# Construction and applications of the manifestly gauge invariant expressions of the solutions of the cosmological perturbation theory

## TL;DR

This paper develops a gauge invariant framework for analyzing cosmological perturbations across multiple times and steps, simplifying the understanding of fluctuation evolution in complex reheating scenarios.

## Contribution

It extends the gauge invariant perturbation theory to many time points and solutions, enabling clearer analysis of multi-step reheating and fluctuation generation processes.

## Key findings

- Derived a formula for changing bases of many time background objects.
- Proved that the dominant fluctuation process near the present erases earlier fluctuations.
- Simplified calculation of present-day Bardeen parameters by focusing on dominant processes.

## Abstract

After giving how to construct the gauge invariant perturbation variable in an arbitrarily high order and in the arbitrary background spacetime, we consider the manifestly gauge invariant theory of the cosmological perturbations in the long wavelength limit with the spatially flat homogeneous isotropic universe being the background spacetime. In the previous paper, the physical laws become manifestly gauge invariant, by writing them in terms of only the single time background/scalar like objects. In the present paper, by extending our definition of the background/scalar like objects from the single time case to the many time case and by writing the solution of the physical law in the form where many time background/scalar like objects are vanishing, the solutions become the manifestly gauge invariant. We derive the formula changing the bases of the many time background/scalar like objects by which we can change the time slices for many times appearing in the solution including the initial time and the final time. We use this formula to our treatment of the evolution of the several slow rolling scalar fields using the $\tau$ function. In the manifestly gauge invariant manner, we discuss the solution of the many step reheating, that is, the reheating with the arbitrarily many energy transfer steps. By using the useful parametrization of the many step reheating, we manifestly prove that in the many step reheating system the prominent fluctuation generation process the nearest to the present time erases all the fluctuations generated by the previous fluctuation generation processes. Therefore in order to know the first/second order Bardeen parameters at the present time it is sufficient to calculate only the effects of the the prominent fluctuation generation processes such as curvaton and the modulated reheating the nearest to the present time.