Directed searches for gravitational waves from ultralight bosons

Gravitational-wave detectors can be used to search for yet-undiscovered ultralight bosons, including those conjectured to solve problems in particle physics, high-energy theory, and cosmology. In particular, ground-based instruments could probe boson masses between ${10}^{-15}\mathrm{eV}$ and ${10}^{-11}\mathrm{eV}$, which are largely inaccessible to other experiments. In this paper, we explore the prospect of searching for the continuous gravitational waves generated by boson clouds around known black holes. We carefully study the predicted waveforms and use the latest-available numerical results to model signals for different black-hole and boson parameters. We then demonstrate the suitability of a specific method (hidden Markov model tracking) to efficiently search for such signals, even when the source parameters are not perfectly known as well as allowing for some uncertainty in theoretical predictions. We empirically study this method’s sensitivity and computational cost in the context of boson signals, finding that it will be possible to target remnants from compact-binary mergers localized with at least three instruments. For signals from scalar clouds, we also compute detection horizons for future detectors (Advanced LIGO, LIGO Voyager, Cosmic Explorer, and the Einstein Telescope). Among other results, we find that, after one year of observation, an Advanced LIGO detector at design sensitivity could detect these sources up to over 100 Mpc, while Cosmic Explorer could reach over $10^4\mathrm{Mpc}$. These projections offer a more complete picture than previous estimates based on analytic approximations to the signal power or idealized search strategies. Finally, we discuss specific implications for the follow-up of compact-binary coalescences and black holes in x-ray binaries. Along the way, we review the basic physics of bosons around black holes, in the hope of providing a bridge between the theory and data-analysis literatures.

Figure 1

Strain amplitude vs frequency for example black hole (scalar cloud). The $x$ axis shows different frequencies at which we might expect a GW signal from a scalar ($l=m=1,n=0$) cloud around a BH with initial mass of $M_i=60M_{\odot }$ and initial spin ${\chi }_i=0.70$. The colored curve shows the corresponding characteristic amplitude, $h_0$ in Eq. (28) assuming $r=5\mathrm{Mpc}$, parametrized by the fine-structure constant, $\alpha$ in Eq. (6), as indicated by the color bar. For reference, the other curve shows the small-$\alpha$ approximation of Eq. (29), including the spin correction responsible for the amplitude turnover. Points to the left of the vertical dotted line have ${\tau }_{\mathrm{inst}}>1\mathrm{yr}$.

Figure 2

Strain amplitude at different GW frequencies vs initial BH mass for fixed initial spin (scalar cloud). Color shows the characteristic strain amplitude, Eq. (28), from a scalar cloud ($l=m=1,n=0$) that would be emitted at different frequencies ($y$ axis), i.e., for different $\alpha$’s (not shown), vs initial BH mass ($x$ axis). The gray dashed line marks the peak amplitude. The source is assumed to lie at $r=5\mathrm{Mpc}$, with initial spin ${\chi }_i=0.70$. For ease of display, we set an arbitrary lower cutoff of $h_0\ge {10}^{-30}$: amplitudes for points in the bottom purple region vanish asymptotically for lower $f$, while amplitudes for points in the top gray region vanish identically since superradiance cannot occur for those parameters. Points below the dotted gray line have ${\tau }_{\mathrm{inst}}>1\mathrm{yr}$. A vertical cross section at $M=60M_{\odot }$ (vertical dotted line) yields Fig. 1.

Figure 3

Strain amplitude at different GW frequencies vs BH initial spin for fixed mass (scalar cloud). Color shows the characteristic strain amplitude, Eq. (28), from a scalar cloud ($l=m=1,n=0$) that would be emitted at different frequencies ($y$ axis), i.e., for different $\alpha$’s (not shown), vs BH initial spin ($x$ axis). The gray dashed line marks the peak amplitude. The source is assumed to lie at $r=5\mathrm{Mpc}$, with mass $M=60M_{\odot }$. For ease of display, we set an arbitrary lower cutoff of $h_0\ge {10}^{-30}$: as can be inferred from Fig. 1, amplitudes for points in the bottom purple region vanish asymptotically for lower $f$, while amplitudes for points in the top gray region vanish identically since superradiance cannot occur for those parameters. A vertical cross section at ${\chi }_i=0.70$ (vertical dotted line) yields Fig. 1.

