Gravitational waves as a probe of dark matter minispikes

Recent studies show that an intermediate mass black hole (IMBH) may develop a dark matter (DM) minihalo according to some BH formation scenarios. We consider a binary system composed of an IMBH surrounded by a DM minispike and a stellar mass object orbiting around the IMBH. The binary evolves due to gravitational pull and dynamical friction from the DM minispike and backreaction from its gravitational wave (GW) radiation which can be detected by future space-borne GW experiments such as eLISA/NGO. We consider a single power-law model for the DM minispike which is assumed to consist of nonannihilating DM particles and derive GW waveforms including the DM effects analytically. We demonstrate that a detection of GWs from such a binary with eLISA/NGO is affected by the DM effects and enables us to measure the DM minispike parameters accurately. For instance, in our reference case originally advocated by Zhao and Silk [Phys. Rev. Lett. 95, 011301 (2005)] and Bertone et al. [Phys. Rev. D 72, 103517 (2005)], we could determine the power-law index $\alpha$ of the DM minispike radial profile with a $1\sigma$ relative error of $\pm 5\times 10^{-6}$ for a GW signal with signal-to-noise ratio 10 and assuming a five-year observation with eLISA. We also investigate how accurately the DM parameters can be determined for various values of the slope of the DM minispike and the masses of the IMBH–stellar mass object binary surrounded by the DM minispike. We find that the power-law index $\alpha$ is measurable at 10% level even for a slightly flatter radial distribution of $\alpha \sim 1.7$. We clarify that the larger masses of the IMBH and the stellar object lead to the worse measurement accuracies of the DM parameters because the number of GW cycles becomes smaller.

Figure 1

The accumulated phase difference $\mathrm{\Delta }\tilde{\mathrm{\Phi }}$ against the power-law index $\alpha$, defined by Eq. (29). In essence, this is the difference between the accumulated phase from GW frequency $f$ and the binary coalescence with and without the DM minispike. Three different curves show $\mathrm{\Delta }\tilde{\mathrm{\Phi }}$ for three different values of $\alpha$. For instance, if detecting a binary GW from $f=0.01\mathrm{Hz}$ to its coalescence, we would observe by a factor of $10^7$ more GW cycles in the case with a $\alpha =7/3$ DM minispike than without any. For this plot, we take $\mu =1M_{\odot }$ and ${\rho }_{\text{sp}}$, $r_{\text{sp}}$, and $M_{\mathrm{BH}}$ are as listed in Table 1.

Figure 2

Initial frequency against the power-law index $\alpha$. We assume that the GW is detected by an eLISA five-year observation. For small $\alpha$, $f_{\mathrm{ini}}$ is almost constant. On the other hand, for large $\alpha$, $f_{\mathrm{ini}}$ drops sharply due mainly to the dynamical friction. For this plot, we take $\mu =1M_{\odot }$ and ${\rho }_{\text{sp}}$, $r_{\text{sp}}$, and $M_{\mathrm{BH}}$ are as listed in Table 1.

Figure 3

Confidence level contours of 68.3%, 95.4% and 99.7% for $S/N=10$ in the case where the initial DM halo has an NFW profile and the final profile has the radial power-law index of $\alpha =7/3$ through an adiabatic growth. We assume ${\rho }_{\text{sp}}$, $r_{\text{sp}}$, and $M_{\mathrm{BH}}$ from Table 1 and $\mu =1M_{\odot }$.

Figure 4

The relative errors of the parameters in the phase $\tilde{\mathrm{\Phi }}(f)$ versus (a) the central BH mass $M_{\mathrm{BH}}$ and (b) the stellar mass object mass $\mu$ for $S/N=10$ and $\alpha =7/3$. For this plot, ${\rho }_{\text{sp}}$ and $r_{\text{sp}}$ are taken from Table 1. The other parameter is fixed to be $\mu =1M_{\odot }$ in the left and $M_{\mathrm{BH}}=10^3{M}_{\odot }$ in the right, respectively. Note that both axes are in the logarithmic scales. The solid line, the dashed line and the dashed-dotted line correspond to $\mathrm{\Delta }\alpha /\alpha$, $\mathrm{\Delta }{c}_{ϵ}/c_{ϵ}$, $\mathrm{\Delta }{M}_c/M_c$ respectively.

Figure 5

The relative errors of the parameters in the phase $\tilde{\mathrm{\Phi }}(f)$ versus the power-law index of the DM profile for $S/N=10$ in the case where the DM minispike harboring the IMBH has a radial power-law profile. The solid line, the dashed line, and the dashed-dotted line corresponds to $\mathrm{\Delta }\alpha /\alpha$, $\mathrm{\Delta }{c}_{ϵ}/c_{ϵ}$, $\mathrm{\Delta }{M}_c/M_c$ respectively. For this plot, $\mu =1M_{\odot }$ and the values of the parameters $M_{\mathrm{BH}}$, ${\rho }_{\text{sp}}$, and $r_{\text{sp}}$ are assumed as in Table 1.

Figure 6

The relative errors of (a) $\alpha$ and (b) $c_{ϵ}$ versus the power-law index of the DM profile for $S/N=10$ in the case where the DM minispike harboring the IMBH has a radially power-law profile. The solid line, the dashed line and the dashed-dotted line corresponds to ${\rho }_{\mathrm{sp}}{r}_{\mathrm{sp}}^{\alpha }$, $0.1{\rho }_{\mathrm{sp}}{r}_{\mathrm{sp}}^{\alpha }$, and $10{\rho }_{\mathrm{sp}}{r}_{\mathrm{sp}}^{\alpha }$, respectively. The values of $r_{\mathrm{sp}}$ and ${\rho }_{\mathrm{sp}}$ are taken from Table 1.

Figure 7

Lower frequency bound which is the minimum frequency the inspiral GW spend in the detector bandwidth $[f_{-},f_{+}]$ for $\mu =1M_{\odot }$ and $M_{\mathrm{BH}}=10^3{M}_{\odot }$.

Figure 8

The number of cycles $N_{\text{cycle}}$ spent in the bandwidth $f\in [f_{\min },f_{\max }]$ for $\mu =1M_{\odot }$ and $M_{\mathrm{BH}}=10^3{M}_{\odot }$. For large $\alpha$, the number of cycles drops sharply because of the DM effect.