Transit dark energy cosmological models in generalized matter-geometry coupling theory using a non-linear form of $f(R,T,L_{m})$ function

Technical Summary: Transit Dark Energy Cosmological Models in Generalized Matter-Geometry Coupling Theory

Problem Statement
The standard $\Lambda$CDM cosmological model, while successful in explaining large-scale structure and the current accelerated expansion of the universe, faces theoretical challenges such as the cosmological constant problem and the need for dark energy as a separate, unexplained component. Modified gravity theories offer an alternative by attributing cosmic acceleration to the interaction between matter and spacetime geometry rather than an exotic energy fluid. Specifically, the paper investigates the $f(R, T, L_m)$ gravity theory, a unified framework proposed by Haghani and Harko that generalizes $f(R)$, $f(R, L_m)$, and $f(R, T)$ theories. The authors aim to explore the cosmological consequences of this theory using a specific non-linear form of the function $f(R, T, L_m)$ and to constrain its parameters against observational data to determine if it can reproduce the observed "transit" universe (a transition from deceleration to acceleration).

Methodology
The study employs a flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime metric. The authors propose a specific non-linear Lagrangian function:
$$f(R, T, L_m) = \alpha R + \beta RT + \gamma RL_m - \eta$$
where $\alpha, \beta, \gamma,$ and $\eta$ are arbitrary coupling constants. $R$ is the Ricci scalar, $T$ is the trace of the energy-momentum tensor, and $L_m$ is the matter Lagrangian density.

1. Theoretical Derivation: The authors derive the modified field equations for this specific $f(R, T, L_m)$ function. They assume a perfect fluid source consisting of matter ($\rho_m$) and radiation ($\rho_r$). By solving these field equations, they obtain an analytical expression for the Hubble function $H(z)$ in terms of the model parameters ($H_0, \Omega_{m0}, \Omega_{r0}, \Omega_\eta, \beta, \gamma$) and redshift $z$.
2. Viability Conditions: Before applying data, the authors establish physical viability conditions for the function, requiring $f > 0$, $\partial f/\partial R > 0$, and $\partial f/\partial \rho > 0$, which impose constraints on the signs of the coupling constants.
3. Observational Constraints: The model parameters are constrained using a Monte Carlo Markov Chain (MCMC) analysis on two joint datasets:
   • Cosmic Chronometers (CC): 31 data points of the Hubble parameter $H(z)$ derived from the differential age method.
   • Pantheon Sample: 1048 data points of apparent magnitudes $m(z)$ for Type Ia Supernovae (SNe Ia) covering redshifts $0.01 \leq z \leq 2.26$.
     The analysis minimizes the $\chi^2$ statistic to find the best-fit values for $H_0, \Omega_{m0}, \Omega_{r0}, \Omega_\eta,$ and $\beta$. Due to parameter degeneracy between $\beta$ and $\gamma$, the authors assume $\beta = \gamma$ for the analysis.

Key Contributions and Results

• Hubble Function and Parameter Estimation: The study successfully derives the Hubble function $H(z)$ for the proposed model. The MCMC analysis yields the following best-fit values (at 1$\sigma$ confidence):
  • CC Dataset: $H_0 = 67.9 \pm 3.1$ km/s/Mpc, $\Omega_{m0} = 0.29 \pm 0.13$, $\Omega_\eta = 0.69^{+0.26}_{-0.16}$, and $\beta = (-0.047 \pm 0.028) \times 10^{-7}$.
  • CC + Pantheon Dataset: $H_0 = 68.6 \pm 1.9$ km/s/Mpc, $\Omega_{m0} = 0.266^{+0.11}_{-0.091}$, $\Omega_\eta = 0.70^{+0.27}_{-0.14}$, and $\beta = (-0.052^{+0.025}_{-0.039}) \times 10^{-7}$.
    The coupling constant $\alpha$ is found to be approximately $1.03$ (CC) and $0.986$ (CC+Pantheon).
• Cosmic Evolution History:
  • Deceleration Parameter ($q$): The model predicts a transition from a decelerated phase ($q > 0$) to an accelerated phase ($q < 0$). The transition redshift is calculated as $z_t \approx 0.5965$ (CC) and $z_t \approx 0.6177$ (CC+Pantheon). The current deceleration parameter is $q_0 \approx -0.54$ to $-0.55$, consistent with an accelerating universe.
  • Equation of State (EoS): The effective EoS parameter $\omega_{eff}$ and the dark energy EoS parameter $\omega_{de}$ are analyzed. The model exhibits a "phantom" behavior where $\omega_{de} \approx -1$ (specifically slightly less than -1, e.g., $-1.000006$), indicating a phantom dark energy scenario arising from the matter-geometry coupling.
  • Density Parameters: The evolution shows a matter-dominated early universe ($\Omega_m \to 1$ as $z \to \infty$) transitioning to a dark energy-dominated late universe ($\Omega_{de} \to 1$ as $z \to -1$). Current values are $\Omega_m \approx 0.3$ and $\Omega_{de} \approx 0.7$.
  • Universe Age: The estimated age of the universe is $t_0 = 13.76^{+0.32}_{-0.14}$ Gyr (CC) and $13.82^{+0.17}_{-0.11}$ Gyr (CC+Pantheon), aligning with standard observational estimates.
• Diagnostic Tests:
  • Om Diagnostic: The Om(z) function shows a non-decreasing trend, confirming the phantom nature of the model, while tending toward a constant slope at late times, indicating a convergence to $\Lambda$CDM behavior.
  • Causality Test: The squared sound speed $c_s^2$ remains within the physical range $[0, 1]$ across the redshift interval, confirming the model's stability and physical acceptability.
  • Energy Conditions: The Null, Weak, and Dominant energy conditions are satisfied. The Strong Energy Condition (SEC) is violated at late times ($z < z_t$), which is necessary to drive the accelerated expansion. The dark sector specifically violates all energy conditions, consistent with a phantom dark energy model.

Significance and Claims
The paper claims to present a viable "transit" dark energy model within the framework of generalized matter-geometry coupling ($f(R, T, L_m)$ gravity). The primary significance lies in demonstrating that a non-linear coupling between matter, curvature, and the matter Lagrangian can naturally generate an accelerated expansion phase without introducing a separate cosmological constant or exotic fluid. The authors assert that the derived model is physically consistent, stable, and compatible with current observational data (CC and Pantheon).

The study highlights that the dark energy in this scenario is not an independent entity but arises from the combined relationship between matter and geometry. The model successfully reproduces the transition from deceleration to acceleration and yields cosmological parameters (Hubble constant, density parameters, age) that are consistent with the $\Lambda$CDM model and recent observational constraints. The authors conclude that this specific non-linear form of $f(R, T, L_m)$ gravity offers a new type of phantom dark energy model that warrants further exploration to understand the history of the universe's evolution.