1. Particle Physics and Cosmology cosmological neutrino abundance

2. relic particles examples: neutrinos neutrinos baryons baryons cold dark matter ( WIMPS ) cold dark matter ( WIMPS )

3. neutrinos neutrino background radiation Ω ν = Σm ν / ( 91.5 eV h 2 ) Ω ν = Σm ν / ( 91.5 eV h 2 ) Σm ν present sum of neutrino masses m ν ≈ a few eV or smaller comparison : electron mass = 511 003 eV comparison : electron mass = 511 003 eV proton mass = 938 279 600 eV proton mass = 938 279 600 eV

4. experimental determination of neutrino mass KATRIN neutrino-less double beta decay double beta decay GERDA

5. experimental bounds on neutrino mass from neutrino oscillations : largest neutrino mass must be larger than 5 10 -2 eV direct tests ( endpoint of spectrum in tritium decay ) electron-neutrino mass smaller 2.3 eV

6. cosmological neutrino abundance How many neutrinos do we have in the present Universe ? How many neutrinos do we have in the present Universe ? neutrino number density n ν neutrino number density n ν for m ν > 10 - 3 eV: for m ν > 10 - 3 eV:

7. estimate of neutrino number in present Universe early cosmology: neutrino numbers from thermal equilibrium “initial conditions” follow evolution of neutrino number until today

8. decoupling of neutrinos ….from thermal equilibrium when afterwards conserved neutrino number density

9. neutrinos in thermal equilibrium

10. decay rate vs. Hubble parameter neutrino decoupling temperature: neutrino decoupling temperature: T ν,d ≈ a few MeV T ν,d ≈ a few MeV

11. hot dark matter particles which are relativistic during decoupling : hot relics hot relics na 3 conserved during decoupling ( and also before and afterwards )

12. neutrino and entropy densities neutrino number density n ν ~ a -3 neutrino number density n ν ~ a -3 entropy density s ~ a -3 entropy density s ~ a -3 ratio remains constant ratio remains constant compute ratio in early thermal Universe compute ratio in early thermal Universe estimate entropy in present Universe estimate entropy in present Universe (mainly photons from background radiation ) (mainly photons from background radiation ) infer present neutrino number density infer present neutrino number density

13. conserved entropy entropy in comoving volume of present size a=1

14. entropy variation from energy momentum conservation :

15. entropy conservation use : S dT + N dμ – V dp = 0 for μ = 0 : dp/dT = S / V = ( ρ + p ) / T dp/dT = S / V = ( ρ + p ) / T adiabatic expansion : dS / dt = 0 adiabatic expansion : dS / dt = 0

16. conserved entropy S = s a 3 conserved S = s a 3 conserved entropy density s ~ a -3 entropy density s ~ a -3

17. neutrino number density and entropy ( = Y ν )

18. present neutrino fraction s( t 0 ) known from background radiation Ω ν = Σm ν / ( 91.5 eV h 2 ) / ( 91.5 eV h 2 ) t ν : time before ( during, after ) decoupling of neutrinos decoupling of neutrinos

19. neutrino density in thermal equilibrium

20. neutrinos neutrino background radiation Ω ν = Σm ν / ( 91.5 eV h 2 ) Ω ν = Σm ν / ( 91.5 eV h 2 ) Σm ν present sum of neutrino masses m ν ≈ a few eV or smaller comparison : electron mass = 511 003 eV comparison : electron mass = 511 003 eV proton mass = 938 279 600 eV proton mass = 938 279 600 eV

21. evolution of neutrino number density σ ~ total annihilation cross section

22. neutrino density per entropy attractive fixed point if Y has equilibrium value

23. conservation of n ν / s in thermal equilibrium in thermal equilibrium after decoupling after decoupling during decoupling more complicated during decoupling more complicated

24. ingredients for neutrino mass bound

25. cosmological neutrino mass bound Σm ν = 91.5 eV Ω ν h 2 or m ν > 2 GeV or neutrinos are unstable other, more severe cosmological bounds arise from formation of cosmological structures

26. cosmological neutrino mass bound cosmological neutrino mass bound is very robust valid also for modified gravitational equations, as long as a) entropy is conserved for T < 10 MeV a) entropy is conserved for T < 10 MeV b) present entropy dominated by photons b) present entropy dominated by photons