# Dynamical system analysis of generalized energy-momentum-squared gravity

## TL;DR

This paper analyzes the dynamics of a modified gravity theory, energy-momentum squared gravity, using dynamical systems to explore its phase space and cosmological implications, including accelerated expansion and matter eras.

## Contribution

It introduces a dynamical system analysis for the $f(R, T^2)$ gravity model, revealing diverse cosmological behaviors and phase space structures depending on model parameters.

## Key findings

- Phase space complexity varies with parameters $m$ and $n$.
- Models can reproduce matter, dark energy, and accelerated expansion epochs.
- Richer phase space allows for multiple cosmological scenarios.

## Abstract

In this work we have investigated the dynamics of a recent modification to the general theory of relativity, the energy-momentum squared gravity model $f(R,\mathbf{T^2})$, where $R$ represents the scalar curvature and $\mathbf{T^2}$ the square of the energy-momentum tensor. By using dynamical system analysis for various types of gravity functions $f(R,\mathbf{T^2})$, we have studied the structure of the phase space and the physical implications of the energy-momentum squared coupling. In the first case of functional where $f(R,\mathbf{T^2})=f_0 R^n(\mathbf{T^2})^m$, with $f_0$ constant, we have shown that the phase space structure has a reduced complexity, with a high sensitivity to the values of the $m$ and $n$ parameters. Depending on the values of the $m$ and $n$ parameters, the model exhibits various cosmological epochs, corresponding to matter eras, solutions associated with an accelerated expansion, or decelerated periods. The second model studied corresponds to the $f(R,\mathbf{T^2})=\alpha R^n+\beta\mathbf{(T^2)}^m$ form with $\alpha, \beta$ constant parameters. In this case, a richer phase space structure is obtained which can recover different cosmological scenarios, associated to matter eras, de--Sitter solutions, and dark energy epochs. Hence, this model represents an interesting cosmological model which can explain the current evolution of the Universe and the emergence of the accelerated expansion as a geometrical consequence.