Thermodynamics in the early universe

1. Thermodynamics in the early universe In equilibrium, distribution functions have the form When m ~ T particles disappear because of Boltzmann supression Decoupled particles: If particles are decoupled from other species their comoving number density is conserved. The momentum redshifts as p ~ 1/a

2. The entropy density of a species with MB statistics is given by In equilibrium, (true if processes like occur rapidly) This means that entropy is maximised when In equilibrium neutrinos and anti-neutrinos are equal in number! However, the neutrino lepton number is not nearly as well constrained observationally as the baryon number It is possible that

3. Leptogenesis (more to come in the last lecture) Conditions for lepto (baryo) genesis (Sakharov conditions) L-violation: Processes that can break lepton number (e.g. ) CP-violation: Asymmetry between particles and antiparticles e.g. has other rate than Non-equilibrium thermodynamics: In equilibrium always applies.

4. Non-equilibrium

5. Different possibilities: Electroweak baryogenesis does not require new physics, does not work! SUSY electroweak baryogenesis requires new physics, does not work in the MSSM GUT baryogenesis requires plausible new physics, it works possible problems with reheating Baryogenesis via leptogenesis requires ”plausible” new physics, it works

6. Thermal evolution after the end of inflation Total energy density

7. Temperature evolution of N(T)

9. In a radiation dominated universe the time-temperature relation is then of the form

10. The number and energy density for a given species, X, is given by the Boltzmann equation Ce[f]: Elastic collisions, conserves particle number but energy exchange possible (e.g. ) [scattering equilibrium] Ci[f]: Inelastic collisions, changes particle number (e.g. ) [chemical equilibrium] Usually, Ce[f] >> Ci[f] so that one can assume that elastic scattering equilibrium always holds. If this is true, then the form of f is always Fermi-Dirac or Bose-Einstein, but with a possible chemical potential.

11. Particle decoupling The inelastic reaction rate per particle for species X is In general, a species decouples from chemical equlibrium when

12. The prime example is the decoupling of light neutrinos (m < TD) After neutrino decoupling electron-positron annihilation takes place (at T~me/3) Entropy is conserved because of equilibrium in the e+- e--g plasma and therefore The neutrino temperature is unchanged by this because they are decoupled and therefore

13. BIG BANG NUCLEOSYNTHESIS The baryon number left after baryogenesis is usually expressed in terms of the parameter h According to observations h ~ 10-10 and therefore the parameter is often used From h the present baryon density can be found as

14. Immediately after the quark-hadron transition almost all baryons are in pions. However, when the temperature has dropped to a few MeV (T << mp) only neutrons and protons are left In thermal equilibrium However, this ratio is dependent on weak interaction equilibrium

15. n-p changing reactions Interaction rate (the generic weak interaction rate) After that, neutrons decay freely with a lifetime of

16. However, before complete decay neutrons are bound in nuclei. Nucleosynthesis should intuitively start when T ~ Eb (D) ~ 2.2 MeV via the reaction However, because of the high entropy it does not. Instead the nucleosynthesis starting point can be found from the condition Since t(TBBN) ~ 50 s << tn only few neutrons have time to decay

17. At this temperature nucleosynthesis proceeds via the reaction network The mass gaps at A = 5 and 8 lead to small production of mass numbers 6 and 7, and almost no production of mass numbers above 8 The gap at A = 5 can be spanned by the reactions

20. ABUNDANCES HAVE BEEN CALCULATED USING THE WELL-DOCUMENTED AND PUBLICLY AVAILABLE FORTRAN CODE NUC.F, WRITTEN BY LAWRENCE KAWANO

21. The amounts of various elements produced depend on the physical conditions during nucleosynthesis, primarily the values of N(T) and h

22. Helium-4: Essentially all available neutrons are processed into He-4, giving a mass fraction of Yp depends on h because TBBN changes with h

23. D, He-3: These elements are processed to produce He-4. For higher h, TBBN is higher and they are processed more efficiently Li-7: Non-monotonic dependence because of two different production processes Much lower abundance because of mass gap

24. Higher mass elements. Extremely low abundances

25. Confronting theory with observations He-4: He-4 is extremely stable and is in general always produced, not destroyed, in astrophysical environments The Solar abundance is Y = 0.28, but this is processed material The primordial value can in principle be found by measuring He abundance in unprocessed (low metallicity) material.

