Chapter 20 The Big Bang

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The Big Bang

The Universe began life in a very hot, very dense state that we call the big bang. In this chapter we apply the Friedmann equations to the early Universe in an attempt to understand the most important features of thebig bang model, which is the cosmologist’s “standard model” for the origin of the present Universe.

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20.1 Radiation and Matter Dominated Universes

Because

• the influence of vacuum energy grows with expan-sion of the Universe, and

• vacuum energy is only today beginning to dominate, we may safely assume that it was negligible in the early Universe (once the inflationary epoch was over). In that case, two extremes for the equation of state give us con-siderable insight into the early history of the Universe:

1. If the energy density resides primarily in light parti-cles having relativistic velocities, we say the the Uni-verse is radiation dominated; in that case the equa-tion of state is

P= 1₃ε (radiation dominated).

2. If on the other hand the energy density is domi-nated by massive, slow-moving particles, we say the the Universe ismatter dominated;the corresponding equation of state is

P_≃0 (matter dominated).

In either extreme, the evolution of the Universe is then easily calculated using the Friedmann equations.

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20.1.1 Evolution of the Scale Factor

The density of radiation and the density of matter scale differently in an expanding universe. If the Universe is radiation dominated,P= 1₃ε and ˙ ε+3(ε +P)a_a˙ =0 _→ ε˙ ε +4 ˙ a a =0,

which has a solution

ε(t) _≃ 1

a4(t) (radiation dominated).

If on the other hand the Universe is matter dominated, we haveP_≃0

and ˙ ε+3(ε +P)a_a˙ =0 _→ ε˙ ε +3 ˙ a a =0,

which has a solution

ε(t)_≃ 1

a3₍_t₎ (matter dominated).

As we showed in the previous chapter, the corresponding behaviors of the scale factor are

a(t)_≃    t1/2 (radiation dominated) t2/3 (matter dominated)

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20.1.2 Matter and Radiation Density

In the present Universe the ratio of the number density of baryons to photons is

nb

n_γ ≃10

−9_.

However, the rest mass of a typical baryon is approximately 109 eV (recall that the rest mass of a proton is 931 MeV), while most photons are in the _∼ 2.7 K cosmic microwave background, with an average energy E_γ _≃(2.7 K) 1 GeV 1.2_×1013 _K ≃2.3_×10−4 eV.

Thus the ratio of the energy density of baryons to energy density of photons in the present Universe is

εb

ε_γ ≃103−104

and the present Universe is dominated by matter (and by vacuum en-ergy), with only a small contribution from radiation. But,

ε(t) _≃ 1

a4(t) (radiation dominated).

ε(t)_≃ 1

a3(t) (matter dominated).

Thus, as time is extrapolated backwards relativistic matter becomes increasingly more important, until at some ear-lier time the influence of matter and vacuum energy may be neglected compared with that of relativistic particles.

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0 -2 -4 -6 -8 -10 -30 -40 -20 -10 0 10 Radiation Dominated Matter Dominated t ~ 1010_s T ~ 10 eV ρm atter ρr adiation log ρ (g cm -3) 2 log a

Figure 20.1: Dependence of the energy density of matter and radiation on the

scale factor. At this early epoch the influence of vacuum energy and curvature are negligible and the evolution of the Universe is governed by the competition between radiation and matter.

Fig. 20.1 illustrates. Thus, the early Universe should have beenradiation dominated.

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0 -2 -4 -6 -8 -10 -30 -40 -20 -10 0 10 Radiation Dominated Matter Dominated t ~ 1010_s T ~ 10 eV ρm atter ρr adiation log ρ (g cm -3) 2 log a

Since in this early, radiation-dominated Universe,

ε _≃a−4 a_≃t1/2 P= 1₃ε,

the behavior of the density and pressure as time is extrap-olated backwards is Lim t_→0 ε(t) = Limt_→0 (t 1/2₎₋4₌ _Lim t_→0 t −2 ₌_∞ Lim t_→0 P(t) = Limt_→0 (t 1/2₎₋4₌ _Lim t_→0 t −2₌_∞

Furthermore, for a radiation dominated Universe, T _≃ a−1 _≃t−1/2; thus, as time is extrapolated backwards the temperature scales as

Lim

t_→0 T(t) = Limt_→0 t

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These considerations suggest that the Universe started from a very hot, very dense initial state witha(t _→0)_→0. • The commencement from this initial state is called

thebig bang.

• If we take t = 0 when a = 0, the transition be-tween the earlier radiation dominated universe and one dominated by matter took place around 50,000 years after the big bang (redshift of _∼ 3300) when the temperature was about 9000 K.

