Physical Foundations of Cosmology

4.6 Beyond the Standard Model 205

Problem 4.30Using this fact and assuming that the hot relics are fermions of
massmψwith negligible chemical potential, verify that their contribution to the
cosmological parameter is

(^) ψh^275 
gψ
g∗
( mψ
19 eV

)

. (4.200)

Heregψ is the total number of degrees of freedom of the hot relics andg∗=
gb+( 7 / 8 )gfis the effective number of bosonic and fermionic degrees of freedom
at freeze-out for all particles which later convert their energy into photons.

Taking three left-handed light neutrino species of the same massmψ,we have
gψ=6 (3 for neutrinos+3 for antineutrinos). The neutrinos decouple at temper-
ature∼O( 1 )MeV.At this time the only particles which contribute tog∗are the
electrons, positrons and photons, andg∗=( 2 +( 7 / 8 )× 4 )= 5. 5 .Therefore, if the
mass of everyneutrino were about 17 eV,then neutrinos would close the uni-
verse. According to observations, the contribution of dark matter to the total density
does not exceed 30% (see, in particular, Chapter 9). Hence thesumof the neutrino
masses should be smaller than 15 eV or, in the case of equal masses,mν<5eV.
The mass bound on hot relics species changes if we assume that they freeze out at
a higher temperature when more relativistic particles are present. For instance, if
decoupling happened before the quark–gluon phase transition, atT∼300 MeV,
theng∗=53 (for the number of degrees of freedom at this time, see (4.33) and do
not forget to include photons and electron–positron pairs). In this case, it follows
from (4.200) thatmψ<151 eV forgψ= 2 .In reality the bound on the masses
is even stronger because hot relics cannot explain all dark matter in the universe;
they can constitute only a subdominant fraction of it. In fact, in models where hot
dark matter dominates, inhomogeneities are washed out by free streaming on all
scales up to the horizon scale at the moment when the relics became nonrelativistic
(see Section 9.2). As a result the large scale structure of the universe cannot be
explained. Therefore, more promising and successful models are those in which
cold relics constitute the dominant part of the dark matter.

Thermal cold relicsCold relicsχ decouple at temperaturesT∗much less than
their massmχ.Therefore, the number densitynχis exponentially suppressed in
comparison with the number density of photons.To deducen∗χat the moment
of decoupling, we simply equate the annihilation rate of relicsχto the Hubble
expansion rate:

n∗χ〈σv〉∗H∗=

(

8 π^3 / 90

) 1 / 2

g ̃^1 ∗/^2 T∗^2 , (4.201)