Constraining a $f(R, L_m)$ Gravity Cosmological Model with Observational Data

We investigate a spatially flat FLRW cosmological model in the framework of modified gravity described by the function $f(R, L_m) = \alpha R + L_m^\beta + \gamma$, where $L_m$ is the matter Lagrangian density. The modified Friedmann equations yield the Hubble parameter as $H(z) = H_0 \sqrt{(1 - \lambda) + \lambda (1 + z)^{3(1 + w)}},$ with the parameters $\lambda = \frac{\gamma}{6\alpha H_0^2} + 1$ and $w = \frac{\beta(n - 2) + 1}{2\beta - 1}$. Using a Bayesian Markov Chain Monte Carlo (MCMC) approach, we constrain the model parameters with recent observational data, including cosmic chronometers, the Pantheon+ Supernovae dataset, Baryon Acoustic Oscillations (BAO), and Cosmic Microwave Background (CMB) shift parameters. The best-fit values are found to be $H_0 = 72.773^{+0.148}_{-0.152}$ km/s/Mpc, $\lambda = 0.289^{+0.007}_{-0.007}$, and $w = -0.002^{+0.002}_{-0.002}$, all quoted at the 1$\sigma$ confidence level.This model predicts a transition redshift of $z_t \approx 0.76$ for the onset of cosmic acceleration and an estimated universe age of 13.21 Gyr. The higher inferred value of $H_0$ compared to the Planck 2018 result offers a potential resolution to the Hubble tension. Additionally, using $\rho_0 = 0.534 \times 10^{-30} \, \text{g/cm}^3$ and assuming $n = 1$, we derive the model constants as $\beta = 1.00201$, $\alpha = 512247$, and $\gamma = -1.215 \times 10^{-29}$. We also evaluate the Bayesian Information Criterion (BIC) to compare the model's performance with that of the standard $\Lambda$CDM model. The small BIC difference ($\Delta \text{BIC} = 0.16$) indicates comparable statistical support for both models. Thus, the $f(R, L_m)$ gravity scenario serves as a consistent and viable alternative to $\Lambda$CDM, potentially addressing open questions in late-time cosmology.