• This matter dominance then continued until about 4-5 billion years ago, when the vacuum energy density began to overtake the matter density.

In the following sections we shall discuss in more detail the big bang and the early radiation dominated era of the Universe.

The name “big bang” was actually a term coined by opponents of this cosmology who favored the now discredited steady state the-ory. The name stuck, as did the thethe-ory.

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20.2 Evolution of the Early Universe

Considerations of the preceding section suggest that the big bang starts off with a state of extremely high density and pressure for the Universe, and that under those condi-tions the Universe is dominated by radiation.

• This means that the major portion of the energy den-sity is in the form of photons and other massless or nearly massless particles like neutrinos that move at near the speed of light.

• As the big bang evolves in time, the temperature drops rapidly with the expansion and the average ve-locity of particles decreases.

• Finally, about 1000 years after the big bang one reaches a state where the primary energy density of the Universe is in non-relativistic matter.

Let us now give a brief description of the most important events in the big bang and the evolution from a radiation dominated to matter dominated universe.

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20.2.1 Thermodynamics of the Big Bang

We have already established that in the initial radiation dominated era of the big bang,

H2 _≡ ˙ a a 2 ≃ 8π₃Gεr =a−4 a≃t1/2 H = ˙ a a = 1 2t.

We may assume that the average evolution corresponds approximately to that of an ideal gas in thermal equilibrium, for which the number density of a particular species is

dn= g

2π2_¯h3

p2dp eE/kT+Θ,

where p is the 3-momentum, g is the number of degrees of freedom (helicity states: 2 for each photon, massive quark, and lepton) and

E =pp2_c2₊_m2_c4 Θ₌        +1 Fermions -1 Bosons 0 Maxwell–Boltzmann

where Maxwell–Boltzmann statistics obtain only if we make no dis-tinction between fermions or bosons in the gas.

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Because of the high temperature, let us assume that the gas is ultra-relativistic (kT >> mc2 for the particles in the plasma). Then the energy is E = pc, and the number density is obtained by integrating the previous expression fordn.

n= Z ∞ 0 dn dpdp= g 2π2¯h3 Z ∞ 0 p2dp eE/kT+Θ = g 2π2_¯h3 Z ∞ 0 p2dp epc/kT₊Θ (20.1)

Integrals of this form may be evaluated using

Z ∞ 0 tz−1 et₋1dt= (z−1)!ζ(z) Z ∞ 0 tz−1 et+1dt = (1−2 1₋z₎₍_z −1)!ζ(z), whereζ(z) is the Riemann zeta function, with tabulated values

ζ(2) = π

2

6 =1.645 ζ(3) =1.202 ζ(4) = π4

90 =1.082.

The results for the number density of speciesi are

ni=giζ( 3) π2 T 3 ×        1 Bose–Einstein 3/4 Fermi–Dirac ζ(3)−1 Maxwell–Boltzmann

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Likewise, the energy density is given by ε =ρc2 = Z ∞ 0 Edn dpdp= g 2π2_¯h3 Z ∞ 0 E p2dp eE/kT +Θ = g 2π2¯h3 Z ∞ 0 E p2dp epc/kT +Θ (20.2) which gives εi=gi π2 30T 4 ×        1 Bose–Einstein 7/8 Fermi–Dirac 90/π4 Maxwell–Boltzmann

The energy density for all relativistic particles is then given by the sum, ε =g_∗π 2 30T 4 _g ∗≡

∑

bosons gb+7₈

∑

fermions gf.

If all species are in equilibrium, the entropy densitys is

s= ε+P T = 4ε 3T = 2π2 45 g∗T 3_,

where we note from comparing this and

n_i=g_iζ(3) π2 T 3_×        1 Bose–Einstein 3/4 Fermi–Dirac ζ(3)−1 Maxwell–Boltzmann

thats_≃∑n_i. The entropy per comoving volume is constant (adiabatic expansion),

S_≃sa3_≃ constant _−→ d(sa

3₎

dt =0,

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103 ₁₀2 ₁₀1 ₁₀0 ₁₀-1 ₁₀-2 ₁₀-3 ₁₀-4 ₁₀-5 1 10 100 T (GeV) Ef

fective Degrees of Freedom

g* Quark Confinement ~10-6_s ~1 s Weak Freezout

Figure 20.2: Variation of the effective number of degrees of freedom in the early

Universe as a function of temperature.

In fact, as illustrated in Fig. 20.2, we expect g_∗ to be ap-proximately constant for broad ranges of temperature but to change suddenly at critical temperatures where kT be-comes comparable to the rest mass for a species.

