Equatorial periodic orbits and gravitational waveforms in a black hole free of Cauchy horizon

TL;DR

We address EMRIs around a black hole that lacks a Cauchy horizon, constructed in Kerr-Schild form via gravitational decoupling and characterized by a hair parameter $\ell$. The work analyzes equatorial geodesics and the effective potential $V_{\mathrm{eff}}=f(r)\left(1+L^2/r^2\right)$, classifies equatorial periodic orbits by the rational ratio $q=\frac{\omega_{\phi}}{\omega_r}-1= w+v/z$, and investigates how $\ell$ shifts MBO and ISCO properties and the corresponding zoom-whirl trajectories. Gravitational waves are computed in the adiabatic/Kludge framework, showing that GW waveforms faithfully capture zoom-whirl features while the phase evolution encodes the hair parameter $\ell$, enabling constraints on the BH solution. The results indicate measurable deviations from Schwarzschild for small $\ell$ and demonstrate that future space-based detectors could use GW phase information to distinguish black holes without Cauchy horizons from standard black holes.

Abstract

In this paper, we study the periodic orbits and gravitational wave radiation in an extreme mass ratio inspiral system, where a stellar-mass object orbits a supermassive black hole without Cauchy horizons. Firstly, by using the effective potential, the marginally bound orbits and the innermost stable circular orbits are investigated. It is found that the radius, orbital angular momentum, and energy increase with the hair parameter for both orbits. Based on these results, we examine one special type of orbit, the periodic orbit, around the black hole without the Cauchy horizon. The results show that, for a fixed rational number, the energy and angular momentum of the periodic orbit increase with the hair parameter. In particular, we observe a significant deviation from the Schwarzschild case for small hair parameter with a large amount of external mass outside the black hole horizon. Moreover, we examine the waveforms in the extreme mass ratio inspiral system to explore the orbital information of the periodic orbits and the constraints on the parameters of the black holes. The results reveal that the gravitational waveforms can fully capture the zoom-whirl behavior of periodic orbits. Moreover, the phase of the gravitational waves imposes constraints on the parameters of the black hole solutions. As the system evolves, the phase shift of the waveforms becomes increasingly significant, with cumulative deviations becoming more pronounced over time. Compared to the Schwarzschild black hole background, the waveform phase will advance for the central supermassive black hole without a Cauchy horizon.

Figures (11)

• Figure 1: (a) The behavior of the metric function $f(r)$ constructed from the internal $m(r)$ and external $\tilde{m}(r)$ mass functions. (b) The dependence of the smooth connection radius $r_{s_{(i,o)}}$ and the horizons $r_{h_{(i,o)}}$ on the dimensionless hair parameter $\ell/\mathcal{M}$.
• Figure 2: The radius $r_{\rm MBO}$ and angular momentum $L_{\rm MBO}$ of MBOs as a function of the hair paraemter $\ell/\mathcal{M}$.
• Figure 3: The radius $r_{\rm ISCO}$, energy $E_{\rm ISCO}$, and angular momentum $L_{\rm ISCO}$ of ISCOs.
• Figure 4: The effective potential $V_{\mathrm{eff}}$ as a function of $r/\mathcal{M}$. The angular momentum $L/\mathcal{M}$ varies from $L_{\rm ISCO}$ to $L_{\rm MBO}$ from bottom to top. The dashed line represents the extremal points of the effective potential. The black hole horizon is located at $r_s=1 \mathcal{M}$ and $r_s=1.995 \mathcal{M}$, respectively.
• Figure 5: Parameter regions for the bound orbits (in shadow) for different $\ell/\mathcal{M}$.