Particle PhysicsParticle Physicsand Cosmologyand Cosmology

cosmological cosmological neutrino abundanceneutrino abundance

relic particlesrelic particles

examples:examples:

neutrinosneutrinos

baryonsbaryons

cold dark matter ( WIMPS )cold dark matter ( WIMPS )

neutrinosneutrinos

neutrino background radiationneutrino background radiation

ΩΩνν = = ΣΣmmνν / ( 91.5 eV h/ ( 91.5 eV h22 ) )

ΣΣmmνν present sum of neutrino massespresent sum of neutrino massesmmνν ≈ a few eV or smaller ≈ a few eV or smaller

comparison : electron mass = 511 003 comparison : electron mass = 511 003 eVeV

proton mass = 938 279 600 eVproton mass = 938 279 600 eV

experimental experimental determination of determination of

neutrino massneutrino massKATRIN neutrino-less KATRIN neutrino-less

double beta decay double beta decay

GERDAGERDA

experimental bounds on experimental bounds on neutrino massneutrino mass

from neutrino oscillations :from neutrino oscillations :

largest neutrino mass must be larger than largest neutrino mass must be larger than 5 105 10-2-2 eV eV

direct tests ( endpoint of spectrum in direct tests ( endpoint of spectrum in tritium decay )tritium decay )

electron-neutrino mass smaller 2.3 eVelectron-neutrino mass smaller 2.3 eV

cosmological neutrino cosmological neutrino abundanceabundance

How many neutrinos do we have in How many neutrinos do we have in the present Universe ?the present Universe ?

neutrino number density n neutrino number density n νν

for m for m νν > 10 > 10 - 3- 3 eV: eV:

estimate of neutrino estimate of neutrino number in present Universenumber in present Universe

early cosmology: early cosmology:

neutrino numbers from thermal neutrino numbers from thermal equilibriumequilibrium

““initial conditions”initial conditions”

follow evolution of neutrino number follow evolution of neutrino number until todayuntil today

decoupling of neutrinosdecoupling of neutrinos

…….from thermal equilibrium when.from thermal equilibrium when

afterwards conserved neutrino afterwards conserved neutrino number densitynumber density

neutrinos in thermal neutrinos in thermal equilibriumequilibrium

decay rate vs. Hubble decay rate vs. Hubble parameterparameter

neutrino decoupling temperature:neutrino decoupling temperature:

TTνν,d ,d ≈ a few MeV≈ a few MeV

hot dark matterhot dark matter

particles which are relativistic during particles which are relativistic during decoupling :decoupling :

hot relicshot relics

nana3 3 conserved during decoupling ( and conserved during decoupling ( and also before and afterwards )also before and afterwards )

neutrino and entropy neutrino and entropy densitiesdensities

neutrino number density nneutrino number density nνν ~ a ~ a -3-3

entropy density s ~ a entropy density s ~ a -3-3

ratio remains constantratio remains constant compute ratio in early thermal Universecompute ratio in early thermal Universe estimate entropy in present Universeestimate entropy in present Universe

(mainly photons from background (mainly photons from background radiation )radiation )

infer present neutrino number densityinfer present neutrino number density

conserved entropyconserved entropy

entropy in comoving volume entropy in comoving volume of present size a=1of present size a=1

entropy variationentropy variation

from energy momentum conservation :from energy momentum conservation :

entropy conservationentropy conservation

use : use : S dT + N dS dT + N dμμ – V dp = 0 – V dp = 0

for for μμ = 0 : = 0 :

dp/dT = S / V = ( dp/dT = S / V = ( ρρ + p ) / T + p ) / T

adiabatic expansion : dS / adiabatic expansion : dS / dt = 0dt = 0

conserved entropyconserved entropy

S = s a S = s a 33 conserved conserved

entropy density s ~ entropy density s ~ a a -3-3

neutrino number density neutrino number density and entropyand entropy

( = Y( = Yνν ) )

present neutrino fractionpresent neutrino fraction

s( ts( t0 0 ) known from background radiation) known from background radiation

ΩΩνν = = ΣΣmmνν

/ ( 91.5 eV / ( 91.5 eV hh22 ) )

ttνν : time before ( during , after ) : time before ( during , after ) decoupling of neutrinosdecoupling of neutrinos

neutrino density in thermal neutrino density in thermal equilibriumequilibrium

neutrinosneutrinos

neutrino background radiationneutrino background radiation

ΩΩνν = = ΣΣmmνν / ( 91.5 eV h/ ( 91.5 eV h22 ) )

ΣΣmmνν present sum of neutrino massespresent sum of neutrino massesmmνν ≈ a few eV or smaller ≈ a few eV or smaller

comparison : electron mass = 511 003 comparison : electron mass = 511 003 eVeV

proton mass = 938 279 600 eVproton mass = 938 279 600 eV

evolution of neutrino evolution of neutrino number densitynumber density

σσ ~ total annihilation cross section ~ total annihilation cross section

neutrino density per neutrino density per entropyentropy

attractive fixed point if Y has equilibrium valueattractive fixed point if Y has equilibrium value

conservation of nconservation of nνν / s/ s

in thermal equilibriumin thermal equilibrium after decouplingafter decoupling during decoupling more complicatedduring decoupling more complicated

ingredients for neutrino ingredients for neutrino mass boundmass bound

cosmological neutrino mass cosmological neutrino mass boundbound

ΣΣmmνν = 91.5 eV = 91.5 eV ΩΩνν hh22

or mor mνν > 2 GeV> 2 GeVor neutrinos are unstableor neutrinos are unstable

other , more severe cosmological bounds other , more severe cosmological bounds arise from formation of cosmological arise from formation of cosmological structuresstructures

cosmological neutrino mass cosmological neutrino mass boundbound

cosmological neutrino mass bound is cosmological neutrino mass bound is very robustvery robust

valid also for modified gravitational valid also for modified gravitational equationsequations, as long as , as long as

a) entropy is conserved for T < 10 MeVa) entropy is conserved for T < 10 MeV b) present entropy dominated by b) present entropy dominated by

photonsphotons