Physical Foundations of Cosmology

4.6 Beyond the Standard Model 207

Problem 4.31Consider the annihilation reaction and its inverse:χχ ̄ f ̄f. As-
suming that the annihilation products f and ̄f always have thermal distribution
with zero chemical potential (because of some “stronger” interactions with other
particles), derive the following equation:

dX

ds

=

〈σv〉

3 H

(

X^2 −Xeq^2

)

, (4.206)

whereX≡nχ/sis theactualnumber ofχparticles per comoving volume andXeq
is the equilibrium value. Total entropy is conserved and therefore the entropy density
sscales asa−^3. Find the approximate solutions of (4.206) for hot and cold relics and
compare them with the numerical solutions. Determine the corresponding freeze-
out number densities and compare the results with the estimates made in this section
(HintIn equilibrium the direct and inverse reactions balance each other exactly, so
that〈σv〉χχ ̄neqχneqχ ̄ =〈σv〉f ̄fneqfneq ̄f.)

Nonthermal relicsIn general, for nonthermal relics, interactions are so weak that
they are never in equilibrium. There is no general formula which describes the
contribution of these relics to the total energy density because this contribution de-
pends on the concrete dynamics. As an example of nonthemal relics we consider the
homogeneous condensate of massive scalar particles and neglect their interactions
with other fields. The homogeneous scalar field satisfies the equation

φ ̈+ 3 Hφ ̇+m^2 φ= 0. (4.207)

To solve it for generic∫ H(t),it is convenient to use the the conformal timeη≡
dt/aas a temporal variable. Introducing the rescaled fieldφ≡u/awe find that
in terms ofu, (4.207) becomes

u′′+

(

m^2 a^2 −

a′′

a

)

u= 0 , (4.208)

where a prime denotes the derivative with respect toη.If

∣

∣a′′/a^3

∣

∣∼H^2 m^2 ,

the first term in the brackets can be neglected and the corresponding approximate
solution of this equation is

ua

(

C 1 +C 2

∫

dη

a^2

)

, (4.209)

whereC 1 andC 2 are integration constants. The dominant mode given by the first
term yieldsφC 1 ∼const.Therefore, while the massmis much smaller than the
Hubble constantH,the scalar field is frozen and its energy density

εφ=^12

(

φ ̇^2 +m^2 φ^2

)

(4.210)

remains constant ifm=const, and resembles the cosmological constant.