Physical Foundations of Cosmology

206 The very early universe

where〈σv〉∗is the thermally averaged product of the annihilation cross-sectionσ
and the relative velocityv. The effective number of degrees of freedomg ̃∗accounts
for all particles which are relativistic atT∗. Alternatively, we know from (3.61),
settingμ=0, that

n∗χgχ

T∗^3

( 2 π)^3 /^2

x^3 ∗/^2 e−x∗, (4.202)

wherex∗≡mχ/T∗. Then, using the estimate〈σv〉∗∼σ∗

√

T∗/mχ,whereσ∗is the
effective cross-section atT=T∗,we find that at freeze-out

x∗ln

(

0. 038 gχg ̃−∗^1 /^2 σ∗mχ

)

 16. 3 +ln

[

gχg ̃∗−^1 /^2

( σ∗

10 −^38 cm^2

)(mχ

GeV

)]

. (4.203)

The relics are cold only ifx∗>O( 1 ); to be definite we takex∗>3. It follows
from (4.203) thatσ∗mχ> 103 g−χ^1 g ̃^1 ∗/^2 (in Planck units) for cold relics. Their energy
density today is

ε^0 χmχn∗χ

s 0

s∗

=

2

g∗

mχn∗χ

(

Tγ 0

T∗

) 3

, (4.204)

whereTγ 0  2 .73 K ands 0 /s∗is the ratio of the present entropy density of radiation
to the total entropy density at freeze-out of those components of matter (with
g∗effective degrees of freedom) which later transfer their entropy to radiation.
Substitutingn∗χfrom (4.201) into (4.204) we finally obtain

(^) χh^275 
g ̃∗^1 /^2
g∗
x∗^3 /^2

(

3 × 10 −^38 cm^2

σ∗

)

. (4.205)

Remarkably, the contribution of cold relics to the cosmological parameter depends
only logarithmically on their mass (throughx∗) and is mainly determined by the
effective cross-sectionσ∗at decoupling.
Weakly interacting massive particles, which have masses between 10 GeV and
a few TeV and cross-sections of approximately electroweak strengthσEW∼ 10 −^38
cm^2 ,are ideal candidates for cold dark matter. Their number density freezes out
whenx∗∼20 and, as is clear from (4.205 ), they may easily contribute the nec-
essary 30% to the total density of the universe and thus constitute the dominant
component of dark matter. Currently, the leading weakly interacting massive parti-
cle candidate is the lightest supersymmetric particle. Most sypersymmetric theories
have a discrete symmetry calledR-parity, under which particles have eigenvalue+ 1
and superparticles−1. R-parity conservation guarantees the stability of the light-
est supersymmetric particle. The lightest supersymmetric particle is most likely a
neutralino, which could be (mostly) a bino or a photino.