The Role of Interferometric Phase in Measuring Black Hole Photon Rings

TL;DR

This work investigates how interferometric phase and amplitude in VLBI observations encode the photon-ring structure around black holes, with a focus on spin-induced displacement between the direct and higher-order lensed images. It combines analytic geometric models, synthetic space-ground VLBI data (EHT/BHEX), and semi-analytic Kerr ray-tracing accretion models to quantify how ring displacement and ring stretching imprint on complex visibilities. The findings show that the interferometric phase is a powerful probe of spin through the relative centroid shift of the first two photon rings (approximately $1\,\mu{\rm as}$ per unit spin for M87*-like systems, yielding substantial phase slopes on long baselines), while amplitude constrains the ring’s shape; non-spacetime emission effects can confound spin signals and hence must be modeled jointly. These results underscore the value of space-based VLBI baselines to jointly constrain black hole spacetime and accretion physics, advancing prospects for precise spin measurements and tests of non-Kerr spacetimes. The work highlights that phase information is indispensable alongside amplitude for robust photon-ring-based inferences in upcoming facilities like BHEX.

Abstract

The Event Horizon Telescope (EHT) captured the first images of a black hole using Very Long Baseline Interferometry (VLBI). In the near future, extensions of the EHT such as the Black Hole Explorer (BHEX) will allow access to finer-scale features, such as a black hole's ''photon ring.'' In the Kerr spacetime, this image structure arises from strong gravitational lensing near the black hole that results in a series of increasingly demagnified images of each emitting region that exponentially converge to a limiting critical curve. Exotic black hole alternatives, such as wormholes, can introduce additional photon rings. Hence, precisely characterizing multi-ring images is a promising pathway for measuring black hole parameters, such as spin, as well as exploring non-Kerr spacetimes. Here, we examine the interferometric response of multi-ring systems using a series of 1) simple geometric toy models, 2) synthetic BHEX and EHT observations of geometric models, and 3) semi-analytic accretion models with ray-tracing in the Kerr spacetime. We find that interferometric amplitude is more sensitive to the shape of the photon ring, while interferometric phase is more sensitive to its displacement, which is most sensitive to black hole spin. We find that for models similar to Messier 87* (M87*), the relative displacement of the first strongly lensed image from the weakly lensed direct image is approximately $1\,μ{\rm as}$ per unit dimensionless spin, yielding an expected phase signature on a 25 G$λ$ baseline of $\sim44^\circ$ per unit spin.

Figures (7)

• Figure 1: Three images of spinning black holes and their associated interferometric amplitude and phase responses. Top: ray-traced images of black holes of high spin oriented into the page (left), low spin oriented out of the page (middle), and high spin oriented out of the page (right). To isolate the effects of spin, we have fixed the viewing inclination at $\theta=17^\circ$, and we assume isotropic emission from a plasma at rest with respect to the local zero angular momentum observer. Middle: the amplitude response for each image. Bottom: the phase response for each image. The spin has a subtle effect on visibility amplitudes and a pronounced effect on visibility phases.
• Figure 2: Interferometric phase response for a single geometric ring, varying its degree of stretching $\epsilon$ (row 1), the displacement of its center $\Delta x$ (row 2), and the combined effects of stretching and shifting (row 3). Vertical stretching in the image corresponds to horizontal stretching in the interferometric response. Shifting in the image introduces a phase slope that grows linearly with the baseline projected along the shift direction, thus replacing the monochromatic slices with a phase that varies smoothly and periodically. For this single-ring case, both shifting and stretching preserve the discontinuous phase jumps between slices.
• Figure 3: Row 1: visibility phases from a single-ring model with different parameters. Row 2: corresponding azimuthal variation (along the Fourier angle $\varphi$) of the phase $\psi$ at two different fixed baseline lengths, chosen to match typical Earth-Earth and Earth-space baselines. The marked circles in row 1 show the fixed baseline lengths $\rho=25\,{\rm G}\lambda$ and $\rho=5\,{\rm G}\lambda$. The longer baseline consistently samples a greater variation in the phase.
• Figure 4: Interferometric phase response of two superimposed rings, holding the $n=0$ ring centered at the origin in the image domain. Row 1: we vary their thickness ratio $\sigma_0/\sigma_1$, with the $n=1$ ring shifted by $\Delta x_1 = 4 \,\mu {\rm as}$ to the right. Row 2: we scale their absolute thickness while holding the thickness ratio fixed at $\sigma_0/\sigma_1=10$, with the $n=1$ similarly displaced. Row 3: we vary the degree of stretching $\epsilon$ for the $n=1$ ring, fixing its absolute thickness at $\sigma_1= 0.5\, \mu {\rm as}$ and without displacing its center. Row 4: we vary the displacement $\Delta x_1$ of the $n=1$, shifting its center to the right. Row 5: we vary both the shift and stretch of the $n=1$ ring. The phase slope along each annular slice grows linearly with the baseline length, becoming more visible at longer baselines. Rows 1-4 preserve the discontinuous phase jumps of the Bessel function. In the last row, however, the mix of the two distortions blurs the discontinuity and allows for a continuous phase slope across slices. The phase jumps are restored at longer baselines.
• Figure 5: Same as \ref{['fig:onering_orbitslice']} but for models with two rings: one with $\sigma_0=5\,\mu{\rm as}$, and a thinner ring with $\sigma_1=0.5\,\mu{\rm as}$. The left panel corresponds to the case where the rings are nearly overlapping and the thinner ring is nearly circular ($\Delta x_1 = 0.1 \,\mu{\rm as}$, $\epsilon=-10^{-4}$). The middle panel has a larger displacement between the rings, and the thinner ring is slightly more stretched ($\Delta x_1 = 0.5 \,\mu{\rm as}$, $\epsilon=-10^{-3}$). The right panel has the greatest displacement and stretching of the thin ring ($\Delta x_1 = 1 \,\mu{\rm as}$, $\epsilon=-10^{-2}$).