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

Abstract: The Event Horizon Telescope (EHT) has provided images of the supermassive black-holes Sgr A$<sup>{*}$ and M87$<sup>{*}$, 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 ${\varrho_0, \tildeα, \tildeν}$, 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 $d_{\text{sh}}\equiv Dθ/M$ measured by the EHT -- $d_{\text{sh}}<sup>{M87*}=11.0\pm1.5$ and $d_{\text{sh}}<sup>{SgrA*}=9.5\pm1.4$ -- 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$<sup>{*}$, adopting the stellar-orbit mass-prior, we find $$\varrho_0 \lesssim 10<sup>{-11},M_\odot/\text{pc}<sup>3$$ at $1σ$ confidence, while for M87$<sup>{*}$ the bounds are weaker due to distance-uncertainties. The Einasto-index $\tildeν$ 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.

• The paper constructs a static, spherically symmetric SMBH–Einasto halo metric to analyze the effect of dark matter on black hole shadow geometry.
• It uses null geodesic analysis and Bayesian parameter estimation to quantify how variations in central density, scale, and profile index influence photon sphere stability and shadow size.
• The study constrains Sgr A*'s dark matter density (ρ₀ ≲ 10⁻¹¹ M☉/pc³) and offers insights into fuzzy/ultralight dark matter models via the solitonic core–halo relation.

Observational Limits on Einasto Dark Matter Parameters from EHT Images of Sgr A* and M87*

Introduction: Event Horizon Telescope Shadows and Dark Matter Halos

This paper systematically addresses the influence of extended, Einasto-profiled dark matter (DM) halos on the spacetime structure near supermassive black-holes (SMBHs), leveraging observational constraints from the Event Horizon Telescope (EHT) shadow images of Sgr A* and M87*. By formulating an analytic, static, spherically symmetric black-hole–halo metric rooted in the Einasto DM distribution and comparing theoretical photon ring observables to EHT data, the authors perform a Bayesian estimation of the allowed DM parameter space, focusing on the central density $\varrho_0$, halo scale $\tilde{\alpha}$, and profile index $\tilde{\nu}$.

The implications are direct: this approach tests the degree to which horizon-scale imaging constrains non-baryonic matter in galactic nuclei, complementing large-scale galactic dynamics and indirect probes of DM. The methodology and results are also applicable to fuzzy/ultralight DM (FDM/ULDM) scenarios through the solitonic core–halo relation.

Theoretical Framework: Black-Hole–Einasto Metric Construction

The core of the analysis is the construction of a spherically symmetric spacetime metric incorporating a SMBH and a surrounding Einasto halo. The Einasto DM density profile is defined as

$$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$

where $\varrho_0$ is the central density, $\tilde{\alpha}$ the scale radius, and $\tilde{\nu}$ the profile index. The enclosed mass is provided by an incomplete gamma function, but for tractability, an accurate Padé-type interpolation is used for the metric construction.

The static, isotropic metric function is then given as

$f(r) = 1 - \frac{2M}{r} + 2M_\infty \tilde{g}(r),$

where $M$ is the black-hole mass, $M_\infty$ encodes the total halo mass, and $\tilde{\alpha}$0 is an analytic function accounting for the halo's influence on curvature. The metric smoothly transitions from pure Schwarzschild near the SMBH to a DM-dominated region at large radii.

Figure 1: The lapse function $\tilde{\alpha}$1 for varying Einasto parameters and black-hole mass, illustrating the interplay between DM profile and horizon structure.

Geodesics, Photon Effective Potential, and Shadow Formation

The analysis proceeds by deriving the effective potential for null geodesics in the constructed metric to identify the location and stability of photon spheres, which characterize the boundary of the black-hole shadow. Variations in $\tilde{\alpha}$2, $\tilde{\alpha}$3, $\tilde{\alpha}$4, and $\tilde{\alpha}$5 modulate the effective potential and shift the radius of the unstable photon orbits.

Figure 3: The photon effective potential in the presence of varying Einasto DM parameters, showing the sensitivity of the photon sphere to the halo density and scale.

