Construction of fuzzy black holes and self-gravitational droplets in exponential f(R) gravity

This paper discusses the construction of non-commutative geometry based self-gravitational droplets or fuzzy black holes (BHs) within the context of exponential-f(R) gravity model , where β is a parameter with units [length2]. Considering the Einasto density profile, a generalization of the Gaussian profile, we extend the noncommutative-inspired mini BH metric to incorporate dark matter (DM) as a constituent of the BH. We investigate various Einasto DM fuzzy objects filled with anisotropic fluid content, considering different values of the shape parameter and rescaled mass. This study investigates the possibility of stable self-gravitational compact objects with fuzzy mass distributions inspired by a well-known DM profile, the Einasto parametrization. These intriguing astrophysical configurations fall into three categories: fuzzy self-gravitational droplets with no event horizon, and fuzzy black holes featuring either one or two horizons. It is important to highlight that all these astrophysical entities are solutions derived from the exponential-f(R) gravity model. The corresponding effective potentials exhibit a significant resemblance to the effective potential of the classical Schwarzschild Black Hole located at the center of the Galaxy. Specifically, we investigate the potential for these black holes to serve as central galactic entities.