Constraining magnetized black-bounce spacetime from HFQPOs data and selecting resonance models via information criteria and Bayesian analysis

Some of the most tantalizing objects in astrophysics refuse to sit still. In the X-ray skies, certain stellar-mass black hole candidates flicker with pairs of nearly periodic oscillations whose frequencies lock into a simple 3:2 ratio. A new theoretical study now argues that a piece of the puzzle may be missing from most analyses: the magnetic field threading the accretion disk, and what it does to charged particles spiraling toward the central object.

The work, published in The European Physical Journal C, focuses on the black-bounce spacetime proposed by Simpson and Visser, often abbreviated as the SV metric. This geometry is remarkable for its economy: a single regularization parameter a determines what kind of object it describes. When a equals zero, the metric reduces to the familiar Schwarzschild solution of general relativity. For values between zero and twice the mass, it describes a regular black hole whose central singularity has been smoothed away. At exactly twice the mass, the geometry becomes a one-way wormhole with a null throat, and beyond that threshold it opens into a fully traversable wormhole. The source of the metric, within general relativity, is a combination of a minimally coupled phantom scalar field and a nonlinear electrodynamics field, making it a popular phenomenological laboratory for ideas about how quantum gravity might soften the singularities predicted by classical theory.

Into this setting the authors, Shining Yang, Jianbo Lu, Mou Xu and Yu Liu, introduce an external, asymptotically uniform magnetic field oriented perpendicular to the equatorial plane, described by the Wald solution. Such fields are physically motivated: accretion disks of ionized plasma generate their own magnetic fields, and the intense gravity of a compact object can amplify ambient galactic fields. Although a weak field barely distorts the geometry of spacetime itself, its effect on the motion of charged matter is anything but negligible. Through the Lorentz force, even a small magnetic field reshapes the effective potential governing a particle’s orbit, and the study shows this reshaping in detail using the Hamilton-Jacobi formulation for a test particle of unit mass.

One of the clearest results concerns the innermost stable circular orbit, the ISCO, the radius inside of which stable orbits cease to exist. In the magnetized SV spacetime, the ISCO of a charged particle migrates inward as the regularization parameter a/M increases, so orbits hug the coordinate center more tightly near wormhole-like configurations than near black-hole-like ones. For a fixed spacetime type, increasing the magnetic field strength also pulls the ISCO inward, with the largest radius occurring when no field is present at all. The direction of the field matters too: for the same absolute field strength, the ISCO radius is consistently smaller when the magnetic parameter is positive than when it is negative. For black-hole-like cases, the field draws circular orbits closer to the event horizon, while for the wormhole case, particle trajectories can approach the wormhole throat arbitrarily closely.

The authors then turn to the oscillation frequencies that lie at the heart of the QPO puzzle. A particle on a circular orbit, when nudged slightly in radius or latitude, oscillates harmonically around that orbit, and these epicyclic frequencies, together with the orbital frequency, are the raw ingredients from which QPO models are built. In the unmagnetized SV spacetime, the latitudinal and orbital frequencies coincide and both exceed the radial frequency outside the ISCO, a hierarchy that constrains which resonance combinations are possible. With a magnetic field switched on, that hierarchy can break. Depending on the sign and magnitude of the field, the latitudinal frequency can equal or even fall below the radial frequency at larger radii, opening the door to resonance configurations that simply cannot occur in the field-free geometry.

This matters because the observed 3:2 twin-peak ratio in three Galactic microquasars, GRS 1915+105, XTE 1550-564 and GRO 1655-40, is widely interpreted as evidence of nonlinear resonance between two oscillation modes in the accretion disk. The canonical epicyclic resonance model comes in several variants, labeled ER1 through ER5, corresponding to different combinations of radial, latitudinal and orbital frequencies and their beat frequencies. In the magnetized spacetime, the altered frequency ordering permits four additional variants, ER6 through ER9, including cases where the upper observed frequency is a sum of the radial and latitudinal frequencies, or where the frequency ratios invert altogether. The magnetic field also shifts the resonance positions, the orbital radii where each variant produces the observed 3:2 ratio, inward as the field strength grows, mirroring the behavior of the ISCO.

