# Analytic Solution of the Starobinsky Model for Inflation

## TL;DR

This paper demonstrates that the Starobinsky inflation model's field equations are integrable and provides explicit analytical solutions using Painlevé series for different spatial curvatures, enhancing understanding of early universe dynamics.

## Contribution

It proves the integrability of the Starobinsky model's equations and derives explicit analytical solutions for zero and nonzero spatial curvature cases.

## Key findings

- Field equations pass the singularity test indicating integrability
- Analytical solutions are obtained using Painlevé series
- Leading-order terms describe the radiation era in early universe

## Abstract

We prove that the field equations of the Starobinsky model for inflation in a Friedmann-Lema\^{\i}tre-Robertson-Walker constitute an integrable system as the field equations pass the singularity test. The analytical solution in terms of a Painlev\'{e} Series for the Starobinsky model is presented for the case of zero and nonzero spatial curvature. In both cases the leading-order term describes the radiation era provided by the corresponding higher-order theory.