On the initial conditions of scalar and tensor fluctuations in $f(R,\phi)$ gravity

TL;DR

This paper analyzes the initial conditions of scalar and tensor fluctuations during inflation in $f(R,)$ gravity, deriving quantization and gauge-invariant initial conditions for perturbations.

Contribution

It provides a detailed derivation of initial conditions for scalar and tensor perturbations in generalized scalar-tensor theories during inflation.

Findings

Scalar metric perturbations are negligible at early times compared to scalar field perturbations.

Derived WKB solutions for perturbations at early times.

Quantized scalar fluctuations and established initial conditions for the generalized Mukhanov-Sasaki equation.

Abstract

We have considered the perturbation equations governing the growth of fluctuations in generalized scalar tensor theory during inflation. we have found that the scalar metric perturbations at very early times are negligible compared with the scalar field perturbation, just like general relativity. At sufficiently early times, when $q/a\gg H$, we have obtained the metric and scalar field perturbation in the form of WKB solutions up to an undetermined coefficient. Then we have quantized the scalar fluctuations and expanded the metric and the scalar field perturbations with the help of annihilation and creation operators of the scalar field perturbation. The standard commutation relations of annihilation and creation operators fix the unknown coefficient. Going over to the gauge invariant quantities which are conserved beyond the horizon, we have obtained the initial condition of the generalized Mukhanov-Sasaki equation. And a similar procedure is performed for the case of tensor metric perturbation.