Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 109
where ρ represents the average cosmological density. Note that this is just one
of the Friedmann–Lemaître equations (Section 1.1, equation 2) where we have
considered k = 0 (flat universe) and we have neglected the cosmological constant
term (which is a reasonable choice at early epochs).
In the perturbed region we consider the metric (Carr, 1975)
ds 2 = dτ 2 − S 2
dr
(τ)
2
1 − ∆ɛr2 + r2 dθ 2 + sin 2 θdφ 2
(205)
where ∆ɛ is the perturbed total energy per unit mass and the time τ is proper
time as measured by comoving observers. Here S(τ) plays the role of a scale
factor for the perturbed region. The evolution of S(τ) can be written in the
form (Carr, 1975)
dS
dτ
2
= 8πG
3 ρ(τ)S(τ)2 − ∆ɛ (206)
where ρ(τ) represents the density in the perturbed region.
Considering that initially the overdense region is comoving with the unperturbed
background we consider τk = tk (here the subscript k denotes a
quantity evaluated when the fluctuation crosses the horizon), Sk = Rk and
(dS/dτ)k =(dR/dt)k. With these choices we obtain for ∆ɛ the expression
∆ɛ = 8πG
3 R2 k (ρk − ρ k) (207)
Inserting this into equation (206) and taking into account that ρk = ρk(1 + δk)
(cf. equation 200) we obtain
2 dS
=
dτ
8πG
ρ(τ)S(τ)
3
2 − ρkR 2
δk
k . (208)
1+δk
The density ρ(τ) can be written as (cf. Section 1.1, equation 10)
ρ(τ) =KsS(τ) −3(1+w)
where Ks is a constant. In the case of ρk(tk) we have
ρk = KkR −3(1+wk)
k
(209)
(210)
where Kk is a constant and wk is the adiabatic index when the fluctuation
crosses the horizon. Inserting expressions (209) and (210) into equation (208)
we obtain
dS
dτ
2
= 8πG
3
Ks
1+δk
1+δk Kk
−
S(τ) 1+3w
Ks
δk
R 1+3wk
k
. (211)
The turnaround point is reached when the perturbed region stops expanding,
i.e., when dS/dτ = 0. Thus, the evaluation of equation (211) at the turnaround
point gives
S 1+3wc
c
= Ks
R
Kk
1+3wk
k
1+δk
δk
(212)