In General Relativity with a canonical scalar field one has
and
, which corresponds
to
. Then the perturbation
corresponds to
. In the spatially flat
gauge (
) this reduces to
, which implies that the perturbation
corresponds to a
canonical scalar field
. In modified gravity theories it is not clear at this stage that the
perturbation
corresponds a canonical field that should be quantized, because Eq. (7.37
) is
unchanged by multiplying a constant term to the quantity
defined in Eq. (7.38
). As we will see in
Section 7.4, this problem is overcome by considering a second-order perturbed action for the theory (6.2
)
from the beginning.
In order to derive the spectrum of curvature perturbations generated during inflation, we introduce the
following variables [315]
During inflation one has , so that
. For the modes deep inside the Hubble
radius (
, i.e.,
) the perturbation
satisfies the standard equation of a canonical field
in the Minkowski spacetime:
. After the Hubble radius crossing (
) during inflation,
the effect of the gravitational term
becomes important. In the super-Hubble limit (
, i.e.,
) the last term on the l.h.s. of Eq. (7.37
) can be neglected, giving the following solution
In the asymptotic past () the solution to Eq. (7.39
) is determined by a vacuum state in
quantum field theory [88], as
. This fixes the coefficients to be
and
,
giving the following solution
We define the power spectrum of curvature perturbations,
Using the solution (7.45http://www.livingreviews.org/lrr-2010-3 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |