The bound orbits and gravitational waveforms of timelike particles around renormalization group improved Kerr black holes

Abstract

In this article, we investigate the bound orbits of the timelike particles and the gravitational waveforms emitted from these orbits around a renormalization group improved Kerr black hole in the framework of the asymptotic safety approach. The running Newton coupling in the metric is characterized by two free quantum parameters $(ω,\,γ)$ arsing from the non-perturbative renormalization group theory and the appropriate cutoff identification, respectively. As expected, the radii of the horizon, the marginally bound orbits and the innermost stable orbit are all decrease as the quantum parameters increase. Under the extreme mass-ratio inspirals approximation the deviation of gravitational waveforms radiated by the periodic orbits from those in the classical Kerr background increases with the two quantum parameter. However, this effect is much smaller in the retrograde case compared to the prograde case. Especially, by comparing the characteristic strain of those gravitational wave with the sensitivity curve of several potential detectors, we find that their characteristic frequencies can fall within the sensitivity ranges of several planned gravitational wave observatories, suggesting that such signals may be detectable with sufficient instrumental sensitivity.

Figures (12)

• Figure 1: The maximum $\omega_{max}$ as a function of $\gamma$ with the spin parameter $a=0.1$ (blue solid), $a=0.5$ (red dashed) and $a=0.9$ (green dotted), respectively. The horizon exists in the parameter region $\omega\in[0,\omega_{max}]$ for fixed $\gamma$.
• Figure 2: The horizon radius $r_0$ of the extremal RGI-Kerr black hole as a function of $\gamma$ with the spin parameter $a=0.1$ (blue solid), $a=0.5$ (red dashed) and $a=0.9$ (green dotted), respectively.
• Figure 3: MBOs $r_{MBO}$ and the corresponded angular momentum $\mathcal{L}_{MBO}$ as functions of $\gamma$ when $a=0.5$. The right superscripts $P$ and $R$ represent the prograde and retrograde orbits, respectively.
• Figure 4: $r_{ISCO}$, $\mathcal{E}_{ISCO}$ and $\mathcal{L}_{ISCO}$ as functions of $\gamma$. The right superscripts $P$ and $R$ represent the prograde and retrograde orbits, respectively. Here $a=0.5$ but for other spin parameters one can find similar behaviors.
• Figure 5: The energy $\mathcal{E}_{m}$ as functions of $\gamma$ with fixed $\mathcal{L}_{m}=3$ (prograde) or $\mathcal{L}_{m}=-4$ (retrograde), where the spin parameter $a=0.5$. The solid lines correspond to prograde motions (right superscripts $P$) while the dashed lines represent retrograde motions (right superscripts $R$).