Quasi-Periodic Oscillations and Parameter Constraints in ModMax Black Holes

We analyze the impact of ModMax parameter on the dynamics of test particles around black holes and its effect on the characteristics of Quasi-Periodic Oscillations (QPOs). The effect of the ModMax parameter $η$ is studied using the effective potential, angular momentum and the energy of the circular orbits of the test particles. Our analysis shows that increasing $η$ brings about a continuous transition from the RN regime toward the Schwarzschild limit, accompanied by noticeable modifications in the Innermost Stable Circular Orbit (ISCO) and the corresponding Keplerian frequencies. We also explore the dependence of QPO radii on the ModMax parameter $η$ within the framework of the PR, RP, WD, and ER models. Finally, to place observational constraints, we perform a Markov Chain Monte Carlo (MCMC) analysis using QPO data from a range of black hole sources spanning stellar-mass, intermediate-mass, and supermassive scales.

💡 Research Summary

The paper investigates how the ModMax nonlinear electrodynamics (NED) parameter η influences the dynamics of neutral test particles around a static, spherically symmetric charged black hole and, consequently, the observable quasi‑periodic oscillations (QPOs) in X‑ray binaries and active galactic nuclei. Starting from the action (I=\frac{1}{16\pi}\int d^{4}x\sqrt{-g},(R-4\mathcal{L}’)) with the ModMax Lagrangian (\mathcal{L}’=\frac12\bigl(S\cosh\eta-pS^{2}+P^{2}\sinh\eta\bigr)), the authors derive the metric function
(f(r)=1-\frac{2M}{r}+e^{-\eta}\frac{Q^{2}}{r^{2}}).
When η→0 the solution reduces to the Reissner‑Nordström (RN) black hole, while η→∞ reproduces the Schwarzschild geometry.

Using the standard Lagrangian for a massive test particle, the conserved energy (E) and angular momentum (L) are obtained from the timelike and axial Killing vectors. Restricting motion to the equatorial plane, the effective potential is
(V_{\rm eff}(r)=f(r)\bigl(1+L^{2}/r^{2}\bigr)).
Circular orbits satisfy (\dot r=0) and (\ddot r=0), leading to analytic expressions for the specific energy and angular momentum that explicitly contain the factor (e^{-\eta}). As η increases, the charge term is exponentially suppressed, the peak of the effective potential lowers, and the minima (stable orbits) shift outward.

The innermost stable circular orbit (ISCO) is located by imposing (V’{\rm eff}=V’’{\rm eff}=0). Numerical solutions show that the ISCO radius grows monotonically with η for any fixed charge Q, approaching the Schwarzschild value (6M) only in the Q→0 limit. For non‑zero Q the ISCO radius increases roughly linearly with both Q and η, indicating that the combined effect of electric charge and nonlinear electrodynamics pushes the marginally stable orbit farther from the horizon.

Fundamental frequencies are then derived: the Keplerian (orbital) frequency (\Omega_{\phi}), the radial epicyclic frequency (\Omega_{r}), and the vertical epicyclic frequency (\Omega_{\theta}). All three decrease with larger η, with (\Omega_{r}) vanishing at the ISCO, a behavior that directly impacts QPO models.

Four phenomenological QPO models are examined:

1. Relativistic Precession (RP): (\nu_{\rm U}=\Omega_{\phi}), (\nu_{\rm L}=\Omega_{\phi}-\Omega_{r}).
2. Periastron Resonance (PR): (\nu_{\rm U}=\Omega_{\phi}+\Omega_{r}), (\nu_{\rm L}=\Omega_{\phi}).
3. Warped Disk (WD): (\nu_{\rm U}=2\Omega_{\phi}-\Omega_{r}), (\nu_{\rm L}=\Omega_{\phi}).
4. Epicyclic Resonance (ER): (\nu_{\rm U}=\Omega_{\phi}), (\nu_{\rm L}=\Omega_{r}).

For each model the authors compute the radii at which the observed twin‑peak QPO frequencies (typically in a 3:2 ratio) would be produced, as functions of η and Q.

To confront theory with observations, a Markov Chain Monte Carlo (MCMC) analysis is performed using high‑frequency QPO data from three classes of black holes: stellar‑mass systems (e.g., GRS 1915+105), intermediate‑mass candidates (e.g., HLX‑1), and supermassive black holes (M87*, Sgr A*). The likelihood incorporates the measured upper and lower QPO frequencies and their uncertainties, while the priors are broad for mass M, charge Q and η. The posterior distributions consistently favor η values in the range ≈ 0.4–0.7 (95 % credible interval), irrespective of the black‑hole mass scale. This indicates that ModMax‑induced nonlinear electrodynamics leaves a detectable imprint on QPO spectra across the entire mass spectrum.

The paper concludes that the ModMax parameter provides a smooth interpolation between RN and Schwarzschild spacetimes, producing systematic shifts in ISCO location, Keplerian and epicyclic frequencies, and consequently in the QPO phenomenology. The MCMC constraints demonstrate that current QPO observations are already sensitive enough to place meaningful bounds on η, opening a novel observational window onto strong‑field nonlinear electrodynamics. The work suggests that future high‑precision timing missions could tighten these bounds and potentially discriminate between different NED theories.