Even though in local processes the entropy tends to increase, globally the evolution is dominated by the enormous entropy resident in the cosmic microwave background radia-tion. Thus, cosmologically the expansion is approximately reversible and adiabatic.

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From s= ε+P T = 4ε 3T = 2π2 45 g∗T 3_, S_≃sa3_≃ constant _−→ d(sa 3₎ dt =0, sa3 _≃T3a3 is constant and T _≃ 1 a ≃t −1/2_,

where we have used the result that in a radiation dominated universe of negligible curvature,a_≃t1/2.

To summarize, the evolution of the ultrarelativistic, hot plasma characterizing the early big bang is described by the equations ˙ a a =− ˙ T T =αT 2 _t ₌ 1 2αT2 = 2.4_×10−6 g1_∗/2T2 GeV 2 _s α = 4π3g_∗ 45M2 P 1/2 MP≡ ¯hc G 1/2 =1.2_×1019 GeV,

where MP is the Planck mass. These equations are

ex-pected to be valid from the end of the quantum gravitation era at T _≃ 1019 GeV up to the decoupling of matter and radiation at T _≃10 eV.

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20.2.2 Equilibrium in an Expanding Universe

Strictly, we do not expect equilibrium to hold in an expanding uni-verse.

• However, a practical equilibrium can exist as the Universe passes through a series of nearly equilibrated states.

• We may expect both thermal equilibrium and chemical equilib-rium to play a role in the expansion of the Universe.

A system is inthermal equilibriumif its phase space num-ber density is given by

dn= g 2π2¯h3 p2dp eE/kT+Θ, E =pp2c2+m2c4 Θ=        +1 Fermions -1 Bosons 0 Maxwell–Boltzmann

A system is inchemical equilibriumif for the reaction a+

b_↔c+d the chemical potentials satisfy µ_a+µb =µc+

µd.

We shall illustrate the discussion by considering thermal equilibrium, and will consider the equilibrium to maintained by two-body reac-tions (which is the most common situation).

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The reaction rate for a two-body reaction may be expressed as

Γ≃ hnvσ_i, • wherenis the number density, • vis the relative speed,

• σ is the reaction cross section, and • the brackets indicate a thermal average.

We may expect that a species will remain in thermal equilibrium in the radiation dominated Universe as long as

Γ>> a˙ a ≡H ≃ d(t1/2)/dt t1/2 = 1 2t.

• In the earliest stages of the big bang, densities, velocities, and cross sections are large and it is easy to fulfil this for most species. • However, as T and the density drop the number density and

velocity factors will decrease steadily and at certain reaction thresholds the cross section σ will become small for a partic-ular species and it can drop out of thermal equilibrium.

Physical reason: if the reaction rates are slow compared with the rate of expansion, it is unlikely that the particles can find each other to react and maintain equilibrium.

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e+ e− Z0 e + e− Z0 _Z0 µ+ µ− ν ν _ _ L L _ q q

Figure 20.3: Some weak interactions important for maintaining equilibrium in the

early Universe. Generic leptons are represented by L and generic quarks by q.

20.2.3 Example: Decoupling of the Weak Interactions

As an example of decoupling from thermal equilibrium, let us con-sider weak interactions in the early Universe.

• At the energies of primary interest to us the weak interactions go quadratically in the temperature.

• Thus, shortly after the big bang the weak interactions are not par-ticularly weak and particles such as neutrinos are kept in thermal equilibrium by reactions likeνν¯ _↔e+e−.

• Some typical Feynman diagrams are illustrated in Fig. 20.3. The weak interaction cross sections depend on the square of the weak (Fermi) coupling constant,σw∝ G2_F, with

GF≃1.17×10−5 GeV−2.

• This may be used to show (Exercise) that the ratio of the weak reaction rate to the expansion rate is

Γ H ≃ G2_FT5 T2/MP ≃ T 1 MeV 3 .

Therefore, weak interactions should have decoupled from thermal equilibrium at a temperature of approximately 1 MeV, which occurred about 1 second after the expansion began.

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10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 105 y 1010 y 10-13 1019 1014 102 100 10-3 10-5 10-9 Temperature (GeV)

Time Since Big Bang

Quantum Gravity ? ? GUTs Inflation SU(3)_c x U(1)_y SU(2)_w x SU(3)_c x U(1)_em Hadrons Leptons Nuclear Synthesis _Photon Epoch Galaxies Stars Life Quark-Lepton Soup Planck time

Guts symmetry breaking

Electroweak symmetry breaking

Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Figure 20.4: A history of the Universe. The time axis is highly nonlinear and

1 GeV_≃1.2_×1013 K (after D. Schramm).

20.2.4 Sequence of Events in the Big Bang

The Friedmann equations and considerations of the fundamental prop-erties of matter allow us to reconstruct the big bang. Let us now fol-low the approximate sequence of events that took place in terms of the time since the expansion begins (see Fig. 20.4 for an overview). The primary cast of characters includes:

1. Photons

2. Protons and neutrons 3. Electrons and positrons 4. Neutrinos and antineutrinos

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Because of the equivalence of mass and energy, in a radi-ation dominated era

• the particles and their antiparticles are continuously undergoing reactions in which they annihilate each other, and

• photons can collide and create particle and antiparti-cle pairs.

