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Thermal History of the Universe and the Cosmic Microwave Background. I. Thermodynamics of the Hot Big Bang Matthias Bartelmann Max Planck Institut für Astrophysik IMPRS Lecture, March 2003 ◦. Part 1: Thermodynamics of the Hot Big Bang 1. assumptions 6. photons and baryons 2. properties of ideal quantum gases 7. physics of recombination 3. adiabatic expansion of ideal gases 8. nucleosynthesis 4. particle freeze-out 9. the isotropic microwave background 5. neutrino background /◦. 1. Assumptions 1. adiabatic expansion 2. thermal equilibrium 3. ideal gases /◦. 1.1. Adiabatic Expansion The Universe expands adiabatically. • isotropy requires it to expand adiathermally: no heat can flow because any flow would define a preferred direction • an adiathermal expansion is adiabatic if it is reversible, but irreversible processes may occur • the entropy of the Universe is dominated by far by the cosmic microwave background, so entropy generation by irreversible processes is negligible /◦. 1.2. Thermal Equilibrium Thermal equilibrium can be maintained despite the expansion. • thermal equilibrium can only be maintained if the interaction rate of particles is higher than the expansion rate of the Universe • the expansion rate of the Universe is highest at early times, so thermal equilibrium may be difficult to maintain as t → 0 • nonetheless, for t → 0, particle densities grow so fast that interaction rates are indeed higher than the expansion rate • as the Universe expands, particle species drop out of equilibrium /◦. 1.3. Ideal Gases Cosmic “fluids” can be treated as ideal gases. • ideal gas: no long-range interactions between particles, interact only by direct collisions • obviously good approximation for weakly interacting particles like neutrinos • even valid for charged particles because oppositely charged particles shield each other • consequence: internal energy of ideal gas does not depend on volume occupied • cosmic “fluids” can be treated as possibly relativistic quantum gases /◦. 2. Properties of Ideal Quantum Gases 1. occupation number 6. entropy density 2. number density 7. results for bosons and fermions 3. energy density 8. numbers 4. grand canonical potential 5. pressure /◦. 2.1. Occupation Number • in thermal equilibrium with a heat bath of temperature T , the phase-space density for a quantum state of energy ε is −1 ε−µ h f i = exp − ±1 kT (“+” for fermions, “−” for bosons) • in thermal equilibrium with a radiation background, the chemical potential µ = 0: Helmholtz free energy F(T,V, N) = E − T S is minimised in equilibrium for a system at constant T and V , so from dF = −SdT − PdV + µdN: ∂F =0=µ ∂N /◦. 2.2. Number Density • particles in volume V , number of states in k-space element: V 3 dk dN = g (2π)3 • g is statistical weight, e.g. spin degeneracy factor • momentum ~p = ~~k, related to energy by p ε(p) = c2 p2 + m2c4 • spatial number density in thermal equilibrium: g n= (2π~)3 Z 0 ∞ 4πp2dp exp[ε(p)/kT ] ± 1 /◦. 2.3. Energy Density • mean energy density: number of states per phase-space cell, times occupation number, times energy per state, integrated over momentum space: Z ∞ g 4πp2 ε(p) dp u= (2π~)3 0 exp[ε(p)/kT ] ± 1 • integrals can most easily be carried out by substituting a geometrical series: Z 0 ∞ xmdx = x e −1 Z 0 ∞ xm e−x dx 1 − e−x Z = ∞ m −x dxx e 0 for fermions, use ∞ ∑e n=0 −nx ∞ = ∑ Z ∞ dxxme−nx = m!ζ(m + 1) ; n=1 0 1 1 2 = − ex + 1 ex − 1 e2x − 1 /◦. 2.4. Grand Canonical Potential • extremal principles yield: (thermodynamical potential) = −kT ln(partition sum) grand canonical potential Φ(T,V, µ) (particle • adequate potential here: numbers can change) • grand-canonical partition sum: ∞ Z= ∑ eµN/kT ∑ N=0 e− ∑ εαNα/kT {Nα |N} • grand-canonical potential: Φ(T,V, µ) = ∓kT gV (2π~)3 Z 0 ∞ h i 2 µ/kT −ε(p)/kT dp4πp ln 1 ± e e Fermions Bosons /◦. 