Bayesian Black Hole Photogrammetry

TL;DR

This work introduces a compact, analytic dual-cone emission model for horizon-scale black hole imaging that enables photogrammetry within Kerr spacetime. By coupling a two-cone, axisymmetric synchrotron emissivity with analytic Kerr ray tracing, the authors perform Bayesian inference directly in the visibility domain, recovering the mass-to-distance ratio $\theta_g$ and emission geometry from interferometric data. The model reproduces time-averaged GRMHD images for both MAD and SANE flows and reveals multimodal posterior structures stemming from degeneracies in image topology, especially under sparse EHT-like coverage. Applied to M87$^*$ data, the method yields a $95\%$ HPDI for $\theta_g=(2.84,3.75)\,\mu{\rm as}$ and an inclination $\theta_o=(11^\circ,24^\circ)$, consistent with independent mass estimates and jet-inclination inferences, while showing that spin remains weakly constrained with current data. The approach offers an efficient pathway to photogrammetric inferences from horizon-scale VLBI data and suggests extensions to polarization and nonaxisymmetric, time-varying models to further constrain black hole parameters.

Abstract

We propose a simple, analytic dual-cone accretion model for horizon scale images of the cores of Low-Luminosity Active Galactic Nuclei (LLAGN), including those observed by the Event Horizon Telescope (EHT). Our underlying model is of synchrotron emission from an axisymmetric, magnetized plasma, which is constrained to flow within two oppositely oriented cones that are aligned with the black hole's spin axis. We show that this model can accurately reproduce images for a variety of time-averaged general relativistic magnetohydrodynamic (GRMHD) simulations, that it accurately recovers both the black hole and emission parameters from these simulations, and that it is sufficiently efficient to be used to measure these parameters in a Bayesian inference framework with radio interferometric data. We show that non-trivial topologies in the source image can result in non-trivial multi-modal solutions when applied to observations from a sparse array, such as the EHT 2017 observations of M87${}^*$. The presence of these degeneracies underscores the importance of employing Bayesian techniques that adequately sample the posterior space for the interpretation of EHT measurements. We fit our model to the EHT observations of M87${}^*$ and find a 95% Highest Posterior Density Interval (HPDI) for the mass-to-distance ratio of $θ_g\in(2.84,3.75)\,μ{\rm as}$, and give an inclination of $θ_{\rm o}\in(11^\circ,24^\circ)$. These new measurements are consistent with mass measurements from the EHT and stellar dynamical estimates (e.g., Gebhardt et al. 2011; EHTC et al. 2019a,b; Liepold et al. 2023), and with the spin axis inclination inferred from properties of the M87${}^*$ jet (e.g., Walker et al. 2018).

Figures (18)

• Figure 1: Schematic depiction of our dual cone emission model. The full model consists of a Kerr black hole (black sphere) surrounded by synchrotron emitting plasma (green arrows) confined to a cone of opening angle $\theta_s$ and threaded by a magnetic field $\vec{B}$ (red field lines). The magnetic field and plasma are taken to be axisymmetric about the equatorial plane. The emission is concentrated at a characteristic radius, $R$, with the profile determined by a double power-law distribution with exponents $p_1$ and $p_2$ (translucent blue cone). The polarization from the synchrotron radiation is parallel transported along null geodesics (yellow and orange ticked trajectories) to generate an image on the screen of an observer at radial infinity. The image generated from the model is the sum of multiple sub-images corresponding to the different trajectories light can take from the source to the observer.
• Figure 2: Comparison of time-averaged MAD (left) and SANE (right) GRMHD simulations of M87$^*$ (top row) with the best-fitting equatorial (2nd row), dual-cone (3rd row), and combination (equatorial + dual cone) models (4th row). These two simulations are representative of the minimal and maximal emission scale heights that are typically seen in GRMHD simulations of M87$^*$. The best-fitting models are determined by maximizing the normalized cross-correlation in the image domain. The text in each panel gives the opening angle of the cone ($\theta_s$), the viewing inclination ($\theta_{\rm o}$), the black hole spin ($a$), and the mass-to-distance ratio ($\theta_g$). The bottom row compares the fitted emissivity functions of our models (colored lines) to the true emissivity of the GRMHD (grayscale). The predicted emissivities are consistent with those of the GRMHD, as is evident by their overlap, which illustrates our model's ability to perform photogrammetry in black hole space times.
• Figure 3: Best-fitting equatorial, dual cone, and combination (equatorial + dual cone) models to a time-averaged retrograde $R_{\text{high}}$$=160$ MAD GRMHD simulation of Sgr A$^*$ viewed at inclinations of $30^\circ$, $50^\circ$, $70^\circ$, and $90^\circ$ (from left to right). The NxCORR score for all models is above 0.85 and increases as the black hole is viewed more face-on. In addition, the parameters of the best-fitting models are within a few percent of the true parameters for both mass and spin at low-to-modest inclinations. As expected, the dual cone model (13 parameters) outperforms the equatorial model (12 parameters) despite having only a single additional parameter. In contrast, the combination model (23 parameters) provides only marginal additional improvement despite nearly doubling the number of parameters. \ref{['tab:best_nxcorr']} lists the parameters for each model.
• Figure 4: Same as \ref{['fig:nxcorr_summary']}, but when the ground truth image and model image are both blurred by a $10\,\mu{\rm as}$ FWHM Gaussian kernel before comparison. The resulting NxCORR values are higher than for the unblurred case, with excellent fidelity for all model types. However, despite the improved image fidelity at this resolution, the best-fitting parameters have significantly larger discrepancies than for the unblurred fits, showing the strong degeneracies that are present when restricted to the current EHT resolution, especially because the photon ring cannot be distinguished from the direct emission.
• Figure 5: Comparison of a time-averaged GRMHD image with the best-fit Dual Cone model, both at native resolution (top) and after being blurred by a $10\mu$as FWHM Gaussian beam (bottom). The simulation corresponds to a retrograde, $R_{\text{high}}$$=160$, MAD simulation of Sgr A$^*$. Columns show the GRMHD image (left), the best-fit model (left-center), horizontal and vertical intensity cross sections (right-center), and residual images as a fraction of peak intensity (right).