Dynamical System Analysis of FLRW Model in f(R,L,T) Theory

Modified gravity theories have been extensively studied recently as viable substitutes for general relativity to deal with cosmological issues like dark energy and late-time cosmic acceleration. In the present work, we investigate the dynamical behavior of the $f(R,L,T)$ gravity model with a scalar field utilizing exponential potential, where $R$ represents the Ricci scalar, $L$ is the Lagrangian density and $T$ is the trace of the energy-momentum tensor. We concentrate on a specific type of modified gravity characterized by $f(R,L,T) =R+αL+βT$, where $α$ and $β$ are positive constants. We study the dynamical behavior and late-time evolution of a cosmological model using a thorough phase-space analysis. We assess important cosmological parameters at the critical places, such as the density parameters corresponding to various cosmic components, the deceleration parameter, and the effective equation of state parameter. The nature of the cosmic phases such as matter-dominated, radiation-dominated, and accelerated expansion eras, described using these quantities.

💡 Research Summary

The paper investigates a simple linear form of the recently proposed f(R,L,T) gravity, namely f(R,L,T)=R+αL+βT, where α and β are positive coupling constants that introduce non‑minimal interactions between geometry, the matter Lagrangian L and the trace T of the energy‑momentum tensor. Working in a spatially flat Friedmann‑Lemaître‑Robertson‑Walker (FLRW) background, the authors supplement the theory with a canonical scalar field ϕ endowed with an exponential potential V(ϕ)=V₀ e^{‑λϕ}, a standard choice for quintessence models. Assuming separate conservation of the matter, scalar and dark‑energy components, they derive the modified Friedmann equations and the Klein‑Gordon equation for ϕ.

To study the highly non‑linear system, they introduce four dimensionless variables: x₁²∝ϕ̇²/H², x₂²∝V/H², x₃²∝ρ_m/H² and x₄²∝ρ_de/H² (the latter containing the α,β‑dependent combination). These variables satisfy the constraint x₁²+x₂²+x₃²+x₄²=1, which is the normalized energy budget. Expressing the Hubble‑rate derivative in terms of the x_i’s yields compact formulas for the deceleration parameter q and the effective equation‑of‑state ω_eff.

By differentiating the x_i’s with respect to N=ln a, the authors obtain an autonomous four‑dimensional dynamical system (Eqs. 31‑34). Setting the right‑hand sides to zero gives eight critical points, labelled A±, B±, C± and D±. The points are interpreted as follows:

• A±: (0,0,±1,0) – pure matter domination (Ω_m=1, Ω_ϕ=Ω_de=0). They give q=½, ω_eff=0, corresponding to a decelerating, dust‑like universe. The eigenvalue spectrum contains both positive and negative values, so A± are saddles.

• B±: (0,0,0,±√{α+(1‑3ω_de)β}/√{α+2β}) – pure dark‑energy domination. Ω_de is a constant ratio of α and β, q=½(1+3ω_de) and ω_eff=ω_de. For ω_de<‑1/3 the point is stable (accelerated expansion), otherwise it is a saddle.

• C±: Points with non‑zero x₁ (scalar kinetic) and x₂ (potential) but vanishing matter and dark‑energy contributions. They exist only when the scalar equation‑of‑state ω_ϕ≠‑1 and depend on λ, κ and ω_ϕ. Their stability hinges on the balance between kinetic and potential terms; in certain parameter ranges they can represent scaling solutions where the scalar mimics the background fluid.

• D±: Points where both scalar and dark‑energy contributions are present (x₁, x₄ ≠0) and require ω_ϕ=1. They correspond to a stiff‑fluid scalar field interacting with the geometrically induced dark energy. Their eigenvalues are generally complex; stability is achieved only for finely tuned α,β,λ values.

Tables 1‑3 in the manuscript list the coordinates, existence conditions, eigenvalues, and the associated cosmological parameters (q, Ω_m, Ω_ϕ, Ω_de, ω_eff) for each fixed point. The linear stability analysis shows that the model naturally contains a matter‑dominated saddle (A±), a dark‑energy dominated attractor (B±) that can drive late‑time acceleration, and a set of scalar‑dominated or mixed points (C±, D±) that may describe early‑time inflationary or scaling epochs.

Overall, the study demonstrates that the simple linear f(R,L,T) model, despite its minimal modification of General Relativity, is capable of reproducing the sequence of cosmological eras—radiation/matter domination followed by accelerated expansion—through appropriate choices of the coupling constants α, β and the scalar potential parameters λ and V₀. The dynamical‑system framework provides a clear, quantitative picture of how the extra matter‑geometry couplings affect the phase‑space structure and the stability of cosmological solutions, offering a useful tool for confronting such theories with observational data.