Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 17
Figure 5: The schematic illustration of the potential of the hybrid (double)
inflation model. This model is characterized by multiple scalar fields. In this
example we have two scalar fields: φ and χ. For large φ we have a situation similar
to the other models with a single field. However, when φ reaches the critical
value φc a different behaviour takes place. In particular, the energy density
during inflation can be much lower than normal while still giving suitably large
density perturbations, and secondly the field φ can be rolling extremely slowly
which is of benefit to particle physics model building (Liddle, 1999; Tsujikawa,
2003).
The inflaton φ is an homogeneous scalar field, whose potential energy V (φ)
leads to the exponential expansion of the Universe. The number of e–folds that
elapsed during inflation (equation 40) can be written also as a function of φ
(e.g. Boyanovsky et al., 2006)
N(φ(t)) = − 1
m 2 pl
φe
φ(t)
V (φ)
V ′ dφ. (41)
(φ)
The energy density and the pressure density of the inflaton can be described,
respectively, as (e.g. Liddle & Lyth, 1993)
ρ = 1
2 ˙ φ 2 + V (φ) (42)
p = 1
2 ˙ φ 2 − V (φ). (43)
Substituting equations (42) and (43) for equations (2) and (4) we get (e.g. Liddle
& Lyth, 1993) 7
H 2 = 8πG
3
1
2 ˙ φ 2
+ V (φ) = 8π
3m2
1
pl 2 ˙ φ 2
+ V (φ)
(44)
7 We have written equation (44) also in terms of the Planck mass mpl =(c/G) 1/2 with
= c = 1.