Asymptotic dynamical analysis of $f(R,T^φ) = R+αT^φ + β(T^φ)^2/2$ cosmology

TL;DR

This work analyzes the asymptotic cosmological dynamics of the modified gravity model $f(R,T^φ)=R+α T^φ+β/2 (T^φ)^2$ by casting the cosmological equations into a compact autonomous system in a flat FLRW background. It identifies a rich set of critical points including matter-dominated saddles, stiff-fluid regimes, and de Sitter–type attractors, with the quadratic trace term $β/2 (T^φ)^2$ playing a central role in enabling late-time acceleration at the background level. Several accelerated fixed points reside in a degenerate scalar sector with $Q_s=0$, making linear perturbation analysis inconclusive there, while a genuinely ghost-free quasi-de Sitter point exists for $α>0$ with $Q_s>0$ and $c_s^2=0$ but as a saddle. The results indicate robustness of background acceleration in trace-coupled $f(R,T^φ)$ cosmologies and highlight the need for beyond-linear perturbation studies to assess full viability and observational compatibility, particularly concerning structure formation.

Abstract

In this work we investigate the asymptotic cosmological dynamics of a modified gravity model based on the $f(R,T^φ)$ theory, where $R$ denotes the Ricci scalar and $T^φ$ is the trace of the stress-energy tensor of a scalar field. Despite the extensive study of $f(R,T)$ gravity, the asymptotic implications of quadratic trace couplings in scalar field cosmology remain largely unexplored. We focus on a specific form given by $ f(R,T^φ) = R + αT^φ+ β(T^φ)^2/2$, in which the parameters $α$ and $β$ control the strength of non-minimal couplings between geometry and matter. We derive the set of cosmological equations for a spatially flat, homogeneous and isotropic universe and construct the autonomous system of first-order differential equations using a compact set of dimensionless variables. This formulation provides a foundation for the qualitative analysis of the asymptotic behavior. We identify and classify all critical points and analyze their stability properties. Finally, the energy conditions and the presence of dynamical instabilities are examined. We study the general scenario $α\neq 0$ and $β\neq 0$, along with the subcases $α= 0$ and $β= 0$, in order to compare with minimally coupled quintessence $α= β= 0$. We find that the quadratic term in $T^φ$ admits late-time accelerated de Sitter-like critical solutions at the background level. However, several accelerated points lie in a degenerate scalar sector with $Q_s=0$, where the standard linear perturbation criteria are inconclusive, while the quasi-de Sitter point with $Q_s>0$ is of saddle type. Therefore, establishing full perturbative viability requires going beyond the linear analysis in the degenerate sector.