Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 3
(e.g Lyth, 1993). The Λ term was introduced by Einstein with the purpose of
constructing a static cosmological model for the Universe. However, with the
discovery of the expansion of the Universe (Slipher, 1917) the model became
obsolete. More recently, a Λ > 0 term was introduced again in order to account
for the remarkable discovery that the expansion of the Universe is, in fact,
accelerating rather than retarding (Section 1.5).
Energy conservation leads to a third equation, which can also be derived
from equations (2) and (3), and is just a consequence of the First Law of Thermodynamics
(e.g. Yao et al., 2006)
˙ρ = −3 ˙ R
(ρ + p). (4)
R
We need also an equation of state (EoS) relating the pressure p to the energy
density ρ at a given epoch. This relation is, in general, non–trivial. However,
in Cosmology, where one deals with dilute gases, the EoS can be written in a
simple linear form (e.g. Carr, 2003; Ryden, 2003)
p = wρ (5)
where the dimensionless quantity w is the so–called adiabatic index. Normally
w is a constant such that 0 ≤ w ≤ 1. If w = 0 we are in the case of a pressureless
matter–dominated universe and, if w = 1 we have a stiff EoS which may be the
case if the universe is dominated by a scalar field 3 (e.g. Harada & Carr, 2005).
In the case of cosmological perturbations the radiation fluid behaves as a
perfect (i.e. dissipationless) fluid, entropy (s) in a comoving volume is conserved
and, one has a reversible process. The isentropic4 sound speed can be written
as (e.g. Schmid et al., 1999)
c 2
∂p
s = = w. (6)
∂ρ
s
In the early hot and dense primordial Universe it is appropriate to assume
an EoS corresponding to a gas composed of radiation and relativistic massive
particles with w =1/3 (e.g. Carr, 2003)
p = ρ
3
which means that, in the case of a radiation–dominated universe, the sound
speed is given by
cs = 1
√ 3 . (8)
3 A scalar field is a field that associates a scalar value to every point in space. On the other
hand, a vector field associates a vector to every point in space. In quantum field theory, a
scalar field is associated with spin 0 particles (scalar bosons) and a vector field is associated
with spin 1 particles (vector bosons).
4 A thermodynamic process that occurs at a constant entropy (s) is sayd to be isentropic.
If it is a reversible process then it is identical to an adiabatic process, i.e., a thermodynamic
process in which there is no energy added or subtracted from the system.
(7)