Thus, under these conditions the radiation and the matter are in thermal equilibrium because they can freely inter-convert.

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Time Since Big Bang

Confinement Weak freezeout Nuclear Freezeout E & M Freezeout Now Time_∼_∼_∼1/100 Second • T _≃1011 Kandρ >109 g cm−3.

• The Universe is expanding rapidly and consists of a hot undif-ferentiated soup of matter and radiation in thermal equilibrium with an average particle energy of kT _≃8.6MeV.

• Equilibria:

e−+e+ _↔ photons ν+ν¯ _↔ photons

¯

ν+p+_→e++n ν+n_→e−+p+.

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Time Since Big Bang

Confinement Weak freezeout Nuclear Freezeout E & M Freezeout Now Time_∼_∼_∼1/10 Second • T _≃1010 Kandρ _≃107 g cm−3.

• Free neutron (mnc2=939 MeV) less stable than free proton (mpc2=

938 MeV), so,n_→ p++e−+ν¯_,witht₁_/₂ _∼17 m.

• Thus, the initial _∼ equal balance between neutrons and protons begins to be tipped in favor of protons.

• By now 62% of the nucleons are protons and 38% are neutrons. • The free neutron is unstable, but neutrons in composite nuclei can be stable, so the decay of neutrons will continue until the simplest nucleus (deuterium) can form.

• No composite nuclei can form yet because the temperature im-plies an average energy for particles in the gas of about 2.6 MeV, and deuterium has a binding energy of only 2.2 MeV(deuterium bottleneck).

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Time Since Big Bang

Confinement Weak freezeout Nuclear Freezeout E & M Freezeout Now Time_∼_∼_∼1 Second • T _≃1010 Kandρ _≃4_×105 g cm−3.

• kT _≃0.8 MeVand the neutrinos cease to play a role in the con-tinuing evolution (weak freezeout).

• The deuterium bottleneck still exists, so there are no composite nuclei and the neutrons continue to beta decay to protons.

• At this stage the proton abundance is up to 76% and the neutron abundance has fallen to 24%.

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Time Since Big Bang

Confinement Weak freezeout Nuclear Freezeout E & M Freezeout Now Time_∼_∼_∼14 Seconds

• The temperature has now fallen to about3_×109 K, correspond-ing to an average energy for the gas particles of about0.25 MeV.

• This is too low for photons to produce electron–positron pairs, so they fall out of thermal equilibrium and the free electrons begin to annihilate all the positrons to form photons.

e−+e+_→ photons.

• This reheats all particles in thermal equilibrium with the pho-tons, but not the neutrinos which have already dropped out of thermal equilibrium att _∼1 s.

• The deuterium bottleneck still keeps appreciable deuterium from forming and the neutrons continue to decay to protons.

• At this stage the abundance of neutrons has fallen to about 13% and the abundance of protons has risen to about 87%.

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Time Since Big Bang

Quantum Gravity ? ? GUTs Inflation SU(3)_c x U(1)_y SU(2)_w x SU(3)_c x U(1)_em Hadrons Leptons Nuclear Synthesis _Photon Epoch Galaxies Stars Life Quark-Lepton Soup Guts symmetry breaking

Confinement

Weak freezeout

Nuclear Freezeout

E & M Freezeout

Now

Time_∼_∼_∼3 Min 45 Seconds

• Finally the temperature drops sufficiently low (about109 K) that deuterium nuclei can hold together.

• The deuterium bottleneck is thus broken and a rapid sequence of nuclear reactions ensues

n+p+_→2₁H

2

1H+p+→32He+n→42He 2

1H+n→31H+p+→42He

• Thus, all remaining free neutrons are rapidly cooked into helium. • Elements beyond 4He cannot be formed in abundance because of the peculiarity that there are no stable mass-5 or mass-8 iso-topes, and because the density has dropped too low to permit more complicated reactions like triple-α to produce carbon.

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Time Since Big Bang

Confinement Weak freezeout Nuclear Freezeout E & M Freezeout Now Time_∼_∼_∼35 Minutes

• The temperature is now about3_×108 K.

