Dynamic signatures of black hole binaries with superradiant clouds

Jun Zhang and Huan Yang

Phys. Rev. D 101, 043020 – Published 25 February 2020

Abstract

Superradiant clouds may develop around a rotating black hole, if there is a bosonic field with Compton wavelength comparable to the size of the black hole. In this paper, we investigate the effects of the cloud on the orbits of nearby compact objects. In particular, we consider the dynamical friction and the backreaction due to level mixing. Under these interactions, the probability of a black hole dynamically capturing other compact objects, such as stellar mass black holes and neutron stars, is generally enhanced with the presence of the cloud. For extreme mass ratio inspirals and binary stellar mass binary black holes, the cloud-induced orbital modulation may be detected by observing the gravitational waveform using space borne gravitational wave detectors, such as LISA. Interestingly within certain range of boson Compton wavelength, the enhanced capture rate of stellar mass black holes could accelerate hierarchical mergers, with a higher-generation merger product being more massive than the mass threshold predicted by supernova pair instability. These observational signatures provide promising ways of searching light bosons with gravitational waves.

Received 15 August 2019
Accepted 10 February 2020

DOI:https://doi.org/10.1103/PhysRevD.101.043020

Physics Subject Headings (PhySH)

Gravitational waves

Astronomical black holes Binary stars

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Jun Zhang ^1,* and Huan Yang^2,3,†

¹Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
²Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
³University of Guelph, Guelph, Ontario N2L 3G1, Canada

^*jun.zhang@imperial.ac.uk
^†hyang@perimeterinstitute.ca

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Vol. 101, Iss. 4 — 15 February 2020

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Images

Figure 1
The value of $x_{p}^{2} I_{DF} [x_{p}, e, \hat{n}]$ with $e = 0$ (blue) and 1 (orange, dotted), assuming an orbit lies in the equator. $x_{p} \equiv α^{2} r_{p} / G M$ with $r_{p}$ being the periapsis of the orbit. The left panels assume the cloud is saturated at $| 211 ⟩$ mode, while the right panels assume the cloud is saturated at $| 322 ⟩$ mode. The upper panels assume $q = M_{*} / M ≪ 1$ , and the lower panels assume $q = 1$ .
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Figure 2
The energy change of the cloud after one passage. We consider parabolic orbits with periapsis $r_{p} = x_{p} α^{- 2} G M$ and $θ_{*} = π / 2$ , and choose the mass ratio $q = 1$ . The upper panel shows the case with a cloud initially saturates at mode $| 211 ⟩$ , while the lower panel shows the case with a cloud initially saturates at mode $| 322 ⟩$ . We show both corotating orbits (in blue) and counterrotating orbits (in orange) considering all modes with $n \leq 7$ (solid lines). We also show the results with different number of modes taken into account ( $n \leq 4$ , 5, 6 with dashed, dash-dotted, and dotted lines). We can see that the result converges as $n$ increases. We mark $I = 0.001$ defined in Eq. (51) with grey horizontal lines in the plots.
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Figure 3
The ratio between energy loss rate caused by dynamical friction and that caused by GW radiation. In the plot, we consider a cloud saturated at $| 211 ⟩$ mode (upper panel) or $| 322 ⟩$ mode (lower panel). The horizontal axis shows the orbital frequency, scaled by the mass of the black hole. The blue/orange dashed line shows the orbital frequency corresponding to a radius of $n^{2} α^{- 2} G M$ , which locates the density peak of the cloud. The shaded region shows the LISA frequency band ( $0.1 Hz - 0.001 Hz$ ), assuming a $10^{5} M_{⊙}$ BH. The gray vertical line denotes the innermost stable circular orbit. The observation band shifts toward the left for smaller BHs.
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Figure 4
The timescales that the GW signal sweeps over the observation band of a LISA-like GW detector, i.e., $0.001 Hz - 0.1 Hz$ . We assumed a binary of two $30 M_{⊙}$ BHs, and show the timescales with different initial frequency $f_{i}$ , and different $α$ (solid lines). We also show the case with no cloud (gray dashed lines) for comparison. In the upper panel, we assume no depletion of the cloud. In the lower panel, we assume the mass of the cloud is suppressed by a factor of $e^{- 4}$ due to a previous depletion [36, 37].
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Figure 5
The phase difference $Δ Φ$ between the case with and without dynamical friction, assuming a binary of two $30 M_{⊙}$ BHs and one year integration time. In the gray region, the signal will sweep the observation band within one year taking into account the dynamical friction. The contours are labeled by $\log_{10} Δ Φ$ . In the lower panel, we assume the mass of the cloud is suppressed by a factor of $e^{- 4}$ due to a previous depletion [36, 37].
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Figure 6
The ratio between angular momentum changing rate caused by level mixing and that caused by GW radiation. We consider a circular orbit with the inclination $i = π / 4$ (as some coupling strength is proportional to $\cos i$ or $\sin i$ ), and show the ratio for the dominant level mixing with $| 211 ⟩$ and $| 322 ⟩$ . $α = 0.1$ for the upper plot and $α = 0.3$ for the lower plot. The gray dashed line shows the orbital frequency corresponding to a radius $4 α^{- 2}$ , which locates the density peak of the cloud. The shaded region shows the LISA frequency band ( $0.1 Hz - 0.001 Hz$ ), assuming a $10^{5} M_{⊙}$ BH. The gray vertical line denotes the innermost stable circular orbit.
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