Modeling Transit Dark Energy in $f(R, L_m)$-gravity

This research paper deals with a transit dark energy cosmological model in $f(R, L_{m})$-gravity with observational constraints. For this, we consider a flat FLRW space-time and have taken a cosmological cosntant-like parameter $\beta$ in our field equations. The model has two energy parameters~ $\Omega_{m0}~ and~ \Omega_{\beta0}$, which govern the mechanism of the universe, in particular its present accelerated phase. To make the model cope with the present observational scenario, we consider three types of observational data set: $46$ Hubble parameter data set, SNe Ia $715$ data sets of distance modulus and apparent magnitude, and $40$ datasets of SNe Ia Bined compilation in the redshift $0\leq z<1.7$. We have approximated the present values of the energy parameters by applying $R^{2}$ and $\chi^{2}$-test in the observational and theoretical values of Hubble, distance modulus, and apparent magnitude parameters. Also, we have measured the approximate present values of cosmographic coefficients $\{H_{0}, q_{0}, j_{0}, s_{0}, l_{0}, m_{0}\}$. It is found that our approximated value-based model fits best with the observational module. We have found that as $t\to\infty$ (or $z\to 0$) then $\{q, j, s, l, m\}\to\{-1, 1, 1, 1, 1\}$. The cosmic age of the present universe is also approximated and comes up to the expectation. Our model shows a transit phase of the present accelerating universe with a deceleration in the past and has a transition point.