1. NEUTRINOS AND COSMOLOGY STEEN HANNESTAD AARHUS UNIVERSITY 11 OCTOBER 2010 e    

2. Where do Neutrinos Appear in Nature?Astrophysical Accelerators Soon ? Big Bang (Today 330 /cm 3 ) Indirect Evidence Indirect Evidence Nuclear Reactors Earth Crust (NaturalRadioactivity) Particle Accelerators Particle Accelerators Earth Atmosphere (Cosmic Rays) Sun Supernovae (Stellar Collapse) SN 1987A SN 1987A (2005) (2005)

3. Lecture 1 (today): Basic cosmology – the Friedmann equation Neutrino thermodynamics and decoupling in the early universe Bounds on massive neutrinos Neutrinos and BBN - Theory vs. Observations Bounds on the number of neutrino species

4. Lectures 2 and 3: Structure formation in the universe Absolute value of neutrino masses How many neutrinos are there? Future observational probes

5. Friedmann-Robertson-Walker Cosmology Line element Reduces to in a homogeneous and isotropic universe a(t): Scale factor, only dynamical variable

6. The Einstein equation However, it reduces to an evolution equation for a(t) (The Friedmann equation) in a homogeneous and isotropic universe is a combination of 10 coupled differential equations since the involved tensors are explicitly symmetric

7.  TOT =  MATTER +  RADIATION   THE TOTAL ENERGY DENSITY THEN BY DEFINITION INCLUDES NON-RELATIVISTIC MATTER, RADIATION AND THE COSMOLOGICAL CONSTANT HOWEVER, THESE TYPES OF ENERGY BEHAVE COMPLETELY DIFFERENTLY AS A FUNCTION OF TIME AND SCALE FACTOR

8. FROM THE ABOVE EQUATION AND THE FACT THAT IT CAN BE SEEN THAT THE EFFECTIVE ’TEMPERATURE’ OF RADIATION SCALES AS

9. A DEFINITION: A QUANTITY WHICH IS CONSISTENTLY USED IS THE REDSHIFT, DEFINED AS FROM THE SCALING OF PHOTON ENERGY IT CAN IMMEDIATELY BE SEEN THAT THE OBSERVED WAVELENGTH OF A PHOTON IS RELATED TO THE SCALE FACTOR OF THE UNIVERSE WHEN IT WAS EMITTED

10. log(  log(a) -40 MM RR  a eq present EVOLUTION OF ENERGY DENSITY WITH SCALE FACTOR

11. The geometry of the universe Open Universe k = -1,  < 1 Flat Universe k = 0,  Closed Universe k =1,  >1 The Friedmann equation can be recast in terms of the density parameter, 

12. An empty universe expands linearly with time Matter acts to slow the expansion, for example If  M +  R >1 then the universe eventually recollapses

13.   M     M   

14. A cosmological constant acts to accelerate the expansion. In general the pressure of an energy density component can be written as For the cosmological constant, w = -1 Any component which has w < -1/3 leads to an accelerated expansion and is referred to as dark energy

15.   M     M   

17. The total contribution to  from baryons Stars:  ~ 0.005 Interstellar gas:  ~ 0.005 Hot gas in clusters:  ~ 0.03

18. The net lepton or baryon asymmetry can be Expressed in terms of the parameter  According to observations  ~ 6 x 10 -10  for baryons can be found from the present baryon density

19. NEUTRINO THERMODYNAMICS AND BIG BANG NUCLEOSYNTHESIS

20. 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

21. In equilibrium, 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 The entropy density of a species with MB statistics is given by This means that entropy is maximised when (true if processes like occur rapidly) It is possible that

22. Conditions for lepto (baryo) genesis (Sakharov conditions) CP-violation:Asymmetry between particles and antiparticles e.g. has other rate than L-violation:Processes that can break lepton number (e.g. ) Non-equilibrium thermodynamics: In equilibrium always applies. A small note on how to generate asymmetry (for leptons or baryons)

23. Non-equilibrium Much more to come in the lecture by Michael Plümacher

24. Thermal evolution of standard, radiation dominated cosmology Total energy density

25. Temperature evolution of g *

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

28. The number and energy density for a given species, X, is given by the Boltzmann equation C e [f]: Elastic collisions, conserves particle number but energy exchange possible (e.g. ) [scattering equilibrium] C i [f]: Inelastic collisions, changes particle number (e.g. ) [chemical equilibrium] Usually, C e [f] >> C i [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.

