Parameter Inference from Black Hole Images using Deep Learning in Visibility Space

The Event Horizon Telescope (EHT) has unveiled the horizon-scale radiation properties of Sagittarius A* (Sgr A*), the supermassive black hole at the center of our galaxy, providing a novel platform for testing gravitational theories by comparing observations with theoretical models. A key next step is to investigate the nature of accretion flows and spacetime structures near black holes by analyzing the time variability observed in EHT data alongside general relativistic magnetohydrodynamic (GRMHD) simulations. We explored the dynamics of accretion flows in spherically symmetric black hole spacetimes with deviations from general relativity utilizing two dimensional GRMHD simulations based on the Rezzolla–Zhidenko parameterized spacetime. This study marks the first systematic investigation into how variability amplitudes in light curves, derived from non-Kerr GRMHD simulations, depend on deviations from the Schwarzschild spacetime. The deviation parameters are consistent with the constraints from weak gravitational fields and the size of Sgr A*’s black hole shadow. We find that the dynamics of accretion flows systematically depend on these parameters. In spacetimes with a deeper gravitational potential, fluid and Alfvén velocities consistently decrease relative to the Schwarzschild metric, indicating weaker dynamical behavior. We also examined the influence of spacetime deviations on radiation properties by computing luminosity fluctuations at 230 GHz using general relativistic radiative transfer simulations, in line with EHT observations. The amplitude of these fluctuations exhibits a systematic dependence on the deviation parameters, decreasing for deeper gravitational potentials compared to the Schwarzschild metric. These features are validated using one of the theoretically predicted metrics, the Hayward metric, a model that describes nonsingular black holes. This characteristic is expected to have similar effects in more comprehensive simulations that include more realistic accretion disk models and electron cooling in the future, potentially aiding in distinguishing black hole solutions that explain the variability of Sgr A*.

Context. The observed spectral energy distribution of an accreting supermassive black hole typically forms a power-law spectrum in the near infrared (NIR) and optical wavelengths, that may be interpreted as a signature of accelerated electrons along the jet. However, the details of acceleration remain uncertain. Aim. In this paper, we study the radiative properties of jets produced in axisymmetric general relativistic magnetohydrodynamics (GRMHD) simulations of hot accretion flows onto underluminous supermassive black holes both numerically and semi-analytically, with the aim of investigating the differences between models with and without accelerated electrons inside the jet. Methods. We assume that electrons are accelerated in the jet regions of our GRMHD simulation. To model them, we modify the electrons’ distribution function in the jet regions from a purely relativistic thermal distribution to a combination of a relativistic thermal distribution and the κ-distribution function (the κ-distribution function is itself a combination of a relativistic thermal and a non-thermal power-law distribution, and thus it describes accelerated electrons). Inside the disk, we assume a thermal distribution for the electrons. In order to resolve the particle acceleration regions in the GRMHD simulations, we use a coordinate grid that is optimized for modeling jets. We calculate jet spectra and synchrotron maps by using the ray tracing code RAPTOR, and compare the synthetic observations to observations of Sgr A*. Finally, we compare numerical models of jets to semi-analytical ones. Results. We find that in the κ-jet models, the radio-emitting region size, radio flux, and spectral index in NIR/optical bands increase for decreasing values of the κ parameter, which corresponds to a larger amount of accelerated electrons. This is in agreement with analytical predictions. In our models, the size of the emission region depends roughly linearly on the observed wavelength λ, independently of the assumed distribution function. The model with κ = 3.5, ηacc = 5–10% (the percentage of electrons that are accelerated), and observing angle i = 30° fits the observed Sgr A* emission in the flaring state from the radio to the NIR/optical regimes, while κ = 3.5, ηacc < 1%, and observing angle i = 30° fit the upper limits in quiescence. At this point, our models (including the purely thermal ones) cannot reproduce the observed source sizes accurately, which is probably due to the assumption of axisymmetry in our GRMHD simulations. The κ-jet models naturally recover the observed nearly-flat radio spectrum of Sgr A* without invoking the somewhat artificial isothermal jet model that was suggested earlier. Conclusions. From our model fits we conclude that between 5% and 10% of the electrons inside the jet of Sgr A* are accelerated into a κ distribution function when Sgr A* is flaring. In quiescence, we match the NIR upper limits when this percentage is <1%.

