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 [573, 390].http://www.livingreviews.org/lrr-2010-3 |
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