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Particle Physics
and Cosmology
cosmological neutrino
abundance
relic particles
examples:

neutrinos

baryons

cold dark matter ( WIMPS )
neutrinos
neutrino background radiation
Ων = Σmν / ( 91.5 eV
2
h
)
Σmν present sum of neutrino masses
mν ≈ a few eV or smaller
comparison :
electron mass =
511 003 eV
proton mass = 938 279 600 eV
experimental determination of
neutrino mass
KATRIN
neutrino-less
double beta decay
GERDA
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
cosmological neutrino abundance


How many neutrinos do we have in the present
Universe ?
neutrino number density n ν
for m ν > 10 - 3 eV:
estimate of neutrino number in
present Universe
early cosmology:
neutrino numbers from thermal equilibrium
“initial conditions”
follow evolution of neutrino number until today
decoupling of neutrinos
….from thermal equilibrium when
afterwards conserved neutrino number density
neutrinos in thermal equilibrium
decay rate vs. Hubble parameter
neutrino decoupling temperature:
Tν,d ≈ a few MeV
hot dark matter
particles which are relativistic during decoupling :
hot relics
na3 conserved during decoupling ( and also before
and afterwards )
neutrino and entropy densities
neutrino number density nν ~ a -3
 entropy density s ~ a -3
 ratio remains constant
 compute ratio in early thermal Universe
 estimate entropy in present Universe
(mainly photons from background radiation )
 infer present neutrino number density

conserved entropy
entropy in comoving volume
of present size a=1
entropy variation
from energy momentum conservation :
entropy conservation
use :
S dT + N dμ – V dp = 0
for μ = 0 :
dp/dT = S / V = ( ρ + p ) / T
adiabatic expansion : dS / dt = 0
conserved entropy
S = s a 3 conserved
entropy density s ~ a
-3
neutrino number density and entropy
( = Yν )
present neutrino fraction
tν : time before ( during , after )
decoupling of neutrinos
Ων = Σmν
/ ( 91.5 eV h2 )
s( t0 ) known from background radiation
neutrino density in thermal
equilibrium
neutrinos
neutrino background radiation
Ων = Σmν / ( 91.5 eV
2
h
)
Σmν present sum of neutrino masses
mν ≈ a few eV or smaller
comparison :
electron mass =
511 003 eV
proton mass = 938 279 600 eV
evolution of neutrino number density
σ ~ total annihilation cross section
neutrino density per entropy
attractive fixed point if Y has equilibrium value
conservation of nν / s



in thermal equilibrium
after decoupling
during decoupling more complicated
ingredients for neutrino mass bound
cosmological neutrino mass bound
Σmν = 91.5 eV Ων
2
h
or mν > 2 GeV
or neutrinos are unstable
other , more severe cosmological bounds arise from
formation of cosmological structures
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
 b) present entropy dominated by photons