Chapter 20 The Big Bang

Chapter 20 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 the big bang model, which is the cosmologist s standard model for the origin of the present Universe. 595

596 CHAPTER 20. THE BIG BANG 20.1 Radiation and Matter Dominated Universes Because the influence of vacuum energy grows with epansion 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 etremes for the equation of state give us considerable insight into the early history of the Universe: 1. If the energy density resides primarily in light particles having relativistic velocities, we say the the Universe is radiation dominated; in that case the equation of state is P = 1 3 ε (radiation dominated). 2. If on the other hand the energy density is dominated by massive, slow-moving particles, we say the the Universe is matter dominated; the corresponding equation of state is P 0 (matter dominated). In either etreme, the evolution of the Universe is then easily calculated using the Friedmann equations.

20.1. RADIATION AND MATTER DOMINATED UNIVERSES 597 20.1.1 Evolution of the Scale Factor The density of radiation and the density of matter scale differently in an epanding universe. If the Universe is radiation dominated, P = 1 3 ε and ε + 3(ε + P)ȧa = 0 ε ε + 4ȧ a = 0, which has a solution ε(t) 1 a 4 (t) (radiation dominated). If on the other hand the Universe is matter dominated, we have P 0 and ε + 3(ε + P)ȧa = 0 ε ε + 3ȧ a = 0, which has a solution ε(t) 1 a 3 (t) (matter dominated). As we showed in the previous chapter, the corresponding behaviors of the scale factor are t 1/2 (radiation dominated) a(t) t 2/3 (matter dominated)

598 CHAPTER 20. THE BIG BANG 20.1.2 Matter and Radiation Density In the present Universe the ratio of the number density of baryons to photons is n b n γ 10 9. However, the rest mass of a typical baryon is approimately 10 9 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 ( ) 1 GeV E γ (2.7 K) 1.2 10 13 2.3 10 4 ev. K Thus the ratio of the energy density of baryons to energy density of photons in the present Universe is ε b ε γ 10 3 10 4 and the present Universe is dominated by matter (and by vacuum energy), with only a small contribution from radiation. But, ε(t) 1 a 4 (t) ε(t) 1 a 3 (t) (radiation dominated). (matter dominated). Thus, as time is etrapolated backwards relativistic matter becomes increasingly more important, until at some earlier time the influence of matter and vacuum energy may be neglected compared with that of relativistic particles.

20.1. RADIATION AND MATTER DOMINATED UNIVERSES 599 10 0 ρradiation Matter Dominated log ρ (g cm -3 ) -10-20 t ~ 10 10 s T ~ 10 ev ρmatter -30 Radiation Dominated -40-10 -8-6 -4-2 0 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 been radiation dominated.

600 CHAPTER 20. THE BIG BANG 10 0 ρradiation Matter Dominated log ρ (g cm -3 ) -10-20 t ~ 10 10 s T ~ 10 ev ρmatter -30 Radiation Dominated -40-10 -8-6 -4-2 0 2 log a Since in this early, radiation-dominated Universe, ε a 4 a t 1/2 P = 1 3 ε, the behavior of the density and pressure as time is etrapolated backwards is Lim t 0 ε(t) = Lim t 0 (t 1/2 ) 4 = Lim t 0 t 2 = Lim P(t) = Lim (t 1/2 ) 4 = Lim t 2 = t 0 t 0 t 0 Furthermore, for a radiation dominated Universe, T a 1 t 1/2 ; thus, as time is etrapolated backwards the temperature scales as Lim t 0 T(t) = Lim t 0 t 1/2 =.

20.1. RADIATION AND MATTER DOMINATED UNIVERSES 601 These considerations suggest that the Universe started from a very hot, very dense initial state with a(t 0) 0. The commencement from this initial state is called the big bang. If we take t = 0 when a = 0, the transition between 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 theory. The name stuck, as did the theory.

