Anisotropic solutions for $R^2$ gravity model with a scalar field

TL;DR

This paper investigates anisotropic solutions in a pure $R^2$ gravity model with a scalar field, revealing singular behavior at zero Ricci scalar and deriving explicit solutions in Einstein and Jordan frames.

Contribution

It provides the first explicit anisotropic solutions for the $R^2$ gravity model with a scalar field using conformal transformations.

Findings

Anisotropic solutions exist only when Ricci scalar does not cross zero.

Explicit solutions are obtained in the Einstein frame.

Solutions in the Jordan frame are constructed in quadratures.

Abstract

We study anisotropic solutions for the pure $R^2$ gravity model with a scalar field in the Bianchi I metric. The evolution equations have a singularity at zero value of the Ricci scalar $R$ for anisotropic solutions, whereas these equations are smooth for isotropic solutions. So, there is no anisotropic solution with the Ricci scalar smoothly changing its sign during evolution. We have found anisotropic solutions using the conformal transformation of the metric and the Einstein frame. The general solution in the Einstein frame has been found explicitly. The corresponding solution in the Jordan frame has been constructed in quadratures.