Ionization of gravitational atoms

Superradiant instabilities may create clouds of ultralight bosons around rotating black holes, forming so-called “gravitational atoms“. It was recently shown that the presence of a binary companion can induce resonant transitions between bound states of these clouds, whose backreaction on the binary’s orbit leads to characteristic signatures in the emitted gravitational waves. In this work, we show that the interaction with the companion can also trigger transitions from bound to unbound states of the cloud—a process that we refer to as “ionization” in analogy with the photoelectric effect in atomic physics. The orbital energy lost in the process overwhelms the losses due to gravitational wave emission and contains sharp features carrying information about the energy spectrum of the cloud. Moreover, we also show that if the companion is a black hole, then the part of the cloud impinging on the event horizon will be absorbed. This “accretion” leads to a significant increase of the companion’s mass, which alters the dynamical evolution and ensuing waveform of the binary. We argue that a combined treatment of resonances, ionization, and accretion is crucial to discover and characterize gravitational atoms with upcoming gravitational-wave detectors.

Figure 1

Schematic illustration of the ionization of the gravitational atom. In the bottom panel, we plot the ratio of the ionization power $P_{\mathrm{ion}}$ and the power lost due to GW emission $P_{\mathrm{GW}}$. We see that the energy loss due to ionization can overwhelm that due to GW emission and hence dominate the binary’s dynamics. The signal has sharp features when the bound state begins to resonate with the continuum, which occurs at specific separations $R_{*}$. Shown also is the density profile of the cloud, $|\psi (R_{*}){|}^2$, for a $|211⟩$ bound state.

Figure 2

Illustration of the spectrum of bound and unbound states of the gravitational atom.

Figure 3

Schematic diagram of an equatorial binary inspiral. The position of the companion with mass $M_{*}$ can be described by the distance between the two black holes, $R_{*}$, and the true anomaly ${\phi }_{*}$, which is the polar angle of the companion in the equatorial plane.

Figure 4

Time dependence of the orbital frequency and its linear approximation.

Figure 5

Schematic illustration of the transitions between a bound state and the continuum.

Figure 6

The imaginary part of the induced energy ${\mathcal{E}}_b(t)$ (top) and the log occupation of the bound state $\log |c_b(t){|}^2$ (bottom) as functions of dimensionless time $\sqrt{\gamma }t$, using both the exact expression (3.9) [blue] and our approximation (3.10) [orange]. Here, we assume that the bound-to-continuum couplings take the form $|\eta (k){|}^2=(k/\mu )/[1+k^4/(81{\mu }^2\gamma )]$.

Figure 7

The ionization power (3.29) as a function of the binary separation $R_{*}$, for $\alpha =0.2$, $q={10}^{-3}$, $M_{\mathrm{c}}=0.01M$, and a cloud in the $|211⟩$ state. We ignore both cloud depletion and the backreaction on the orbital dynamics (see Sec. 5). In the top panel, we normalize both curves by the peak ionization power of the counter-rotating orbit; so-called arbitrary units. In the bottom panel, we have normalized each curve by $P_{\mathrm{GW}}$, the energy lost due to GW emission (2.19). We see that the energy lost due to ionization, whose overall amplitude is controlled by the mass of the cloud $M_{\mathrm{c}}|c_b(t){|}^2$, can dominate over GW emission.

Figure 8

Cartoon illustrating the accretion of the boson cloud by the companion black hole. As explained in the text, the cloud will respond rapidly and replenish the local density behind the companion.

Figure 9

Schematic illustration of the near-field and far-field expansions, where $r_{\pm }$ are the inner and outer horizons of the black hole. The two asymptotic solutions are matched in the overlap region, $M_{*}\ll r\ll 1/k$.

Figure 10

Mass accretion rate of a Schwarzschild black hole computed analytically—from (4.14)—and numerically for a scalar field with $2\mu M_{*}={10}^{-4}$. Shown for comparison is also the accretion rate for particles given by (4.1).

Figure 11

Evolution of the separation $R_{*}$, for $M={10}^4{M}_{\odot }$ and $\alpha =0.2$, with initial values of $R_{*}=400M$, $q={10}^{-3}$ and $M_{\mathrm{c}}/M=0.01$ in a $|211⟩$ state. Shown are the results for both corotating ($+$) and counter-rotating ($-$) orbits. The vacuum system, where no cloud is present, is shown for comparison. We see that accretion and ionization significantly reduce the merger time.

Figure 12

Fractional changes of the mass of the companion $M_{*}$ and the mass of the cloud $M_{\mathrm{c}}$, for $M={10}^4{M}_{\odot }$ and $\alpha =0.2$, with initial values of $R_{*}=400M$, $M_{*}={10}^{-3}M$. Shown are the results for three different initial values of $M_{\mathrm{c}}$. All curves refer to corotating orbits and a $|211⟩$ bound state.

Figure 13

Evolution of the GW frequency as a function of the remaining time to merger, $t-t_{\mathrm{m}}$, for $M={10}^4{M}_{\odot }$ and $\alpha =0.2$, with initial values of $R_{*}=400M$, $q={10}^{-3}$, and $M_{\mathrm{c}}/M={10}^{-3}$ in a $|211⟩$ state. The central region of the range shown on the $y$ axis corresponds to a few millihertz, falling inside the LISA sensitivity band. The “kinks” at separations $R_{*}^{(g)}$ correspond to the discontinuities in the ionization power, see Fig. 7.

Figure 14

The real [blue] and imaginary [orange] parts of the modulating function $\mathcal{I}(\tilde{\epsilon },\tau )$, for several values of the dimensionless time $\tau$. For large negative values of $\tau$, the integrand of (a5) is highly suppressed for $\tilde{\epsilon }\in [0,\infty )$. For large positive times $\tau \gg 1$, the integrand oscillates rapidly when $\tilde{\epsilon }\in [0,1]$, slowing down when $\tilde{\epsilon }\sim 1$, and is then again highly suppressed for $\tilde{\epsilon }\gg 1$.

Figure 15

The dimensionless ratio $|{\gamma }^{-1/2}\mathrm{Im}{\mathcal{E}}_b(R_{*})|$ as a function of the orbital separation $R_{*}$, using our approximation (3.28) as an estimate, for an inspiral with $q={10}^{-3}$ and $\alpha =0.2$, where $\gamma$ is the instantaneous chirp rate $\gamma ={\ddot{\phi }}_{*}(t)$, defined in (2.20) with ${\mathrm{\Omega }}_0^2{R}_{*}^3=(1+q)M$.

Figure 16

The radial zero mode density ${\lim }_{k\to 0}|k^{-1/2}{R}_{k;1}(r){|}^2$ compared to several bound state densities, all with orbital angular momentum $\ell =1$. Here, $r_{\mathrm{c}}=(\mu \alpha {)}^{-1}$ is the typical radius of the cloud, and we have normalized each density so that it has unit maximum. Ignoring the overall normalization, the zero-mode wave function can also be thought of as the limit of the bound state wave functions as $n\to \infty$.