208 The very early universe
As the Hubble constant becomes smaller than the mass, and eventuallyH^2
m^2 ,we can neglect the second term inside the brackets in (4.208). The WKB
solution of the simplified equation is then
u∝(ma)−^1 /^2 sin(∫
madη)
, (4.211)
and correspondingly
φ∝m−^1 /^2 a−^3 /^2 sin(∫
mdt)
. (4.212)
Note that this solution is also valid for a slowly varying massm.Substituting this
into (4.210) we find, in the leading order, that the energy density of the scalar field
decreases asma−^3 ; hence it behaves as dust-like matter (p0). This is easy to
understand: after the value of the Hubble constant drops below that of the mass,
the scalar field, which was frozen before, starts to oscillate and can be interpreted
as a Bose condensate of many cold particles of massmwith zero momentum. For
a slowly varying mass the particle number density, which is proportional toεφ/m,
decays asa−^3 and the total particle number is conserved.
Using these results, we can easily calculate the current energy density of the
scalar field (in Planck units):
ε^0 φm 0(εφ
m)
∗s 0
s∗O( 1 )
g ̃∗^3 /^4
g∗m 0
m^1 ∗/^2φin^2 Tγ^30 , (4.213)wherem 0 is its mass at present,m∗is the mass at the moment whenH∗m∗and
φinis the initial value of the scalar field when it was still frozen. For the case of
constant mass (m 0 =m∗), the contribution of this field to the total energy density
is
(^) φh^275 ∼
g ̃∗^3 /^4
g∗
( m 0
100 GeV
) 1 / 2 ( φin
3 × 109 GeV
) 2
. (4.214)
Thus, tuning two parameters, the mass and the initial value of the scalar field,
we have a straightforward “explanation” for the observed cold dark matter in the
universe.
AxionsAs previously mentioned, axions are an attractive nonthermal relic candi-
date. The axion field is introduced to solve the strongCPproblem. Because the
strong coupling constant is large at low energies, topological transitions are not sup-
pressed in quantum chromodynamics. Therefore, one expects that the true quantum
chromodynamics vacuum is theθvacuum, which is a superposition of vacua with