Impact of neglecting center-of-mass acceleration in parameter estimation of stellar-mass black holes

TL;DR

This work studies how a tertiary body, such as a supermassive black hole, can impart center-of-mass acceleration on a merging stellar-mass BBH, imprinting a Doppler-like phase shift in its gravitational-wave signal. The authors model the CoM effect by reparameterizing time with $t \rightarrow t + \alpha(t - T)$ and adding a $\Delta \Psi^{\mathrm{CoM}}_{\mathrm{3PN}}$ term to a TaylorF2 waveform, where $\alpha$ depends on the tertiary mass and separation via $\alpha = 9.9 \times 10^{-12}\,\mathrm{s}^{-1} \frac{M_3}{M_\odot} (\frac{1\ \mathrm{AU}}{r})^2$, and explore detectability with CE and ET. Using the Fisher information matrix and the Cutler–Vallisneri formalism, they quantify statistical uncertainties and systematic biases in parameters like the chirp mass $M_c$ and symmetric mass ratio $\eta$ for binaries with total masses $5$–$30\,M_\odot$ at 500 Mpc and spins $(\chi_1,\chi_2)=(0.2,0.1)$. They find that neglecting CoM effects can yield biases exceeding statistical errors for $\alpha \sim 10^{-9}$–$10^{-10}\ \mathrm{s}^{-1}$ (CE/ET), especially for low-mass and asymmetric binaries; including CoM allows constraining $\alpha$, with ET achieving $\delta\alpha \sim 9\times10^{-11}\ \mathrm{s}^{-1}$ for certain configurations, and CE probing roughly $\alpha \in [10^{-8},10^{-5}]\ \mathrm{s}^{-1}$. These results imply that 3G detectors can probe BBH environments near SMBHs and help distinguish formation channels, while emphasizing the necessity of environmental effects in waveform modeling to avoid biases in population inferences.

Abstract

A tertiary body near a coalescing binary can imprint its influence on the gravitational waves (GWs) emitted by that binary in the form of center-of-mass (CoM) acceleration. An example of such a scenario is a binary black hole (BBH) merging near a supermassive black hole, which is touted to occur frequently. The limited low-frequency sensitivity of current GW detectors makes it challenging to detect these effects, as the associated waveform phase remains elusive. However, next-generation (3G) detectors such as Cosmic Explorer (CE) and Einstein Telescope (ET), with improved sensitivity at lower frequencies, are expected to be capable of capturing such signatures. In our study, we focus on the stellar-mass BBHs and explore the parameter space where the CoM acceleration will play a dominant role affecting parameter inference of the binary. We demonstrate that an unaccounted CoM acceleration of a low-mass binary with a total mass of $5\, \rm{M}_{\odot} $ can lead to significant systematic biases, exceeding statistical errors in the estimation of the chirp mass and symmetric mass ratio when the CoM parameter $α$ is as small as $\sim 10^{-9}$ and $10^{-10}$ $\rm{s}^{-1}$ for CE and ET, respectively. We also find that asymmetric binaries are more susceptible to systematic bias when CoM acceleration is neglected. When the effect of CoM acceleration is included in the GW phase, then $α= 10^{-7} \rm s^{-1}$ can be constrained with a bound of $10^{-9} (10^{-11})\, \rm s^{-1}$ for CE (ET). Our study thus highlights the crucial implications of considering the presence of a tertiary body in the GW emitted by a stellar-mass BBH when observed in 3G detectors.

Figures (4)

• Figure 1: The plot shows the allowed upper value of $\alpha$ i.e. $\alpha_{max}$ for different binaries considered in our study when observed through CE and ET. Note that $\alpha_{max}$ is an order of magnitude lower for a particular binary in ET in comparison to CE.
• Figure 2: The plot shows ratio of absolute value of systematic bias over statistical one for $\rm ln M_c$ for different values of $\alpha$. Binary black holes of different mass and mass ratio are considered with aligned spin magnitudes $(0.2, 0.1)$ at a distance of 500 Mpc. The systematic error dominates over statistical for lowest value of $\alpha$ being $\sim 10^{-9} \rm s^{-1}$ for CE and $\sim 10^{-10} \rm s^{-1}$ for ET.
• Figure 3: The plot shows ratio of absolute value of systematic bias over statistical one for $\rm ln \eta$ for different values of $\alpha$. Binary black holes of different mass and mass ratio are considered with aligned spin magnitudes $(0.2, 0.1)$ at a distance of 500 Mpc. The systematic error dominates over statistical for lowest value of $\alpha$ being $\sim 10^{-9} \rm s^{-1}$ for CE and $\sim 10^{-10} \rm s^{-1}$ for ET.
• Figure 4: The plot shows the fractional error on $\rm ln \alpha$ for different values of $\alpha$. Binary black holes of different mass and mass ratio are considered with aligned spin magnitudes $(0.2, 0.1)$ at a distance of 500 Mpc.