This work summarizes a program intended to unify three burgeoning branches of the high-energy astrophysics of relativistic jets: general relativistic magnetohydrodynamic (GRMHD) simulations of ever-increasing dynamical range, the microphysical theory of particle acceleration under relativistic conditions, and multiwavelength observations resolving ever-decreasing spatiotemporal scales. The process, which involves converting simulation output into time series of images and polarization maps that can be directly compared to observations, is performed by (1) self-consistently prescribing models for emission, absorption, and particle acceleration and (2) performing time-dependent polarized radiative transfer. M87 serves as an exemplary prototype for this investigation due to its prominent and well-studied jet and the imminent prospect of learning much more from Event Horizon Telescope (EHT) observations this year. Synthetic observations can be directly compared with real observations for observational signatures such as jet instabilities, collimation, relativistic beaming, and polarization. The simplest models described adopt the standard equipartition hypothesis; other models calculate emission by relating it to current density or shear. These models are intended for application to the radio jet instead of the higher frequency emission, the disk and the wind, which will be subjects of future investigations.

The M87 jet is extensively examined by utilizing general relativistic magnetohydrodynamic (GRMHD) simulations, as well as the steady axisymmetric force-free electrodynamic (FFE) solution. Quasi-steady funnel jets are obtained in GRMHD simulations up to the scale of ∼100 gravitational radii (r g) for various black hole (BH) spins. As is known, the funnel edge is approximately determined by the following equipartitions: (i) the magnetic and rest-mass energy densities and (ii) the gas and magnetic pressures. Our numerical results give an additional factor that they follow the outermost parabolic streamline of the FFE solution, which is anchored to the event horizon on the equatorial plane. We also show that the matter-dominated, nonrelativistic corona/wind plays a dynamical role in shaping the funnel jet into the parabolic geometry. We confirm a quantitative overlap between the outermost parabolic streamline of the FFE jet and the edge of the jet sheath in very long baseline interferometry (VLBI) observations at ∼(101–105)r g, suggesting that the M87 jet is likely powered by the spinning BH. Our GRMHD simulations also indicate a lateral stratification of the bulk acceleration (i.e., the spine-sheath structure), as well as an emergence of knotty superluminal features. The spin characterizes the location of the jet stagnation surface inside the funnel. We suggest that the limb-brightened feature could be associated with the nature of the BH-driven jet, if the Doppler beaming is a dominant factor. Our findings can be examined with (sub)millimeter VLBI observations, giving a clue for the origin of the M87 jet.

The Event Horizon Telescope Collaboration (EHTC) observed the Galactic centre source Sagittarius A$^\ast$ (Sgr A$^{\ast }$) and used emission models primarily based on single ion temperature (1T) general relativistic magnetohydrodynamic (GRMHD) simulations. This predicted emission is strongly dependent on a modelled prescription of the ion-to-electron temperature ratio. The most promising models are magnetically arrested disc (MAD) states. However, nearly all MAD models exhibit larger temporal fluctuations in radiative 230 GHz emission compared to observations. This limitation possibly stems from the fact that the actual temperature ratio depends on microphysical dissipation, radiative processes, and other effects not captured in ideal fluid simulations. Therefore, we investigate the effects of two-temperature (2T) thermodynamics in MAD GRMHD simulations of Sgr A$^{\ast }$, where the temperatures of both species are evolved. We find that the 230 GHz synchrotron flux variability more closely matches historical observations when we include the 2T treatment compared to 1T simulations. For the low accretion rates of Sgr A$^{\ast }$, a common assumption is to neglect radiative cooling. However, we find that the radiative cooling of electrons – via synchrotron, inverse Compton, and bremsstrahlung processes – reduces the electron temperature in the inner disc, where the EHT observes, by about 10 per cent, which, in turn, decreases both the (sub)millimetre synchrotron flux and its temporal fluctuations compared to uncooled simulations.

In general relativistic magnetohydrodynamic (GRMHD) simulations, accreted magnetic flux on the black hole horizon episodically decays, during which magnetic reconnection heats up the plasma near the horizon, potentially powering high-energy flares like those observed in M87* and Sgr A*. We study the mm observational counterparts of such flaring episodes in very high resolution GRMHD simulations. The change in 230 GHz flux during the expected high energy flares depends primarily on the efficiency of accelerating γ ≳ 100 (Te ≳ 1011 K) electrons. For models in which the electrons are heated to Te ∼ 1011 K during flares, the hot plasma produced by reconnection significantly enhances 230 GHz emission and increases the size of the 230 GHz image. By contrast, for models in which the electrons are heated to higher temperatures (which we argue are better motivated), the reconnection-heated plasma is too hot to produce significant 230 GHz synchrotron emission, and the 230 GHz flux decreases during high energy flares. We do not find a significant change in the mm polarization during flares as long as the emission is Faraday thin. We also present expectations for the ring-shaped image as observed by the Event Horizon Telescope during flares, as well as multiwavelength synchrotron spectra. Our results highlight several limitations of standard post-processing prescriptions for the electron temperature in GRMHD simulations. We also discuss the implications of our results for current and future observations of flares in Sgr A*, M87*, and related systems. Appendices contain detailed convergence studies with respect to resolution and plasma magnetization.

