Phys. Rev. D 102, 063020 (2020) - Characterizing the continuous gravitational-wave signal from boson clouds around Galactic isolated black holes

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Characterizing the continuous gravitational-wave signal from boson clouds around Galactic isolated black holes

Sylvia J. Zhu, Masha Baryakhtar, Maria Alessandra Papa, Daichi Tsuna, Norita Kawanaka, and Heinz-Bernd Eggenstein

Phys. Rev. D 102, 063020 – Published 21 September 2020

Abstract

Ultralight bosons can form large clouds around stellar-mass black holes via the superradiance instability. Through processes such as annihilation, these bosons can source continuous gravitational-wave signals with frequencies within the range of LIGO and Virgo. If boson annihilation occurs, then the Galactic black hole population will give rise to many gravitational signals; we refer to this as the ensemble signal. We characterize the ensemble signal as observed by the gravitational-wave detectors; this is important because the ensemble signal carries the primary signature that a continuous wave signal has a boson annihilation origin. We explore how a broad set of black hole population parameters affects the resulting spin-0 boson annihilation signal and consider its detectability by recent searches for continuous gravitational waves. A population of $10^{8}$ black holes with masses up to $30 M_{⊙}$ and a flat dimensionless initial spin distribution between zero and unity produces up to 1000 signals loud enough in principle to be detected by these searches. For a more moderately spinning population, the number of signals drops by about an order of magnitude, still yielding up to 100 detectable signals for some boson masses. A nondetection of annihilation signals at frequencies between 100 and 1200 Hz disfavors the existence of scalar bosons with rest energies between $2 \times 10^{- 13}$ and $2.5 \times 10^{- 12} eV$ . Finally, we show that, depending on the black hole population parameters, care must be taken in assuming that the continuous wave upper limits from searches for isolated signals are still valid for signals that are part of a dense ensemble: Between 200 and 300 Hz, we urge caution when interpreting a null result for bosons between $4 \times 10^{- 13}$ and $6 \times 10^{- 13} eV$ .

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Received 26 March 2020
Accepted 31 August 2020

DOI:https://doi.org/10.1103/PhysRevD.102.063020

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Axions Gravitational wave detection Gravitational wave sources Gravitational waves

Astronomical black holes Hypothetical particles

Gravitation, Cosmology & AstrophysicsParticles & Fields

Authors & Affiliations

Sylvia J. Zhu ^1,2,3,*, Masha Baryakhtar ^4,†, Maria Alessandra Papa ^1,2,5,‡, Daichi Tsuna^6,7,§, Norita Kawanaka ^8,9, and Heinz-Bernd Eggenstein ^1,2

¹Max-Planck-Institut für Gravitationsphysik, Callinstraβe 38, D-30167 Hannover, Germany
²Leibniz Universität Hannover, D-30167 Hannover, Germany
³DESY, D-15738 Zeuthen, Germany
⁴Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, New York 10003, USA
⁵University of Wisconsin–Milwaukee, Milwaukee, Wisconsin 53201, USA
⁶Research Center for the Early Universe (RESCEU), The University of Tokyo, Hongo, Tokyo 113-0033, Japan
⁷Department of Physics, School of Science, The University of Tokyo, Hongo, Tokyo 113-0033, Japan
⁸Department of Astronomy, Graduate School of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
⁹Hakubi Center, Yoshida Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

^*sylvia.zhu@desy.de
^†mbaryakhtar@nyu.edu
^‡maria.alessandra.papa@aei.mpg.de
^§tsuna@resceu.s.u-tokyo.ac.jp

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Vol. 102, Iss. 6 — 15 September 2020

