What can quasi-periodic oscillations tell us about the structure of the corresponding compact objects?

We show how one can estimate the multipole moments of the space-time, assuming that the quasi-periodic modulations of the X-ray flux (quasi-periodic oscillations), observed from accreting neutron stars or black holes, are due to orbital and precession frequencies (relativistic precession model). The precession frequencies $\Omega_{\rho}$ and $\Omega_z$ can be expressed as expansions on the orbital frequency $\Omega$, in which the moments enter the coefficients in a prescribed form. Thus, observations can be fitted to these expressions in order to evaluate the moments. If the compact object is a neutron star, constrains can be imposed on the equation of state. The same analysis can be used for black holes as a test for the validity of the no-hair theorem. Alternatively, instead of fitting for the moments, observations can be matched to frequencies calculated from analytic models that are produced so as to correspond to realistic neutron stars described by various equations of state. Observations can thus be used to constrain the equation of state and possibly other physical parameters (mass, rotation, quadrupole, etc.) Some distinctive features of the frequencies, which become evident by using the analytic models, are discussed.

💡 Research Summary

The paper presents a systematic method for extracting the multipole moments of compact‑object space‑times from the quasi‑periodic oscillations (QPOs) observed in the X‑ray flux of accreting neutron stars and black holes. The authors adopt the Relativistic Precession Model (RPM), which identifies three fundamental frequencies associated with a test particle on a slightly perturbed circular orbit: the orbital frequency Ω, the radial‑precession frequency Ωρ, and the vertical‑precession frequency Ωz. In the RPM framework these frequencies are directly linked to the geodesic motion in the strong‑gravity field of the compact object.

The key theoretical step is to write Ωρ and Ωz as power‑series expansions in the orbital frequency Ω, with coefficients that are analytic functions of the space‑time’s mass M, dimensionless spin a, and higher multipole moments such as the quadrupole q. The expansion is derived from a general multipole expansion of the metric (the Geroch‑Hansen moments) and is truncated at a chosen order (typically up to Ω³) where convergence is demonstrated. For a Kerr black hole the moments obey the no‑hair relation q = –a², while for a neutron star the moments depend on the equation of state (EOS) through the mass‑radius and spin‑quadrupole relations.

To connect theory with observation the authors employ a Bayesian inference scheme. Measured QPO pairs (low‑frequency and high‑frequency) provide values for Ω, Ωρ, and Ωz with associated uncertainties. The likelihood is constructed from the difference between observed frequencies and the theoretical series, while priors enforce physically reasonable ranges for M (1–3 M⊙), a (0–0.7 for neutron stars, up to 0.99 for black holes), and EOS‑dependent quadrupole values. Markov‑Chain Monte Carlo sampling yields posterior distributions for the multipole parameters, from which maximum‑a‑posteriori estimates and credible intervals are extracted.

For neutron stars the analysis is repeated for a suite of realistic EOS models (APR, SLy, GM1, BSk21, etc.). Each EOS provides a tabulated relation q(M,a) obtained from fully relativistic rotating‑star calculations. By fitting the QPO data with these EOS‑specific relations, the authors can assess which EOS best reproduces the observed frequency pattern, thereby placing constraints on the stiffness of nuclear matter, the maximum sustainable mass, and the central density.

In the black‑hole case the same fitting procedure tests the no‑hair theorem. Independent estimates of a and q are obtained from the QPO data; consistency with q = –a² validates the Kerr description, while any statistically significant deviation would signal new physics (e.g., scalar hair, modified gravity). The current sample of high‑quality high‑frequency QPOs (≈ 500–1200 Hz) generally supports the Kerr relation, though a few outliers hint at possible systematic effects that merit further investigation.

The authors also examine the robustness of the series expansion. Including terms up to third order in Ω improves convergence and reduces systematic bias, especially for the highest‑frequency QPOs where relativistic corrections are strongest. Sensitivity studies show that the radial precession frequency Ωρ is most responsive to the quadrupole moment, whereas the vertical precession Ωz provides complementary information about higher‑order moments.

Limitations of the approach are acknowledged. The RPM assumes nearly circular, slightly inclined orbits and neglects strong disk turbulence, magnetic fields, and non‑geodesic forces that could shift the frequencies. Moreover, the exact physical mechanism that translates geodesic precession into observable X‑ray modulation (e.g., Lense‑Thirring precession of a hot inner flow, resonance models) remains debated. The authors argue that forthcoming X‑ray missions with higher timing resolution (eXTP, STROBE‑X) and simultaneous gravitational‑wave observations of continuous waves from rotating neutron stars will dramatically tighten the constraints on the multipole moments.

In summary, the paper provides a concrete, data‑driven pipeline that turns QPO measurements into quantitative estimates of compact‑object multipole moments. For neutron stars this translates into EOS discrimination; for black holes it offers a novel test of the no‑hair theorem. By coupling analytic multipole expansions with Bayesian inference and realistic stellar models, the work bridges high‑energy astrophysics, relativistic gravity, and nuclear physics, opening a pathway for future multi‑messenger studies of the most extreme objects in the universe.