Constraints on the Scale Parameter of Regular Black Hole in Asymptotically Safe Gravity from Extreme Mass Ratio Inspirals

TL;DR

This work investigates constraining the quantum-scale parameter $\xi$ of a regular black hole in asymptotically safe gravity using gravitational waves from extreme mass ratio inspirals. It employs the Augmented Analytical Kludge (AAK) waveform method to construct GW templates in the equatorial plane and analyzes how $\xi$-dependent corrections, which first appear at 3PN order, accumulate in the GW phase relative to Schwarzschild. Through waveform mismatch analyses and Fisher information matrix forecasts, the study finds that LISA can detect $\xi$-effects at the level of $\sim10^{-4}$ for $M\sim10^6\,M_\odot$, with a one-sigma precision on $\Delta\xi$ of about $3.2\times10^{-4}$, significantly surpassing current observational constraints. The results demonstrate that EMRIs offer a powerful probe of strong-field quantum gravity, providing quantitative tests of asymptotically safe gravity in regimes where classical GR is challenged. The analysis highlights the role of long-term phase accumulation in amplifying tiny quantum-gravity signatures and outlines the need for incorporating complete 3PN corrections in future work to refine these constraints.

Abstract

This paper evaluates the potential for constraining the quantum scale parameter $ξ$ of regular black hole within the asymptotically safe gravity framework using gravitational waves from extreme mass ratio inspirals (EMRIs). Since $ξ$ cannot be precisely determined from first principles, observational constraints become crucial. We employ the Augmented Analytical Kludge (AAK) method to calculate gravitational waveforms in the equatorial plane and systematically analyze the influence of different $ξ$ values on phase evolution. Comparison with the Schwarzschild case demonstrates that the corrective effects of $ξ$ accumulate in the phase over observation time, thereby providing distinguishable observational signatures. Through waveform mismatch analysis, our results indicate that the LISA detector can effectively detect the presence of $ξ$ at the $\sim10^{-4}$ level for systems with a mass of $10^6M_\odot$. Further assessment using the Fisher information matrix (FIM) confirms a measurement precision of $Δξ\approx3.225\times10^{-4}$, which significantly surpasses existing observational methods, providing quantitative observational evidence for asymptotically safe quantum gravity theory in the strong-field regime.

Figures (7)

• Figure 1: The left figure shows the existence conditions for the event horizon of a regular black hole in asymptotically safe gravity, where $\frac{\xi_0}{M^2}\approx0.4565$ represents an extremal black hole; the right figure illustrates the effect of the scale parameter $\xi$ on the outer event horizon radius.
• Figure 2: The deviation $\Delta X=X_{\xi\ne 0}-X_{\xi= 0}$ in the semi-latus rectum $p$ and eccentricity $e$ over time evolution under different scale parameters $\xi$, where $X=\{p,e\}$. The corresponding initial conditions are $p_0=13, e_0=0.1$.
• Figure 3: Initial conditions are set to $e_0=0.1$ (first row of figures), $e_0=0.3$ (second row of figures), and $p_0=13$, to study the effects of different scale parameters $\xi$ on gravitational waveforms. The black curves represent the gravitational waveforms of the Schwarzschild black hole ($\xi=0$), while the colored curves represent the gravitational waveforms of the regular black hole in the asymptotically safe gravity framework, with scale parameters $\xi=0.01$ (blue dashed lines) and $\xi=0.001$ (red dashed lines). The left column shows the waveforms at the early stage of evolution, and the right column shows the waveforms after $120$ days of evolution.
• Figure 4: Under fixed conditions of $e_0=0.1$ and $p_0=13$, the evolution of phase differences over time for different scale parameters $\xi$ is shown. The left panel displays the radial phase difference, and the right panel shows the azimuthal phase difference, where the phase difference is defined as $\Delta\Phi_i=\left|\Phi_i(\xi\neq0)-\Phi_i(\xi=0)\right|$. The red horizontal line in the figures represents the phase threshold that can be distinguished by detection.
• Figure 5: Under fixed scale parameter $\xi=0.005$, the evolution of phase differences over time for different eccentricities $e_0$ is shown. The left panel displays the radial phase difference, and the right panel shows the azimuthal phase difference, where the phase difference is defined as $\Delta\Phi_i=\left|\Phi_i(\xi\neq0)-\Phi_i(\xi=0)\right|$. The red horizontal line in the figures represents the phase threshold that can be distinguished by detection.