To determine which of these ten variants the data actually favor, the team performed a chi-squared fit of the theoretical frequencies to the observed twin-peak frequencies from the three microquasars, treating the regularization parameter, the magnetic parameter and three resonance radii as free parameters. They then ranked the models with the Akaike Information Criterion, which balances goodness of fit against model complexity, and cross-checked the ranking with the Bayesian Information Criterion and a full Bayesian evidence calculation using the Jeffreys scale. The verdict was strikingly lopsided. The ER8 model, in which the upper frequency equals the sum of the latitudinal and radial frequencies and the lower frequency equals the radial frequency, emerged as the best model with an AIC of 12.623 and received strong support. The ER3 model earned moderate evidence, ER6 and ER7 showed considerably less support, and the remaining variants were essentially unsupported by the data. The BIC and Bayes factor analyses returned broadly consistent conclusions, with the Bayesian comparison providing decisive evidence for ER8 over six of its rivals.

For the models the data do support, the constraints on the spacetime itself are intriguing. The regularization parameter is bounded at 0 less than or equal to a less than 0.736 at the 68 percent confidence level, a range that sits squarely within the regular black hole regime of the SV family rather than the wormhole regime. The authors note that the preference for smaller values of a may hint at quantum effects regularizing the classical singularity, though this remains an interpretive suggestion rather than a demonstrated result. Translating the fitted dimensionless magnetic parameter into physical field strengths, assuming electrons or protons as the ionized test particles, yields best-fit values of roughly 10^-5 to 10^-4 Gauss for electrons and 10^-2 to 10^-1 Gauss for protons. The electron estimates are consistent with measured large-scale Galactic magnetic fields and with prior estimates for microquasars modeled in Kerr spacetime, while the proton estimates in some models are comparable to the magnetic field at Earth’s surface.

Perhaps the most pointed result is the negative control. When the authors repeated the entire fitting procedure for the SV spacetime without any magnetic field, the best-performing resonance model, ER3, landed at an AIC difference of 23.357 relative to the magnetized best fit, and since all models in that comparison share the same number of free parameters, every field-free variant fared no better. On the standard interpretation scales, a difference that large renders a model essentially irrelevant to the data. In other words, the unmagnetized SV geometry cannot simultaneously account for the HFQPOs of all three microquasars, a failure consistent with earlier findings that harmonic geodesic models struggle to explain these sources jointly, whereas the magnetized version can.

The study carries the usual caveats of theoretical model-fitting work. The real magnetic environments of microquasars are neither perfectly uniform nor globally simple, and the Wald solution is an approximation that captures the field magnitude in a local region. The precise mechanism generating HFQPOs remains unresolved, and the resonance framework is one hypothesis among several, including relativistic precession, tidal disruption and warped disk models. What the analysis demonstrates is a methodological point with real force: when competing QPO models are confronted with data through information criteria and Bayesian evidence, the ranking is decisive, and in this case the data speak clearly for a magnetized, regular black-bounce geometry with a specific resonance recipe.

Subject of Research: Constraining a magnetized Simpson-Visser black-bounce spacetime and selecting resonance models using high-frequency quasi-periodic oscillation data from three microquasars.

Article Title: Constraining magnetized black-bounce spacetime from HFQPOs data and selecting resonance models via information criteria and Bayesian analysis

Article References: Yang, S., Lu, J., Xu, M., & Liu, Y. (2026). Constraining magnetized black-bounce spacetime from HFQPOs data and selecting resonance models via information criteria and Bayesian analysis. The European Physical Journal C, 86(10), Article 1233. https://doi.org/10.1140/epjc/s10052-026-16491-5

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16491-5

Keywords: black-bounce spacetime, regular black holes, Simpson-Visser metric, high-frequency quasi-periodic oscillations, microquasars, magnetic fields, epicyclic resonance, Akaike Information Criterion, Bayesian analysis, innermost stable circular orbit, charged particles, X-ray binaries