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

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

31. There are small corrections to this because neutrinos still interact slightly when electrons and positrons annihilate (neutrino heating) Additional small effect from finite temperature QED effect (O(  )) ~ 1% In total the neutrino energy density gets a correction of Dicus et al. ’82, Lopez & Turner ’99 + several other papers

32. Upper limit on the mass of light neutrinos: For light neutrinos, m << T dec, the present day density is Assuming that the three active species have the same mass A conservative limit (  h 2 <1) on the neutrino mass is then For any of the three active neutrino species

33. If MB statistics is used one finds (as long as elastic scattering equilibrium holds) by integrating the Boltzmann equation This is the standard equation used for WIMP annihilation (e.g. massive neutrinos, neutralinos, etc) This equation is usually not analytically solvable (Riccati equation), but is trivial to solve numerically. For a particle with standard weak interactions one finds that the species decouples from chemical equilibrium when Mass limits on very massive neutrinos (m >> T D ):

34. As long as a species is close to equilibrium with Comparing this to the Hubble rate yields

36. Dirac neutrinos Majorana neutrinos This means that m i is the mass of the annihilation product (usually the most massive final state available)

37. This leads to a lower bound (the Lee-Weinberg bound) on very massive neutrinos TDTD

38. Because of the stringent bound from LEP on neutrinos lighter than about 45 GeV This bound is mainly of academic interest. However, the same argument applies to any WIMP, such as the neutralino (very similar to Majorana neutrino).

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

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

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

42. However, before complete decay neutrons are bound in nuclei. Nucleosynthesis should intuitively start when T ~ E b (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(T BBN ) ~ 50 s <<  n only few neutrons have time to decay

43. 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

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

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

48. Helium-4: Essentially all available neutrons are processed into He-4, giving a mass fraction of Y p depends on  because T BBN changes with 

49. D, He-3: These elements are processed to produce He-4. For higher , T BBN 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

50. Higher mass elements. Extremely low abundances

51. Confronting theory with observations 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. He-4 is extremely stable and is in general always produced, not destroyed, in astrophysical environments He-4:

52. Extragalactic H-II regions

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

54. Aver et al. 2010

56. 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

57. 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

58. Pettini et al 2010 The discrepancy has been ”resolved” in favour of a low deuterium value of roughly

59. 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

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

61. 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

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

63. 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

64. N  = 3 N  = 4 N  = 2 Since The helium production is very sensitive to N

65. n-p changing reactions 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 More electron neutrinos shifts the balance towards more neutrons, and a higher relativistic energy density can be accomodated

66. Ways of producing more electron neutrinos: 1) A neutrino chemical potential 2) Decay of massive particle into electron neutrinos Increasing  e decreases n/p so that N can 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:

67.  e = 0.1  e = 0.2 Allowed region Allowed for  e = 0

68. 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     this occurs at T ~ 10 MeV >> T BBN, leading to complete equilibration before decoupling. For e    the amount of equilibration depends completely on the Solar neutrino mixing parameters.

69. Lunardini & Smirnov, hep-ph/0012056 (PRD), Dolgov et al., hep-ph/0201287 Abazajian, Beacom & Bell, astro-ph/0203442, Wong hep-ph/0203180 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 Dolgov et al., hep-ph/0201287 SH 02

70. For the LMA solution the bound becomes equivalent to the electron neutrino bound for all species Dolgov et al. ’02 See also Serpico & Raffelt 05

71. 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 ….

72. PERSPECTIVES Neutrino thermodynamics and decoupling is well understood in the standard model. Big bang nucleosynthesis provides a powerful probe of neutrino physics beyond the standard model. However, many of the abundance measurements are dominated by systematics which need to be better understood Deuterium measurements provide by far the most sensitive probe of the baryon density and is (now) consistent with CMB. More on this later!