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Figure 1
Gravitational fine structure constant $α \equiv G M_{BH} μ_{b} / (ℏ c)$ . Given the superradiance condition, the $(n, ℓ, m) = (0, 1, 1)$ bound state cannot form for $α \geq 0.5$ , which is shown in gray.
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Figure 2
Critical spin $χ_{c}$ for the $(n, ℓ, m) = (0, 1, 1)$ bound state. The critical spin is a function of the gravitational fine structure constant $α$ and the bound state quantum numbers. A cloud of bosons with rest energy $μ_{b}$ will form around a black hole of mass $M_{BH}$ and initial spin $χ_{i}$ only if $χ_{i} > χ_{c}$ [Eq. (4)].
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Figure 3
Maximum characteristic strain $h_{0, peak}$ for a rapidly rotating black hole as a function of black hole mass $M_{BH}$ and boson mass $μ_{b}$ at a distance of $d = 1 kpc$ . The peak strain scales as $1 / d$ . The value of $h_{0}$ increases with $α$ until $α \sim 0.35$ ; the quadrupolar gravitational-wave power—and thus the observed strain—changes nonmonotonically at large $α$ (Sec. pp1-s4).
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Figure 4
The gravitational-wave signal amplitude decay half-time $τ_{GW}$ [Eq. (14)] ranges from longer than the Hubble time for lighter bosons to less than 100 yr for heavier bosons. The longer signals from lighter bosons are weaker compared to shorter signals from heavier bosons [Eq. (13)]. Like $h_{0}$ , $τ_{gw}$ is a nonmonotonic function of $α$ above $α ≳ 0.35$ (Sec. pp1-s4).
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Figure 5
Bottom panel: The leading-order gravitational-wave frequency $f_{GW}^{0}$ is proportional to the boson rest energy $μ_{b}$ [Eq. (9)]. Top panel: Systems with heavier black holes (orange curves) produce GW signals with lower frequencies than do systems with lighter black holes (blue curves) due to the larger gravitational binding energy. Systems with larger spins (dashed curves) result in slightly higher frequencies than systems with smaller spins (solid curves) due to a positive spin-orbit energy. The two solid curves—corresponding to $χ_{i} = 0.5$ —stop at intermediate boson masses; for heavier bosons, $χ_{c} > 0.5$ and the systems never form. The contribution from the cloud self-energy is of the order of $10^{- 3}$ or less of the black hole’s gravitational potential energy.
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Figure 6
Top-down (top panel) and edge-on (bottom panel) view of the Galaxy, showing a random sample of 30,000 black holes from our simulated population. The disk and bulge components are visually distinct. Our simulations assume axial symmetry, so the results do not change if the position of the Sun is chosen arbitrarily on the circle of distance $\sim 8 kpc$ from the Galactic Center (orange circle). As explained in the text, we exploit this freedom to simulate $10^{8}$ black holes starting from a set of $10^{6}$ . The orange star marks one sample position and the arrow shows its velocity in the disk, tangential to the circle.
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Figure 7
Radial velocities of the black holes of Fig. 6 as a function of their distance from the Sun. The distribution is symmetric around zero km/s and is widest at 8 kpc, corresponding to the distance of the Galactic Center, as the bulk velocity in the bulge is uniformly distributed in direction. In contrast, since the bulk velocity in the disk is in the $ϕ$ direction within the Galactic plane, the radial velocity of black holes in the disk (i.e., not near 8 kpc) has smaller magnitude. The scatter in velocities at large distances from the Sun is due to the black holes that have wandered into the halo and no longer follow the bulk motion of the disk.
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Figure 8
Most black holes in our simulation are located close to the Galactic Center and at low Galactic latitudes. These regions tend to be metal rich, which can limit the maximum black hole mass $M_{BH, \max}$ that can form from stars.
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Figure 9
The distribution of signal strains as a function of source frequency (gray dots) and observed frequency with Doppler shift (blue dots). The strains are selected for signals with $d \geq 1 kpc$ to facilitate comparison to the strain as a function of source frequency for a system at a distance $d = 1 kpc$ : the upper envelope for two sets of black hole spin and age parameters is shown in orange. The black vertical lines show the boundaries of the source frequency distribution: the vertical dotted line shows the rest mass frequency $f_{GW}^{0}$ , and the thick and thin vertical lines correspond to $f_{GW, source}$ for ( $20 M_{⊙}$ , $χ_{i} = 0.999$ ) and ( $20 M_{⊙}, χ_{i} = 0.5$ ), respectively, yielding successively more negative potential energy corrections. (In the left panel, the two solid vertical lines lie close to each other.)