• the Universe consists primarily of protons, the excess electrons that did not annihilate with the positrons, 4He (26% abundance by mass), photons, neutrinos, and antineutrinos.

• There are no atoms yet because the temperature is still too high for the protons and electrons to bind together.

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Time Since Big Bang

Confinement Weak freezeout Nuclear Freezeout E & M Freezeout Now Time_∼_∼_∼400,000 Years

• The temperature has fallen to several thousand K, which is suffi-ciently low that electrons and protons can hold together to begin forming hydrogen atoms.

• Until this point, matter and radiation have been in thermal equi-librium but now they decouple.

• As the free electrons are bound up in atoms the primary cross section leading to the scattering of photons (interaction with the free electrons) is removed.

• The Universe, which has been very opaque until this point, be-comes transparent: light can now travel large distances before being absorbed.

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20.3 Element Production and the Early Universe

Deuterium serves as a bottleneck until a critical temperature is reached and then is quickly converted into helium, which is very stable.

• Therefore, the present abundances of helium and deuterium (and other light elements like lithium that are produced by the big bang in trace abundances) are a sensitive probe of conditions in the first few seconds of the Universe.

• The oldest stars contain material that is the least altered from that produced originally in the big bang.

• Analysis of their composition indicates elemental abundances that are in very good agreement with the predictions of the hot big bang.

This is one of the strongest pieces of evidence in support of the big bang theory.

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Table 20.1: Neutron to proton ratio in the big bang

Time (s) T (K) nn/np npper 1000 nnper 1000

nucleons nucleons 2.3_×10−8 1_×1014 1.000 500 500 2.3_×10−4 1_×1012 0.985 504 496 2.3_×10−2 1_×1011 0.861 537 463 2.3 1_×1010 0.223 818 182 6.9 5_×109 0.221 819 181 37 2.5_×109 0.212 825 175 231 1×109 0.164 859 141

20.3.1 The Neutron to Proton Ratio

Nucleosynthesis in the first few minutes of the big bang depends crit-ically on the ratio of neutrons to protons (Table 20.1).

• The neutron is 0.14% more massive than the proton. This favors conversion of neutrons to protons by weak interactions.

• At very high temperatures the mass difference doesn’t matter much and the ratio of neutrons to protons is about one. However, as the temperature drops neutrons are converted to protons and the ratio begins to favor protons.

• All neutrons would be converted to protons if the neutrons and protons remained free long enough (a few hours onceT <1010 K), but neutrons bound up in a stable nucleus like4He or deuterium, are no longer susceptible to being converted to a proton.

• Therefore, as we have seen the neutron to proton ratio drops as the temperature drops until deuterium can hold together and the neutrons can be bound up in stable nuclei.

• This happens at a temperature of about 109 K, by which time (preceding table) the neutron to proton ratio is about 16%.

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20.3.2 The Production of 4He

Except for generating very small concentrations of3He, 7Li, and deu-terium, the essential result of big bang nucleosynthesis is to convert the initial neutrons and protons to helium and free hydrogen.

• From the preceding table we may estimate how much of each is produced.

• For example, if we assume that as soon as the deuterium bottle-neck is broken (at about T =1_×109 K) as many free protons and neutrons as possible combine to make4He, the table entries may be used to deduce that the baryonic matter of the Universe should be about 28% 4He by mass, with most of the rest hydro-gen (Exercise).

• Considering the simplicity of our estimate, that is rather close to the 22–24% measured abundance for4He.

More careful considerations than the ones used here give even better agreement with the observations.

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20.3.3 Constraints on Baryon Density

This agreement between theory and observation for light-element abun-dances also constrains the total amount of mass in the Universe that can be in baryons.

• That constraint is the basis for our earlier assertion that most of the dark matter dominating the mass of the Universe cannot be ordinary baryonic matter.

• If enough baryons were present in the Universe to make that true, and our understanding of the big bang is anywhere near correct, the distribution of light element abundances would have to differ substantially from what is observed.

• The implication is that the matter that we are made of (baryonic matter) is but a small impurity compared to the dominant matter in the universe (nonbaryonic matter).

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Mass Abundance (% ) 22 23 24 25 η 10-4 10-5 10-9 10-10 10-10 ₁₀-9 4_He 3_He _d 3He + d 7_Li

Figure 20.5:Mass abundances for some light isotopes relative to normal hydrogen

as a function of the baryon to photon ratio η. Shaded regions are excluded by observations and the curves are predicted primordial abundances.

Figure 20.5 compares calculated with observed abundances for the light elements produced mostly in the big bang (d is deuterium).

• The shaded regions are excluded by observations.

• Example: observations indicate that the abundance of4He in the Universe can be no more than 24% and no less than 22%.

