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

Abstract: We investigate a spatially flat FLRW cosmological model within modified gravity, defined by the function ( f(R, L_m) = \alpha R + L_m^\beta + \gamma ), where ( L_m ) is the matter Lagrangian density. The resulting Friedmann equations yield the Hubble parameter: [ H(z) = H_0 \sqrt{(1 - \lambda) + \lambda (1 + z)^{3(1 + w)}}, ] with ( \lambda = \frac{\gamma}{6\alpha H_0^2} + 1 ) and ( w = \frac{\beta(n - 2) + 1}{2\beta - 1} ). The parameter ( n ), linked to the equation of state ( p = (1 - n)\rho ), allows the model to account for different cosmological epochs. Using a Bayesian MCMC method, we constrain the model parameters with data from cosmic chronometers, the Pantheon+ Supernovae sample, BAO, and CMB shift parameters. The best-fit values obtained are ( H_0 = 72.773^{+0.148}_{-0.152} \, \text{km/s/Mpc} ), ( \lambda = 0.289^{+0.007}_{-0.007} ), and ( w = -0.002^{+0.002}_{-0.002} ), all at the 1(\sigma) confidence level. The model predicts a transition redshift ( z_t \approx 0.76 ) for the onset of acceleration and an estimated universe age of 13.21 Gyr. The relatively higher value of ( H_0 ) compared to Planck 2018 suggests a possible resolution to the Hubble tension. Assuming ( \rho_0 = 0.534 \times 10^{-30} \, \text{g/cm}^3 ) and ( n = 1 ), the constants are found to be ( \beta = 1.00201 ), ( \alpha = 512247 ), and ( \gamma = -1.215 \times 10^{-29} ). We also compute the Bayesian Information Criterion (BIC) to compare this model to the standard (\Lambda)CDM. A small BIC difference (( \Delta \text{BIC} = 0.16 )) indicates similar statistical support. Thus, the ( f(R, L_m) ) model stands as a consistent and viable alternative to (\Lambda)CDM, offering insights into late-time cosmology.