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Figure 10
The ensemble signals for the “standard” black hole population (Table 1) and boson mass (left to right) $μ_{b} = 2 \times 10^{- 13}, 4 \times 10^{- 13}, 8 \times 10^{- 13}$ , and $20 \times 10^{- 13} eV$ . Top row: Each point represents the annihilation signal from an individual system, illustrating the range of strains and frequencies of the signals. The dashed horizontal red line corresponds to the approximate upper limit at that frequency reported by recent all-sky continuous wave searches [20, 21, 47, 97]. The thick dashed vertical red line marks the values of $f_{GW}^{0}$ , and the dotted red line shows $f_{GW}$ calculated using the mode of the $M_{BH}$ distribution (Fig. 11). Bottom row: Distribution of strains as a function of distance and black hole mass. Smaller, whiter circles correspond to lighter black holes (minimum $5 M_{⊙}$ ) and larger bluer circles correspond to heavier black holes (maximum $20 M_{⊙}$ ); the differences in circle sizes are exaggerated for visual effect and do not scale linearly with $M_{BH}$ . The horizontal dashed red line again represents current search upper limits.
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Figure 11
Each of the four panels shows the mass $M_{BH}$ and initial spin $χ_{i}$ of the black holes that produce the ensemble signals for the four boson masses from Fig. 10, assuming the standard black hole population (Table 1). In each panel, the scatterplot shows $M_{BH}$ and $χ_{i}$ for the systems that are detectable with the current search sensitivities (blue circles with orange outlines) and a factor of 2 improvement in sensitivity (blue circles). The histograms show the distributions of $M_{BH}$ (top panels) and $χ_{i}$ (right panels) for these sets of systems. The black holes are assigned spins uniformly distributed in [0, 1], and only systems satisfying Eq. (4) will form clouds. The upper-left set of plots ( $μ_{b} = 2 \times 10^{- 13} eV$ ) contains no signals at these values of $h_{0}$ , suggesting that a larger increase in sensitivity is necessary to detect the annihilation signals produced by the lightest bosons.
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Figure 12
Each of the four panels shows the age $τ_{BH}$ and distance $d$ of the black holes that produce the ensemble signals for the four boson masses in Fig. 10, assuming the standard black hole population (Table 1). In each panel, the scatterplot shows $τ_{BH}$ and $d$ for the systems that are detectable with the current search sensitivities (blue circles with orange outlines) and a factor of 2 improvement in sensitivity (blue circles). The histograms show the distributions of $τ_{BH}$ (top panels) and $d$ (right panels) for these sets of systems. As in Fig. 11, the upper-left set of plots ( $μ_{b} = 2 \times 10^{- 13} eV$ ) contains no signals at these values of $h_{0}$ .
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Figure 13
The number of signals above $h_{0} > 10^{- 25}$ depends on the maximum of the initial spin distribution, $χ_{i, \max}$ . For a cloud to form, the initial spin must be larger than the critical spin [Eq. (4)]. Since $χ_{c}$ increases with $α$ , this also results in fewer signals above a given strain for heavier bosons. The contours show Gaussian statistical uncertainties. There are no signals with amplitude $h_{0} > 10^{- 25}$ for $μ_{b} = 2 \times 10^{- 13} eV$ for any value of $χ_{i, \max}$ .
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Figure 14
For a given boson mass, a fraction of black holes is born with spins high enough to support cloud formation (filled markers). Gravitational-wave emission stops when the second level is fully populated, so a subset of the initial clouds are still emitting today (unfilled markers). We consider the five black hole populations of Table 1: standard (black circles), heavy (blue circles), moderate spins (gray squares), heavy with moderate spins (blue squares), and pessimistic spin (gray triangles) populations. The percentage of systems still emitting today decreases rapidly with increasing boson mass; at $μ_{b} = 4 \times 10^{- 12} eV$ , only up to one in every $10^{6}$ black holes in the Galaxy is in a currently emitting system.
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Figure 15
The number of signals with intrinsic amplitude above a given $h_{0}$ value. The signal number is generally largest at $μ_{b} = 8 \times 10^{- 13} eV$ ( $f_{GW} \approx 400 Hz$ ) for the standard black hole population (Table 1). At higher boson masses, the number of signals decreases because both the signal half-time [Eq. (15)] and the signal cutoff time due to the formation of the second level [Eq. (17)] decrease with increasing frequency. The contours show Gaussian statistical uncertainties, an underestimate at a small signal number.
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Figure 16
