Deci-Hz gravitational waves from the self-interacting axion cloud around a rotating stellar mass black hole (2024)

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  Figure 1

  Left: the behavior of $M(\mu -{\omega }_{nlm,R})$ as the function of $\mu M$, where ${\omega }_{nlm,R}$ denotes the real part of the frequency ${\omega }_{nlm}$. The red and the blue solid curve correspond to the $l=m=1$ and $l=m=2$ fundamental superradiant modes, respectively. The corresponding dotted curves are the ones with the nonrelativistic approximation Eq.(15). The spin of the central black hole is set to $a/M=0.99$. Right: the similar plot for the imaginary part of the frequencies ${\omega }_{nlm}$.

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  Figure 2

  The real part of the radial mode function, $R_{lm\omega }$ as a function of $r/M$. The red and blue solid curves, respectively, correspond to the $l=m=1$ and $l=m=2$ superradiant modes without any approximation, while the dashed curves are the counterparts in the nonrelativistic approximation. The spin of the black hole and the mass of the axion are set to $a/M=0.99$ and $\mu M=0.42$, respectively. The phase of the mode functions is chosen so that there is no imaginary part for large $r$.

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  Figure 3

  Dissipative processes induced by the self-interaction. Each line represents an energy level. The left panel corresponds to the dissipation of the axion due to the absorption by the black hole, and the right panel corresponds to the dissipation due to the radiation to infinity. The particles in the cloud $i$ and $j$ make a transition to the one in the cloud $k$ and the dissipative mode specified by the superscript $\mathcal{H}$ and $\mathcal{I}$.

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  Figure 4

  An example of the time evolution of the normalized axion cloud mass $M_{\mathrm{cl},i}$. For clarity, we recovered the normalization of the cloud mass. The red solid, blue dashed, black dotted, and green dash-dotted curves correspond to the $l=m=1$, 2, 3, and 4 modes, respectively. The initial black hole mass and black hole spin are $M_{\mathrm{BH},\mathrm{ini}}=10M_{\odot }$ and ${\chi }_{\mathrm{ini}}=0.99$, respectively. We take the axion mass such that ${\alpha }_{\mathrm{ini}}\equiv \mu M_{\mathrm{BH},\mathrm{ini}}=0.2$ and the axion decay constant to be $F_a={10}^{-3}$.

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  Figure 5

  The same figure as Fig.4 but with $F_a={10}^{-1}$(top) and ${10}^{-7}$(bottom), respectively.

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  Figure 6

  The same figure as Fig.4 but with ${\alpha }_{\mathrm{ini}}=0.1$ (top) and 0.4 (bottom), respectively.

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  Figure 7

  The parameter region where the effective growth rate for the $l=m=3$ (${\omega }_{3,I}^{\mathrm{eff}}$) and the $l=m=4$ (${\omega }_{4,I}^{\mathrm{eff}}$) are positive in the $(\mu M_{\mathrm{BH}},\chi )$-plane. The blue and orange region corresponds to ${\omega }_{3,I}^{\mathrm{eff}}>0$ and ${\omega }_{4,I}^{\mathrm{eff}}>0$, respectively. In the most of the overlapping region, ${\omega }_{3,I}^{\mathrm{eff}}>{\omega }_{4,I}^{\mathrm{eff}}$ holds.

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  Figure 8

  The time evolution of the black hole spin parameters $\chi$ as functions of time. Top: the red solid, blue dashed, black dotted, and green dash-dotted curves correspond to the decay constant $F_a={10}^{-1},{10}^{-3},{10}^{-5}$, and ${10}^{-7}$, respectively. The axion mass is chosen to satisfy ${\alpha }_{\mathrm{ini}}=\mu M_{\mathrm{BH},\mathrm{ini}}=0.1$. Middle: the same figure as the top one but with ${\alpha }_{\mathrm{ini}}=0.2$. Bottom: the same figure as the top one but with ${\alpha }_{\mathrm{ini}}=0.4$.

