Asymptotic solutions in f(R)-gravity

TL;DR

This paper investigates the asymptotic cosmological solutions in $R + eta R^{N}$-gravity, revealing how late-time behavior depends on matter properties and higher-curvature effects, with new oscillatory solutions introduced.

Contribution

The study introduces a novel set of asymptotic oscillatory solutions in $f(R)$-gravity using a new averaging method, and analyzes their stability and late-time behavior.

Findings

Late-time behavior depends on the sign of $eta - eta_{crit}$.

Oscillatory solutions exist for $N>2$ and influence the universe's evolution.

Differences between $N=2$ and $N>2$ cases are highlighted.

Abstract

We study cosmological solutions in $R + \beta R^{N}$-gravity for an isotropic Universe filled with ordinary matter with the equation of state parameter $\gamma$. Using the Bogolyubov-Krylov-Mitropol'skii averaging method we find asymptotic oscillatory solutions in terms of new functions, which have been specially introduced by us for this problem and appeared as a natural generalization of the usual sine and cosine. It is shown that the late-time behaviour of the Universe in the model under investigation is determined by the sign of the difference $\gamma-\gamma_{crit}$ where $\gamma_{crit}=2N/(3N-2)$. If $\gamma < \gamma_{crit}$, the Universe reaches the regime of small oscillations near values of Hubble parameter and matter density, corresponding to General Relativity solution. Otherwise higher-curvature corrections become important at late times. We also study numerically basins of attraction for the oscillatory and phantom solutions, which are present in the theory for $N>2$. Some important differences between $N=2$ and $N>2$ cases are discussed.