Testing Symmergent gravity through the shadow image and weak field photon deflection by a rotating black hole using the M87$^*$ and Sgr. A$^*$ results

In this paper, we study rotating black holes in symmergent gravity, and use deviations from the Kerr black hole to constrain the parameters of the symmergent gravity. Symmergent gravity induces the gravitational constant $G$ and quadratic curvature coefficient $c_{\rm O}$ from the flat spacetime matter loops. In the limit in which all fields are degenerate in mass, the vacuum energy $V_{\rm O}$ can be wholly expressed in terms of $G$ and $c_{\rm O}$. We parametrize deviation from this degenerate limit by a parameter ${\hat \alpha}$ such that the black hole spacetime is dS for ${\hat \alpha} < 1$ and AdS for ${\hat \alpha} > 1$. In constraining the symmergent parameters $c_{\rm O}$ and ${\hat \alpha}$, we utilize the EHT observations on the M87* and Sgr. A* black holes. We investigate first the modifications in the photon sphere and shadow size, and find significant deviations in the photonsphere radius and the shadow radius with respect to the Kerr solution. We also find that the geodesics of time-like particles are more sensitive to symmergent gravity effects than the null geodesics. Finally, we analyze the weak field limit of the deflection angle, where we use the Gauss-Bonnet theorem for taking into account the finite distance of the source and the receiver to the lensing object. Remarkably, the distance of the receiver (or source) from the lensing object greatly influences the deflection angle. Moreover, $c_{\rm O}$ needs be negative for a consistent solution. In our analysis, the rotating black hole acts as a particle accelerator and possesses the sensitivity to probe the symmergent gravity.