Probing stellar binary black hole formation in galactic nuclei via the imprint of their center of mass acceleration on their gravitational wave signal

Multifrequency gravitational wave (GW) observations are useful probes of the formation processes of coalescing stellar-mass binary black holes (BBHs). We discuss the phase drift in the GW inspiral waveform of the merging BBH caused by its center-of-mass acceleration. The acceleration strongly depends on the location where a BBH forms within a galaxy, allowing observations of the early inspiral phase of Laser Interferometer Gravitational Wave Observatory (LIGO)-like BBH mergers by the Laser Interferometer Space Antenna (LISA) to test the formation mechanism. In particular, BBHs formed in dense nuclear star clusters or via compact accretion disks around a nuclear supermassive black hole in active galactic nuclei would suffer strong acceleration, and produce large phase drifts measurable by LISA. The host galaxies of the coalescing BBHs in these scenarios can also be uniquely identified in the LISA error volume, without electromagnetic counterparts. A nondetection of phase drifts would rule out or constrain the contribution of the nuclear formation channels to the stellar-mass BBH population.

Figure 1

The relative SNR of the deviation in the GW inspiral waveform caused by the CoM acceleration $\epsilon$, for different combinations of the redshifted chirp mass $M_{\mathrm{cz}}$ and the frequency $f_{\min }$ when the LISA observation begins. The corresponding time to coalescence ${\tau }_{\mathrm{c}}$ is indicated in the figure. The LISA observation is limited to the duration $\delta t=5\mathrm{yr}$. The relative SNRs saturate at $\epsilon >{\epsilon }_{\mathrm{crit}}$ ($\delta {\mathrm{\Psi }}_{\mathrm{acc}}\gtrsim 1$) shown by open circles.

Figure 2

The $1\sigma$ errors $\mathrm{\Delta }Y/Y$ from LISA alone (blue solid) and $\mathrm{LISA}+\text{LIGO}$ (i.e., assuming that the coalescence time $t_c$ has been fixed by LIGO; blue dashed). For our fiducial case ($f_{\min }=0.014\mathrm{Hz}$, $\epsilon =3\times {10}^4$, $z=0.05$ and $\eta =0.25$), nonzero acceleration can be detected, i.e., $\mathrm{\Delta }Y/Y<1$, for merging BBHs with $35{\mathrm{M}}_{\odot }\lesssim M_{\mathrm{cz}}\lesssim 63{\mathrm{M}}_{\odot }$. The LISA SNR (purple) and the inverse of the numerically computed $\rho (\delta h)$ due to the CoM acceleration (red) are shown. The green curve shows the Fisher error assuming all parameters but $Y$ are fixed: for $M_{\mathrm{cz}}\gtrsim 35{\mathrm{M}}_{\odot }$ it coincides with the inverse of $\rho (\delta h)$, validating the Fisher analysis; for lower $M_{\mathrm{cz}}$ the Fisher analysis does not capture the saturation effect discussed below Eq. (9) and shown in Fig. 1.

Figure 3

Contours of the marginalized $1\sigma$ error $\mathrm{\Delta }Y/Y$ in the $f_{\min }–M_{\mathrm{cz}}$ parameter space, provided by LISA alone (top left panel) and $\mathrm{LISA}+\text{LIGO}$ (top right panel) assuming a five year mission, and LISA alone (bottom left panel) and $\mathrm{LISA}+\text{LIGO}$ (bottom right panel) assuming a 10 year mission. The solid curves indicate constant times to coalescence. Merging BBHs with $2\mathrm{yr}\lesssim {\tau }_c\lesssim 5$ (or 10) yrs provide the best combinations of $f_{\min }$ and $M_{\mathrm{cz}}$ to probe the CoM acceleration.

Figure 4

GW phase drift detection conditions in the $\epsilon –d_L$ parameter space. The four solid curves mark marginalized $1\sigma$ errors $\mathrm{\Delta }Y/Y<0.1$ (orange), $<0.3$ (red), $<0.5$ (blue) and $<1.0$ (green). The two horizontal dotted lines show a maximum distance ($D<640\mathrm{Mpc}$) for a total SNR $\rho (h)\ge 8$ and a minimum distance ($D>60\mathrm{Mpc}$) to find a BBH at $f\simeq 0.014\mathrm{Hz}$ (i.e., $N_{\mathrm{m}}\equiv 4\pi n_{\mathrm{m}}{d}_L^3/3>1$). The maximum distance saturates and the Fisher analysis is invalid for large accelerations $\epsilon >{\epsilon }_{\mathrm{crit}}$, marked by an arrow [see below Eq. (9)].