Primordial non-Gaussianity in the cosmological perturbations - CBPF

3 Entropy
Throughout most of the history of the universe the reaction rates of particles in the thermal bath
were much greater than the expansion rate of the universe and local thermal equilibrium (LTE) was
attained. In this case the entropy per comoving volume element remains constant. The entropy in a
comoving volume provides a very useful quantity during the expansion of the universe. The second
law of theormodynamics as applied to a comoving volume elementof unit coordinate volume and
physical volume V = a 3 , implies that (we assume small chemical potentials)
The integrability condition
relates the energy density and pressure
or, equivalently,
We therefore obtain from Eq. (118) that
dS = 1
T
T dS = d(ρV ) + P dV = d[(ρ + P )V ] − V dP. (118)
∂ 2 S
∂T ∂V = ∂2 S
∂V ∂T
T dP
dT
dP =
(119)
= ρ + P, (120)
ρ + P
T
dT (ρ + P )V
d[(ρ + P )V ] − (ρ + P )V = d
T 2 T
That is, up to an additional constant, the entropy per comoving volume is
Reacll that the first law (energy conservation) can be written as
Thus substituting (121) in Eq. (124), we obtain
dT. (121)
+ const. . (122)
3 (ρ + P )
S = a . (123)
T
d[(ρ + P )V ] = V dP. (124)
(ρ + P )V
d
= 0. (125)
T
This implies that in thermal equilibrium the entropy per comoving volume V , S, is conserved. It is
useful to define the entropy density s as
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