7 Perturbations Generated During Inflation

Let us consider scalar and tensor perturbations generated during inflation for the theories (6.2View Equation) without taking into account the perfect fluid (SM = 0). In f (R) gravity the contribution of the field ϕ such as δϕ is absent in the perturbation equations (6.11View Equation) – (6.16View Equation). One can choose the gauge condition δF = 0, so that ℛ δF = ψ. In scalar-tensor theory in which F is the function of ϕ alone (i.e., the coupling of the form F (ϕ)R without a non-linear term in R), the gauge choice δϕ = 0 leads to ℛ = ψ δϕ. Since δF = F δϕ = 0 ,ϕ in this case, we have ℛ δF = ℛ δϕ = ψ.

We focus on the effective single-field theory such as f (R) gravity and scalar-tensor theory with the coupling F (ϕ)R, by choosing the gauge condition δϕ = 0 and δF = 0. We caution that this analysis does not cover the theory such as ℒ = ξ(ϕ)R + αR2 [500], because the quantity F depends on both ϕ and R (in other words, δF = F δϕ + F δR ,ϕ ,R). In the following we write the curvature perturbations ℛ δF and ℛ δϕ as ℛ.

 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 f(R ) = αRn (n > 0)
  7.3.2 The model f(R ) = R + R2∕ (6M 2)
  7.3.3 The power spectra in the Einstein frame
 7.4 The Lagrangian for cosmological perturbations

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