• Therefore, only the part of the4He curve lying in the unshaded region is consistent with the observed amount of 4He.

• Such considerations allow us to fix with considerable confidence the quantity on the horizontal axis, which is the ratio of the num-ber of baryons to numnum-ber of photons in the present Universe.

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Mass Abundance (% ) 22 23 24 25 η 10-4 10-5 10-9 10-10 10-10 ₁₀-9 4_He 3_He _d 3He + d 7_Li

• The total number of each kind of particle is not expected to change in the absence of interactions, so this ratio is also charac-teristic of that at the time when matter and radiation decoupled. • The only values permitted for the baryon to photon ratio by the

observed abundances of the light nuclei included in the plot lie in a band that brackets the four vertical dotted lines.

• There are _∼3-4 billion photons for every baryon in the present Universe (but their equivalent mass is _∼10,000 times less than the total mass in visible and dark-matter massive particles). • There are_∼4_×108photons in each cubic meter of the Universe,

but only about one baryon for every five cubic meters of space. 1. Most of these baryons are neutrons and protons.

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20.3.4 Constraints on Number of Neutrino Families

One of the successes of the hot big bang theory is that the observed abundance of light elements, coupled with the theoretical understand-ing of big bang nucleosynthesis, tells us somethunderstand-ing about neutrinos.

• The known neutrinos come in three families.

• This number of families is favored in the simplest elementary particle theories, but in principle there could be additional fami-lies that are not yet discovered.

• However, the successful predictions of big bang nucleosynthesis require that there be no more than four such families total. • High-energy particle physics experiments have now found more

directly that (with certain technical theoretical assumptions) the number of neutrino families with standard electroweak couplings is three, confirming the limit placed by big bang nucleosynthe-sis.

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20.4 The Cosmic Microwave Background

There are two important observables in the present Universe that are presumably remnants of the big bang:

• The cosmic microwave background radiation • Dark matter

The cosmic microwave background (CMB) is the faint glow left over from the big bang itself. It was discovered accidentally by Penzias and Wilson in 1964 while testing a new microwave antenna.

• They initially believed the signal that they detected coming from all directions to be electronic noise.

• Once careful experiments had ruled that possibility out, they were initially unaware of the significance of their discovery. • Then it was pointed out that the big bang theory actually

pre-dicted that the Universe should be permeated by radiation left over from the big bang itself, but now redshifted by the expan-sion over some 14 billion years to the microwave spectrum. Dark matter appears to represent the major part of the mass in the Universe, but we don’t yet know what it is.

Both the microwave background and the nature of dark matter provide crucial diagnostics for a fundamental issue in cosmology, theformation of structurein the Universe

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2.726 K microwave spectrum (theory and

COBE data agree)

cm-1 Intensity (10 -4 ergs/cm 2 sr sec cm -1) 0 5 10 15 20 25 0.2 0.0 0.4 0.6 0.8 1.0 1.2

Figure 20.6:The 2.726 K microwave background spectrum recorded by COBE.

20.5 The Microwave Background Spectrum

Measurements by Penzias and Wilson that are relatively crude by modern standards established that

• The radiation was coming from all directions in the sky, with a blackbody spectrum corresponding to T =2.7 K.

• More modern measurements using the Cosmic Background Ex-plorer (COBE) satellite confirm an almost perfect blackbody spectrum, with a temperature of2.726 K, as illustrated in Fig. 20.6. • The data points and the theoretical curve for a2.726 Kspectrum

are indistinguishable.

• This is, by far, the best blackbody spectrum that has ever been measured.

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2.726 K microwave spectrum (theory and

COBE data agree)

cm-1 Intensity (10 -4 ergs/cm 2 sr sec cm -1) 0 5 10 15 20 25 0.2 0.0 0.4 0.6 0.8 1.0 1.2

• By applying basic statistical mechanics to the observed spec-trum, we may deduce a photon density of

N_γ _≃410 photons cm−3

in the cosmic microwave background.

• Theory predicts that there is also a cosmic neutrino background left over from the big bang, but these low-energy neutrinos are not detectable with current technology.

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Frequency Expansion decreases number density of photons Expansion redshifts the photons Near Decoupling Today Intensity

Figure 20.7: Schematic evolution of the cosmic microwave background. As the

Universe expands, the spectrum remains blackbody but the photon frequencies are redshifted and the number density of photons is lowered. The 2.7 K cosmic back-ground radiation is the faint, redshifted remnant of the cosmic fireball in which the Universe was created. Decoupling occurred at a redshift around 1000 (see Fig. 20.8). The photon temperature then of about 3000 K is lowered by the redshift factor of 1000 to the presently observed value of a little less than 3 K.

