Primordial non-Gaussianity in the cosmological perturbations - CBPF

8πGN
ρia
3
2 = cia −(1+3wi)
, (76)
for some constant ci. Then the Friedman equation takes the more explicit form
˙a 2 =
i
cia −(1+3wi) − k + Λ
3 a2 . (77)
In addition to the vacuum energy (and pressure), there are typically two other kinds of matter which
are relevant in our approximation, namely matter in the form of dust and radiation. Denoting the
corresponding constants by cm and cr respectively, the Friedman equation that we will be dealing
with takes the form
˙a 2 = cm
a
cr Λ
+ − k +
a2 3 a2 , (78)
illustrating the qualitatively different conntributions to the time-evolution. One can then charac-
terise the different eras in the evolution of the universe by which of the above terms dominates, i.e.
gives the leading contribution to the equation of motion for a. This already gives some insight into
the physics of the situation. We will call a universe matter-diminated if the piece cm/a dominates;
radiation-dominated if the piece cr/a 2 dominates; curvature-dominated if the piece k dominates and
vacuum-dominated if the piece Λa 2 dominates. As mentioned before, for a long time it was believed
that our present universe is purely matter dominated while recent observations appear to indicate
that contributions from both matter and the cosmological constant are non-negligible.
Here are some immediate consequences of the Friedman equation (78):
1. No matter how small cr is, provided that it is non-zero, for sufficiently small values of a that
term will dominate and one is in the radiation dominated era. In that case, one finds the
characteristic behaviour
2. If matter dominates, one finds the characteristic behaviour
3. For a general equation of state w = −1, one finds
˙a 2 = cr
a 2 ⇒ a(t) ∼ t1/2 (RD). (79)
˙a 2 = cm
a ⇒ a(t) ∼ t2/3 (MD). (80)
2
a(t) ∼ t 3(1+w) . (81)
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