Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 108
spectrum of fluctuations we must consider instead δ(r); which can be defined
as (e.g. Musco et al., 2005)
δ(r) =
ρ(r, t) − ρ
ρ
(199)
where ρ(r, t) represents the density evolution inside a region of radius r and ρ
represents the average cosmological density. It may be useful to write this last
expression in the form
ρ = ρ(1 + δ). (200)
Each perturbation δ(r) can be written as a Fourier series defined in a comoving
box much bigger than the observable Universe (e.g. Liddle & Lyth, 1993)
δ(r) =
δke ik.r (201)
k
where k represents the wavenumber. Each physical scale λ(t) is defined by some
wavenumber k and evolves with time according to (Blais et al., 2003; Bringmann
et al., 2002)
λ(t) =2π R(t)
k
(202)
where R(t) is the scale factor (Section 1.6). The name ‘scale’ is appropriate
because features with size r are dominated by wavenumbers of order k ∼ r −1
(e.g. Liddle & Lyth, 1993). For a given physical scale, the horizon crossing time
tk is conventionally defined by (e.g. Blais et al., 2003; Bringmann et al., 2002)
ck = R(tk)H(tk) (203)
where H is the Hubble parameter (Section 1.1). This corresponds to the time
when that scale reenters the Hubble radius which will inevitably happen after
inflation for scales that are larger than the Hubble radius at the end of inflation
(e.g. Blais et al., 2003; Bringmann et al., 2002).
If there is a perturbation associated with the scale entering the horizon at
time tk and if that perturbation is large enough, then it will begin to collapse
into a PBH at a later time tc >tk. We refer to this instant tc as the turnaround
point.
For a perturbation of a fixed size, its collapse cannot begin before it goes
through the cosmological horizon. The size of a PBH when it forms, therefore,
is related to the horizon size, or, equivalently, to the horizon mass MH (equation
30) when the collapsing perturbation enters the horizon.
Next we determine the relation between the size of the overdense region
at turnaround Sc(tc) and the scale factor at horizon crossing Rk(tk). In the
unperturbed region we consider the FLRW metric (Section 1.1). The evolution
of the scale factor, for a FLRW universe can be written in the form (Carr, 1975)
dR
dt
2
= 8πG
3 ρ(t)R(t)2
(204)