Crucially, the shadow radius seen by a distant observer is computed as

$\tilde{\alpha}$6

where $\tilde{\alpha}$7 is the photon-sphere radius obtained by solving $\tilde{\alpha}$8. The authors demonstrate that the shadow size is most sensitive to $\tilde{\alpha}$9 and $\tilde{\nu}$0, with weaker dependence on the profile slope $\tilde{\nu}$1 and the scale radius $\tilde{\nu}$2.

Figure 2: The computed shadow radius under Einasto DM modifications, quantifying deviations from standard Kerr/Schwarzschild expectations as a function of DM parameters.

Energy Emission and Orbits: Theoretical Insight

Although Hawking radiation from astrophysical SMBHs is negligible observationally, its calculation in the modified background provides insight into how DM alters near-horizon geometry and associated quantum emission. The halo parameters dampen the energy emission rate, while increasing black-hole mass amplifies it, in line with expectations from the geometry of the photon sphere and surface gravity.

Figure 4: The influence of Einasto DM on the theoretical energy emission rate spectrum of the SMBH horizon.

The interplay of DM parameters on stable/unstable orbit structure and the phase space of photon geodesics is further elucidated, supporting the global dynamics of light in these composite spacetimes.

Figure 5: Phase portrait of null geodesics and sample photon trajectories, showing photon capture, critical, and escape classes in the Einasto-modified metric.

Shadow Constraints from EHT Data: Bayesian Parameter Estimation

Employing the EHT measurements of Sgr A* and M87* shadow diameters and masses, the authors perform a Bayesian analysis in the $\tilde{\nu}$3 space. The EHT angular shadow diameters are recast as dimensionless units $\tilde{\nu}$4 (distance-normalized), yielding:

• $\tilde{\nu}$5
• $\tilde{\nu}$6

With mass priors derived independently from stellar dynamics, stringent constraints can be placed on $\tilde{\nu}$7 for Sgr A*, and weaker limits for M87* due to larger distance and mass uncertainties. Notably, the analysis reveals:

• For Sgr A*: $\tilde{\nu}$8 at $\tilde{\nu}$9 confidence.
• For M87*: Constraints are less restrictive, dominated by uncertainties in $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$0 and $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$1.

The allowed region in parameter space forms a degeneracy band, reflecting joint sensitivity of shadow size to both $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$2 and DM halo properties.

Figure 6: Black-hole shadow radius and its dependence on Einasto halo parameters and $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$3, with $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$4 and $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$5 EHT shadow constraints illustrated.

Figure 7: Posterior confidence regions in the $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$6, $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$7, and $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$8 planes consistent with observed EHT shadow sizes for Sgr A

and M87*.*

Implications: Dark Matter Models and Future Developments

The strong upper limit on $$\varrho(r) = \varrho_0 \exp\left[-\left(\frac{r}{\tilde{\alpha}}\right)^{1/\tilde{\nu}}\right],$$9 for Sgr A* rules out high-density DM spikes in the immediate vicinity of the Galactic center, complementing stellar orbital and gravitational wave constraints. The weak sensitivity to $\varrho_0$0 indicates that current shadow precision constrains the mass enclosed on horizon scales, rather than the detailed slope of the DM profile.

The results also apply to ULDM/FDM scenarios via the solitonic core relation, yielding $\varrho_0$1 eV for core radii of order kpc—covering parameter space largely independent of those probed by Lyman-$\varrho_0$2 forest or large-scale dynamics.

Conclusion

This paper rigorously connects horizon-scale imaging of SMBHs with precision constraints on the properties of inner DM halos parameterized by the Einasto profile. The analytic framework developed provides a robust test of DM models using the cleanest direct gravity probe available: shadow geometry from EHT observations. The main findings include a robust upper limit of $\varrho_0$3 for Sgr A*, a quantitative delineation of parameter degeneracies, and an explicit demonstration that the DM-induced perturbations on shadow size are detectable, albeit subdominant compared to the SMBH mass effect. The work opens avenues for using increasingly precise EHT and next-generation VLBI data to distinguish and constrain DM models at previously inaccessible physical scales.

Future extensions—incorporating spin (Kerr+halo metrics), axisymmetric DM structures, and joint multiwavelength constraints—will further refine these results and may critically test exotic DM models and deviations from general relativity in the strong-field regime.