Gravitational Wave Signatures from Periodic Orbits around a non-commutative inspired black hole surrounded by quintessence

TL;DR

This paper investigates gravitational wave signatures from periodic orbits of a test particle around a noncommutative geometry–inspired black hole surrounded by quintessence. It combines the zoom–whirl taxonomy to classify periodic orbits with a numerical kludge approach to compute gravitational waveforms in the adiabatic, extreme mass-ratio regime. The results show that the noncommutative parameter Theta and the quintessence field significantly alter orbital dynamics and produce distinctive GW modulations, with characteristic strains peaking in the millihertz band suitable for space-based detectors like LISA. These findings suggest that future space missions could constrain noncommutative gravity effects in strong-field spacetimes by analyzing EMRI waveforms and their spectral content. Overall, the work highlights a promising avenue for testing quantum gravity-inspired modifications to general relativity using gravitational waves.

Abstract

We study gravitational wave emission from periodic orbits of a test particle around a noncommutative-inspired black hole surrounded by quintessence. Using the zoom-whirl taxonomy, which is characterized by three topological numbers $(z, w, v)$, we classify these orbits and calculate several representative gravitational waveforms for certain periodic orbits. We find that the noncommutative parameter $Θ$ and the quintessence field significantly modify both the orbital structure and the emitted waveforms. In particular, increasing $Θ$ leads to a phase shift and a change in amplitude in the waveform, while higher zoom numbers produce more complicated substructures. The characteristic strain spectra peak in the millihertz range, lying within the sensitivity band of the LISA detector. Moreover, the presence of the quintessence field introduces significant modifications to these waveforms, imprinting measurable deviations that could be tested or constrained by future space-based gravitational wave detectors. These results suggest that future space-based gravitational wave missions could probe or constrain noncommutative effects in strong gravitational fields.

Figures (7)

• Figure 1: Periodic orbits around a non-commutative-inspired black hole surrounded by quintessence with an equation-of-state parameter $\omega = -2/3$. The non-commutative parameter is set to $\Theta = 0.01$ and the particle energy to $E = 0.94$. 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 2: Periodic orbits for various $(z, w, v)$ combinations around a non-commutative-inspired black hole surrounded by quintessence with $\omega = -2/3$. Here, the non-commutative parameter is increased to $\Theta = 0.02$ while keeping the particle energy fixed at $E = 0.94$. Increasing $\Theta$ slightly modifies the orbit shape, leading to broader zoom regions and altered precession characteristics.
• Figure 3: Gravitational waveforms (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 $M = 10^{7} M_{\odot}$. The non-commutative parameter is $\Theta = 0.01$ and $E = 0.94$. Distinct zoom–whirl phases in the orbital motion are reflected in the modulation of the waveform amplitude and frequency.
• Figure 4: Gravitational waveforms from a test object with $m=10 M_\odot$ around periodic orbits $(1,2,0)$: blue, $(2,1,1)$: green, and $(3,2,2)$: red, around a supermassive black hole with mass $M=10^7 M_\odot$. The value of parameter $\Theta=0.02$ and energy is fixed at $E=0.94$. The left and right panels correspond to plus and cross polarizations, respectively.
• Figure 5: Fourier spectra $|\tilde{h}_{+,\times}(f)|$ corresponding to the time-domain waveforms shown in Fig. \ref{['gwpolar1']} for $\Theta = 0.01$. The spectral peaks correspond to characteristic frequencies of the zoom–whirl orbits, showing distinct harmonic structures related to the orbital parameters $(z, w, v)$.