Focusing conditions for extended teleparallel gravity theories

TL;DR

This paper derives focusing conditions for extended teleparallel gravity theories, specifically $f(T)$ models, analyzing their implications for cosmological models and comparing with $f(R)$ and $f(Q)$ theories.

Contribution

It introduces focusing conditions for $f(T)$ gravity and examines their behavior in cosmological models, highlighting differences from or similarities to other modified gravity theories.

Findings

Certain $f(T)$ models satisfy weak and strong focusing conditions.

Focusing behavior varies across different $f(T)$ cosmological models.

Similar focusing properties are observed in $f(R)$ and $f(Q)$ theories.

Abstract

In the context of extended theories of teleparallel gravity $f(T)$ we derive the focusing conditions for a one-parameter dependent congruence of timelike auto-parallels of the Levi-Civita connection. We also consider the $f(T)$ field equations for a general metric tensor before moving on to consider a spatially flat Robertson-Walker space-time. Following this, we study the expansion rate for a one-parameter dependent congruence of timelike auto-parallel curves of the Levi-Civita connection. Given the fact that test particles follow auto-parallels of the Levi-Civita connection, the torsion-free Raychaudhuri equation is used in order to determine the desired focusing conditions. Finally we study the obtained focusing conditions for three $f(T)$ paradigmatic cosmological models and discuss the satisfaction or violation of these conditions. Through this, we find $f(T)$ models that allow for the weak and strong focusing conditions to be satisfied or violated. It is mentioned that this behaviour can also be found in the so-called $f(R)$ and $f(Q)$ theories.