Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 24
When the Universe becomes dominated by dark energy (Λ > 0) the Hubble
parameter becomes constant in time (cf. expression 19)
Λ
H(t) =c . (65)
3
Since inflation only lasts a few e–folds (see Section 1.3), the Hubble parameter
can be taken as a constant during this period (e.g. Huang, 2007; Narlikar &
Padmanabhan, 1991). From equation (40) we have
H(t) = N(te)
. (66)
te
Considering the normalisation (61) we can determine the proportionality constant
in expression (19), yielding, for the scale factor of a Universe dominated
by a positive cosmological constant, the result 9
R(t) = exp c
Λ
3
(t − t0) , tSN ≤ t ≤ t0 (67)
where t0 is the present time (i.e. the age of the Universe) and tSN is the age
of the universe at matter–Λ equality (corresponding to the instant when the
expansion starts to accelerate). The dark energy domination is preceded by a
matter–dominated stage which started when photons decoupled from matter.
During matter domination the scale factor behaves according to expression (18).
Considering that R(t) is a continuous function of time we will write, for the
matter–dominated stage
Λ
R(t) = exp c 3 (tSN
2/3 t − t0)
, teq ≤ t ≤ tSN (68)
tSN
where teq is the age of the Universe at radiation–matter equality. Before that
time, the Universe was radiation–dominated up to the end of inflation at some
instant t = te. During radiation domination the scale factor behaves according
to expression (17). During the period (te ≤ t ≤ teq) the Universe experienced
two phase transitions during which it might have been, for brief instants, dust–
like (Section 2). When one goes backwards in time the first phase transition
is the QCD. Considering that t+ corresponds to the age of the Universe at the
end of the QCD we write
Λ
R(t) = exp c 3 (tSN
2/3 1/2 teq t
− t0)
, t+ ≤ t ≤ teq (69)
tSN teq
9 To our best knowledge, this sequence of formulae (equations 67–77) has never been deducted
in the literature, although common-knowledge.