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 . If () one has
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 , as . This fixes the coefficients to be and , giving the following solution
We define the power spectrum of curvature perturbations, [573, 390].
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