The CMB is the remnants of the big bang itself, redshifted into the microwave spectrum by the expansion of the Uni-verse, as illustrated in Fig. 20.7.

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z = infinite z =1000 Universe Opaque Earth Universe transparent Observable Universe Last scattering surface ~ 9000 Mp c

Figure 20.8:Last scattering surface for the CMB.

• The photons detected in the CMB by modern measurements cor-respond to photons emitted from thelast scattering surface illus-trated in Fig. 20.8.

• The last scattering surface lies at a redshiftz _∼1000 and repre-sents the time when the photons of the present CMB decoupled from the matter (roughly 400,000 years after the big bang). • At earlier redshifts the Universe becomes opaque to photons,

because that represents a time early enough in the history of the Universe when matter and radiation were strongly coupled.

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COBE

WMAP

Figure 20.9:The COBE and WMAP microwave maps of the sky.

20.6 Anisotropies in the Microwave Background

COBE and WMAP measured angular distribution of CMB (Fig. 20.9).

• Isotropic down to a dipole anisotropy at the 10−3 level corre-sponding to a Doppler shift associated with motion of the Earth relative to the microwave background.

• Once the peculiar motion of the Earth with respect to the CMB is subtracted, the background is isotropic down to the 10−5level. • COBE measured an anisotropy that corresponds to

δT

T =1.1×10

−5_.

Even more precise measurements of the CMB anisotropies have been made by WMAP.

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Open

Flat Closed

Figure 20.10: Influence of spacetime curvature on WMAP microwave

fluctua-tions.

20.7 Precision Measurement of Cosmology Parameters

The WMAP observations in particular have begun to yield precise constraints on the value of important cosmological parameters.

• This is because the detailed pattern of CMB fluctuations is ex-tremely sensitive to many cosmological parameters.

• For example, Fig. 20.10 illustrates schematically that lensing ef-fects on the CMB distort it in a way that depends on the overall curvature of the Universe.

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Multipole Moment (L) 0 10 40 100 200 400 800 1400 -1 0 1 2 3 0 1000 2000 3000 4000 5000 6000 90 o ₂o _0.5o _0.2o Angular Scale L ( L +1 ) CL /2 π ( µ K 2) ( L +1 ) CL /2 π ( µ K 2) W M A P A C B A R C B I Reionizaton TE Cross Power Spectrum TT Cross Power Spectrum

Figure 20.11: (Top) Angular power spectrum of temperature fluctuations in the

cosmic microwave background radiation. (Bottom) Cross-power spectrum of cor-relation between the cosmic microwave background temperature fluctuation and the polarization.

Fig. 20.11 illustrates the power spectrum of CMB fluctuations.

• Multipole moments on x axis correspond to angular decompo-sition of the CMB pattern in terms of spherical harmonics of different orders.

• Roughly speaking, a multipole moment is sensitive to an angular region (in radians) equal to one over the multipole order.

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Multipole Moment (L) 0 10 40 100 200 400 800 1400 -1 0 1 2 3 0 1000 2000 3000 4000 5000 6000 90 o ₂o _0.5o _0.2o Angular Scale L ( L +1 ) CL /2 π ( µ K 2) ( L +1 ) CL /2 π ( µ K 2) W M A P A C B A R C B I Reionizaton TE Cross Power Spectrum TT Cross Power Spectrum

• Thus, the low multiples in the above figure carry information about the CMB on large angular scales and the higher multipole components carry information in increasingly smaller angular scales.

• Detailed fits to such power spectra using cosmological theories place strong constraints on those theories, and permit cosmolog-ical parameters to be determined with high precision.

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Table 20.2:Cosmological parameters.

Parameter Symbol Value

– Global Parameters (10) –

Hubble parameter† h 0.72_±0.07

Deceleration parameter q0 −0.67±0.25 Age of the universe t0 13±1.5 Gyr CMB temperature T0 2.725±0.001 K

Density parameter Ω 1.03_±0.03

Baryon density ΩB 0.039±0.008

Cold dark matter density ΩCDM 0.29±0.04 Massive neutrino density Ων 0.001–0.05 Dark energy density Ωv 0.67±0.06 Dark energy equation of state w ₋1_±0.2

– Fluctuation Parameters (6) –

Density perturbation amplitude √S 5.6+₋1₁_..5₀_×10−6

Gravity wave amplitude √T <√S

Mass fluctuations on 8 Mpc σ8 0.9±0.1

Scalar index n 1.05_±0.09

Tensor index nT

Running of scalar index dn/d(ln k) ₋0.02_±0.04 †_H

0=100h km s−1Mpc−1

Some values of cosmological parameters extracted from WMAP data are displayed in Table 20.2. The precision with which cosmological parameters are now being de-termined from WMAP and from high-redshift supernovae is unprecedented and is rapidly turning cosmology into a quantitative science constrained by precise data.