602 CHAPTER 20. THE BIG BANG 20.2 Evolution of the Early Universe Considerations of the preceding section suggest that the big bang starts off with a state of etremely high density and pressure for the Universe, and that under those conditions the Universe is dominated by radiation. This means that the major portion of the energy density 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 epansion and the average velocity 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.

20.2. EVOLUTION OF THE EARLY UNIVERSE 603 20.2.1 Thermodynamics of the Big Bang We have already established that in the initial radiation dominated era of the big bang, (ȧ ) 2 H 2 8πG a 3 ε r = a 4 a t 1/2 H = ȧ a = 1 2t. We may assume that the average evolution corresponds approimately to that of an ideal gas in thermal equilibrium, for which the number density of a particular species is dn = g p 2 dp 2π 2 h 3 e E/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 = +1 Fermions p 2 c 2 + m 2 c 4 Θ = -1 Bosons 0 Mawell Boltzmann where Mawell Boltzmann statistics obtain only if we make no distinction between fermions or bosons in the gas.

604 CHAPTER 20. THE BIG BANG Because of the high temperature, let us assume that the gas is ultrarelativistic (kt >> mc 2 for the particles in the plasma). Then the energy is E = pc, and the number density is obtained by integrating the previous epression for dn. n = 0 dn dp dp = g 2π 2 h 3 0 = g 2π 2 h 3 0 p 2 dp e E/kT + Θ p 2 dp e pc/kt + Θ (20.1) Integrals of this form may be evaluated using 0 t z 1 e t dt = (z 1)!ζ(z) 1 0 t z 1 e t + 1 dt = (1 21 z )(z 1)!ζ(z), where ζ(z) is the Riemann zeta function, with tabulated values ζ(2) = π2 6 = 1.645 ζ(3) = 1.202 ζ(4) = π4 90 = 1.082. The results for the number density of species i are ζ(3) 1 Bose Einstein n i = g i π 2 T 3 3/4 Fermi Dirac ζ(3) 1 Mawell Boltzmann where now we introduce h = c = k = 1 units.

20.2. EVOLUTION OF THE EARLY UNIVERSE 605 Likewise, the energy density is given by ε = ρc 2 = 0 E dn dp dp = g 2π 2 h 3 = g 2π 2 h 3 0 0 E p 2 dp e E/kT + Θ E p 2 dp e pc/kt + Θ which gives π 2 1 Bose Einstein ε i = g i 30 T 4 7/8 Fermi Dirac 90/π 4 Mawell Boltzmann (20.2) The energy density for all relativistic particles is then given by the sum, π 2 ε = g 30 T 4 g g b + 8 7 g f. bosons fermions If all species are in equilibrium, the entropy density s is s = ε + P T = 4ε 3T = 2π2 45 g T 3, where we note from comparing this and ζ(3) 1 Bose Einstein n i = g i π 2 T 3 3/4 Fermi Dirac ζ(3) 1 Mawell Boltzmann that s n i. The entropy per comoving volume is constant (adiabatic epansion), S sa 3 constant provided that g does not change. d(sa 3 ) dt = 0,

606 CHAPTER 20. THE BIG BANG Effective Degrees of Freedom g* 100 10 Quark Confinement ~10-6 s ~1 s Weak Freezout 1 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 10-5 T (GeV) 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 epect g to be approimately constant for broad ranges of temperature but to change suddenly at critical temperatures where kt becomes 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 radiation. Thus, cosmologically the epansion is approimately reversible and adiabatic.

20.2. EVOLUTION OF THE EARLY UNIVERSE 607 From s = ε + P T = 4ε 3T = 2π2 45 g T 3, S sa 3 constant d(sa 3 ) dt = 0, sa 3 T 3 a 3 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 t 1/2. To summarize, the evolution of the ultrarelativistic, hot plasma characterizing the early big bang is described by the equations ȧ a = Ṫ T = αt 2 t = 1 2.4 10 6 = 2αT 2 g 1/2 GeV 2 s T 2 ( 4π 3 ) 1/2 ( ) g 1/2 hc α = 45M 2 M P = 1.2 10 19 GeV, P G where M P is the Planck mass. These equations are epected to be valid from the end of the quantum gravitation era at T 10 19 GeV up to the decoupling of matter and radiation at T 10 ev.

