Thermodynamics in Cosmology Nucleosynthesis

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1 Thermodynamics in Cosmology Nucleosynthesis Thermodynamics Expansion Evolution of temperature Freeze out Nucleosynthesis Production of the light elements Potential barrier Primordial synthesis calculations Primordial abundance measurements 4 He, 2 H, 3 He, 7 Li Comparison with theory 1

2 History of the universe 2

3 The universe is expanding Universe expansion Hubble recession of distant galaxies To first approximation v = Hr Cosmic microwave background Abundance of primordial elements Homogeneous and isotropic universe:scale parameter a t Best way to look at expansion: comoving coordinates r = a( t ) x Friedman equation physical coordinates comoving coordinates ( ) = scale parameter Hubble "constant"= H = da t a( t) The sum of kinetic energy and potential energy is constant (Newtonian mechanics) Consider spherical shell of constant comoving radius r ( ) 3 ax 1 2 m ( a x ) 2 Gm 4π ρ ax 3 ( ) True in General Relativity also 2 a = 8π a 3 Gρ 1 a 2 R + Λ 2 General 3 relativity ( )/ dt a = t a t = constant ( ) ( ) r = Hr + a( t) x recession velocity peculiar velocity Cosmological constant Curvature

4 Energy Density of Ultra Relativistic Gases Generalization Important for behavior of early universe (energy density =>expansion) Suppose that particles are non degenerate (µ<<τ) Density of energy Bosons x 3 dx 0 e x = π u = 0 with f ( ε ) εd( ε) f( ε)dε u = g 2 1 exp ε ±1 τ τ 4 π 2 c x 3 dx e x ±1 D( ε)dε = g ε 2 dε 2 π 2 c 3 3 g is the spin multiplicity Fermions x 3 dx 0 e x = π 4 15 g 2 π c 3 k BT ( ) 4 = g 2 a BT 4 u = g bosons g fermions 2 T 4 Effective number of degrees of freedom for a relativistic fermion = g x 7/8! a B 7 g 8 2 π 2 ( 15 3 c 3 k BT) 4 = 7 g 8 2 a BT 4 4

5 Evolution of the Temperature Entropy per unit comoving volume has to be constant in a homogeneous/isotropic universe (no exchange of heat!) Initially dominated by relativistic particles σ σ V g 3 T 3 V = a 3 ( t)v comoving = s comoving a t V comoving No change of number of degrees of freedom u = g * a B 2 T 4 Stefan - Boltzmann Simple interpretation: comoving number density is constant, energy is redshifted (in agreement with General Relativity). True for all relativistic species even if they have dropped out of thermal equilibrium. Change of number of degrees of freedom e.g., Temperature does not fall as rapidly ( ) 3 g * T 3 = constant T g * 1 3 a( t) 1 T a( t) 1 e + + e - γ +γ : e ± disappear when T < 300MeV 5 ( ) logt t log a( t)

6 Equilibrium Non relativistic case (m>>t) By integrating on state density, we have seen that the number density of non relativistic species i in thermal equilibrium with photons is n i = 1 1 V exp ε exp m ic 2 + ε K µ i s s µ i τ D( ε k )dε k τ ± 1 n i = g i Statistical equilibrium m i k B T 2π 2 Nuclear element A,Z 3 2 m c 2 µ exp i i k B T We have kept explicitly mass energy Common origin of energy Exothermic A + reaction B C + D µ A + µ B = µ C + µ D e.g. Law of Mass Action: n A n B = n C n D n QAn QB ( exp m A + m B m C m D )c 2 n QC n QD τ exp ( µ A / T ) = n A 2π 2 g A m A k B T 3/ 2 ( ) = exp Zµ p + A Z exp m A c 2 / T [( ( )µ n )/ k B T ] = n p 2 2π 2 m p k B T 3 / 2 Z n n 2 2π 2 m n k B T 3/ 2 A Z [ ] exp ( Zm p + ( A Z )m )c 2 / k n B T 6

