A Modified Gravitational Theory of the Matter Sector of the Type $φ(R,T)\mathcal{L}_{m}$

TL;DR

The paper introduces a modified gravity model in which the Einstein-Hilbert geometry is untouched while the matter Lagrangian is weighted by a density-dependent function $φ(T)$, producing high-density corrections without new geometric degrees of freedom. For a nonlinear ansatz $φ(T)=1+\frac{βT}{1+γT}$ and a barotropic fluid, the authors derive modified Friedmann equations with effective sources that produce a smooth, finite-density bounce ($M(ρ_b)=0$, $\dot H(ρ_b)>0$) and recover $Λ$CDM in the infrared. They analyze energy conditions and perturbative stability, finding WEC and DEC preserved while NEC/SEC are briefly violated near the bounce, and show $c_s^2>0$ and subluminal throughout; stability regions in parameter space are broad. The framework acts as an EFT-inspired, density-dependent renormalization of the matter sector, yielding a robust singularity-resolving mechanism without modifying the geometric sector, and it naturally reduces to GR with $Λ$CDM at low densities. This positions the model as a principled proof of concept for singularity resolution within a controlled subset of $f(R,\mathcal{L}_m,T)$ theories.

Abstract

We investigate a modified gravity framework where the geometric Einstein--Hilbert sector remains untouched while the matter Lagrangian is weighted by a nontrivial function $φ(T)$ of the energy--momentum trace. Unlike $f(R,T)$ or $f(R,\mathcal L_m)$ theories, this construction alters how matter curves spacetime without introducing extra geometric degrees of freedom, thereby remaining consistent with local tests of gravity. Physically, the factor $φ(T)\mathcal L_m$ can be interpreted as an effective renormalization of the matter sector, relevant at high densities and smoothly reducing to GR at low densities. Within this setup we identify a robust window in parameter space leading to a smooth and nonsingular cosmological bounce, with bounded density, finite $\dot H>0$ at the bounce, and preservation of the infrared limit where $Λ$CDM is recovered. This mechanism provides a natural route to singularity resolution while retaining the empirical successes of standard cosmology.

Figures (6)

• Figure 1: Schematic relation between GR and different modified gravity frameworks. From the Einstein--Hilbert baseline (top), one may either modify the geometric sector as in $f(R,T)$ gravity, introduce mixed curvature--matter couplings as in $f(R,\mathcal{L}_m)$ gravity, or keep Einstein geometry untouched while weighting the matter sector by $\phi(T)$, as in the present proposal.
• Figure 2: Conceptual origin of nonconservation in different frameworks. In GR the standard conservation law holds. In $f(R,T)$ and $f(R,\mathcal{L}_m)$ the violation of $\nabla^\mu T_{\mu\nu}=0$ is tied to the geometric sector, while in the present model it arises solely from the matter weighting $\phi(T)$, with the Einstein--Hilbert geometry left intact.
• Figure 3: Effective energy conditions in the $\phi(T)\mathcal{L}_m$ model. WEC and DEC are preserved, while NEC and SEC approach zero in the neighborhood of the bounce (vertical dashed line).
• Figure 4: Cosmological dynamics in the $\phi(T)\mathcal{L}_m$ model for representative parameters. Panel (a): the energy density $\rho(t)$ remains finite and reaches a maximum at the bounce. Panel (b): the Hubble parameter $H(t)$ crosses zero with positive slope, confirming a smooth nonsingular bounce. Panel (c): the scale factor $a(t)$ exhibits a finite minimum, replacing the Big Bang singularity. Panel (d): phase portrait $H$ versus $\rho$, showing the bounce as the turning point where $H=0$ at finite $\rho$.
• Figure 5: Stability diagnostics of the $\phi(T)\mathcal{L}_m$ model. Left: the deviation function $Q(t)=A(\rho)/N(\rho)$, which quantifies the departure from the GR continuity law. Near the bounce $Q(t)\neq 1$, signaling matter--geometry energy exchange, while at late times $Q\to 1$, recovering GR. Right: the effective sound speed squared $c_s^2(t)=\dot p_{\rm eff}/\dot \rho_{\rm eff}$. In the physical parameter window $c_s^2$ remains positive (and close to $w$), ensuring perturbative stability across the bounce.