Observational Limits on Einasto Dark Matter Parameters from Event Horizon Telescope Images of Sgr A$^{*}$ and M87$^{*}$
The Event Horizon Telescope (EHT) has provided images of the supermassive black-holes Sgr A$^{*}$ and M87$^{*}$, 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 ${\varrho_0, \tilde{\alpha}, \tilde{\nu}}$, 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 $d_{\text{sh}}\equiv D\theta/M$ measured by the EHT -- $d_{\text{sh}}^{M87*}=11.0\pm1.5$ and $d_{\text{sh}}^{SgrA*}=9.5\pm1.4$ -- 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$^{*}$, adopting the stellar-orbit mass-prior, we find $\varrho_0 \lesssim 10^{-11},M_\odot/\text{pc}^3$ at $1\sigma$ confidence, while for M87$^{*}$ the bounds are weaker due to distance-uncertainties. The Einasto-index $\tilde{\nu}$ 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.