2.5. Pressure • Helmholtz free energy: F(T,V, N) = U − T S • grand-canonical potential: Φ(T,V, µ) = F − µN = U − T S − µN • thermodynamical Euler relation: • it follows: Φ Φ = −PV ⇒ P = − V • example: relativistic boson gas in thermal equilibrium; ε = cp, µ = 0, π2 (kT )4 PB = g 90 (~c)3 U = T S − PV + µN /◦. 2.6. Entropy Density • Helmholtz free energy: dF(T,V, N) = −SdT − PdV + µdN • example: entropy density for relativistic boson gas in thermal equilibrium 2 • grand-canonical potential: S 2π s = = gk V 45 kT ~c 3 dΦ(T,V, µ) = −SdT − PdV − Ndµ • thus, the entropy is: ∂Φ S=− ∂T /◦. 2.7. Results for Bosons and Fermions n u P s relativistic Bosons Fermions 3 3 gF ζ(3) kT nB gB 2 4 gB π ~c π2 (kT )4 7 gF gB uB 3 30 (~c) 8 gB π2 (kT )4 uB 7 gF gB = PB 90 (~c)3 3 8 gB 3 2 2π kT 7 gF gBk sB 45 ~c 8 gB nonrelativistic 3/2 kT 2 e−kT /mc g 2π~ 3 n kT 2 n kT /◦. 2.8. Numbers note: 1 eV = 1.6 × 10−12 erg correspond to kT = 1.16 × 104 K • • • 3 3 T kT nB = 10gB cm−3 = 1.6 × 1013gB cm−3 K eV 4 4 T erg kT erg −3 uB = 3.8 × 10−15gB = 2.35 × 10 g B K cm3 eV cm3 3 3 sB T kT −3 13 = 36gB cm = 5.7 × 10 gB cm−3 k K eV /◦. 3. Adiabatic Expansion of Ideal Gases 1. temperature 2. density 3. matter-radiation equality /◦. 3.1. Temperature • for relativistic boson or fermion gases: E u P= = 3 3V • first law of thermodynamics, dE + PdV = 0, then implies: dE = −PdV = 3d(PV ) ⇒ P ∝ V −4/3 • since P ∝ T 4 for relativistic gases: T ∝ V −1/3 ∝ a−1 (a is cosmological scale factor) • for non-relativistic ideal gas: T ∝ PV ∝ V −5/3+1 ∝ a−2 (adiabatic index is γ = 4/3) • for non-relativistic ideal gas: γ = 5/3 /◦. 3.2. Density • non-relativistic gases: rest mass energy kinetic energy mass density: ⇒ ρ ∝ V −1 ∝ a−3 • density of relativistic particles drops faster than that of non-relativistic particles: • dilution due to volume expansion, plus energy loss due to redshift • relativistic gases: mass density u ρ = 2 ∝ T 4 ∝ a−4 c /◦. 3.3. Matter-Radiation Equality • today’s matter density in the Universe: 3H02 ρ0 = Ω0 = 1.9 × 10−29 Ω0h2 g cm−3 8πG • today’s radiation density in the Universe: T = 2.73 K ⇒ ρR,0 = u −34 −3 = 4.7 × 10 g cm c2 • radiation domination: −3 ρR(aeq) = ρR,0a−4 = ρ a 0 eq eq = ρ(aeq ) ⇒ aeq = ρR,0 = 2.5 × 10−5 (Ω0h2)−1 ρ0 /◦. 4. Particle Freeze-Out 1. interaction rates 2. freeze-out conditions 3. particle distributions after freeze-out /◦. 4.1. Interaction Rates • expansion timescale: • thermal equilibrium maintained by two-body interactions, collision rate for one particle species is texp ∼ (Gρ)−1/2 ρ is density of dominant fluid • in early Universe, dominates, thus ρ∝a −4 ⇒ texp ∝ a radiation 2 expansion timescale is shortest at early times and increases as a2 Γ ∝ n ∝ T 3 ∝ a−3 • thus, the collision time scale is tcoll ∝ Γ−1 ∝ a3 • for a → 0, tcoll decreases faster than texp, so equilibrium can be maintained /◦. 4.2. Freeze-Out Conditions • continuity equation in absence of collisions: • continuity equation now: ṅ + 3Hn = −Γn + S ṅ + 3Hn = 0 • collision rate: Γ = hσvi n • source term from thermal particle creation: • introduce comoving number density N = a3n: d ln N Γ =− d ln a H NT 1− N • freeze-out can occur if Γ H: particle number N cannot adapt S = hσvi n2T /◦. 4.3. Particle Distributions After Freeze-Out • relativistic particles: n ∝ T 3 ⇒ N = a3n = const. • from freeze-out equation: d ln N = 0 ⇒ N = NT d ln a independent of Γ! • relativistic particles maintain thermal distribution even after freeze-out • non-relativistic particles, comoving number density: 2 mc NT ∝ T −3/2 exp − kT • once T mc2, NT drops rapidly, hence NT N, and d ln N Γ =− →0 d ln a H particle production stops: freezeout /◦. 5. Neutrino Background 1. neutrino decoupling 2. electron-positron annihilation 3. photon heating /◦. 5.1. Neutrino Decoupling • neutrinos are kept in thermal equilibrium by the weak interaction ν + ν̄ ↔ e+ + e− • this interaction freezes out when the temperature drops to Tν ∼ 1010.5 K ∼ 2.7 MeV • neutrinos are ultrarelativistic at that “time”, i.e. their comoving number density is that of an ideal, relativistic fermion gas /◦. 