In the absence of the matter fluid, Eq. (2.16) translates into
Using the relation , the spectral index (7.57) of tensor perturbations is given by
Let us consider the inflation model: (). From the discussion given in Section 3.1 the slow-roll parameters () are constants:
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: .
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
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. Taking the typical value , the mass is constrained to be ). The tensor-to-scalar ratio (7.65) can be estimated as . We note that a minimally coupled field with the potential in Einstein gravity (chaotic inflation model ) 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.
Let us consider the power spectra in the Einstein frame. Under the conformal transformation , the perturbed metric (6.1) is transformed as
Using the above transformation law, one can easily show that the curvature perturbation in f (R) gravity is invariant under the conformal transformation:
The equivalence of the curvature perturbation between the Jordan and Einstein frames also holds for scalar-tensor theory with the Lagrangian [411, 240]. For the non-minimally coupled scalar field with [269, 241] the spectral indices of scalar and tensor perturbations have been derived by using such equivalence [366, 590].
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