Images of black holes encode both astrophysical and gravitational properties. Detecting highly lensed features in images can differentiate between these two effects. We present an accretion disk emission model coupled to the Adaptive Analytical Ray Tracing (AART) code that allows a fast parameter space exploration of black hole photon ring images produced from synchrotron emission from 10 to 670 GHz. As an application, we systematically study several disk models and compute their total flux density, average radii, and optical depth. The model parameters are chosen around fiducial values calibrated to general relativistic magnetohydrodynamic (GRMHD) simulations and observations of M87*. For the parameter space studied, we characterize the transition between optically thin and thick regimes and the frequency at which the first photon ring is observable. Our results highlight the need for careful definitions of photon ring radius in the image domain, as in certain models the highly lensed photon ring is dimmer than the direct emission at certain angles. We find that at low frequencies the ring radii are set by the electron temperature, while at higher frequencies the magnetic field strength plays a more significant role, demonstrating how multifrequency analysis can also be used to infer plasma parameters. Lastly, we show how our implementation can qualitatively reproduce multifrequency black hole images from GRMHD simulations when adding time variability to our disk model through Gaussian random fields. This approach provides a new method for simulating observations from the Event Horizon Telescope and the proposed Black Hole Explorer space mission.

The Event Horizon Telescope (EHT) collaboration, an Earth-size sub-millimetre radio interferometer, recently captured the first images of the central supermassive black hole in M87. These images were interpreted as gravitationally lensed synchrotron emission from hot plasma orbiting around the black hole. In the accretion flows around low-luminosity active galactic nuclei such as M87, electrons and ions are not in thermal equilibrium. Therefore, the electron temperature, which is important for the thermal synchrotron radiation at EHT frequencies of 230 GHz, is not independently determined. In this work, we investigate the commonly used parametrized ion-to-electron temperature ratio prescription, the so-called R−β model, considering images at 230 GHz by comparing with electron-heating prescriptions obtained from general-relativistic magnetohydrodynamical (GRMHD) simulations of magnetized accretion flows in a Magnetically Arrested Disc (MAD) regime with different recipes for the electron thermodynamics. When comparing images at 230 GHz, we find a very good match between images produced with the R−β prescription and those produced with the turbulent- and magnetic reconnection-heating prescriptions. Indeed, this match is on average even better than that obtained when comparing the set of images built with the R−β prescription with either a randomly chosen image or with a time-averaged one. From this comparative study of different physical aspects, which include the image, visibilities, broad-band spectra, and light curves, we conclude that, within the context of images at 230 GHz relative to MAD accretion flows around supermassive black holes, the commonly used and simple R−β model is able to reproduce well the various and more complex electron-heating prescriptions considered here.

Accretion of magnetized gas on compact astrophysical objects such as black holes (BHs) has been successfully modeled using general relativistic magnetohydrodynamic (GRMHD) simulations. These simulations have largely been performed in the Kerr metric, which describes the spacetime of a vacuum and stationary spinning BH in general relativity (GR). The simulations have revealed important clues to the physics of accretion flows and jets near the BH event horizon and have been used to interpret recent Event Horizon Telescope images of the supermassive BHs M87* and Sgr A*. The GRMHD simulations require the spacetime metric to be given in horizon-penetrating coordinates such that all metric coefficients are regular at the event horizon. Only a few metrics, notably the Kerr metric and its electrically charged spinning analog, the Kerr–Newman metric, are currently available in such coordinates. We report here horizon-penetrating forms of a large class of stationary, axisymmetric, spinning metrics. These can be used to carry out GRMHD simulations of accretion on spinning, nonvacuum BHs and non-BHs within GR, as well as accretion on spinning objects described by non-GR metric theories of gravity.