608 CHAPTER 20. THE BIG BANG 20.2.2 Equilibrium in an Epanding Universe Strictly, we do not epect equilibrium to hold in an epanding universe. However, a practical equilibrium can eist as the Universe passes through a series of nearly equilibrated states. We may epect both thermal equilibrium and chemical equilibrium to play a role in the epansion of the Universe. A system is in thermal equilibrium if its phase space number density is given by dn = g p 2 dp 2π 2 h 3 e E/kT + Θ, E = +1 Fermions p 2 c 2 + m 2 c 4 Θ = -1 Bosons 0 Mawell Boltzmann A system is in chemical equilibrium if for the reaction a+ b c+d the chemical potentials satisfy µ a + µ b = µ c + µ d. We shall illustrate the discussion by considering thermal equilibrium, and will consider the equilibrium to maintained by two-body reactions (which is the most common situation).

20.2. EVOLUTION OF THE EARLY UNIVERSE 609 The reaction rate for a two-body reaction may be epressed as Γ nvσ, where n is the number density, v is the relative speed, σ is the reaction cross section, and the brackets indicate a thermal average. We may epect that a species will remain in thermal equilibrium in the radiation dominated Universe as long as Γ >> ȧ a H d(t1/2 )/dt t 1/2 = 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 particular species and it can drop out of thermal equilibrium. Physical reason: if the reaction rates are slow compared with the rate of epansion, it is unlikely that the particles can find each other to react and maintain equilibrium.

610 CHAPTER 20. THE BIG BANG e + µ + Z 0 e + _ ν L Z 0 Z 0 q e µ e ν _ L _ 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 Eample: Decoupling of the Weak Interactions As an eample of decoupling from thermal equilibrium, let us consider 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 particularly weak and particles such as neutrinos are kept in thermal equilibrium by reactions like ν ν 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, σ w G 2 F, with G F 1.17 10 5 GeV 2. This may be used to show (Eercise) that the ratio of the weak reaction rate to the epansion rate is Γ H G2 F T 5 ( ) T 3 T 2. /M P 1 MeV Therefore, weak interactions should have decoupled from thermal equilibrium at a temperature of approimately 1 MeV, which occurred about 1 second after the epansion began.

20.2. EVOLUTION OF THE EARLY UNIVERSE 611 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Figure 20.4: A history of the Universe. The time ais is highly nonlinear and 1 GeV 1.2 10 13 K (after D. Schramm). 20.2.4 Sequence of Events in the Big Bang The Friedmann equations and considerations of the fundamental properties of matter allow us to reconstruct the big bang. Let us now follow the approimate sequence of events that took place in terms of the time since the epansion 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

612 CHAPTER 20. THE BIG BANG Because of the equivalence of mass and energy, in a radiation 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 antiparticle pairs. Thus, under these conditions the radiation and the matter are in thermal equilibrium because they can freely interconvert.

20.2. EVOLUTION OF THE EARLY UNIVERSE 613 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Time 1/100 Second T 10 11 K and ρ > 10 9 g cm 3. The Universe is epanding rapidly and consists of a hot undifferentiated soup of matter and radiation in thermal equilibrium with an average particle energy of kt 8.6 MeV. Equilibria: e + e + photons ν + ν photons ν + p + e + + n ν + n e + p +. The number of protons is about equal to the number of neutrons.

614 CHAPTER 20. THE BIG BANG 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Time 1/10 Second T 10 10 K and ρ 10 7 g cm 3. Free neutron (m n c 2 = 939 MeV) less stable than free proton (m p c 2 = 938 MeV), so, n p + + e + ν, with t 1/2 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 implies 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).

