Confront $f(R,T)=\mathcal{R}+\beta T$ modified gravity with the massive pulsar PSR J0740+6620

Many physically inspired general relativity (GR) modifications predict significant deviations in the properties of spacetime surrounding massive neutron stars. Among these modifications is $f(\mathcal{R}, \mathbb{T})$, where $\mathcal{R}$ is the Ricci scalar, $\mathbb{T}$ represents the trace of the energy-momentum tensor, the gravitational theory that is thought to be a neutral extension of GR. Neutron stars with masses above 1.8 $M_\odot$ expressed as radio pulsars are precious tests of fundamental physics in extreme conditions unique in the observable universe and unavailable to terrestrial experiments. We obtained an exact analytical solution for spherically symmetric anisotropic perfect-fluid objects in equilibrium hydrostatic using the frame of the form of $f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta \mathbb{T}$ where $\beta$ is a dimensional parameter. We show that the dimensional parameter $\beta$ and the compactness, $C=\frac{ 2GM}{Rc^2}$ can be used to express all physical quantities within the star. We fix the dimensional parameter $\beta$ to be at most. (Here ${\mathrm \kappa^2}$ is the coupling constant of Einstein which is figured as $\kappa^2=\frac{8\pi G}{c^4}$, the Newtonian constant of gravitation is denoted as $G$ while $c$ represents the speed of light.) $\beta_1=\frac{\beta}{\kappa^2}= 0.1$ in positive values through the use of observational data from NICER and X-ray Multi-Mirror telescopes on the pulsar PSR J0740+6620, which provide information on its mass and radius.