a shows that the data points (from COBE Satellite, 1989) fit blackbody spectrum of this temperature perfectly!

CBR is found to be isotropic and homogeneous (when the earth’s motion is subtracted) – hence proving perfectly Einstein’s Cosmological Principle!

Note: Discovery of CBR is, together with discovery of Hubble Law, regarded as the strongest proof to support the hot big bang theory.

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Small Temperature/Density Fluctuations:

Although COBE was not sensitive enough to detect them, later balloon experiments (BOOMERANG; MAXIMA (1998)) found small inhomogeneity, small temperature and density fluctuations T~ 10-4 K (i.e., hot spots) and - important as the origin of stars and galaxies. See Section III-3c(ii).

III-3c. Dark Matter, Dark Energy, and Fate of the Universe (Main Ref.: Lecture notes; FK Sec. 17-1, 17-8, 19-6, 22-4, 23-4 & 8, 25-6 & 7)

(i) Dark Matter: (Ref.: Lecture notes; FK Sec. 22-4, 23-8)

Dark Matter (sometimes called `missing mass’) = all matter not observable in all wavelengths of em radiation (i.e. light/photons).

Candidates:

MACHO(= massive compact halo objects) – e.g., brown dwarfs, black holes;

WIMPS (= weakly interacting massive particles) – e.g., neutrinos, heavy

neutrinos, axions.

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Fig. III-79: The Rotation Curves of Four Spiral Galaxies

Without dark matter

FK

Detections: can estimate density from dynamics – e.g., galaxy rotation curves (motion of stars and gas around the center of a galaxy) – See Fig. III-79.

Note: Boundary of a galaxy including

dark matter much larger than the

visible galaxy boundary. Also,

clusters of galaxies are filled with dark

matter – can tell from gravitational

lensing of distant quasars by clusters

of galaxies in the foreground

Conclusion: Dark matter ~90% of all matter!

See class notes for the details.

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(ii) Matter and Energy Content and Shape of the Universe: (Ref.: Lecture notes; FK Sec. 25-6)

•The shape of the universe indicates its matter and energy content•The curvature of the universe as a whole depends on how the combined average mass density ρ0 compares to a critical density ρc

•Definitions:

Combined average mass density of the universe 0: = the sum of the average mass densities of matter, radiation, and any other form of energy.

•Critical Density C : = critical density which is = the combined average mass density of the flat universe

the universe is flat when C = 0 .

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• Note:

C = 3 H02 / ( 8 G ) = 9.2 x 10–27 kg/m3. Eqn(III-23)

where H0 is Hubble Constant, G is Gravitational Constant.

• Density Parameter 0 : defined as:

0 = 0 / C . Eqn(III-24)

Curvature of Space:

Density parameter tells whether the space is spherical (positive curvature) and the universe is closed; space is flat (zero curvature) and the universe is flat; or space is hyperbolic (negative curvature) and the universe is open.

See Fig. III-80 and Table III-4. See class notes for the details.

27Fig. III-80:The Geometry of the Universe

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Table III-4 The Geometry and Average Density of the Universe

What is the curvature of our space?

•Ans: It is nearly FLAT,

from CBR observation, i.e.: 0 ~ 1. Eqn(III-25)

• Can find it by measuring the size of the fine structure in CBR.

*Big bang theory predicts that the true angular size of the structure r0 should be ~ 1o.

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Fig. III-49 (= FK Fig. 28-16) : CBR and the Curvature of Space

Fig. III-49 (= FK Fig. 28-16) : CBR and the Curvature of Space

Fig. III-49 (= FK Fig. 28-16) : CBR and the Curvature of Space

Fig. III-81:CBR and the Curvature of Space

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Observations of temperature variations in the cosmic microwave background indicate that the universe is flat or nearly so, with a combined average mass density equal to the critical density

Fig. III-82:

31Fig. III-83: WMAP

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• If the measured size r > r0 the universe is closed. If the measured size r < r0 the universe is open. If the measured size r = r0 the universe is flat.

Result:

r ~ 1o ~ r0 Eqn(III-26)

See Fig. III-81.See class notes for the details.

(iii) Dark Energy: (Ref.: Lecture notes; FK Sec. 25-7)

Including dark matter, we find that the total matter density parameter

m = m / C = 0.2 – 0.4 (*). Eqn(III-27)

(*) Obtained by dividing Eqn(III-21) by Eqn(III-23).

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Note: m includes all visible mass (galaxies, stars, planets, etc)

+ Dark Matter.

However, CBR observation gives: 0 ~ 1. Eqn(III-28)

Remember: C = 9.2 x 10–27 kg/m3, Eqn(III-23) now

m = (2 – 4) X 10–27 kg/m3, Eqn (III-21) now

and R = 4.6 X 10–31 kg/m3 Eqn (III-20b) now.

So, m >> R now m >> R. now.