20.2. EVOLUTION OF THE EARLY UNIVERSE 615 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Time 1 Second T 10 10 K and ρ 4 10 5 g cm 3. kt 0.8 MeV and the neutrinos cease to play a role in the continuing evolution (weak freezeout). The deuterium bottleneck still eists, 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%.

616 CHAPTER 20. THE BIG BANG 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Time 14 Seconds The temperature has now fallen to about 3 10 9 K, corresponding to an average energy for the gas particles of about 0.25 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 photons, but not the neutrinos which have already dropped out of thermal equilibrium at t 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%.

20.2. EVOLUTION OF THE EARLY UNIVERSE 617 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Time 3 Min 45 Seconds Finally the temperature drops sufficiently low (about 10 9 K) that deuterium nuclei can hold together. The deuterium bottleneck is thus broken and a rapid sequence of nuclear reactions ensues n+ p + 2 1 H 2 1 H+ p+ 3 2 He+n 4 2 He 2 1 H+n 3 1 H+ p+ 4 2 He Thus, all remaining free neutrons are rapidly cooked into helium. Elements beyond 4 He cannot be formed in abundance because of the peculiarity that there are no stable mass-5 or mass-8 isotopes, and because the density has dropped too low to permit more complicated reactions like triple-α to produce carbon.

618 CHAPTER 20. THE BIG BANG 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Time 35 Minutes The temperature is now about 3 10 8 K. the Universe consists primarily of protons, the ecess electrons that did not annihilate with the positrons, 4 He (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.

20.2. EVOLUTION OF THE EARLY UNIVERSE 619 10 19 Planck time Temperature (GeV) 10 14 10 2 10 0 10-3 10-5 10-9 10-13? Quantum Gravity? GUTs Inflation Guts symmetry breaking SU(3) c SU(2) w U(1) y Quark-Lepton Soup Electroweak symmetry breaking SU(3) c U(1) em Confinement Hadrons Leptons Weak freezeout Nuclear Synthesis Nuclear Freezeout Photon Epoch E & M Freezeout Galaies Stars Life Now 10-43 s 10-35 s 10-11 s 10-6 s 1 s 3 min 10 5 y 10 10 y Time Since Big Bang Time 400,000 Years The temperature has fallen to several thousand K, which is sufficiently low that electrons and protons can hold together to begin forming hydrogen atoms. Until this point, matter and radiation have been in thermal equilibrium 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, becomes transparent: light can now travel large distances before being absorbed.

620 CHAPTER 20. THE BIG BANG 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.

20.3. ELEMENT PRODUCTION AND THE EARLY UNIVERSE 621 Table 20.1: Neutron to proton ratio in the big bang Time (s) T (K) n n /n p n p per 1000 n n per 1000 nucleons nucleons 2.3 10 8 1 10 14 1.000 500 500 2.3 10 4 1 10 12 0.985 504 496 2.3 10 2 1 10 11 0.861 537 463 2.3 1 10 10 0.223 818 182 6.9 5 10 9 0.221 819 181 37 2.5 10 9 0.212 825 175 231 1 10 9 0.164 859 141 20.3.1 The Neutron to Proton Ratio Nucleosynthesis in the first few minutes of the big bang depends critically 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 once T < 10 10 K), but neutrons bound up in a stable nucleus like 4 He 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 10 9 K, by which time (preceding table) the neutron to proton ratio is about 16%.

622 CHAPTER 20. THE BIG BANG 20.3.2 The Production of 4 He Ecept for generating very small concentrations of 3 He, 7 Li, and deuterium, 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 eample, if we assume that as soon as the deuterium bottleneck is broken (at about T = 1 10 9 K) as many free protons and neutrons as possible combine to make 4 He, the table entries may be used to deduce that the baryonic matter of the Universe should be about 28% 4 He by mass, with most of the rest hydrogen (Eercise). Considering the simplicity of our estimate, that is rather close to the 22 24% measured abundance for 4 He. More careful considerations than the ones used here give even better agreement with the observations.