Figure 4

Optimal strain frequency and amplitude vs initial BH mass for different initial spins (scalar cloud). Frequency (top) and characteristic amplitude (bottom) of the strain produced by the best-possible cloud (best-possible $\alpha$) as a function of initial BH mass. Different curves correspond to different initial spins, showing that higher spins result in stronger emission. We assume that the source is situated at $r=5\mathrm{Mpc}$, and that the scalar cloud is dominated by the fastest level ($l=m=1,n=0$). The intersection of the ${\chi }_i=0.70$ line with a vertical cut at $M=60M_{\odot }$ (dotted vertical line) give the amplitude and frequency of the peak in Fig. 1 (dotted horizontal lines).

Figure 5

Optimal strain amplitude vs initial BH parameters (scalar cloud). Color gives the characteristic strain amplitude emitted by the best-possible cloud matched to a BH with the indicated initial mass ($x$ axis) and spin ($y$ axis). Horizontal cuts yield the curves shown in the bottom panel of Fig. 4. The intersection of the dotted white lines ($M_i=60M_{\odot }$, ${\chi }_i=0.70$) corresponds to the peak of Fig. 1. Gray lines mark ${\tau }_{\mathrm{inst}}=1\mathrm{day}$ and ${\tau }_{\mathrm{inst}}=1\mathrm{yr}$ for reference.

Figure 6

Signal growth and duration timescales for example BH (scalar cloud). Curves show the signal duration (purple line, top), Eq. (15), and growth (orange line, bottom), Eq. (22), timescales for a scalar cloud ($l=m=1,n=0$) as a function of the fine-structure constant $\alpha$ from Eq. (6). The BH is assumed to have an initial mass of $M=60M_{\odot }$ and spin of $\chi =0.70$. The vertical dashed line marks the value $\alpha$ that yields peak emission for such BH, for which ${\tau }_{\mathrm{inst}}=39$ days and ${\tau }_{\mathrm{GW}}=7.5\times {10}^3\mathrm{yr}$. Note that values of $\alpha >0.2$ preclude superradiance given this spin; cf. Eq. (7).

Figure 7

Superradiant-instability timescale (scalar cloud). Color shows the characteristic growth time, Eq. (15), for a scalar cloud ($l=m=1,n=0$) as a function of BH mass ($x$ axis) and fine-structure constant ($y$ axis). The BH is assumed to have an initial spin of $\chi =0.70$. The highlighted contours correspond to ${\tau }_{\mathrm{inst}}=1\mathrm{day}$ (top, thin line) and ${\tau }_{\mathrm{inst}}=1\mathrm{yr}$ (bottom, thick line). The vertical dotted line gives the instability timescales for $M=60M_{\odot }$. The horizontal dashed line corresponds to the value of $\alpha$ that yields optimal GW emission for a BH with initial spin $\chi =0.70$. Note that values of $\alpha >0.2$ preclude superradiance given this spin; cf. Eq. (7).

Figure 8

Signal duration timescale (scalar cloud). Color shows the characteristic signal duration, Eq. (22), for a scalar cloud ($l=m=1,n=0$) as a function of BH mass ($x$ axis) and fine-structure constant ($y$ axis). The BH is assumed to have an initial spin of $\chi =0.70$. The vertical dotted line gives the instability timescales for $M=60M_{\odot }$. The horizontal dashed line corresponds to the value of $\alpha$ that yields optimal GW emission for a BH with initial spin $\chi =0.70$. Note that values of $\alpha >0.2$ preclude superradiance given this spin; cf. Eq. (7).

