Gravitational waves with dark matter minispikes: The combined effect

It was shown that the dark matter (DM) minihalo around an intermediate mass black hole (IMBH) can be redistributed into a cusp, called the DM minispike. We consider an intermediate-mass-ratio inspiral consisting of an IMBH harbored in a DM minispike with nonannihilating DM particles and a small black hole (BH) orbiting around it. We investigate gravitational waves (GWs) produced by this system and analyze the waveforms with the comprehensive consideration of gravitational pull, dynamical friction and accretion of the minispike and calculate the time difference and phase difference caused by it. We find that for a certain range of frequency, the inspiralling time of the system is dramatically reduced for smaller central IMBH and large density of DM. For the central IMBH with ${10}^5{M}_{\odot }$, the time of merger is ahead, which can be distinguished by LISA, Taiji and Tianqin. We focus on the effect of accretion and compare it with that of gravitational pull and friction. We find that the accretion mass is a small quantity compared to the initial mass of the small BH and the accretion effect is inconspicuous compared with friction. However, the accumulated phase shift caused by accretion is large enough to be detected by LISA, Taiji, and Tianqin, which indicate that the accretion effect can not be ignored in the detection of GWs.

Figure 1

The time difference with different central IMBH. The horizontal axis is the time without DM minispike $t_0$, the vertical axis is the time difference $\mathrm{\Delta }t$. We take ${\mu }_0=10M_{\odot }$, ${\rho }_{\mathrm{sp}}=226M_{\odot }/{\mathrm{pc}}^3$, $r_{\mathrm{sp}}=0.54\mathrm{pc}$. The solid line $\alpha =7/3$, the dashed line $\alpha =2.0$, the dash-dot line $\alpha =1.5$. In panel (a) $M_{\mathrm{BH}}={10}^3{M}_{\odot }$ and $f_c=0.1\mathrm{Hz}$, in panel (b) $M_{\mathrm{BH}}={10}^4{M}_{\odot }$ and $f_c=0.1\mathrm{Hz}$, in panel (c) $M_{\mathrm{BH}}={10}^5{M}_{\odot }$ and $f_c=f_{ISCO}$.

Figure 2

The phase difference with different central IMBH. The horizontal axis is the initial frequency $f$ and the vertical axis is the phase difference. We take ${\mu }_0=10M_{\odot }$, ${\rho }_{\mathrm{sp}}=226M_{\odot }/{\mathrm{pc}}^3$, $r_{\mathrm{sp}}=0.54\mathrm{pc}$. The solid line $\alpha =7/3$, the dashed line $\alpha =2.0$, the dash-dot line $\alpha =1.5$. In panel (a) $M_{\mathrm{BH}}={10}^3{M}_{\odot }$ and $f_c=0.1\mathrm{Hz}$, in panel (b) $M_{\mathrm{BH}}={10}^4{M}_{\odot }$ and $f_c=0.1\mathrm{Hz}$, in panel (c) $M_{\mathrm{BH}}={10}^5{M}_{\odot }$ and $f_c=f_{ISCO}$.

Figure 3

The phase difference by accretion with different central IMBH. We take ${\mu }_0=10M_{\odot }$, ${\rho }_{\mathrm{sp}}=226M_{\odot }/{\mathrm{pc}}^3$, $r_{\mathrm{sp}}=0.54\mathrm{pc}$. Panel (a): $M_{BH}=10^3{M}_{\odot }$ and $f_c=0.1$ Hz; Panel (b): $M_{\text{BH}}=10^4{M}_{\odot }$ and $f_c=0.1$ Hz; Panel (c): $M_{\text{BH}}=10^5{M}_{\odot }$ and $f_c=f_{\text{ISCO}}$

Figure 4

The proportion of phase difference caused by accretion to total dephasing. Curves for different $\alpha$ overlap in panels (b, c). We take ${\mu }_0=10M_{\odot }$, ${\rho }_{\mathrm{sp}}=226M_{\odot }/{\mathrm{pc}}^3$, $r_{\mathrm{sp}}=0.54\mathrm{pc}$. Panel (a): $M_{\text{BH}}=10^3{M}_{\odot }$ and $f_c=0.1$ Hz; Panel (b): $M_{\text{BH}}=10^4{M}_{\odot }$ and $f_c=0.1$ Hz, Panel (c): $M_{\text{BH}}=10^5{M}_{\odot }$ and $f_c=f_{\text{ISCO}}$.

Figure 5

Top panel: the phase difference caused by accretion with different density ${\rho }_{\mathrm{sp}}$. We set $M_{\mathrm{BH}}={10}^{4}M$, ${\mu }_0=10M$, $r_{\mathrm{sp}}=0.5\mathrm{pc}$ and $\alpha =2.0$. Bottom panel: the phase difference caused by accretion with different masses ${\mu }_0$ of the small black holes. We set $M_{\mathrm{BH}}={10}^{4}M$,$r_{\mathrm{sp}}=0.5\mathrm{pc}$, ${\rho }_{\mathrm{sp}}=226M/{\mathrm{pc}}^3$ and $\alpha =2.0$.