Probing regular black holes with sub-Planckian curvature through periodic orbits and their gravitational wave radiation

TL;DR

This work investigates regular black holes with sub-Planckian curvature and a Minkowskian core (RSBH) as potential astrophysical objects by focusing on their geodesic structure, photon spheres, and gravitational-wave signatures from EMRIs. It analyzes how the quantum-gravity parameter $α$ modifies the effective potential, MBOs, ISCOs, and the photon sphere, and uses observational data from M87$^*$ and Sgr A$^*$ to assess viable ranges for $α$. The authors classify periodic orbits via the triplet $(z,w,ν)$ and compute corresponding gravitational waves using a numerical kludge approach, revealing that $α$ induces notable phase shifts and amplitude changes in the zoom–whirl GW patterns. These results suggest that EMRIs and their GW signals could serve as probes of regular black-hole spacetimes and help constrain quantum-gravity corrections to spacetime structure with upcoming detectors.

Abstract

Extreme mass-ratio inspirals (EMRIs) are among the key targets for future space-based gravitational wave detectors. The gravitational waveforms emitted by EMRIs are highly sensitive to the orbital dynamics of the small compact object, which in turn are determined by the geometry of the underlying spacetime. In this paper, we explore the de- tectability of regular black holes with sub-Planckian curvature, which can be interpreted as regularized versions of the Schwarzschild black hole (RSBH). To do so, we begin by ana- lyzing the metric and geodesics, determining the effective potential, and investigating the marginally bound orbits and the innermost stable circular orbits for timelike particles. Our analysis reveals that orbital radius, angular momentum, and energy significantly depend on the model parameter α for both orbits. Our main aim is to focus on the influence of the model parameter on a specific kind of orbit, the periodic orbit, surrounding a supermassive RSBH. The findings show that, for a constant rational integer, α has a significant impact on the energy and angular momentum of the periodic orbit. Utilising the numerical kludge method, we further investigate the gravitational waveforms of the small celestial body over various periodic orbits. The waveforms display discrete zoom and spin phases within a complete orbital period, influenced by the RSBH parameter α. As the system evolves, the phase shift in the gravitational waveforms grows progressively more pronounced, with cumulative deviations amplifying over time. With the ongoing advancements in space- based gravitational wave detection systems, our results will aid in leveraging EMRIs to probe and characterize the RSBH properties.

Figures (17)

• Figure 1: Plot of the metric function $A(r)$ for different values of $\alpha$. The comparison includes the Bardeen BH, the full RSBH solution (solid curves), and its asymptotic approximation (dashed curves).
• Figure 2: The Plot indicates the dependence of the different radii and the critical impact parameter on the variable $\alpha$. The event horizon radius (Red solid line), Cauchy horizon radius (Red dashed line), photon sphere radius (Pink solid line), and the critical impact parameter (Cyan solid line) are all plotted.
• Figure 3: Behaviour of scalar invariants in RSBH spacetime as a function of $r$ for varying values of the regularization parameter $\alpha$. Panels from left to right correspond to the Ricci scalar, to the Squared Ricci tensor, and to the Kretschmann scalar, respectively.
• Figure 4: The radius $r_{\text{MBO}}$ (left panel) and angular momentum $L_{\text{MBO}}$ (right panel) of MBOs around the RSBH as functions of the deviation parameter $\alpha$.
• Figure 5: The radius $r_{\text{ISCO}}$ (first panel), energy $E_{\text{ISCO}}$ (second panel), and angular momentum $L_{\text{ISCO}}$ (third panel) of ISCOs around the RSBH as functions of the deviation parameter $\alpha$.