The mean density of signals above a given $h_{0}$ . The mean density is largest for $μ_{b} = 5 \times 10^{- 13} eV$ ( $f_{GW} \approx 250 Hz$ ) for the standard black hole population (Table 1). For heavier bosons, the overall density decreases; the number of signals above a given $h_{0}$ drops due to the combination of effects described in Fig. 15, while the frequency range spanned by the ensemble signal continues to increase. The contours show the uncertainty $\sqrt{N} / Δ f$ , where $N$ is the number of signals and $Δ f$ is the frequency range spanned by the $N$ signals. Note that this does not take into account the additional uncertainty from $Δ f$ , which is itself correlated with $N$ .
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Figure 17
The maximum density of signals above a given $h_{0}$ is up to an order of magnitude larger than the mean density (Fig. 16). Frequency ranges with sufficiently high signal densities can cause complications for standard searches for continuous wave signals (Sec. 5). The contours show the uncertainty $\sqrt{N} / Δ f$ , where $N$ is the number of signals and $Δ f = 0.01 Hz$ is the size of the frequency bin over which the maximum density is calculated.
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Figure 18
Number of signals with amplitudes above the detectability level of recent all-sky searches for continuous gravitational waves [20, 21, 22], for bosons in the mass range $1 \times 10^{- 13}$ to $4 \times 10^{- 12} eV$ . The Falcon search [20, 21] provides 95% confidence strict upper limits on continuous wave emission with near-zero frequency derivative between 20 and 600 Hz. These upper limits are valid for any sky position and polarization of the source, even the ones with the most unfavorable coupling to the detectors. In contrast, the upper limits from the Freq Hough search [22] hold for 95% of the entire population. For this reason, even at the same sensitivity, the Freq Hough upper limits are lower than the Falcon upper limits. We use the Falcon upper limits for bosons with $μ_{b} < 1.2 \times 10^{- 12} eV$ and the Freq Hough upper limits for bosons with $μ_{b} \geq 1.2 \times 10^{- 12} eV$ . We consider the standard (black circles) and heavy (blue circles) black hole populations, as well as the moderate (gray squares), heavy moderate (blue squares), and pessimistic (gray triangles) spin populations. The fluctuations are due to stationary lines and other artifacts in the detectors, which decrease the continuous wave search sensitivity in nearby frequency bins.
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Figure 19
The $x$ axes show the signal frequency binned with the resolution of the two searches: $2.4 \times 10^{- 4} Hz$ for the Freq Hough search and $1.7 \times 10^{- 5} Hz$ for the Falcon search. From the bottom panel going up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude. The main reason why there are fewer signals per frequency bin in the Falcon searches [20, 21] than in the Freq Hough search [22] is that the frequency bin of Freq Hough is significantly larger (by a factor of 14) than the frequency bin of Falcon. The maximum initial spin of the black hole population is taken to be 1.
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Figure 20
Same plots as those in Fig. 19 but with the assumption that the maximum initial spin of the black hole population is 0.5.
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Figure 21
The amplitude spectral density of the LIGO O2 data alone (dashed red line) and of an ensemble signal (top green points) assuming that $μ_{b} = 7 \times 10^{- 13} eV$ and a Galactic black hole population with maximum mass $20 M_{⊙}$ and maximum initial spin of 1. The time baseline assumed is 4096 s, as used by the Freq Hough search [22].
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Figure 22
The power emitted in the $\tilde{ℓ} = \tilde{m} = 2$ spherical harmonics mode of the gravitational radiation, normalized by $(M_{BH, i} / M_{cloud}) G_{N}$ to be a dimensionless quantity. The initial black hole spin assumed here is $χ_{i} = 0.99$ ; the curves for other values of $χ_{i}$ are very similar and primarily differ in their maximum value of $α_{i}$ . The power for $α_{i} > 0.1$ is calculated numerically [Eq. (a16)] and is suppressed for some high spins, resulting in a “dip” near $α \sim 0.4$ . This directly affects the shape of $h_{0}$ and $τ_{gw}$ .
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Figure 23
The spin-up $\dot{f}$ due to the decreasing binding energy of the cloud depends on $μ_{b}$ and $M_{BH}$ [Eq. (16)]. For both the standard and heavy black hole populations, the largest value of $\dot{f}$ is $6 \times 10^{- 13} Hz / s$ .
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Figure 24
If black holes are born with faster natal kicks (100 rather than $50 km / s$ ), more of them have sufficiently high speeds to escape the gravitational pull of the Galactic bulge and disk. This causes a slight deficit of black holes at small Galactic radii, which also means there are fewer BHs close to Earth, decreasing on average the number of loud signals.
