Direct Determination of Supermassive Black Hole Properties with Gravitational-Wave Radiation from Surrounding Stellar-Mass Black Hole Binaries

A significant number of stellar-mass black-hole (BH) binaries may merge in galactic nuclei or in the surrounding gas disks. With purposed space-borne gravitational-wave observatories, we may use such a binary as a signal carrier to probe modulations induced by a central supermassive BH (SMBH), which further allows us to place constraints on the SMBH’s properties. We show in particular the de Sitter precession of the inner stellar-mass binary’s orbital angular momentum (AM) around the AM of the outer orbit will be detectable if the precession period is comparable to the duration of observation, typically a few years. Once detected, the precession can be combined with the Doppler shift arising from the outer orbital motion to determine the mass of the SMBH and the outer orbital separation individually and each with percent-level accuracy. If we further assume a joint detection by space-borne and ground-based detectors, the detectability threshold could be extended to a precession period of $\sim 100\mathrm{yr}$.

Figure 1

Periods of various dynamical processes. Approximately, below the solid trace an effect is detectable with joint detection by space-borne and ground-based detectors. Below the dashed trace, the effect can be constrained with a single TianGO-like detector. We set the lower $y$ limit in each panel to $a_o(1-e_o)=9M_3$. The decay of the outer orbit and hence the shaded region with $a_o/4{\dot{a}}_o<5\mathrm{yr}$ is discarded. The triple stability [42] is always satisfied in the upper panel and its boundary is similar to the shaded region in the lower one.

Figure 2

Cartoon illustrating the geometry of the problem. Note the amplitudes of vectors are chosen only for visualization purpose.

Figure 3

Sample waveforms shown in $2\sqrt{f}|\tilde{h}|$. The olive trace includes variation in the detector’s orientation only, while the purple ones further include the dS precession. The initial frequency $f^{(0)}$ is chosen such that the binary merges in $T_{\mathrm{obs}}=5\mathrm{yr}$ and the dot symbols indicate the instant 0.1 yr prior to the merger. We assumed $(D_L,{\overline{\theta }}_S,{\overline{\varphi }}_S,{\overline{\theta }}_J,{\overline{\varphi }}_J)=(1\mathrm{Gpc},33°,147°,75°,150°)$ and $(P_{\text{dS}},{\lambda }_L)=(2.7\mathrm{yr},45°)$.

Figure 4

PE results assuming a simple-precession problem (no Doppler phase of the outer orbit). We fix the inner binary to have $(M_1,M_2,f^{(0)})=(50M_{\odot },50M_{\odot },12\mathrm{mHz})$. The source’s sky location in the solar frame is $(D_L,{\overline{\theta }}_S,{\overline{\varphi }}_S)=(1\mathrm{Gpc},33°,147°)$ and the orientation of the outer orbit is $({\overline{\theta }}_J,{\overline{\varphi }}_J)=(75°,150°)$. Note the dS precession is detectable if $P_{\text{dS}}\lesssim 10\mathrm{yr}$ if the source is detected by TianGO alone.

Figure 5

PE results combining both the dS precession and the Doppler phase shift. We assumed an opening angle between the inner and outer orbits of ${\lambda }_L=45°$ and other parameters are the same as in Fig. 4. We also plotted the line of $P_{\text{dS}}=10\mathrm{yr}$. The grey regions have $\mathrm{\Delta }{\lambda }_L>{\lambda }_L$ and therefore are excluded. Note $M_3$ can be constrained to 10% if $P_{\text{dS}}\simeq 10\mathrm{yr}$ with TianGO alone. The error decreases as $M_3$ increases thanks to the additional information provided by the amplitude of the Doppler phase.