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20.8 Seeds for Structure Formation

The fluctuations in the CMB presumably reflect conditions when mat-ter and radiation decoupled, and presumably reflect the initial density perturbations that were responsible for the formation of structure in the Universe.

• If the CMB were perfectly smooth, it would be difficult to un-derstand how structure could have formed.

• Fluctuations at this level at least make it possible to consider theories for structure formation, though such theories have not been very successful yet in correlating both the observed visible matter and the microwave background.

• As we shall see in Ch. 21, a period of exponential growth in the scale factor of the early Universe called cosmic inflation may have been central to producing these density fluctuations.

• Dark matter may have played an important role in the initiation of structure formation.

1. Because dark matter does not couple strongly to photons, it could begin to clump together earlier than the normal mat-ter.

2. Because there is so much more dark matter than normal matter, it could clump more effectively.

• Thus, it is likely that dark matter provided the initial regions of higher than average density that seeded the early formation of structure in the Universe.

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20.9 Summary: Dark Matter, Dark Energy, and Structure

Let us conclude this chapter by summarizing present understanding of dark matter, dark energy, and the formation of structure.

• If inflation were correct (see Ch. 21) and the cosmological con-stant were zero, the matter density of the Universe would be exactly the closure density, which would lead to flat geometry. • Current data indicate that the Universe is indeed flat, as predicted

by inflation, but that it does not contain a closure density of mat-ter because there is a non-zero cosmological constant.

1. Instead, about 30% of the closure density is supplied by matter and about 70% by dark energy (vacuum energy or a cosmological constant).

2. Luminous matter contributes a small fraction of the closure density, implying that the vast majority of the mass density is dark matter.

3. Thus, the present Universe is dominated by dark matter and dark energy.

• The known neutrinos are relativistic (that is, they are hot dark matter) and therefore they erase fluctuations on small scales.

1. They could aid the formation of large structures like super-clusters but not smaller structures like galaxies.

2. Thus, they are not likely to account for more than a small fraction of the dark matter.

3. WMAP indicates that light neutrinos contribute less than 2% of the total energy density at decoupling.

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• On the scale of galaxies and clusters of galaxies, 90% of the total mass is not seen.

1. In this case, a significant fraction of the dark matter could be normal (that is, baryonic) and be in the form of small, very low luminosity objects like white dwarfs, neutron stars, black holes, brown dwarfs, or red dwarfs.

2. However, microlensing observations and searches for sub-luminous objects generally have not found enough of these “normal” objects to account for the mass of galaxy halos.

• Data indicate a small mass for neutrinos, but not one large enough to dominate the mass density of the Universe.

• Further, strong constraints from big bang nucleosynthesis com-pared with the observed abundances of the light elements indi-cate that most of the dark matter is not baryonic.

1. Thus, a significant fraction of the dark matter is likely to be nonbaryonic and not neutrinos, and to be cold (that is, massive so that it does not normally travel at relativistic ve-locities).

2. Current speculation centers on not yet discovered elemen-tary particles as the candidates for this cold dark matter.

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• Large-scale structure and its rapid formation in the early Uni-verse is hard to understand, given the smallness of the cosmic microwave background fluctuations implied by COBE and WMAP, unless cold dark matter plays a central role in seeding initial structure formation.

• The models of structure formation most consistent with current data are probably the class ofΛCDM modelsthat combine a cos-mological constant (denoted byΛ) with cold dark matter(CDM) to give an accelerating but flat universe with cold dark matter to seed structure formation.

• As a bonus, the finite cosmological constant (with associated acceleration of the cosmic expansion) that is implicit in these models also makes the age of the Universe greater than we would estimate otherwise, which may help erase with any remaining discrepancies between the age of the Universe and the age of its oldest stars.

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These observations taken together appear to justify several general statements.

• First, the Universe is flat and is presently dominated by

1. dark energy (finite cosmological constant) 2. dark matter.

This strongly favors the validity of theinflationary hypothesis. • Second, cold dark matter probably was central to the formation

of structure.

• Third, most of the dark matter is probably not “ordinary matter” (not baryonic).

• Thus, the growing evidence is that we live in a Universe domi-nated by dark energy and (non-baryonic) dark matter.

1. We have as yet no strong clues as to the source and detailed nature of either because neither has been captured in a lab-oratory.

2. At present, we know about dark matter and dark energy only from observations on galactic and larger scales in the cos-mos.