20.3. ELEMENT PRODUCTION AND THE EARLY UNIVERSE 623 20.3.3 Constraints on Baryon Density This agreement between theory and observation for light-element abundances 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).

624 CHAPTER 20. THE BIG BANG 25 24 23 4 He 22 Mass Abundance (%) 10-4 10-5 10-9 3 He d 3 He + d 7 Li 10-10 10-10 η 10-9 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 ecluded 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 ecluded by observations. Eample: observations indicate that the abundance of 4 He in the Universe can be no more than 24% and no less than 22%. Therefore, only the part of the 4 He curve lying in the unshaded region is consistent with the observed amount of 4 He. Such considerations allow us to fi with considerable confidence the quantity on the horizontal ais, which is the ratio of the number of baryons to number of photons in the present Universe.

20.3. ELEMENT PRODUCTION AND THE EARLY UNIVERSE 625 25 24 23 4 He 22 Mass Abundance (%) 10-4 10-5 10-9 3 He d 3 He + d 7 Li 10-10 10-10 η 10-9 The total number of each kind of particle is not epected to change in the absence of interactions, so this ratio is also characteristic 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 10 8 photons 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. 2. Most photons are in the cosmic microwave background.

626 CHAPTER 20. THE BIG BANG 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 understanding of big bang nucleosynthesis, tells us something 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 families 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 eperiments 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 nucleosynthesis.

20.4. THE COSMIC MICROWAVE BACKGROUND 627 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 eperiments 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 predicted that the Universe should be permeated by radiation left over from the big bang itself, but now redshifted by the epansion 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, the formation of structure in the Universe

628 CHAPTER 20. THE BIG BANG Intensity (10-4 ergs/cm 2 sr sec cm -1) 1.2 1.0 0.8 0.6 0.4 0.2 2.726 K microwave spectrum (theory and COBE data agree) 0.0 0 5 10 15 20 25 cm -1 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 Eplorer (COBE) satellite confirm an almost perfect blackbody spectrum, with a temperature of 2.726 K, as illustrated in Fig. 20.6. The data points and the theoretical curve for a 2.726 K spectrum are indistinguishable. This is, by far, the best blackbody spectrum that has ever been measured.

20.5. THE MICROWAVE BACKGROUND SPECTRUM 629 Intensity (10-4 ergs/cm 2 sr sec cm -1) 1.2 1.0 0.8 0.6 0.4 0.2 2.726 K microwave spectrum (theory and COBE data agree) 0.0 0 5 10 15 20 25 cm -1 By applying basic statistical mechanics to the observed spectrum, 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.

630 CHAPTER 20. THE BIG BANG Near Decoupling Intensity Epansion decreases number density of photons Today Epansion redshifts the photons Frequency Figure 20.7: Schematic evolution of the cosmic microwave background. As the Universe epands, the spectrum remains blackbody but the photon frequencies are redshifted and the number density of photons is lowered. The 2.7 K cosmic background 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 epansion of the Universe, as illustrated in Fig. 20.7.

20.5. THE MICROWAVE BACKGROUND SPECTRUM 631 Universe Opaque z = infinite Universe transparent Observable Universe Earth z =1000 ~ 9000 Mpc Last scattering surface Figure 20.8: Last scattering surface for the CMB. The photons detected in the CMB by modern measurements correspond to photons emitted from the last scattering surface illustrated in Fig. 20.8. The last scattering surface lies at a redshift z 1000 and represents 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.

632 CHAPTER 20. THE BIG BANG 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 corresponding 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 5 level. 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.

20.7. PRECISION MEASUREMENT OF COSMOLOGY PARAMETERS 633 Open Flat Closed Figure 20.10: Influence of spacetime curvature on WMAP microwave fluctuations. 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 etremely sensitive to many cosmological parameters. For eample, Fig. 20.10 illustrates schematically that lensing effects on the CMB distort it in a way that depends on the overall curvature of the Universe.

