Bianchi I spacetimes in $f\left(Q\right)$-gravity

TL;DR

This paper explores Kasner and Kasner-like solutions within $f(Q)$-gravity, analyzing their properties, conditions for existence, and the impact of different connections on anisotropic universe evolution.

Contribution

It introduces Kasner-like solutions in $f(Q)$-gravity, identifies conditions for their existence, and performs a phase-space analysis of anisotropic dynamics.

Findings

Kasner-like solutions exist when the nonmetricity scalar $Q$ vanishes.

Kasner relations are valid only in the coincidence gauge.

Different connections influence the evolution of anisotropies.

Abstract

The Kasner spacetimes are exact solutions that are self-similar and of significant interest because they can describe the dynamic evolution of kinematic variables near the singularity of the Mixmaster universe. The chaotic behavior of the Mixmaster universe is related to the Kasner relations. This study investigates Kasner and Kasner-like solutions in symmetric teleparallel $f\left( Q\right) $-gravity. We consider three families of connections for the spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker. We found that Kasner-like solutions are present in the theory when the nonmetricity scalar $Q$ vanishes. Kasner-like relations are introduced only for the field equations of the connection defined in the coincidence gauge. Finally, we perform a detailed phase-space analysis to understand the dynamics of $f\left( Q\right) $-gravity in the evolution of anisotropies and the effects of different connections on this evolution.