Can X-ray quasi-periodic oscillations determine the spacetime of an accreting compact object? Kerr deviations, equatorial reflection symmetry, and the presence of a horizon

Quasi-periodic oscillations (QPOs) are narrow peaks in the X-ray power spectra of accreting neutron-star and black-hole binaries whose frequencies are measured to sub-percent precision. Because the geodesic orbital and epicyclic frequencies of test particles depend only on the background metric, QPO timing has long been proposed as a tool for strong-field spacetime metrology: testing the Kerr metric and the no-hair relation, probing breaking of equatorial reflection symmetry, and distinguishing horizon-bearing from horizonless compact objects. All strands remain open: every published QPO-based constraint on non-Kerr deformation parameters is weak and rests on a contested QPO-identification model; no dedicated study of QPO signatures of reflection-symmetry breaking exists; the very-high-frequency QPO horizon test for non-singular objects awaits spin, detectability, and inversion analysis; and neutron-star-calibrated precession models have not been extended to black holes.

Background

Accreting compact objects in low-mass X-ray binaries show quasi-periodic oscillations (QPOs): peaks in the Fourier power spectrum of their X-ray light curves, with frequencies from Hz up to the kilohertz range. The high-frequency QPOs are widely thought to originate in the innermost accretion flow, where gravity is strong. In the relativistic precession model (RPM), introduced by Stella and Vietri in 1999, a pair of kHz QPOs and a low-frequency QPO are identified with the orbital (Keplerian) frequency and the radial and vertical epicyclic frequencies of a test particle on a nearly circular orbit — the frequencies of small radial and latitudinal oscillations about that orbit. For structureless test particles these frequencies are purely geodesic quantities fixed by the background metric, so measuring them is, in principle, a direct measurement of the spacetime geometry rather than of the accretion physics.

This opens three distinct routes to probing the central object's spacetime with QPO timing. First, the no-hair theorem of general relativity states that an isolated, stationary, vacuum black hole is fully described by the Kerr metric, whose multipole moments are all fixed by mass and spin alone; measuring a third quantity, such as the quadrupole moment, or constraining parameters of phenomenological non-Kerr metrics (e.g. the quasi-Kerr metric with a free quadrupole, or the Ghasemi-Nodehi-Bambi deformation parameters $b_i$), tests this hypothesis. Johannsen and Psaltis (2011) formulated the QPO version of this test, and a series of works has since fit deformed metrics to QPO data — notably Ghasemi-Nodehi, Lu, Chen and Yang (2020), who found that QPOs of the black-hole candidate GRO J1655-40 constrain only one of the Ghasemi-Nodehi-Bambi parameters, and that simulated samples of about ten sources with percent-level frequency precision would be needed to constrain four. Allahyari and Shao (2021) found a stubborn spin-quadrupole degeneracy in the same source, and Rink, Caiazzo and Heyl (2022), extending the RPM to the Kerr-Newman-de Sitter metric (the rotating, electrically charged black-hole solution, further generalized to include a cosmological constant), obtained only weak deviation constraints. The recurring obstacle is that the physical identification of the observed QPOs (RPM versus epicyclic resonance models versus others) and the emission radius are not established, so any metric inference inherits the assumptions of the chosen QPO model. A recent line of work addresses this model problem itself: Luongo, Muccino and collaborators propose a macroscopic precession model (MPM) in which disk inhomogeneities are spinning test bodies governed by the Mathisson-Papapetrou-Dixon (MPD) equations — the equations of motion of extended test bodies whose spin couples to spacetime curvature — adding spin-curvature corrections to the epicyclic frequencies; applied to eight neutron-star binaries it removes an apparent Schwarzschild-de Sitter degeneracy of the RPM without modifying the metric, but it has not yet been applied to black-hole systems.

Second, the Kerr spacetime is symmetric under reflection across its equatorial plane (a $Z_2$ symmetry). Datta and Mukherjee (2021) showed that in a general stationary axisymmetric vacuum spacetime this reflection symmetry is tied to the very existence of circular equatorial orbits: if the symmetry is broken, neither stable nor unstable circular orbits can lie on the equatorial plane. Testing equatorial reflection symmetry therefore tests the Kerr hypothesis, and the authors suggested that X-ray time variability — QPOs from the innermost flow — could carry imprints of the off-equatorial dynamics induced by symmetry breaking, while explicitly deferring the observational study. Related probes of reflection-symmetry breaking have since been developed through other channels (gravitational waves from LISA extreme-mass-ratio inspirals, black-hole shadow and disk images, spectral line profiles, thick-disk tori), not through QPO timing.

Third, QPOs may distinguish objects with a horizon from horizonless ones. Boos and Wunsch (2025, published JCAP 2026) studied static, spherically symmetric non-singular black-hole metrics modified by a regulator length scale $L$; for $L \gtrsim GM$ the horizon disappears, leaving a compact horizonless object that still possesses a stable circular orbit at radii far inside the innermost stable circular orbit (ISCO) of the corresponding black hole. Matter on these regulator-induced orbits emits very-high-frequency QPOs (VHFQPOs, roughly 1 kHz for a $10 M_\odot$ object to 25 kHz for $2 M_\odot$) which, unlike in the black-hole case, can escape to infinity; the absence of such signals would therefore indicate the presence of a horizon. The same paper flags what remains unresolved: the effect of the central object's angular momentum, instrumental detectability with existing or planned X-ray observatories, a model-independent frequency inversion (the truncated small-radius expansion overestimates the frequencies, so the result is an upper bound), and alternative regulator prescriptions such as mass-dependent $L(M)$.

Problem Statement

The shared question, posed independently in each of the four source works, is whether QPO timing observations can be inverted into decisive constraints on the spacetime geometry of an accreting compact object — that is, constraints that survive the current model uncertainty in how QPO frequencies are produced. Concretely, the shared question is posed in the following linked facets, each of which is explicitly left open in its own source:

1. Metric inversion and the no-hair test. Determine whether sets of measured QPO frequencies (pairs or triplets, possibly together with broad-band timing features) can constrain deformation parameters of non-Kerr metrics — or equivalently measure a third spacetime multipole such as the quadrupole — at a level that is decisive rather than conditional on a specific QPO-identification model and emission radius. Related open questions raised in the literature are whether the spin-quadrupole degeneracy found by Allahyari and Shao can be broken, what measurement precision and sample size are actually achievable (the multi-object constraint forecasts assume percent-level frequencies on about ten systems), and whether extensions of the RPM to the spin-curvature-corrected macroscopic precession model change which metric parameters are recoverable — including whether the MPM, calibrated on neutron-star binaries, can be extended to black-hole systems as its authors propose.

2. Reflection symmetry. Establish how equatorial reflection-symmetry-breaking parameters translate into observable QPO timing signatures from the innermost accretion flow, and whether current or planned X-ray observations can constrain them — the dedicated study that Datta and Mukherjee's circular-orbit no-go result calls for.

3. Horizon versus horizonless structure. Determine whether the predicted very-high-frequency QPOs from regulator-induced innermost stable orbits around non-singular horizonless objects are a practical horizon test: how the central object's angular momentum shifts the predicted frequencies, whether existing or planned observatories can detect or exclude them in realistic systems, how to invert observed frequencies into properties of the compact object without relying on the truncated small-radius expansion, and how alternative (e.g. mass-dependent) regulator prescriptions modify the signal.

The three facets are distinct open problems, each left open by its own source. The questions enumerated inside a facet fall within that facet's scope rather than standing as separate problems.

References

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