634 CHAPTER 20. THE BIG BANG 6000 Angular Scale 90 o 2 o 0.5 o 0.2 o L(L+1)CL/2π (µk 2 ) 5000 4000 3000 2000 1000 TT Cross Power Spectrum WMAP ACBAR CBI 0 (L+1)CL/2π (µk 2 ) 3 2 1 0 Reionizaton TE Cross Power Spectrum -1 0 10 40 100 200 400 800 1400 Multipole Moment (L) Figure 20.11: (Top) Angular power spectrum of temperature fluctuations in the cosmic microwave background radiation. (Bottom) Cross-power spectrum of correlation between the cosmic microwave background temperature fluctuation and the polarization. Fig. 20.11 illustrates the power spectrum of CMB fluctuations. Multipole moments on ais correspond to angular decomposition 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.

20.7. PRECISION MEASUREMENT OF COSMOLOGY PARAMETERS 635 6000 Angular Scale 90 o 2 o 0.5 o 0.2 o L(L+1)CL/2π (µk 2 ) 5000 4000 3000 2000 1000 TT Cross Power Spectrum WMAP ACBAR CBI 0 (L+1)CL/2π (µk 2 ) 3 2 1 0 Reionizaton TE Cross Power Spectrum -1 0 10 40 100 200 400 800 1400 Multipole Moment (L) 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 cosmological parameters to be determined with high precision.

636 CHAPTER 20. THE BIG BANG Table 20.2: Cosmological parameters. Parameter Symbol Value Global Parameters (10) Hubble parameter h 0.72 ± 0.07 Deceleration parameter q 0 0.67 ± 0.25 Age of the universe t 0 13 ± 1.5 Gyr CMB temperature T 0 2.725 ± 0.001 K Density parameter Ω 1.03 ± 0.03 Baryon density Ω B 0.039 ± 0.008 Cold dark matter density Ω CDM 0.29 ± 0.04 Massive neutrino density Ω ν 0.001 0.05 Dark energy density Ω v 0.67 ± 0.06 Dark energy equation of state w 1 ± 0.2 Fluctuation Parameters (6) Density perturbation amplitude S 5.6 +1.5 Gravity wave amplitude T 1.0 10 6 < S Mass fluctuations on 8 Mpc σ 8 0.9 ± 0.1 Scalar inde n 1.05 ± 0.09 Tensor inde Running of scalar inde dn/d(ln k) 0.02 ± 0.04 H 0 = 100h km s 1 Mpc 1 n T Some values of cosmological parameters etracted from WMAP data are displayed in Table 20.2. The precision with which cosmological parameters are now being determined from WMAP and from high-redshift supernovae is unprecedented and is rapidly turning cosmology into a quantitative science constrained by precise data.

20.8. SEEDS FOR STRUCTURE FORMATION 637 20.8 Seeds for Structure Formation The fluctuations in the CMB presumably reflect conditions when matter 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 understand 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 eponential 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 matter. 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.

638 CHAPTER 20. THE BIG BANG 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 constant were zero, the matter density of the Universe would be eactly 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 matter 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 superclusters but not smaller structures like galaies. 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.

20.9. SUMMARY: DARK MATTER, DARK ENERGY, AND STRUCTURE 639 On the scale of galaies and clusters of galaies, 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 subluminous objects generally have not found enough of these normal objects to account for the mass of galay 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 compared with the observed abundances of the light elements indicate 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 velocities). 2. Current speculation centers on not yet discovered elementary particles as the candidates for this cold dark matter.

640 CHAPTER 20. THE BIG BANG Large-scale structure and its rapid formation in the early Universe 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 of ΛCDM models that combine a cosmological constant (denoted by Λ) 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 epansion) 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.

20.9. SUMMARY: DARK MATTER, DARK ENERGY, AND STRUCTURE 641 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 the inflationary 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 dominated 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 laboratory. 2. At present, we know about dark matter and dark energy only from observations on galactic and larger scales in the cosmos.