Gravitational Waves from Resonant Transitions of Tidally Perturbed Gravitational Atoms

TL;DR

The paper investigates gravitational waves produced directly by resonant transitions in a gravitational-atom cloud around a spinning black hole embedded in a binary. It develops a formalism that maps GA transitions to a Landau-Zener problem and derives analytic expressions for the GW strain and spectrum from the time varying GA quadrupole, then evaluates detectability with a future space based detector. A parameter space study shows potential DECIGO detections for intermediate mass black hole hosts with alpha around 0.3, M around 150 M_sun, and boson masses near 10^-13 eV within about 100 kpc, but realistic spin uncertainties and annihilation considerations render detections challenging. The work provides templates for transition GWs and clarifies the conditions under which GA transitions can leave observable imprints, highlighting both the scientific value and the practical hurdles of these exotic signals.

Abstract

Light bosons can form a gravitational atom (GA) around a spinning black hole through the superradiance process. Considering the black hole to be part of a binary system, the tidal potential of the companion periodically perturbs the GA such that an ``atomic'' transition occurs between two of its energy eigenstates. The resonant transition is modeled by the Landau-Zener system, where the orbital frequency of the companion determines the relevant transition. In this work, we study a novel quasi-monochromatic gravitational wave signal originating directly from the level transition of the GA in a binary system. We derive the analytical formulae of both the strain waveform and frequency spectrum of the signal. We further investigate the GA-binary systems that can have a large signal-to-noise ratio in the milli-Hz to deci-Hz frequency band. Using the future space-based gravitational wave observatory DECIGO, we find the signal-to-noise ratio is $\mathcal{O}(10-200)$ for the fine-structure constant $α\simeq 0.3$, host black hole mass $M= 150M_\odot$ and boson mass $μ\simeq 10^{-13} \rm eV$ at a distance within 100 kpc. Given astrophysical uncertainties about the black hole's initial spin, the degeneracy with other monochromatic signals and the small merger rate at those distances, we conclude that the detection of the signal would be challenging.

Figures (12)

• Figure 1: Superradiant timescale in years for the $|211\rangle$ state for two different initial black hole spins as a function of the $\alpha$ parameter. The dashed lines are the non-relativistic approximations \ref{['eqn:sr rates']}. The black hole mass is set to $150 M_{\odot}$.
• Figure 2: The adiabatic and non-adiabatic transitions for $z=20$ (bold) and $z=0.2$ (pale). The horizontal dashed blue line shows the residual population $e^{-2 \pi z}$ of the initial state for the second case.
• Figure 3: Comparison of the numerical solution (solid) of \ref{['eqn:Hamiltonian']} and the analytical results in \ref{['eqn: d1^2 gamma>> 1/dt']} (dashed). We have chosen $z=1$ and $\frac{\eta}{|\Gamma|} = 0.25$. The horizontal grey dashed line is the asymptotic value $e^{-2 \pi z}$.
• Figure 4: (left) The plus polarization of the GW signal \ref{['eqn: plus strain']} versus time for the $|211\rangle \rightarrow |21\textrm{-}1\rangle$ transition. We have chosen $\alpha = 0.3$, $q=1/150$, and $M = 150 M_{\odot}$. The red dashed line is set at $t_{\rm max} = - z |\Gamma|/\gamma$. The inset shows the oscillations around $t_{\rm max}$. (right) The frequency spectrum $|\tilde{h}(f)|/h_{0}$ versus frequency for the same parameters. The solid dark blue line is given by the stationary phase approximation, while the pale blue line is obtained by the short-time Fourier transform. The red dashed vertical line shows the peak frequency of the signal \ref{['eqn:peak freq']}, while the inset shows the fast oscillations around that frequency. The black arrow shows the frequency width, which is equal to $\gamma T_{\rm obs}$.
• Figure 5: Contours of $\Delta t/t_{\rm merger}$ in the $\alpha-q$ plane for two different transitions.