### 7.3 The spectra of perturbations in inflation based on f (R) gravity

Let us study the spectra of scalar and tensor perturbations generated during inflation in metric f (R) gravity. Introducing the quantity , we have and
Since the field kinetic term is absent, one has in Eqs. (7.42) and (7.49). Under the conditions (), the spectral index of curvature perturbations is given by .

In the absence of the matter fluid, Eq. (2.16) translates into

which gives for . Hence we obtain [315]
From Eqs. (7.50) and (7.60), the amplitude of is estimated as

Using the relation , the spectral index (7.57) of tensor perturbations is given by

which vanishes at first-order of slow-roll approximations. From Eqs. (7.58) and (7.63) we obtain the tensor-to-scalar ratio

#### 7.3.1 The model ()

Let us consider the inflation model: (). From the discussion given in Section 3.1 the slow-roll parameters () are constants:

In this case one can use the exact results (7.48) and (7.56) with and given in Eqs. (7.42) and (7.54) (with ). Then the spectral indices are

If we obtain the scale-invariant spectra with and . Even the slight deviation from leads to a rather large deviation from the scale-invariance. If , for example, one has , which does not match with the WMAP 5-year constraint:  [367].

#### 7.3.2 The model

For the model , the spectrum of the curvature perturbation shows some deviation from the scale-invariance. Since inflation occurs in the regime and , one can approximate . Then the power spectra (7.63) and (7.58) yield

where we have employed the relation .

Recall that the evolution of the Hubble parameter during inflation is given by Eq. (3.9). As long as the time at the Hubble radius crossing () satisfies the condition , one can approximate . Using Eq. (3.9), the number of e-foldings from to the end of inflation can be estimated as

Then the amplitude of the curvature perturbation is given by
The WMAP 5-year normalization corresponds to at the scale  [367]. Taking the typical value , the mass is constrained to be
Using the relation , it follows that . Hence the spectral index (7.62) reduces to
For we have , which is in the allowed region of the WMAP 5-year constraint ( at the 68% confidence level [367]). The tensor-to-scalar ratio (7.65) can be estimated as
which satisfies the current observational bound  [367]. We note that a minimally coupled field with the potential in Einstein gravity (chaotic inflation model [393]) gives rise to a larger tensor-to-scalar ratio of the order of . Since future observations such as the Planck satellite are expected to reach the level of , they will be able to discriminate between the chaotic inflation model and the Starobinsky’s f (R) model.

#### 7.3.3 The power spectra in the Einstein frame

Let us consider the power spectra in the Einstein frame. Under the conformal transformation , the perturbed metric (6.1) is transformed as

We decompose the conformal factor into the background and perturbed parts, as
In what follows we omit a bar from . We recall that the background quantities are transformed as Eqs. (2.44) and (2.47). The transformation of scalar metric perturbations is given by
Meanwhile vector and tensor perturbations are invariant under the conformal transformation (, , ).

Using the above transformation law, one can easily show that the curvature perturbation in f (R) gravity is invariant under the conformal transformation:

Since the tensor perturbation is also invariant, the tensor-to-scalar ratio in the Einstein frame is identical to that in the Jordan frame. For example, let us consider the model . Since the action in the Einstein frame is given by Eq. (2.32), the slow-roll parameters and vanish in this frame. Using Eqs. (7.49) and (3.27), the spectral index of curvature perturbations is given by
where we have ignored the term of the order of . Since in the slow-roll limit (), Eq. (7.78) agrees with the result (7.72) in the Jordan frame. Since in the Einstein frame, Eq. (7.59) gives the tensor-to-scalar ratio
where the background equations (3.21) and (3.22) are used with slow-roll approximations. Equation (7.79) is consistent with the result (7.73) in the Jordan frame.

The equivalence of the curvature perturbation between the Jordan and Einstein frames also holds for scalar-tensor theory with the Lagrangian  [411240]. For the non-minimally coupled scalar field with  [269241] the spectral indices of scalar and tensor perturbations have been derived by using such equivalence [366590].