Late-time acceleration without a vacuum term in ${f(R,L_m)}$ gravity: scaling deSitter dynamics and parameter constraints

TL;DR

This work analyzes late-time cosmic acceleration within f(R,L_m) gravity, focusing on two realizations: Case A with a vacuum-like term γ and a nonlinear matter term βρ_m^n, and Case B with a linear βρ_m plus a nonlinear γρ_m^n contribution. Through a bounded phase-space (dynamical-systems) analysis, Case A requires γ to drive acceleration and yields a standard radiation→matter→de Sitter sequence only for n≳4/5, while Case B supports a genuine scaling de Sitter attractor for 0<n<1/2, enabling acceleration without an explicit cosmological constant. Background observational constraints using CC, Union3, DESI BAO, and a BBN prior favor n≈1.08±0.05 in Case A and n≈0.05±0.10 in Case B, with Case B residing in the accelerating window yet remaining ΛCDM-compatible at the background level. The study identifies necessary stability conditions (f_R>0 and f_{L_m}>0, luminal tensor propagation) and calls for a dedicated perturbation-level analysis to assess growth and lensing, which would decisively test the viability of the dynamical acceleration mechanism in Case B.

Abstract

We investigate late-time cosmic acceleration in $f(R,L_m)$ gravity driven by nonlinear matter contributions, focusing on the class $f(R,L_m)=R/2+c_1 L_m+c_n L_m^{n}+c_0$ with the explicit choice $L_m=ρ_m$ and an uncoupled radiation sector. We analyze two realizations: (i) Case A: $f(R,L_m)=R/2+βρ_m^{n}+γ$, where $γ$ acts as a vacuum term, and (ii) Case B: $f(R,L_m)=R/2+βρ_m+γρ_m^{n}$, where the nonlinear sector can mimic dark energy without an explicit cosmological constant. For each case, we construct a bounded autonomous system, classify all critical points and their stability, and compute cosmographic diagnostics. The phase-space analysis shows that Case A reproduces the standard radiation$\to$matter$\to$de~Sitter sequence only for $n\gtrsim 4/5$, with acceleration essentially enforced by the vacuum term. In contrast, Case~B admits a qualitatively distinct and phenomenologically appealing branch: for $0<n<1/2$ the system possesses a physical \emph{scaling} de~Sitter future attractor inside the bounded simplex, yielding radiation$\to$matter$\to$acceleration with $q=-1$ and $ω_{\rm eff}=-1$ and without introducing $c_0$. We confront both models with background data (CC, Union3, DESI BAO, plus a BBN prior on $Ω_b h^2$) using nested sampling and perform model comparison via Bayesian evidence and AIC/BIC. The full data combination constrains $n=1.08\pm0.05$ in Case A and $n=0.05\pm0.10$ in Case B (68\% CL), the latter lying within the accelerating window while remaining statistically consistent with $Λ$CDM kinematics at the background level. We also record minimal consistency conditions for stability (tensor no-ghost and luminal propagation) and motivate a dedicated perturbation-level analysis as the next step to test growth and lensing observables.

Figures (4)

• Figure 1: Phase-space portraits for the system (\ref{['SDxy']}) in the $(x,y)$ plane for Case A. Left: representative value $\tfrac{1}{2}<n<\tfrac{4}{5}$ ($n=0.7$). Center:$n=0.85$ (near the transition $n=\tfrac{4}{5}$). Right:$n=1$ ($\Lambda$CDM limit). Blue-shaded regions indicate physically viable matter densities ($0\leq\Omega_m\leq 1$). Gray-shaded regions correspond to $-1\leq q\leq 0$, representing accelerated expansion. Arrows denote the direction of evolution. Critical points are marked to highlight the dynamical behavior for different $n$.
• Figure 2: Phase–space portraits of the system (\ref{['DS-pol-linxA1']})–(\ref{['DS-pol-linyA1']}) for representative $n$. Top-left: $n=0$ ($\Lambda$CDM limit). Top-center: $n=0.15$ (viable accelerating regime $0<n<1/2$). Top-right: $n=0.75$ (non-accelerating regime $1/2<n<4/5$). Bottom-left: $n=0.85$ (nonlinear attractor regime $4/5<n<1$). Bottom-center: $n=1.1$ (future matter attractor, $1<n\le 4/3$). Bottom-right: $n=1.5$ (asymptotic matter–dominated attractor, $n>4/3$).
• Figure 3: Marginalized posterior distributions for the parameters of the Nonlinear Matter Coupling with Vacuum Term model (left) and the Nonlinear Matter Terms as Dynamical Dark Energy Mimickers model (right), at 1-$\sigma$ and 2-$\sigma$ confidence level. The vertical dashed lines represent the case of the standard $\Lambda$CDM model.
• Figure 4: Functional posterior of the, left, dark matter contribution $\Omega_{dm,0} u(z)$ and, right, the Non-Lineal term $\Omega_{NL,0} u^n(z)$, using the data set combination CC+Union3+DESI. The solid red lines indicate the $\Lambda$CDM model. The 68% (1$\sigma$) and 95% (2$\sigma$) confidence intervals are plotted as black lines.