Figure 9

HMM sample tracking paths. Injected $f_0(t)$ (light curves) and optimal Viterbi paths (dark curves) for the injected signals with (a) weaker random walk $|\delta f|\le 0.1\mathrm{\Delta }f$ and (b) stronger random walk $|\delta f|\le 0.5\mathrm{\Delta }f$. The top panels show the random walk $\delta f$ added to the injected signals at each step, which is too small to be seen by eye in the bottom panel of (a). The horizontal axis is in units of HMM steps with each step spanning for $T_{\text{drift}}=8$ days. Good matches are obtained in both (a) and (b) with ${ϵ}_{f_0}=0.16\mathrm{\Delta }f$ and $0.50\mathrm{\Delta }f$, respectively. Injection parameters are in Table 2, and the injected signal strain is $h_0=5\times {10}^{-26}$.

Figure 10

Receiver operator characteristic curves for the injections with parameters in Table 2. The four curves (from top to bottom) correspond to the four representative wave strains $h_0/{10}^{-26}=5$, 4, 3, and 2. The horizontal and vertical axes indicate the false alarm probability $P_{\mathrm{a}}$ and detection probability $1-P_{\mathrm{d}}$, respectively. Each curve is based on 200 realizations with randomly chosen polarization and inclination angles and initial phase.

Figure 11

Sensitivity vs GW frequency for different detectors. Value of $h_0^{95\%}$ marginalized over source orientation for design aLIGO (gray line), LIGO Voyager (yellow line), Cosmic Explorer (purple line), and the Einstein Telescope (red line). All curves assume one year of continuous observation by a single detector of the indicated type.

Figure 12

Detection horizons for scalar clouds. Maximum-detectable luminosity distances (color) for optimal scalar clouds ($l=m=1$, $n=0$, $\tilde{l}=\tilde{m}=2$) around BHs with the indicated initial mass ($x$ axis) and spin ($y$ axis) for different detectors. White contour lines indicate the values of the corresponding boson rest-energy $\mu /\mathrm{eV}$. The shaded region (top left) marks parameters that would yield signals evolving prohibitively fast (${\dot{f}}_{\det }>{10}^{-8}$) for existing search methods, based on Eq. (33). The dotted white lines highlight the mass and spin ($60M_{\odot }$, 0.70) of the GW150914-like example discussed repeatedly in the main text. Values correspond to HMM tracking for one year of continuous observation by a single detector, accounting for signal redshifts and variability in maximum $T_{\text{drift}}$ allowed by the expected signal; cf. Eq. (36).

Figure 13

[Horizon scaling with $T_{\text{drift}}$] Horizon scaling with $T_{\text{drift}}$. Advanced LIGO horizon distance ($y$ axis) vs coherence window $T_{\text{drift}}$ (bottom $x$ axis) and maximum allowed frequency derivative $|{\dot{f}}_{\max }|$ (top $x$ axis). Each curve corresponds to one of the systems in Table 1, labeled by $(M_i,{\chi }_i)$, and shows the $T_{\text{drift}}^{1/4}$ scaling of Eq. (38). The dashed curve highlights the GW150914-like example. Vertical dotted lines mark the maximum $T_{\text{drift}}$ allowed by the $\dot{f}$ expected from each system, which is also the value assumed in Table 5 and Fig. 12 (circles). We assume $T_{\mathrm{obs}}=1\mathrm{yr}$. The shaded region cannot be explored with existing methods.

Figure 14

Sky resolution. Color shows detection efficiency ($1-P_d$, for $P_d$ the false-dismissal probability) as a function of offsets in right ascension ($x$ axis) and declination ($y$ axis) with respect to the true location for injections with $h_0=5\times {10}^{-26}$ (left) and $h_0=2\times {10}^{-25}$ (right). All other injection parameters are as in Table 2 and search settings are shown in Table 3. The left (right) plot was interpolated from a square grid with 5 (7) sky locations on each side.