On the order reduction

TL;DR

This paper extends the order reduction technique to higher perturbative approximations, analyzes its convergence conditions, and demonstrates its application to inflationary cosmology, revealing convergence in strong coupling and slow-roll regimes.

Contribution

It introduces an iterative extension of order reduction, analyzes its convergence properties, and applies it to Starobinsky's inflationary model.

Findings

Order reduction does not converge in weak coupling limit.

Converges in strong coupling regimes.

Successfully applied to inflationary model in slow-roll regime.

Abstract

In this work we present an extension of the technique of the order reduction to higher perturbative approximations in an iterative fashion. The intention is also to analyze more carefully the conditions for the validity of the order reduction technique. With this in mind, a few simple situations in which the iterative order reduction converges analytically to the exact solutions are presented as examples. It is discovered that the order reduction as a perturbative iterative technique does not converge in the weak coupling limit as most of the known perturbative schemes, at least when applied to these examples. Also, considering these specific examples, the convergence of the order reduction occurs in strong coupling regimes. As a more realistic case, the order reduction is applied to Starobinsky's inflationary model is presented. It is verified that the method converges to the inflationary solution in the slow-roll regime.