Primordial non-Gaussianity in the cosmological perturbations - CBPF

where V ′ (φ) = (dV (φ)/dφ). Note, in particular, the appearance of the friction term 3H ˙ φ: a scalar
field rolling down its potential suffers a friction due to the expansion of the universe.
We can write the energy-momentum tensor of the scalar field
Tµν = ∂µφ∂νφ − gµν L.
The corresponding energy density ρφ and pressure density Pφ are
T00 = ρφ = ˙ φ2 2
Tii = Pφ = ˙ φ2 2
(∇φ)2
+ V (φ) + , (175)
2a2 (∇φ)2
− V (φ) − . (176)
6a2 Notice that, if the gradient term were dominant, we would obtain Pφ = −ρφ/3, not enough to drive
inflation. We can now split the inflaton field in
φ(t) = φ0(t) + δφ(x, t),
where φ0 is the ‘classical’ (infinite wavelength) field, that is the expectation value of the inflaton field
on the initial isotropic and homogeneous state, while δφ(x, t) represents the quantum fluctuations
around φ0. as for now, we will be only concerned with the evolution of the classical field φ0. This
separation is justified by the fact that quantum fluctuations are much smaller than the classical
value and therefore negligible when looking at the classical evolution. To not be overwhelmed by
the notation, we will keep indicating from now on the classical value of the inflaton field by φ. The
energy-momentum tensor becomes
If
we obtain the following condition
T00 = ρφ = ˙ 2
φ0
2 + V (φ0) (177)
Tii = Pφ = ˙ φ 2 0
2 − V (φ0). (178)
V (φ0) ≫ ˙ φ 2 0
Pφ −ρφ
From this simple calculation, we realize that a scalar field whose energy is dominant in the universe
and whose potential energy dominates over the kinetic term gives inflation! Inflation is driven by
the vacuum energy of the inflaton field.
6.5 Slow-roll conditions
Let us now quantify better under which circumstances a scalar field may give rise to a period of
inflation. The equation of motion of the field is
˙φ0 + 3H ˙ φ0 + V ′ (φ0) = 0 (179)
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