Recent observations by the Event Horizon Telescope (EHT) of supermassive black holes M87* and Sgr A* offer valuable insights into their space–time properties and astrophysical conditions. Utilizing a library of model images ($\sim 2$ million for Sgr A*) generated from general-relativistic magnetohydrodynamic (GRMHD) simulations, limited and coarse insights on key parameters such as black hole spin, magnetic flux, inclination angle, and electron temperature were gained. The image orientation and black hole mass estimates were obtained via a scoring and an approximate rescaling procedure. Lifting such approximations, probing the space of parameters continuously, and extending the parameter space of theoretical models is both desirable and computationally prohibitive with existing methods. To address this, we introduce a new Bayesian scheme that adaptively explores the parameter space of ray-traced, GRMHD models. The general relativistic radiative transfer code IPOLE is integrated with the EHT parameter estimation tool THEMIS. The pipeline produces a ray-traced model image from GRMHD data, computes predictions for very long baseline interferometric (VLBI) observables from the image for a specific VLBI array configuration and compares to data, thereby sampling the likelihood surface via a Markov chain Monte Carlo scheme. At this stage we focus on four parameters: accretion rate, electron thermodynamics, inclination, and source position angle. Our scheme faithfully recovers parameters from simulated VLBI data and accommodates time-variability via an inflated error budget. We highlight the impact of intrinsic variability on model fitting approaches. This work facilitates more informed inferences from GRMHD simulations and enables expansion of the model parameter space in a statistically robust and computationally efficient manner.

Perturber objects interacting with supermassive black hole accretion disks are often invoked to explain observed quasiperiodic behavior in active galactic nuclei (AGN). We present global, 3D general relativistic magnetohydrodynamic (GRMHD) simulations of black holes on inclined orbits colliding with magnetically arrested thick AGN disks using a binary black hole spacetime with mass ratio 0.1. We do this by implementing an approximate time-dependent binary black hole metric into the GRMHD Athena++ code. The secondary enhances the unbound mass outflow rate 2–4 times above that provided by the disk in quasiperiodic outbursts, eventually merging into a more continuous outflow at larger distances. We present a simple analytic model that qualitatively agrees well with this result and can be used to extrapolate to unexplored regions of parameter space. We show self-consistently for the first time that spin–orbit coupling between the primary black hole spin and the binary orbital angular momentum causes the accretion disk and jet directions to precess significantly (by 60°–80°) on long timescales (e.g., ∼20 times the binary orbital period). Because this effect may be the only way for thick AGN disks to consistently precess, it could provide strong evidence of a secondary black hole companion if observed in such a system. Besides this new phenomenology, the time-average properties of the disk and accretion rates onto the primary are only marginally altered by the presence of the secondary, consistent with our estimate for a perturbed thick disk. This situation might drastically change in cooled thin disks.

Context. Active galactic nuclei (AGNs) exhibit a wide range of black hole masses and inflow/outflow properties. It is now possible to probe regions close to the event horizons of nearby supermassive black holes (SMBHs) using very long baseline interferometry (VLBI) with earth-sized baselines, as performed by the Event Horizon Telescope (EHT). Aims. This study explores the emission properties of accretion and outflows near the event horizon of both low-mass and high-mass SMBHs. Using resistive general relativistic magnetohydrodynamic (GR-MHD) simulations, we model AGNs with thin Keplerian disks. This contrasts with widely studied models featuring thick disks, such as magnetically arrested disks (MADs) or the standard and normal evolution (SANE) scenario. Methods. Our GR-MHD models serve as simplified representations to study disk-jet-wind structures. These simulations are postprocessed and ray-traced, using constraints of black hole mass and observed spectral energy distributions (SEDs). Thermal synchrotron emission generated near the event horizon is used to create emission maps, which are analysed by separating accretion and outflow components to determine their contributions to the total intensity. Results. Whether the emission appears optically thick or thin at a given frequency depends on its position relative to the synchrotron SED peak. At 230 GHz, low-mass SMBHs appear optically thicker than high-mass ones, even at lower accretion rates. Doppler beaming affects the brightness of emission from outflows with changing viewing angles in low-mass systems. Conclusions. Eddington ratios from our models align with those inferred by the EHTC for M87 and SgrA* using thicker MAD/SANE models. Although thin disks are optically thicker, their spectral properties make high-mass systems appear optically thinner at 230 GHz-ideal for probing GR effects like photon rings. In contrast, low-mass systems remain optically thicker at these frequencies because of synchrotron self-absorption, making outflow emissions near the horizon more pronounced. However, distinguishing these features remains challenging with current EHT resolution.

