Primordial non-Gaussianity in the cosmological perturbations - CBPF

The second fact, that S = g∗ST 3 a 3 = constant, implies that the temperature of the universe
evolves as
T ∼ g −1/3
∗S a−1 . (133)
When g∗S is constant one gets the familiar result T ∼ a−1 . The factor g −1/3
∗S enters because whenever
a particle species becomes non-relativistic and disappears from the plasma, its entropy is transferred
to the other relativistic particles in the thermal plasma causing T to decrease slightly less slowly
(sometimes it is said, but in a wrong way, that the universe slightly reheats up).
Part III
The inflationary cosmology
In this chapter we will discuss the inflationary universe. As we will come out along the way, inflation
is responsible not only for the observed homogeneity and isotropy of the universe, but also for
its inhomogeneities. Furthermore, inflation links the quantum mechanical microphysics to the the
macrophysics of the universe as a whole. It is a beautiful example of connection between high energy
physics and cosmology.
Before launching ourselves into the description of inflation, we would like to go back to the
concept of conformal time which will be useful in the next sections. The conformal time τ is defined
through the following relation
dτ = dt
. (134)
a
The metric ds2 = −dt2 + a2 (t)dx2 then becomes
ds 2 = a 2 (τ) −dτ 2 + dx 2 . (135)
The reason why τ is called conformal is manifest from Eq. (135): the corresponding FRW line element
is conformal to the Minkowski line element describing a static four dimensional hypersurface.
Any function f(t) satisfies the rule
f(t) ˙ = f ′ (τ)
,
a(τ)
(136)
f(t) ˙ = f ′′ (τ)
a2 (τ) − H f ′ (τ)
a2 ,
(τ)
(137)
where a prime now indicates differentation wrt to the conformal time τ and
In particular we can set the following rules
H = a′
a
34
. (138)