Primordial non-Gaussianity in the cosmological perturbations - CBPF

2 Standard cosmology
All of the discussions in the previous section concerned the kinematics of a universe described by a
FRW. The dynamics of the expanding universe only appeared implicitly in the time dependence of
the scale factor a(t). To make this time dependence explicit, one must solve for the evolution of the
scale factor using the Einstein equations
Rµν − 1
2 gµνR = 8πGNTµν + Λgµν, (41)
where Tµν is the stress-energy tensor for all the fields present (matter, radiation, and so on) and we
have also included the presence of a cosmological constant. With very minimal assumptions about
the right-hand side of the Einstein equations, it is possible to proceed without detailed knowledge
of the properties of the fundamental fields that contribute to the stress tensor Tµν.
2.1 The stress-energy momentum tensor
To be consistent with the symmetries of the metric, the total stress-energy tensor tensor must be
diagonal, and by isotropy the spatial components must be equal. The simplest realization of such a
stress-energy tensor is that of a perfect fluid characterized by a time-dependent energy density ρ(t)
and pressure P (t)
T µ ν = (ρ + P )u µ uν + P δ µ ν = diag(−ρ, P, P, P ), (42)
where u µ = (1, 0, 0, 0) in a comoving coordinate system. This is precisely the energ-ymomentum
tensor of a perfect fluid. The four-vector uµ is known as the velocity field of the fluid, and the
comoving coordinates are those with respect to which the fluid is at rest. In general, this matter
content has to be supplemented by an equation of state. This is usually assumed to be that of a
barytropic fluid, i.e. one whose pressure depends only on its density, P = P (ρ). The most useful
toy-models of cosmological fluids arise from considering a linear relationship between P and ρ of the
type
P = wρ , (43)
where w is known as the equation of state parameter. Occasionally also more exotic equations of
state are considered. For non-relativistic particles (NR) particles, there is no pressure, pNR = 0, i.e.
wNR = 0, and such matter is usually referred to as dust. The trace of the energy-momentum tensor
is
T µ µ = −ρ + 3P. (44)
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