26. Extragalactic H-II regions

27. I Zw 18 is the lowest metallicity H-II region known

28. Olive, Skillman & Steigman

29. Izotov & Thuan Most recent values:

30. Deuterium: Deuterium is weakly bound and therefore can be assumed to be only destroyed in astrophysical environments Primordial deuterium can be found either by measuring solar system or ISM value and doing complex chemical evolution calculations OR Measuring D at high redshift The ISM value of can be regarded as a firm lower bound on primordial D

31. 1994: First measurements of D in high-redshift absorption systems A very high D/H value was found Carswell et al. 1994 Songaila et al. 1994 However, other measurements found much lower values Burles & Tytler 1996 Burles & Tytler

32. The discrepancy has been ”resolved” in favour of a low deuterium value of roughly Burles, Nollett & Turner 2001

33. Li-7: Lithium can be both produced and destroyed in astrophysical environments Production is mainly by cosmic ray interactions Destruction is in stellar interiors Old, hot halo stars seem to be good probes of the primordial Li abundance because there has been only limited Li destruction

34. Li-abundance in old halo stars in units of Spite plateau Molaro et al. 1995

36. Li-7 abundances are easy to measure, but the derivation of primordial abundances is completely dominated by systematics

37. There is consistency between theory and observations All observed abundances fit well with a single value of eta This value is mainly determined by the High-z deuterium measurements The overall best fit is Burles, Nollett & Turner 2001

38. This value of h translates into And from the HST value for h One finds

39. BOUND ON THE RELATIVISTIC ENERGY DENSITY (NUMBER OF NEUTRINO SPECIES) FROM BBN The weak decoupling temperature depends on the expansion rate And decoupling occurs when N(T) is can be written as

40. Since The helium production is very sensitive to Nn Nn = 4 Nn = 3 Nn = 2

41. The bound using most recent deuterium measurements (Abazajian 2002)

42. Sterile neutrinos: If there are nA-ns oscillations in the early universe at T > Tdec,n then DNn>0 However, DNn<1 always (corresponding to 1 extra neutrino flavour)

43. FLAVOUR DEPENDENCE OF THE BOUNDS Muon and tau neutrinos influence only the expansion rate. However, electron neutrinos directly influence the weak interaction rates, i.e. they shift the neutron-proton ratio n-p changing reactions More electron neutrinos shifts the balance towards more neutrons, and a higher relativistic energy density can be accomodated

44. Ways of producing more electron neutrinos: 1) A neutrino chemical potential 2) Decay of massive particle into electron neutrinos Increasingxne decreases n/p so thatNncan be much higher than 4 Yahil ’76, Langacker ’82, Kang&Steigman ’92, Lesgourgues&Pastor ’99, Lesgourgues&Peloso ’00, Hannestad ’00, Orito et al. ’00, Esposito et al. ’00, Lesgourgues&Liddle ’01, Zentner & Walker ’01 BBN alone:

45. Allowed region xne= 0.2 xne= 0.1 Allowed for xne= 0

46. If oscillations are taken into account these bounds become much tighter: Oscillations between different flavours become important once the vacuum oscillation term dominates the matter potential at a temperature of For nm-nt this occurs at T ~ 10 MeV >> TBBN, leading to complete equilibration before decoupling. For ne-nm,t the amount of equilibration depends completely on the Solar neutrino mixing parameters.

47. SH 02 Dolgov et al., hep-ph/0201287 If all flavours equilibrate before BBN the bound on the flavour asymmetry becomes much stronger because a large asymmetry in the muon or tau sector cannot be masked by a small electron neutrino asymmetry Lunardini & Smirnov, hep-ph/0012056 (PRD), Dolgov et al., hep-ph/0201287 Abazajian, Beacom & Bell, astro-ph/0203442, Wong hep-ph/0203180

48. For the LMA solution the bound becomes equivalent to the electron neutrino bound for all species Dolgov et al. ’02

49. Using BBN to probe physics beyond the standard model Non-standard physics can in general affect either Expansion rate during BBN extra relativistic species massive decaying particles quintessence …. The interaction rates themselves neutrino degeneracy changing fine structure constant ….

50. Example: A changing fine-structure constant a Many theories with extra dimensions predict 4D-coupling constants which are functions of the extra-dimensional space volume. Webb et al.: Report evidence for a change in a at the Da/a ~ 10-5 level in quasars at z ~ 3 BBN constraint: BBN is useful because a changing a would change all EM interaction rates and change nuclear abundance Bergström et al: Da/a < 0.05 at z ~ 1012