5.2. Electron-Positron Annihilation • the reaction e+ + e− ↔ 2γ is suppressed once temperature drops below • the entropy of the annihilating electron-positron gas heats the photons, but not the neutrinos • the microwave background is heated above the neutrino background by the annihilation T ∼ 2mec2 ≈ 1 MeV ≈ 1010 K • thus, electrons and positrons annihilate shortly after neutrinos freeze out /◦. 5.3. Photon Heating • entropy before annihilation equals entropy after annihilation: • since s ∝ T 3, s0e+ + s0e− + s0γ = sγ • temperature after annihilation: • before annihilation, Te+ = Te− = Tγ ≡ T thus s0e+ = s0e− 7 3 2 · + 1 (T 0) = T 3 8 7 0 = sγ 8 0 T= 11 4 1/3 T 0 ≈ 1.4 T 0 • temperature of ν background today: (ge+ = ge− = gγ = 2) Tν,0 = 1.95 K /◦. 6. Photons and Baryons 1. baryon number 2. photon-baryon ratio /◦. 6.1. Baryon Number • number density of baryons today: ρB ΩB 3H02 = = 1.1 × 10−5 ΩBh2 cm−3 nB ≈ mp mp 8πG • measurements: abundance ratios of light elements, microwave background (see later), gas in galaxy clusters: ΩBh2 ≈ 0.025 • only ∼ 10% − 20% of the matter in the Universe is contributed by baryons • baryon number density ∝ a−3 ∝ T 3 /◦. 6.2. Photon-Baryon Ratio • number density of photons today: nγ = 407 cm−3 • scales with temperature like T 3 like baryon number density, hence their ratio is independent of time • there is one baryon for about a billion photons in the Universe • photon entropy completely dominates entropy of the Universe • very important recombination for process of • constant baryon-photon-ratio: nB η= = 2.7 × 10−8 ΩBh2 nγ /◦. 7. Physics of Recombination 1. recombination process 2. Saha equation 3. recombination temperature 4. two-photon recombination 5. thickness of the recombination shell 6. expectation on radiation spectrum /◦. 7.1. Recombination Process • as the temperature drops, electrons and protons can combine to form atoms: e− + p+ ↔ H + γ freezes out • canonical partition function Z, F = −kT ln Z with N N ZeNe Zp p ZHH Z= Ne! Np! NH! • How does recombination proceed, i.e. how does the electron density change with T ? • baryon number NB = Np + NH, electron number Ne = Np, thus NH = NB − Ne • solution: minimisation of Helmholtz free energy F(T,V, N) • Stirling: ln N! ≈ N ln N − N /◦. 7.2. Saha Equation • from ∂F/∂Ne = 0: • chemical potential: Ne2 ZeZp = NB − Ne ZH µ = 0 ⇒ µe + µp = µH • ionisation potential: • partition function for single species: 4πgV Z= (2π~)3 Z ∞ (me + mp − mH)c2 = χ = 13.6 eV dpp2e(µ−ε)/kT 0 • x = Ne/NB, equation: • nonrelativistic limit: ε = mc2 + 2 p 2m nB = NB/V ; Saha x2 (2πmekT )3/2 −χ/kT = e 3 1−x (2π~) nB /◦. 7.3. Recombination Temperature • nB is the baryon number density; we had • 1/η is huge number, so kT χ required for x to be small 3 • putting x = 0.1 yields kTrec ≈ 0.3 eV, or Trec ≈ 3500 K • insert this into Saha equation: • high photon number density delays recombination considerably nB = ηnγ = η 2 x 0.26 ≈ 1−x η 2ζ(3) π2 2 mec kT kT ~c 3/2 e−χ/kT • want x 1, hence x2/(1 − x) ≈ x2 /◦. 7.3. Recombination Temperature /◦. 7.4. Two-Photon Recombination • direct hydrogen recombination produces energetic photons; last step to ground state is Lyman-α (2P → 1S); 3 hν ≥ ELyα = χ = 10.2 eV 4 • abundant Ly-α photons can reionise the cosmic gas • How can recombination proceed at all? • production of lower-energy photons: forbidden transition 2S → 1S, requires emission of two photons; this process is slow • recombination proceeds at a somewhat slower rate than predicted by the Saha equation • photons cannot be lost as in clouds; energy loss due to cosmic expansion is slow /◦. 