7 Formation of a bound object: Product is more bound e.g. Recombination e + p H 0 => Saha equation Nucleosynthesis Equilibrium(2) n H 0 = gh n p n e g p g e A Z p + A Z ( ) n µ A = Zµ p + ( A Z)µ n B H = ( m e + m c m H 0 )c 2 m e k B T 2π exp B H k B T n A n Z p n = g 2π 2 A Z A 2 A A 3/2 n m N k B T 3( A 1)/2 ( ) B A = Zm p + A Z exp B A / k B T ( )m n ( ) m A c2 where we have used in prefactor m A A m p m n = m N Decay/annihilation of massive object e.g. dark matter particle χ + χ q + q m χ >> m q similar expressions but with B < 0 7

8 Freese out Freeze Out we will have a similar equilibrium concentration. But not necessarily reached because the universe is expanding. The dynamic evolution of the density of say A is given by comoving density a( t) 3 n A Freeze out comoving density a( t) 3 n A Freeze out exp B kt Bound object (nuclei) exp B kt Annihilation (e.g. dark matter) log( T) log( T) We need enough time: reaction rate should be much bigger than expansion rate. dn A = Γ dt A B + C n A + Γ B+ C A n B n C where the Γ' s are the reaction rates Not necessarily true: - density decreases - temperature decreases More difficulty for going above potential barriers => rates usually decrease => Freeze out 8

9 Potential barrier Potential energy of nucleus (e.g. 4 He) potential barrier <- Coulomb repulsion,centrifugal barrier Potential radius 2 H + 2 H 4 He + γ Binding energy (28MeV) E pot + E kin = constant Need enough energy to penetrate barrier + emission of e.g. photon 3 temperature regimes At high temperature, nucleus cannot exist <- dissociation At low energy: nucleus is stable but not enough energy to be formed Intermediate: nucleus can be formed and is stable enough to survive Large abundance of 4 He => universe was hot 9

10 Primordial Nucleosynthesis Binding energies 2 H = 2.22 MeV 3 H = 6.92 MeV 2 He = 7.72 MeV 4 He = 28.3 MeV 12 C = 92.2 MeV A number of nuclear reactions: 2 body p + n 2 H + γ 2 H + 2 H 4 He +γ 2 H + 2 H 3 He + n 3 He + 2 H 4 He + p 3 He + 4 H 7 Be + γ 3 H + 2 H 4 He + n 3 He + 4 He 7 Li + γ 2 H + 2 H 3 H + p 3 H + 2 H 4 He + n 10

11 Primordial Nucleosynthesis n depleted by low T Freeze out 2H bottleneck Dependent on expansion rate and Ω b 11

12 Primordial Nucleosynthesis Time evolution 4 He very much more bound than 2 H => higher equilibrium concentration at low temperature But cannot be reached because we have to go through 2-body reactions: deuterium bottleneck! Really starts at 0.150MeV 1 minute neutrons decay (half life time 10.6 minutes) everything is over in 5 minutes: stops below carbon Higher A elements will have to be produced in stars and supernovae => Final abundance Depends on Concentration of p and n Expansion rate, and therefore on the number of relativistic species (neutrinos) 12

13 He4 Metal poor dwarf emission-line galaxies Isotov et al. Olive and Steigman 13

14 Deuterium Line of sight of Q Keck Hires spectra by Tytler et al. 14

15 Baryonic Density Fields &Sarkar Astro-ph/

16 Nucleosynthesis: A remarkable story Excellent agreement between theory and observations => Universe was indeed very hot Very uniform quark hadron phase transition Experimentally, need better measurements of deuterium Theoretically, still problems with 3 He (basically gave up),li Very tight constraints Nucleosynthesis Ω b h 2 = ±.002 Recently confirmed by Cosmic Microwave Background Ω b h 2 = ±.0008 => Ω b = ± rms dominated by uncertainty on H o implies that dark matter is non baryonic Ω m 0.25 >> Ω b 0.04 e.g. CMBR Ω m h 2 = >> Ω bh 2 = , Three light neutrino families! D.Schramm et al. 15 years before confirmation by accelerator Very tight constraint on number of massless particles. 16

17 A surprising but consistent picture Ω Λ Ω matter 17

18 The Number of Neutrino Families 18

19 Conclusion Link between nuclear physics at small scale and the universe at large scale Now attempt to explore the links between particle physics /quantum gravity and the universe Infinitely small <-> infinitely large (Pascal: XVIIth century) Inflation: quantum origin of large scale structure Trieste Lecture 1 7/10/06 19 B.

20 Conclusion A taste of how statistical physics can be put at the service of frontier science Looking beyond: Camille Flammarion 20