Quantum black hole ringdown

Author(s)

Topaloglou, Konstantinos, Cuoco, Elena, Casadio, Roberto

Abstract

The ringdown of a black hole following a merger is a potential candidate for revealing the signatures of quantum gravity in the emerging gravitational waves. In quantum theory, black holes are expected to have a discrete area and energy spectrum, which conflicts with the classical notion of an horizon that absorbs all infalling perturbations. We propose that the quantum black hole dissipates the energy it cannot absorb by emitting ``soft'' gravitons that carry away the energy difference between the energy of the infalling perturbation and the quantum transition energy. We find that the ringdown spectrum is consequently enriched with low-frequency components, and that there exists a weak low-frequency flux that persists for a timescale much longer than the ringdown itself.

Figures

Potential for massless scalar perturbations in the outer Schwarzschild spacetime in tortoise coordinates $r_*=r+\Rh\,\ln(r/\Rh-1)$ (perturbations of spin $s=2$ experience the similar Regge-Wheeler and Zerilli potentials). Asymptotic particle states are defined at $r_*^{(\infty)}=r_*(\Rh+L)$ with $\lp\ll L\ll \Rh$ with purely ingoing boundary condition at $r_*(\Rh)=-\infty$, so that the scattering occurs in the region $-\infty<r_*<r_*^{(\infty)}$.
Caption Potential for massless scalar perturbations in the outer Schwarzschild spacetime in tortoise coordinates $r_*=r+\Rh\,\ln(r/\Rh-1)$ (perturbations of spin $s=2$ experience the similar Regge-Wheeler and Zerilli potentials). Asymptotic particle states are defined at $r_*^{(\infty)}=r_*(\Rh+L)$ with $\lp\ll L\ll \Rh$ with purely ingoing boundary condition at $r_*(\Rh)=-\infty$, so that the scattering occurs in the region $-\infty<r_*<r_*^{(\infty)}$.
Transmission factor for the Zerilli potential obtained by integrating a stationary mode across the barrier (black markers), and by simulating $\sigma = 40$ wavepackets scattering on the barrier (blue line). The transmission factor $T$ in both cases is defined as the squared modulus of the ratio of the transmitted wave amplitude to the incident wave amplitude.
Caption Transmission factor for the Zerilli potential obtained by integrating a stationary mode across the barrier (black markers), and by simulating $\sigma = 40$ wavepackets scattering on the barrier (blue line). The transmission factor $T$ in both cases is defined as the squared modulus of the ratio of the transmitted wave amplitude to the incident wave amplitude.
Emitted spectrum for the probability profiles~\eqref{eq:prob_profile_1} (Gaussian) and~\eqref{eq:prob_profile_2} ($n^2\cdot\,$Gaussian) weighted by the transmission factor $T$ from Fig.~\ref{fig:transmission_factor} (solid lines) compared to the same spectra not weighted by $T$ (dotted lines) for $\alpha=2$. In both cases the weighted spectrum is heavily concentrated at high frequencies while the lower components are significantly attenuated with respect to the unweighted spectra, but the leftmost peaks will persist for substantially longer times than the QNM decay times. Spectral lines are plotted as Lorentzian profiles with full width at half-maximum equal to $10^{-2}$ times the decay rate for illustrative purposes.
Caption Emitted spectrum for the probability profiles~\eqref{eq:prob_profile_1} (Gaussian) and~\eqref{eq:prob_profile_2} ($n^2\cdot\,$Gaussian) weighted by the transmission factor $T$ from Fig.~\ref{fig:transmission_factor} (solid lines) compared to the same spectra not weighted by $T$ (dotted lines) for $\alpha=2$. In both cases the weighted spectrum is heavily concentrated at high frequencies while the lower components are significantly attenuated with respect to the unweighted spectra, but the leftmost peaks will persist for substantially longer times than the QNM decay times. Spectral lines are plotted as Lorentzian profiles with full width at half-maximum equal to $10^{-2}$ times the decay rate for illustrative purposes.
Lowest dissipation frequency $\omega_{\rm min}$ corresponding to the fundamental QNM with $n=0$ for a qBH with $M = 50\, M_\odot$ (left panel) and the corresponding decay timescale $t_{\rm d}$ (right panel) as functions of the quantisation parameter $\alpha$ in Eq.~\eqref{eq:Bek_muk_quantum}.
Caption Lowest dissipation frequency $\omega_{\rm min}$ corresponding to the fundamental QNM with $n=0$ for a qBH with $M = 50\, M_\odot$ (left panel) and the corresponding decay timescale $t_{\rm d}$ (right panel) as functions of the quantisation parameter $\alpha$ in Eq.~\eqref{eq:Bek_muk_quantum}.
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