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Figure 25
The ratio of the sum of $1 / d$ for all the black holes within a distance $d$ provides a comparison of the number of signals between the two populations. Here we plot the ratio for the population with faster versus slower kicks. Overall, this ratio is smaller than unity and is decreasing toward shorter distances; the ratio is affected by small number statistics at distances much less than 1 kpc. The lower signal number is consistent with the fact that the population of black holes with faster kicks in general produces fewer signals above a given $h_{0}$ (Fig. 25).
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Figure 26
The loudest signals in an ensemble are in general produced by the closest black holes. A black hole population with slightly faster natal kicks produces a slight deficit of boson clouds close to Earth (Fig. 24), resulting in fewer signals at larger values of $h_{0}$ . This effect is greater for heavier bosons, as the louder signals are preferentially produced by closer black holes.
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Figure 27
The ratio of maximum signal density above a given $h_{0}$ between the faster and slower kick populations is consistent with unity.
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Figure 28
The ratio of mean signal density above a given $h_{0}$ between the faster and slower kick populations reflects the number of signals above a given $h_{0}$ (Fig. 26). In general, black holes with faster natal kicks produce signals with larger Doppler shifts, thereby increasing the frequency range spanned by the ensemble and decreasing the mean signal density further.
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Figure 29
Increasing the number of light black holes ( $M_{BH} < 5 M_{⊙}$ ) while maintaining the same number of black holes effectively reduces the number of black holes at all other masses. Therefore, the number of signals above a given $h_{0}$ decreases by a factor of 2 for most of the boson masses. The exception is the heaviest boson ( $μ_{b} = 3.5 \times 10^{- 12} eV$ ), for which the number of signals is larger with the light black hole population for signals with $h_{0} < 3 \times 10^{- 25}$ .
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Figure 30
The ratio of the number of signals above a given $h_{0}$ for the heavy ( $M_{BH, \max} = 30 M_{⊙}$ ) versus standard ( $M_{BH, \max} = 20 M_{⊙}$ ) populations. The number of signals for the lighter bosons ( $μ_{b} = 2 \times 10^{- 13}$ and $5 \times 10^{- 13} eV$ ) increases in going from the standard to the heavy black hole population, while the number of signals for the heavier bosons stays approximately the same (Sec. 5a). For $μ_{b} = 2 \times 10^{- 13} eV$ , the signals from the heavy black hole population extend to strains of $h_{0} \approx 10^{- 24}$ (indicated by the light, thick bar) while the signals from the standard black hole population only have $h_{0} \leq 10^{- 25}$ .
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Figure 31
The ratio of the mean signal density above a given $h_{0}$ for the heavy versus standard black hole populations is approximately unity for the five heavier boson masses. For the lightest mass ( $μ_{b} = 2 \times 10^{- 13} eV$ ), the mean density is over 10 times larger for the heavier black hole population.
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Figure 32
For the six boson masses plotted here, the ratio of the maximum signal density above a given $h_{0}$ for the heavy versus standard black hole populations is similar to the ratio of the mean signal density. The maximum signal density for the lightest boson ( $μ_{b} = 2 \times 10^{- 13} eV$ ) is over 100 times larger for the heavier black hole population.
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Figure 33
Mass and spin properties of the heavy black hole population. For the two heavier boson masses ( $μ_{b} = 8 \times 10^{- 13}$ and $2 \times 10^{- 12} eV$ , the black holes that produce the potentially detectable ensemble signals have similar properties to the black holes that produce the signals in Fig. 10 (the standard population) due to the fact that systems with large values of $α$ have dramatically shortened lifetimes. For the two lighter boson masses ( $μ_{b} = 2 \times 10^{- 13}$ and $4 \times 10^{- 13} eV$ ), the addition of black holes with $M_{BH} > 20 M_{⊙}$ greatly increases the number of detectable signals.
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Figure 34
Ages and distances of the heavy black hole population. As in Fig. 33, the addition of black holes with $M_{BH} > 20 M_{⊙}$ increases the number of detectable systems for the lighter bosons ( $μ_{b} = 2 \times 10^{- 13}$ and $4 \times 10^{- 13} eV$ ) but has little to no effect on the heavier bosons.
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Figure 35