So, 0 = m + + R ~ m + Eqn(III-29) now,

which means:

0 = m + + R ~ m + Eqn(III-30) now,

where is dark energy density, and

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= / C = dark energy density parameter. Eqn(III-31)

From Eqns(III-27), (III-28), and (III-30), we conclude that:

= 0. 6 – 0.8 = 60 – 80%! Eqn(III-32)

Note: can be due to Einstein’s , cosmological constant, which is more like `Antigravity’

Conclusion: A majority (60 – 80%) of the total ingredient of the universe is dark energy! Out of the rest (total matter combined), most (~ 90%) are dark matter!

(iv) Fate of the Universe ( Ref.: Lecture notes; FK Sec. 17-1, 17-8, 19-6, 22-4, 23-4, 25-7)

Distance Measurment – Summary only: (Ref.: Lecture notes; FK Sec.17-1, 17-8, 19-6, 23-4)

How to measure distance?

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Summary: The closest can be measured by the direct parallax method (FK Sec. 17-1), and the next closest by spectroscopic parallax with the help of the HR Diagram (FK Sec. 17-8).

Method with Standard Candles: To measure distance to objects further away, such as galaxies, we adopt Method with a `standard candle’.

*Standard Candle in a Galaxy: = an object (e.g., a star) in the galaxy whose absolute luminosity (i.e., absolute magnitude) is known by some other means.

•Then, by measuring the apparent magnitude and with the known absolute magnitude, can find distance (see FK Sec. 17-3). Distance to closer galaxies can be found by using Population II RR Lyrae and Population I Cepheid variables (see FK Sec. 19-6) as the standard candles. For those further away Tully-Fisher method is used (see FK Sec.23-4). Those furthest away can be found by adopting Type Ia supernovae as the standard candle. See Fig. III-84 for the summary.

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Fig. III-84 :The Distance Ladder

Future of the Universe: (Ref.: Lecture notes; FK Sec.22-4, 25-7)

•Earlier, we assumed that Hubble constant H0 is constant with time. We noted that it is essentially constant from FK Fig. 23-17.

However, the data used for this figure are all near-by galaxies, which means that H0, and hence the rate of expansion of the universe, has been constant in recent years

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Fig. III-85: Cosmological Expansion

(a)

(b)

How about the distant past? To

answer that question, we must

measure the distance to very

remote galaxies and quasars.

That has been done, most recently,

by using Type Ia supernovae as

standard candles.

• Note: on the distance d vs receding

velocity vC diagram, slow expansion

means steeper line than the rapid

expansion, and hence, if the

expansion of the universe has

speeded up, the d vs vc curve will look like the blue curve (i.e., steeper earlier)

in Fig. III-85, while it will look like the green curve (i.e., less steep earlier)

if the expansion has slowed down.

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Observations of distant supernovae reveal that we live in an accelerating universe

Observations of galaxy

Clusters suggest that

the average density of

matter in the universe is

about 0.27 of the critical

density. The The remaining contribution to the average density is called dark energy

Measurements of Type Ia supernovae in distant galaxies show that the expansion of the universe is speeding up

Fig. III-86: Varying Rates of Cosmic Expansion

39Fig. III-87: Very Distant Supernovae

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The most recent data from Type Ia supernova observations are shown in Fig. III-88.

Black curve (upper curve) – flat universe with dark energy, the expansion speeding up, with

0 = 1 = m + ; m = 0.24, = 0.76. Eqn(III-33)

Fits the data the best.

Blue curve refers to the flat universe with 0 = m = 1, no dark energy

This may be due to the presence of dark energy in the form of a cosmological constant, which provides a pressure that pushes the universe outward

The conclusion: Comparison of the data and theoretical models tells that the universe is not only expanding, but also that the expansion is speeding up!

Note:The best fits to various observational constraints (see Fig. III-89) gives: 0 = 1.02 (+-) 0.02, m = 0.27 (+-) 0.04,

= 0.73 (+-) 0.04. Eqn(III-34)

See class notes for the details.

41Fig. III-88:The Hubble Diagram for Distant Supernovae

42Fig. III-89 : Limits on the Nature of the Universe

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What is Dark Energy?: Non-zero means we need dark energy, and its contribution to the content of the universe is ~ 76%, according to the supernova data. What is this dark energy? One promising candidate is Einstein’s antigravity expressed by the cosmological constant , which is constant.

Then, since both radiation and matter density decrease with time, while dark energy density is constant, the dark energy now dominates the universe, although earlier it was not important (see Fig. III-75). So, Einstein’s is now needed, after all!

Einstein’s only mistake is that he predicted it to be much smaller, than what we now require from supernova observational data.

Note: there are other speculations about the nature of the dark

energy, but none of them, so far, got beyond speculation.

*Fate of the Universe:

Study Fig. III-90 and class notes.

Conclusion: the future universe will be dark and cold!

44Fig. III-90(a):Earlier Evolution of the Universe

45Fig. III-90(b):Future Evolution of the Universe