Taxonomy of periodic orbits and gravitational waves in a deformed Schwarzschild black hole spacetime

TL;DR

The paper analyzes geodesic motion of a test particle around a deformed Schwarzschild (DSK) black hole, introducing a rational orbital taxonomy $(z,w,v)$ to classify periodic orbits and examining how deformation parameter $\alpha$ modifies circular orbits, including a photon ring that vanishes at $\alpha=8/5$ and dual MBO/MSCO branches. It catalogs periodic orbits in inner and outer regions, showing deformation-driven changes to perihelion/apastron and richer topologies than in Schwarzschild. Gravitational waves are computed via a numerical Kludge approach under an adiabatic approximation, revealing phase shifts and modest amplitude changes with increasing $\alpha$ and a growing mismatch relative to Schwarzschild that could be detectable by future space-based detectors. Overall, the work demonstrates that the DSQ (DSK) deformation imprints on EMRI-like dynamics and GW signatures, providing a potential observational test for non-Kerr spacetimes.

Abstract

In this paper, we investigate periodic orbits of test particles around a deformed Schwarzschild black hole and the resulting gravitational waves. Firstly, we examine the properties of circular orbits and find that circular orbits could disappear when the deformation is large enough. Then, using an orbital taxonomy, we characterize various periodic orbits with a set of triples, which describes the zoom-whirl behaviours. We also calculate the gravitational waveform signals generated by different periodic orbits, revealing the influence of the deformation on the gravitational wave, which can be potentially picked up by future space-based detectors.

Figures (8)

• Figure 1: Different circular orbits with changes in deformation parameter $\alpha$. The red line, the blue line and the green line represent the radius of MSCOs, photon orbits and MBOs, respectively. The Schwarzschild case corresponds to $\alpha=0$.
• Figure 2: Different periodic orbits are presented with deformation parameter $\alpha$ fixed at $1.8$ and the angular momentum $L_{z}$ fixed at $\sqrt{13.814985530588238}$. The triplet array and energy is shown together with corresponding orbit in every subgraph.
• Figure 3: Different orbits are presented with deformation parameter $\alpha$ fixed at $1.8$ and the angular momentum $L_{z}$ fixed at $\sqrt{13.814985530588238}$. The energy is shown below in every subgraph. It clearly shows the unstable behavior passing through the horizon.
• Figure 4: Different periodic orbits are presented with the angular momentum $L_{z}$ fixed at $\sqrt{12}$. The values of deformation parameter $\alpha$ are $1.8955944840993961$, $1.8$ and $1.7537887487646788$, respectively. The values of the triplet array for each orbit are plotted also.
• Figure 5: Different periodic orbits are presented with the fixed angular momentum $L_{z}=3.9$. The values of deformation parameter $\alpha$ are 0 and -1. The values of the triplet array for each orbit are plotted also.