Signatures of Quantum-Corrected Black Holes in Gravitational Waves from Periodic Orbits

Abstract

We investigate gravitational wave emission from periodic timelike orbits of a test particle around a loop quantum gravity-inspired Schwarzschild black hole. The spacetime is characterised by a holonomy-correction parameter that modifies the radial metric component while preserving asymptotic flatness and the classical location of the horizon. The bound geodesics are systematically classified using the zoom--whirl representation labelled by three integers $(z,w,v)$. Gravitational waveforms are computed within a numerical framework that combines exact geodesic motion with the quadrupole approximation, which is suitable for extreme mass ratio inspirals. We demonstrate that the quantum corrections lead to distinct phase shifts, amplitude variations, and modifications to the harmonic structure of the waveforms, with increasingly complex features for orbits with larger zoom numbers. The corresponding frequency spectra and characteristic strain peak, which fall within the millihertz band, are within the sensitivity ranges of space-based detectors such as LISA, Taiji, and TianQin. For specific orbital configurations and values of the quantum-correction parameter, the characteristic strain exceeds the projected detector noise, indicating potential observability. Our results demonstrate that gravitational waves from periodic orbits provide a sensitive probe of quantum-corrected black hole spacetimes in the strong-field regime.

Figures (8)

• Figure 1: The figure demonstrates the dependence of the rational number $q$ on the energy (left panel) and orbital angular momentum (right panel) for different values of the parameter $q_c$. Here, we set $L=3.73$ and $E=0.95$ for the left and right panels, respectively.
• Figure 2: Periodic orbits around a quantum-corrected black hole. The particle energy is fixed at $E = 0.95$. Each trajectory corresponds to a different set of zoom–whirl–vertex numbers $(z, w, v)$, illustrating the geometric complexity and structure of the bound periodic orbits.
• Figure 3: Periodic orbits for various $(z, w, v)$ combinations around a quantum-corrected black hole. Here, we fixed the angular momentum at $L = 3.73$.
• Figure 4: GWforms (plus and cross polarizations) generated by a test particle of mass $m = 10M_{\odot}$ in periodic orbits characterized by $(z, w, v) = (1,2,0)$ (blue), $(2,1,1)$ (green), and $(3,2,2)$ (red) around a supermassive black hole of mass $M = 10^{6} M_{\odot}$. The quantum-correction parameter is $q_c = 0.6$ and $E = 0.95$. Distinct zoom–whirl phases in the orbital motion are reflected in the modulation of the waveform amplitude and frequency.
• Figure 5: Gravitational waveforms from a test object with $m=10 M_\odot$ around periodic orbits with quantum-correction parameter $q_c=0.6$: blue, $1.0$: green, and $1.5$: red, around a supermassive black hole with mass $M=10^6 M_\odot$. The zoom-whirl value of the periodic orbit is $(1,2,0)$, and energy is fixed at $E=0.95$. The left and right panels correspond to plus and cross polarizations, respectively.