7.5. Thickness of Recombination Shell • recombination is not instantaneous; “time” interval between beginning and end? • time: measured in terms of scale factor a, or redshift 1 + z = a−1 • optical depth recombination shell: τ= through Z • scattering probability for photons when travelling from z to z − dz: p(z) dz = e−τ dτ dz dz • probability distribution e−τdτ/dz is well described by Gaussian with mean z̄ ∼ 1100 and standard deviation δz ∼ 80 nexσTdr (σT: Thomson cross section) /◦. 7.5. Thickness of Recombination Shell /◦. 7.6. Expectation on Radiation Spectrum • last-scattering shell has finite width: CMB photons received today were released at different redshifts • cosmic plasma cooled during recombination, photons were released at different mean temperatures: T = T0 (1 + z), thus δT = T0δz ≈ 200 K • photons were redshifted after release: those released earlier, i.e. from hotter plasma, were redshifted by larger amount: E = E0 (1 + z) • these effects cancel if plasma temperature depends on scale factor like T ∝ a−1; then: Planck spectrum of single temperature expected /◦. 8. Nucleosynthesis 1. formation of the light elements 2. Gamow criterion 3. prediction of the microwave background temperature /◦. 8.1. Formation of the Light Elements • high temperature in the early Universe allows nuclear reactions like in stars; density is much lower, so higher temperatures are required: Tnuc ∼ 109 K • before: neutrons and protons formed; equilibrium through p + e− ↔ n + ν , p + ν̄ ↔ n + e+ • relative abundance freezes out once weak interaction becomes too slow; Tfreeze−out ∼ 1.4 × 1010 K afterwards, free neutrons decay • nucleosynthesis proceeds through strong interactions until these freeze out, e.g. deuterium formation n + p ↔ D + γ stops at TD ∼ 7.9 × 108 K /◦. 8.2. Gamow Criterion • deuterium fusion is the most important step towards the fusion of higher elements; e.g. Helium: D + p → 3He + γ , 3He + n → 4He • deuterium needs to be produced in sufficient abundance for higher elements to form, but if all neutrons are immediately locked up into deuterium, no higher elements can form either • George Gamow noticed in 1948 that deuterium formation has to proceed at “just right” rate: nB hσvit ∼ 1 , i.e. collision rates for baryons should not be too small or too large /◦. 8.3. Prediction of CMB Temperature • for deuterium formation, t ∼ 3 min • baryon density drops like T 3; thus: • from a theoretical estimate for hσvi, Gamow estimated the baryon density at deuterium formation: nB ∼ 1018 cm−3 nB = nB,0 nB,0 ∼ 1.1 × 10−5 ΩBh2 cm−3 TD T0 3 • using TD = 7.9 × 108 K: • today’s baryon density is T0 = nB,0 nB 1/3 TD ∼ 4 K CMB temperature is predicted by the Big-Bang model! /◦. 9. The Isotropic Microwave Background 1. temperature and spectrum 2. limits on chemical potential and Compton parameter 3. the dipole /◦. 9.1. Temperature and Spectrum • microwave background was detected, but not recognised, by Penzias and Wilson in 1965 • NASA’s Cosmic Background Explorer (COBE) precisely measured its spectrum (and other things, see below) • its temperature is T0 = (2.726 ± 0.002) K • its spectrum is the best black-body spectrum ever measured /◦. 9.2. Limits on µ and y • finite width of the last-scattering surface: it is important that the CMB has a black-body spectrum • shape of the spectrum allows constraints on the chemical potential: |µ| ≤ 9 × 10−5 • hot gas between the last-scattering surface and us can distort the spectrum through Compton scattering • Compton-y parameter: kT y= mec2 Z neσTdl (typical energy change scattering probability) times • constraint from COBE-FIRAS spectrum strictly contrains heat input in young Universe: y ≤ 1.5 × 10−5 /◦. 9.3. Dipole • “freely falling” observers in the Universe define comoving coordinates • the CMB is at rest with respect to this coordinate frame • Earth moves around the Sun, Sun around the centre of the Galaxy, Galaxy within the Local Group etc. • COBE has measured the dipole: v = (371 ± 1) km s−1 towards l = (264.3 ± 0.2)◦ , b = (48.1 ± 0.1)◦ • motion causes temperature dipole: v T (θ) = T0 1 + cos θ + O (v2/c2) c /◦.