Practical reconstruction of a black hole photon ring's curve shape and intensity from interferometric visibility data

General relativity predicts that a narrow photon ring on a black hole image carries a universal long-baseline interferometric signature: at each baseline angle, the complex visibility is a sum of oscillating terms whose frequencies are the projected positions of the curve points where the baseline direction is normal to the ring, with amplitudes set by the local integrated intensity and radius of curvature. This signature contains, in principle, the full curve shape and intensity profile. The explicit in-principle inversion was supplied within months by a companion paper, and a vigorous parametric program — angle-dependent diameters fitted to the Kerr-predicted 'circlipse' form — now underpins the shape-measurement science case of the proposed Black Hole Explorer space-VLBI mission. What remains open is the completion of that program: a stable, validated inversion from realistic noisy and sparsely sampled visibilities — using the visibility phase to recover the full curve and intensity, or, when only the amplitude is available, the parity-even part of the curve with a quantified statement of the residual odd-function freedom — and a rigorous treatment of non-generic curve points where the curvature vanishes and the universal scaling law fails.

Background

General relativity predicts that light emitted near a black hole is strongly lensed into a sequence of increasingly narrow, demagnified 'photon rings' on the image, which converge to a critical curve — the image of the photon shell, the set of bound photon orbits. For a Kerr black hole — the unique stationary, axisymmetric family of rotating black holes in general relativity, parametrized by mass and spin — this critical curve is fixed entirely by the mass, spin, and observer inclination, so its precise shape is a clean target for tests of strong-field gravity. Black-hole imaging uses very long baseline interferometry (VLBI): each pair of telescopes measures one complex Fourier component of the sky brightness, called the visibility $V(u,\varphi)$, where $u$ is the baseline length projected perpendicular to the line of sight and $\varphi$ its position angle.

Johnson et al. (2020) showed that a narrow circular ring of diameter $d$ and width $w \ll d$ has a 'universal' signature on intermediate baselines $1/d \ll u \ll 1/w$: the visibility oscillates with period $2/d$ and decays as $u^{-1/2}$, with the emission profile confined to the oscillation amplitudes. Gralla (2020) generalized this to a ring of arbitrary shape $\vec r_0(s)$ and transverse-integrated intensity $\mathcal{I}(s)$. A stationary-phase evaluation of the Fourier integral — the standard asymptotic method for oscillatory integrals whose phase is stationary at isolated points — shows that each baseline angle receives contributions only from the curve points where the baseline direction is normal to the curve: $V(u,\varphi) \approx \sum_I \mathcal{I}_I\sqrt{\mathcal{R}_I}\, u^{-1/2} e^{-i\pi S_I/4} e^{-2\pi i z_I u}$, where $z_I$ is the projected position of the point, $\mathcal{R}_I$ its radius of curvature, and $S_I = \pm 1$ a sign. As $\varphi$ sweeps $[0,\pi)$, the frequencies $z_I(\varphi)$ trace out the projected position function, which completely encodes the curve, while the amplitudes encode $\mathcal{I}\sqrt{\mathcal{R}}$.

Gralla and Lupsasca (2020) supplied the explicit inverse formula reconstructing the curve from its projected position function, completing the in-principle argument. Two obstructions separate this from practice. First, the full projected position function (equivalently, both the projected diameter $d_\varphi$ and centroid $C_\varphi$) requires the visibility phase, whereas near-term observations most reliably measure the amplitude alone; for a convex curve the amplitude determines only the projected diameters ('widths'), the parity-even part of the shape (the part invariant under reversing one coordinate axis), leaving a known degeneracy by one odd function on the circle. Second, the derivation assumes finite nonzero curvature; at points where the radius of curvature diverges the decay law changes to $u^{-1/n}$ for a tangency of order $n$, and a straight segment produces no decay at all.

Subsequent work recast the Kerr prediction as a four-parameter 'circlipse' functional form for $d_\varphi$ (Gralla, Lupsasca, and Marrone 2020) and built the mission case for the Black Hole Explorer (BHEX), a proposed space-VLBI extension of the Event Horizon Telescope — the globe-spanning millimeter-VLBI array that imaged M87* and Sgr A* — whose shape-measurement goal rests on extracting $d_\varphi$ from sparse, noisy amplitude samples on space-to-ground baselines. Noise-robust inference of the diameter function from synthetic visibilities has been demonstrated, but always at this parametric level; the full curve and intensity inversion has never been implemented on realistic data, and the singular-direction analysis has never been redone.

Problem Statement

Provide a practical, validated inversion that takes a realistic, noisy, sparsely sampled measurement of the complex visibility $V(u,\varphi)$ of a black hole photon ring — or of its amplitude alone, which determines only the parity-even part of the curve, as will be available to near-term space-VLBI missions — and returns a stable estimate of the underlying curve shape $\vec r_0(s)$ and transverse-integrated intensity profile $\mathcal{I}(s)$, with quantified uncertainties. This single objective comprises two jointly supporting parts:

(i) an implementable, noise-robust reconstruction algorithm — the in-principle inverse formula of Gralla and Lupsasca (2020) exists but has never been turned into a validated estimator — that either uses visibility phase (addressing the difficulty of absolute phase tracking on space baselines) or quantifies and partially resolves the amplitude-only degeneracy: the measured widths $d_\varphi$ then determine only the parity-even part of a convex curve and one odd-function freedom remains, so the amplitude-only branch must return that parity-even part with quantified uncertainties and report the residual odd-function freedom it leaves unresolved;

(ii) a rigorous analysis of the non-generic curve points at which the radius of curvature $\mathcal{R}$ diverges or the curve contains straight segments: at a tangency of order $n$ the long-baseline visibility decays as $u^{-1/n}$ rather than $u^{-1/2}$, and a straight segment produces a non-decaying oscillatory visibility; the derivation of the universal signature must be completed for these directions, and their impact on parameter estimation and their incorporation in (or principled exclusion from) reconstruction algorithms must be established.

The target is a reconstruction procedure whose stability and accuracy are demonstrated on realistic simulated or actual interferometric data, together with the rigorous singular-direction analysis.

References

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