Primordial non-Gaussianity in the cosmological perturbations - CBPF

and for k = +1, thus Λ > 0, one has
a±(t) =
3
Λ e±√ Λ/3 t , (98)
3
a±(t) =
Λ cosh Λ/3 t. (99)
This is also known as the de Sitter universe. It is a maximally symmetric (in space-time) solution
of the Einstein equations with a cosmological constant and thus has a metric of constant curvature.
But we know that such a metric is unique. Hence the three solutions with λ > 0, for k = 0, ±1 must
all represent the same space-time metric, only in different coordinate systems. This is interesting
because it shows that de Sitter space is so symmetric that it has space-like slicings by three-spheres,
by three-hyperboloids and by three-planes. The solution for k = −1 involves sin Λ/3t for Λ < 0
and sinh Λ/3t for Λ > 0. The former is known as the anti de Sitter universe.
Part II
Equilibrium thermodynamics
Today the radiation, or relativistic particles, in the universe is comprised of the 2.75 K microwave
photons and the three cosmic seas of about 1.96 K relic neutrinos. because the early universe was
to a good approximation in thermal equilibrium, there should have been other relativistic particles
present, with comparable abundances. Before going on to discuss the early RD phase, we will quickly
review some basic thermodynamics.
The number density n, energy density ρ and pressure P of a dilute, weakly interacting gas of
particles with g internal degrees of freedom is given in terms of its phase space distribution function
f(p)
n =
ρ =
P =
g
(2π) 3
g
(2π) 3
g
(2π) 3
d 3 p f(p),
d 3 p E(p) f(p),
d 3 p |p|2
f(p), (100)
3E(p)
where E 2 = |p| 2 + m 2 . For a species in kinetic equilibrium, the phase occupancy f is given by the
familiar Fermi-Dirac or Bose-Einstein distributions
f(p) = [exp(E − µ)/T ± 1] −1 , (101)
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