1 Lecture-03 The Thermal History of the universe Ping He ITP.CAS.CN

2 3.1 Thermal history to study: (2) (3) Degree of freedom ~ T, t (4) Decouple & relic background (5) Nucleosynthesis (6) Baryogenesis They are typical events in the early Universe (1) T ～ t [temperature ～ time]

3 Piston : L=a(t)r 3.2. Equilibrium Thermodynamics a(t)r Quasi – static : →Thermal equilibrium Reaction rate Expansion rate (Eq-3.1)

4 This analysis can be applied to cosmology Eq-2.1 is the kernel of this lecture Thermal equilibrium : Coupling : (1) (2) ~ A, C equilibrium ~ A, B equilibrium A B C Coupling mode : (1) AC (2) AB

5 3.3 Distribution Function in Thermal Equilibrium g: spin-degeneracy factor (inner degree of freedom) phase-space distribution function (occupancy function ) x p

6 If equilibrium : (relativistic) Chemical potential If chemical equilibrium : From Eq-2.2 & Eq-2.3

7 p1p1 p2p2 x y

8 From Eq-2.6, Eq-2.7, Eq-2.8 we have The above is the general form for relativistic & quantum cases. In kinetic equilibrium:

9 We used Chemical potential 3.4 Distributions as a function of E

10 Specifically (1) relativistic limit, and non-degeneracy (2) non-relativistic In above calculation, we used the fact that Maxwell-Boltzmann

11 (3) For non-degenerate,relativistic species average energy /particle For a non-relativistic species

The excess of fermions over its antiparticle From thermodynamics and statistical dynamics for photon From Eq-2.19, we have

13 The net (the excess of ) Fermion number density (relativistic) (non-relativistic) For proton Most of the particle species have

Degrees of freedom From eq-2.15 From eq-2.16 At the early epoch of the Universe, T is very high. All are in Relativistic. Non-relativistic exponentially decrease  negligible

15 Here, the effective degrees of freedom: (1) T<1 MeV

16 From eq-2.24 (2) 1 MeV<T<100 MeV (3) T>300 GeV 3 generations quarks & leptons 1 complex Higgs doublet. 8 gluons,

17

Time ~Temperature when radiation dominated

Evolution of Entropy (unit coordinate volume) More, In the expanding Universe, 2nd law of thermodynamics

20 From Eq-2.26 with Eq-2.27 Up to an additive constant, the entropy per commoving volume From And with Eq-2.26, we have:

21 The entropy per commoving volume is consented during the expansion of the Universe. Entropy density s For most of the history of the Universe density of physical volume dominated by relativistic particles

22 For photon number-density Physical entropy density scales as Commoving entropy density is conserved

23 From eq-2.32 Boson - relativistic Fermions - relativistic non - relativistic

24 Define (relativistic) (non-relativistic) If the number of a given species in a commoving volume is not changing, i·e, particles of that species are not being created or destroyed, then remains constant Commoving number density If no baryon non-conserving mechanism, then So, with eq-2.31, we have:

25 is conserved, after annihilations at T=0.5MeV (t ~ 4sec) is constant, so More over So, the temperature of the Universe evolves as: Whynot just When annihilation, there is a change in conserved, so change  T change Explanation:

Decoupling When A is decoupled for massless Assume a massless particle decouples at time temperature when the scale factor was, the phase-space distribution at decoupling is given by the equilibrium distribution: (1) After decoupling, the energy of each particle is red-shifted by the expansion:

27 In addition so for decoupled massless species, while the others still couple with each other, so the temperature scales as:

28 (2) Massive particle decoupling, Decouples at So

29 Summary: In both cases log(p) log(f)

30 (3) general cases For a species that decouples when it is semi-relativistic The phase-space distribution does not maintain an equilibrium distribution. In the absence of interactions: You cannot find a simple relation, for So the equilibrium distribution cannot be maintained

Brief Thermal History of the Universe *Some famous events* Key: the interaction rate per particle The correct way to evolve particlen distributions is to integrate the Boltzmann equation Neutrino decoupling

32 When when

33 For Relic neutrino background

34 With this value of, we have: And

Matter-Radiation Equality In above calculation, we have used:

Photon Decoupling and Recombination Thomson cross-station Radiation-Matter decoupling number density of free electrons number density of free hydrogen number density of free protons (charge neutrality) In thermal equilibrium, at

37 B: binding energy of hydrogen, Define : the fractional ionization (ionization degree) : neutral. Ionization=0 : ionization total baryon-to-photon ratio From eq-2.52 the equilibrium ionization fraction

38 Depends on

39 When recombination, matter–dominated Decoupling: Summary :

The baryon number of the Universe baryon number density B is defined to be the baryon number of the Universe since the epoch of annihilation photon density

41 The primordial nucleosynthesis constrains to the interval that is So the entropy of the Universe is enormous !!! To compare with, in a star, the entropy is entropy per baryon

42 Primordial nucleosynthesis A human age: one day 100 years

43 Key points: (1) (2) (3) phase-space distribution function entropy is concerned interaction rate per particle thermal equilibrium decouple Based on argument: Qualitative and semi-quantitative Full-quantitative treatment: solve collisional Boltzmann Equation

44 References E.W. Kolb & M.S. Turner, The Early Universe, Addison-Wesley Publishing Company, 1993 T. Padmanabhan, Theoretical Astrophysics III: Galaxies and Cosmology, Cambridge, 2002