The Event Horizon Telescope (EHT) has produced images of two supermassive black holes, Messier 87* (M 87*) and Sagittarius A* (Sgr A*). The EHT collaboration used these images to indirectly constrain black hole parameters by calibrating measurements of the sky-plane emission morphology to images of general relativistic magnetohydrodynamic (GRMHD) simulations. Here, we develop a model for directly constraining the black hole mass, spin, and inclination through signatures of lensing, redshift, and frame dragging, while simultaneously marginalizing over the unknown accretion and emission properties. By assuming optically thin, axisymmetric, equatorial emission near the black hole, our model gains orders of magnitude in speed over similar approaches that require radiative transfer. Using 2017 EHT M 87* baseline coverage, we use fits of the model to itself to show that the data are insufficient to demonstrate existence of the photon ring. We then survey time-averaged GRMHD simulations fitting EHT-like data, and find that our model is best-suited to fitting magnetically arrested disks, which are the favored class of simulations for both M 87* and Sgr A*. For these simulations, the best-fit model parameters are within ∼10% of the true mass and within ∼10° for inclination. With 2017 EHT coverage and 1% fractional uncertainty on amplitudes, spin is unconstrained. Accurate inference of spin axis position angle depends strongly on spin and electron temperature. Our results show the promise of directly constraining black hole spacetimes with interferometric data, but they also show that nearly identical images permit large differences in black hole properties, highlighting degeneracies between the plasma properties, spacetime, and, most crucially, the unknown emission geometry when studying lensed accretion flow images at a single frequency.

We present a framework for characterizing the spatiotemporal power spectrum of the variability expected from the horizon-scale emission structure around supermassive black holes, and we apply this framework to a library of general relativistic magnetohydrodynamic (GRMHD) simulations and associated general relativistic ray-traced images relevant for Event Horizon Telescope (EHT) observations of Sgr A*. We find that the variability power spectrum is generically a red-noise process in both the temporal and spatial dimensions, with the peak in power occurring on the longest timescales and largest spatial scales. When both the time-averaged source structure and the spatially integrated light-curve variability are removed, the residual power spectrum exhibits a universal broken power-law behavior. On small spatial frequencies, the residual power spectrum rises as the square of the spatial frequency and is proportional to the variance in the centroid of emission. Beyond some peak in variability power, the residual power spectrum falls as that of the time-averaged source structure, which is similar across simulations; this behavior can be naturally explained if the variability arises from a multiplicative random field that has a steeper high-frequency power-law index than that of the time-averaged source structure. We briefly explore the ability of power spectral variability studies to constrain physical parameters relevant for the GRMHD simulations, which can be scaled to provide predictions for black holes in a range of systems in the optically thin regime. We present specific expectations for the behavior of the M87* and Sgr A* accretion flows as observed by the EHT.

The Event Horizon Telescope (EHT) has produced images of two supermassive black holes, Messier~87* (M 87*) and Sagittarius~A* (Sgr A*). The EHT collaboration used these images to indirectly constrain black hole parameters by calibrating measurements of the sky-plane emission morphology to images of general relativistic magnetohydrodynamic (GRMHD) simulations. Here, we develop a model for directly constraining the black hole mass, spin, and inclination through signatures of lensing, redshift, and frame dragging, while simultaneously marginalizing over the unknown accretion and emission properties. By assuming optically thin, axisymmetric, equatorial emission near the black hole, our model gains orders of magnitude in speed over similar approaches that require radiative transfer. Using 2017 EHT M 87* baseline coverage, we use fits of the model to itself to show that the data are insufficient to demonstrate existence of the photon ring. We then survey time-averaged GRMHD simulations fitting EHT-like data, and find that our model is best-suited to fitting magnetically arrested disks, which are the favored class of simulations for both M 87* and Sgr A*. For these simulations, the best-fit model parameters are within ${\sim}10\%$ of the true mass and within ${\sim}10^\circ$ for inclination. With 2017 EHT coverage and 1\% fractional uncertainty on amplitudes, spin is unconstrained. Accurate inference of spin axis position angle depends strongly on spin and electron temperature. Our results show the promise of directly constraining black hole spacetimes with interferometric data, but they also show that nearly identical images permit large differences in black hole properties, highlighting degeneracies between the plasma properties, spacetime, and most crucially, the unknown emission geometry when studying lensed accretion flow images at a single frequency.