Apparent and emergent dark matter around a Schwarzschild black hole

Abstract

Inspired by the two different dark matter frameworks that were studied recently: one that arises from the non-local effects of entanglement entropy as emergent gravity (characterized by the parameter $ξ(M)$, and zero-point scale length $l$), and one from dark energy viewed as a superconducting medium (characterized by $η(M)$, and screening length parameter $λ_{\rm G}$), two black hole solutions spherically surrounded with these dark matter models were derived. The effect of these two frameworks on SMBH was analyzed through the resulting deviations in the null regions and the black hole shadow. In addition, constraints to the parameters $ξ$ and $η$ (at $3σ$ level) were found using the available EHT data for Sgr. A* and M87*. These constraints allow one to deduce the effective mass $M$, which causes uncertainties in the measurement. On the other hand, if the effective mass is known, one can also deduce the constants associated with $ξ$, and $η$. The former framework also introduces an Appell function, a hypergeometric function of two variables that separately allows the analysis of macroscopic and (hypothetical) microscopic black holes. This first framework was found to decrease the radii of the null regions respective to the Schwarzschild counterpart. The shadow radius, however, behaves reversibly. The result of the numerical analysis for the latter framework revealed increased values for the photonsphere and shadow radii. Remarkably, the study also showed that for SMBHs, the amplifying effects of $λ_{\rm G}$ are stronger than the scale length $l$. Finally, results of constraints, as an example, for the upper bound in $ξ$ for M87* indicated that the effective mass causing the deviation was around $2.4\times 10^{20} M_{\odot}$ given that the observed Milgrom's constant is $5.4\times 10^{-10} \text{ m/s}^2$.