### 7.1 Curvature perturbations

Since and in Eq. (6.12) we obtain
Plugging Eq. (7.34) into Eq. (6.11), we have
Equation (6.14) gives
where we have used the background equation (6.7). Plugging Eqs. (7.34) and (7.35) into Eq. (7.36), we find that the curvature perturbation satisfies the following simple equation in Fourier space
where is a comoving wavenumber and
Introducing the variables and , Eq. (7.37) reduces to
where a prime represents a derivative with respect to the conformal time .

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]

where . Then the quantity can be expressed as
If the parameter is constant, it follows that  [573]. If () one has
Then the solution to Eq. (7.39) can be expressed as a linear combination of Hankel functions,
where and are integration constants.

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

where and are integration constants. The second term can be identified as a decaying mode, which rapidly decays during inflation (unless the field potential has abrupt features). Hence the curvature perturbation approaches a constant value after the Hubble radius crossing ().

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 (7.45), we obtain the power spectrum [317]
where we have used the relations for and . Since the curvature perturbation is frozen after the Hubble radius crossing, the spectrum (7.47) should be evaluated at . The spectral index of , which is defined by , is
where is given in Eq. (7.42). As long as () are much smaller than 1 during inflation, the spectral index reduces to
where we have ignored those terms higher than the order of ’s. Provided that the spectrum is close to scale-invariant (). From Eq. (7.47) the power spectrum of curvature perturbations can be estimated as
A minimally coupled scalar field in Einstein gravity corresponds to , and , in which case we obtain the standard results and in slow-roll inflation [573390].