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  Figure 9

  Frequencies of gravitational waves $f_{\mathrm{gw}}$ emitted from the axion condensates as functions of the axion mass $\mu$. The blue circle corresponds to the pair annihilation of the $l=m=1$ fundamental mode. The yellow square, green diamond, red triangle, purple reversed triangle, brown open circle, and light blue open square correspond to the level transition signal of $|322⟩\to |211⟩$, $|433⟩\to |211⟩$, $|433⟩\to |322⟩$, $|544⟩\to |322⟩$, $|311⟩\to |211⟩$, $|411⟩\to |211⟩$, respectively. Here, $|nlm⟩$ corresponds to the mode labeled by $n$, $l$, and $m$. Although overtones do not appear in our calculation, we show them for reference. The mass and spin of the black hole are fixed at $M_{\mathrm{BH}}=10M_{\odot }$ and $\chi =0.99$.

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  Figure 10

  Left: the energy flux of the gravitational waves from various processes. Each point corresponds to the same process in Fig.9. (Right) The amplitude of the gravitational waves from various processes. Again, each point corresponds to the same process in Fig.9. The black hole is placed at $r=1\mathrm{kpc}$.

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  Figure 11

  Normalized masses of the axion clouds in the quasistationary configuration. The red solid, blue dotted, black dashed, and green dot-dashed curves correspond to the mass of the $l=m=1$, 2, 3 and 4 cloud when the condensate is in the quasistationary configuration. The spin of the black hole is fixed at $\chi =0.99$(top) and 0.7(bottom), respectively.

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  Figure 12

  An example of the time evolution of the gravitational wave amplitude from an axion condensate. The red solid, blue dashed, and black dotted curves correspond to the time dependence of the gravitational wave amplitudes for the pair annihilation signal from the $l=m=1$ cloud, the level transition signals from the $l=m=2$ cloud to the $l=m=1$ cloud, and from the $l=m=3$ cloud to the $l=m=1$ cloud, respectively. We fix the initial black hole mass and spin at $10M_{\odot }$ and 0.99. The axion mass and decay constant are $\mu M_{\mathrm{BH},\mathrm{ini}}=0.15$ and $F_a={10}^{-3}$. The gravitational wave amplitude is estimated from the calculation setting the black hole spin to $\chi =0.99$.

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  Figure 13

  Upper left: the dependence of the peak amplitude of the gravitational wave from the pair annihilation signal of the $l=m=1$ cloud on axion mass $\mu$ and decay constant $F_a$. The parameters of the black hole are taken to be the ones similar to Cygnus X-1, $M_{\mathrm{BH},\mathrm{ini}}=14.8M_{\odot },{\chi }_{\mathrm{ini}}=0.99$, and $d=1.83\mathrm{kpc}$. We set a cutoff at $h_{\mathrm{gw}}={10}^{-28}$. The upper ticks correspond to the $\mu M_{\mathrm{BH},\mathrm{ini}}$. Upper right: the dependence of the duration of the pair annihilation signal, estimated by the FWHM, on $\mu$ and $F_a$. The parameters of the black hole are the same as the upper left panel. Lower left: The similar figure as the upper left figure but with the initial black hole parameters similar to the black hole of GW170817. Namely, $M_{\mathrm{BH},\mathrm{ini}}=3M_{\odot },{\chi }_{\mathrm{ini}}=0.7$, and $d=40\mathrm{Mpc}$. We take the cutoff to be $h_{\mathrm{gw}}={10}^{-34}$. Lower right: the same figure as the upper right figure but with the same parameters as the lower left figure.

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  Figure 14

  The same figures as Fig.13, but with the gravitational waves from the level transition between the $l=m=2$ and the $l=m=1$ clouds.

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  Figure 15

  The same figures as Fig.13, but with the gravitational waves from the level transition between the $l=m=3$ and the $l=m=1$ clouds.

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  Figure 16

  The same figures as Fig.13, but with the gravitational waves from the level transition between the $l=m=4$ and the $l=m=2$ clouds.

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  Figure 17

  Left: the blue solid and orange dotted curves show the amplitude of the axion wave normalized by the decay consent $F_a$ from the processes $F_{221^{*}}^{\mathcal{I}}$ and $F_{231^{*}}^{\mathcal{I}}$, respectively. We set the black hole mass to $M_{\mathrm{BH}}=10M_{\odot }$ and the spin to $\chi =0.99$. We put the black hole to be 1kpc away from us. Right: same as the left panel but with number density flux.

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