Observational Limits on Einasto Dark Matter Parameters from Event Horizon Telescope Images of Sgr A${}^{\ast }$ and M87${}^{\ast }$

Abstract

The Event Horizon Telescope (EHT) has provided images of the supermassive black-holes Sgr A${}^{\ast }$ and M87${}^{\ast }$, enabling direct tests of gravity. Any extended mass-distribution, such as a dark-matter halo, perturbs null-geodesics in the photon-ring regime, making shadow-measurements a probe of inner-halo structure. In this work, we investigate static, spherically-symmetric black-holes surrounded by Einasto-type dark-matter halos and derive constraints from EHT shadow-data. Starting from the Einasto density-profile with parameters ${\varrho }_0,\tilde{\hat{I}}\pm ,\tilde{\hat{I}}½$, we construct a metric-function $f(r)=1-2M/r+2M_{\infty }\tilde{g}(r)$ that interpolates between the black-hole horizon and the asymptotic halo, following the approach of Xu et al. (2018) but adapted specifically to the Einasto scenario. We analyze the photon-potential, null-geodesics, and shadow-radius as functions of black-hole mass $M$, in the non-spinning limit. Using the dimensionless shadow-diameter $d_{\text{sh}}\equiv D\hat{I}¸/M$ measured by the EHT -- $d_{\text{sh}}^{M87\ast }=11.0\pm 1.5$ and $d_{\text{sh}}^{SgrA\ast }=9.5\pm 1.4$ -- we perform Bayesian parameter-estimation to identify allowed regions in the Einasto parameter-space. Combined with the independently-measured black-hole masses from stellar-dynamics, our results place constraints on the inner dark-matter distribution: for Sgr A${}^{\ast }$, adopting the stellar-orbit mass-prior, we find ${\varrho }_0\lesssim 10^{-11},M_{\odot }/{\text{pc}}^3$ at $1\ddot{I}ƒ$ confidence, while for M87${}^{\ast }$ the bounds are weaker due to distance-uncertainties. The Einasto-index $\tilde{\hat{I}}½$ is weakly-constrained, indicating that EHT precision primarily limits the mass enclosed near the photon-sphere rather than the profile-slope. Future EHT observations will refine these constraints and distinguish between competing dark-matter descriptions.

17 pages, 7 figures, accepted for publication in The Astrophysical Journal