Let us consider scalar and tensor perturbations generated during inflation for the theories (6.2) without taking into account the perfect fluid (). In f (R) gravity the contribution of the field such as is absent in the perturbation equations (6.11) – (6.16). One can choose the gauge condition , so that . In scalar-tensor theory in which is the function of alone (i.e., the coupling of the form without a non-linear term in ), the gauge choice leads to . Since in this case, we have .

We focus on the effective single-field theory such as f (R) gravity and scalar-tensor theory with the coupling , by choosing the gauge condition and . We caution that this analysis does not cover the theory such as [500], because the quantity depends on both and (in other words, ). In the following we write the curvature perturbations and as .

7.1 Curvature perturbations

7.2 Tensor perturbations

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

7.3.1 The model ()

7.3.2 The model

7.3.3 The power spectra in the Einstein frame

7.4 The Lagrangian for cosmological perturbations

7.2 Tensor perturbations

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

7.3.1 The model ()

7.3.2 The model

7.3.3 The power spectra in the Einstein frame

7.4 The Lagrangian for cosmological perturbations

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