Microlensing Black Hole Shadows-II: Constraining Primordial Black Hole Dark Matter using the photon rings of M87 and Sgr A*

TL;DR

This work extends a microlensing framework to constrain primordial black hole dark matter by tracking time-dependent distortions in the photon rings of supermassive black holes, using Sgr A* and M87*. It models PBH populations as DM spikes and NFW halos (including the Milky Way foreground) and computes the microlensing event rates and detection thresholds for upcoming EHT-like arrays. The key finding is that photon-ring asymmetries provide the most sensitive observable, enabling constraints on the PBH fraction $f_{\rm PBH}$ over a broad mass range ($10^{-5}\,M_\odot$ to $10^{6}\,M_\odot$), with M87* offering the strongest bounds (down to $f_{\rm PBH} \sim 10^{-2}$–$5\times 10^{-2}$ for certain masses and resolutions). These constraints complement existing microlensing limits and demonstrate the potential of high-resolution black-hole imaging to probe dark matter in parameter spaces inaccessible to traditional methods.

Abstract

The resolution of photon rings of Sgr~A$^*$ and M87 is the next milestone of upcoming EHT-like interferometries. We extend the formalism developed in our previous work~\cite{Verma:2023hes} to constrain primordial black hole (PBH) dark matter using microlensing-induced distortions of black hole shadows. Building upon the theoretical framework for microlensing of photon rings, we apply this methodology to both Sgr A* and M87, considering multiple PBH populations: (i) PBH dark matter spikes around central supermassive black holes, (ii) NFW halo contributions in the Milky Way and M87 galaxies, and (iii) foreground Milky Way PBH dark matter affecting M87* observations. The microlensing signal manifests as a time-dependent asymmetry and deformation of the photon ring, providing the most sensitive observable for lensing effects. We assess the detectability of these signatures with future EHT-like interferometers. Our analysis reveals that M87* provides the strongest constraints on PBH dark matter. We show that the absence of photon-ring asymmetries in observations with angular resolution of order $0.1\,μ{\rm as}$ can constrain PBHs in the mass range $10^{-5}\,M_\odot \lesssim M_{\rm PBH} \lesssim 10^{6}\,M_\odot$, with maximal sensitivity near $M_{\rm PBH}\sim10^{3}\,M_\odot$, for PBH dark matter fractions as small as $f_{\rm PBH}\sim10^{-2}$.

Figures (6)

• Figure 1: We demonstrate the phenomena of microlensing of the photon ring. As an illustration of the phenomena, we have taken the true boundary of the shadow of M87.
• Figure 2: Shadow observables as a function of the lens impact parameter $\xi/theta_E$. Plotted are the normalized centroid displacement $\lvert O C\rvert / R$, the normalized asymmetry $A/R$, and the relative change in radius $( \langle R_L \rangle - R)/R$ as the lens approaches the line of sight.
• Figure 3: Geometric configuration for microlensing of black hole shadows. Left panel: Side view showing the lensing tube extending from the observer to the black hole shadow at distance $D_s$. The tube has cross-sectional radius $R_{\rm th} = \xi_{\rm th} D_l$ at lens distance $D_l$, defining the region within which a PBH lens can induce detectable shadow distortions. A PBH moving with transverse velocity $\vec{v}_{\perp}$ will cause microlensing when its trajectory intersects the tube. Right panel: View in the lens plane showing the detection cross-section. The photon ring (red circle, radius $R$) marks the shadow boundary. Any PBH with impact parameter $|\vec{\xi}| < \xi_{\rm th}$ (within the green dashed circle) will induce asymmetry exceeding the detection threshold $\sigma_A$. The PBH trajectory (purple dotted line) shows the lens motion through the sensitive region with velocity $\vec{v}_{l\perp}$.
• Figure 4: Predicted microlensing event rates for PBHs near SgrA* (left) and M87* (right) due to PBH lenses in the foreground. Different dark matter density profiles shown in different colors: density spike (red), standard NFW (blue), and MilkyWay NFW (green) in the case of M87. The thick, solid black curve in each panel represents the envelope of combined density profiles. Different line styles show different possible telecsope resolution $\sigma/R$. The spike profile yields the highest event rates, reaching values of order $10^{1}$--$10^{2}$ events per year for $M_{\rm PBH}\sim 10^{3}$--$10^{5}\,M_\odot$ in the case of M87
• Figure 5: Right panel: Parameter space for the SgrA dark‐matter spike at fixed $\sigma_A / R = 10^{-4}$ that gives rise to the total number of microlensing events per year $>1$. The horizontal axis shows the spike radius $R_{\rm sp}$ in kiloparsecs, and the vertical axis shows the inner density slope $\gamma_{\rm sp}$. Iso‐mass contours for primordial black holes are overlaid, each labeled by the PBH mass ranging from $10^{-4}$-$10^{4}$$M_\odot$. The range of $R_{\rm sp}$ is determined from the fact that the mass enclosed in the volume of spike radius. Similarly, right panel: shows the parameter space for the M87 dark‐matter spike. Iso‐mass contours for primordial black holes are overlaid, each labeled by the PBH mass ranging from $10^{-6}$-$10^{6}$$M_\odot$. The dashed horizontal line marks the slope $\gamma_{\rm sp}\simeq 7/3$ expected from adiabatic SMBH growth, while the dashed vertical line indicates the maximum allowed spike radius set by the requirement that the enclosed spike mass remains consistent with the host halo.