The $x$ axes show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon, $1.7 \times 10^{- 5} Hz$ [20, 21]. From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 36
The $x$ axes show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon [20, 21], $1.7 \times 10^{- 5} Hz$ . From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 37
The $x$ -axis plots show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon [20, 21], $1.7 \times 10^{- 5} Hz$ . From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 38
The $x$ axes show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon [20, 21], $1.7 \times 10^{- 5} Hz$ . From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 39
The $x$ axes show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon, $1.7 \times 10^{- 5} Hz$ [20, 21]. From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 40
The $x$ axes show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon, $1.7 \times 10^{- 5} Hz$ [20, 21]. From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 41
The $x$ axes show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon, $1.7 \times 10^{- 5} Hz$ [20, 21]. From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 42
The $x$ axes show the signal frequency binned with the resolution of the two searches: (left panels) Freq Hough, $2.4 \times 10^{- 4} Hz$ [22] and (right panels) Falcon, $1.7 \times 10^{- 5} Hz$ [20, 21]. From the bottom panel up, the $y$ axis shows the number of detectable signals in each bin, their maximum intrinsic amplitude, and their average amplitude.
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Figure 43
The amplitude spectral density of the LIGO O2 data alone (dashed red line) and of an ensemble signal (top green points) assuming that $μ_{b} = 3 \times 10^{- 13} eV$ and a Galactic black hole population with maximum mass $20 M_{⊙}$ (top plots) and max initial spin of 0.3, 0.5 and 1, and maximum mass $30 M_{⊙}$ (bottom plots) and maximum initial spins of 0.5 and 1. The time baseline assumed is 4096 s, as used by the Freq Hough search in this frequency range [22].
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Figure 44
The amplitude spectral density of the LIGO O2 data alone (dashed red line) and of an ensemble signal (top green points) assuming that $μ_{b} = 4 \times 10^{- 13} eV$ and a Galactic black hole population with maximum mass $20 M_{⊙}$ (top plots) and maximum initial spins of 0.3, 0.5, and 1, and maximum mass $30 M_{⊙}$ (bottom plots) and maximum initial spins of 0.5 and 1. The time baseline assumed is 4096 s, as used by the Freq Hough search in this frequency range [22].
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Figure 45
The amplitude spectral density of the LIGO O2 data alone (dashed red line) and of an ensemble signal (top green points) assuming that $μ_{b} = 7 \times 10^{- 13} eV$ and a Galactic black hole population with maximum mass $20 M_{⊙}$ (top plots) and maximum initial spins of 0.3, 0.5, and 1, and maximum mass $30 M_{⊙}$ (bottom plots) and maximum initial spins of 0.5 and 1. The time baseline assumed is 4096 s, as used by the Freq Hough search in this frequency range [22].
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Figure 46
The amplitude spectral density of the LIGO O2 data alone (dashed red line) and of an ensemble signal (top green points) assuming that $μ_{b} = 1.5 \times 10^{- 12} eV$ and a Galactic black hole population with maximum mass $20 M_{⊙}$ or $30 M_{⊙}$ and maximum initial spins of 0.5 and 1. The time baseline assumed is 1800 s, close to what is used by the Freq Hough search in this frequency range [22]. The distinct excess at low frequencies arises from an exceptionally young and nearby black hole.
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Figure 47
The amplitude spectral density of the LIGO O2 data alone (dashed red line) and of an ensemble signal (top green points) assuming that $μ_{b} = 3 \times 10^{- 12} eV$ and a Galactic black hole population with maximum mass $20 M_{⊙}$ or $30 M_{⊙}$ and maximum initial spins of 0.5 and 1. The time baseline assumed is 1800 s, close to what is used by the Freq Hough search in this frequency range [22].
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Physical Review D

covering particles, fields, gravitation, and cosmology

Characterizing the continuous gravitational-wave signal from boson clouds around Galactic isolated black holes

Sylvia J. Zhu, Masha Baryakhtar, Maria Alessandra Papa, Daichi Tsuna, Norita Kawanaka, and Heinz-Bernd Eggenstein

Phys. Rev. D 102, 063020 – Published 21 September 2020

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Physical Review D

covering particles, fields, gravitation, and cosmology

Characterizing the continuous gravitational-wave signal from boson clouds around Galactic isolated black holes

Sylvia J. Zhu, Masha Baryakhtar, Maria Alessandra Papa, Daichi Tsuna, Norita Kawanaka, and Heinz-Bernd Eggenstein

Phys. Rev. D 102, 063020 – Published 21 September 2020

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