Anatomy of parameter-estimation biases in overlapping gravitational-wave signals: detector network

TL;DR

Overlapping gravitational-wave signals are inevitable for next-generation detector networks and can bias parameter estimates. The authors extend the single-detector bias anatomy to networks by introducing the bias integral $J$ and separating geometric from intrinsic parameter effects, deriving how extrinsic angles $(\hat{\boldsymbol{n}},\psi,\iota,t_c,\phi_c)$ and intrinsic parameters $(\mathcal{M}, \eta, d_L)$ shape the network biases, with $J^{\mathrm{net}}=\sum_D J_D$ and a stationary-phase description for $\bar{J}_\alpha(f^{\rm spa})$. Using a three-detector network, they show time delays and detector orientations can constructively or destructively combine biases, finding that about 40–50% of overlaps yield larger network biases than a single detector across a range of $\Delta t_c$. Population analyses reveal that despite higher SNR, networks can magnify biases relative to statistical uncertainties, underscoring the need for joint parameter estimation or subtraction strategies and bias-aware diagnostics based on the bias integral.

Abstract

With the significantly improved sensitivity and a wider frequency band, the next-generation gravitational-wave (GW) detectors are anticipated to detect $\sim 10^5$ GW signals per year with durations from hours to days, leading to inevitable signal overlaps in the data stream. While a direct fitting for all signals may be challenging, extracting only one signal will be biased by its overlap with other signals. From this perspective, understanding how the biases arise from the overlapping and their dependence on the signal parameters is crucial for developing effective algorithms. In this work, we extend the anatomy of biases in single-detector cases (Wang et al. 2024) to a detector network. Specifically, we examine how the biases of the chirp mass, symmetric mass ratio, luminosity distance, and coalescence time depend on the source's sky position and orientation, as well as on the coalescence time and phase. We propose a new quantity, named the bias integral, as a useful tool, and establish relationship between the biases in a single detector and that in the entire network, with explicit dependence on extrinsic parameters. Using a 3-detector network as an example, we further explore the potential of a network to suppress biases due to the detectors' different locations and orientations. We find that location generally has a smaller effect than orientation, and becomes significant only when the time separation between signals is below sub-seconds. Through a population-level simulation over the extrinsic parameters, we find that nearly half of overlapping signals will lead to larger biases in the network compared to a single detector, highlighting the need to cope with overlapping biases in a detector network.

Figures (10)

• Figure 1: Generation of the bias integral $J_{D}$ in the $D$-th detector from the averaged bias integral $\bar{J}$, showing the roles of the geometric parameters $\{\hat{\bm{n}},\psi,\iota\}$ according to Eq. \ref{['eq:bias integral from\nJ_bar']}. The gray arrow represents $\bar{J}$ as a function of the coalescence time difference $\Delta t_c = t_c^{(2)}-t_c^{(1)}$ between two signals. The orange arrow considers the time delay $\tau_D = -{\bm r}_D \cdot \hat{\bm{n}}/c$ of the $D$-th detector from the Earth center, which shifts the variable from $\Delta t_c$ to $\Delta t_c + \Delta \tau_D$. Blue color shows quantities coming from the detector response, where $a_D$ and $\varphi_D$ satisfy $a_De^{i\varphi_D} = F^D_+(1+\cos^2 \iota)/2-iF^D_\times\cos \iota$ with $F^D_+$ and $F^D_\times$ being the pattern functions of the $D$-th detector which are functions of $\hat{\bm{n}}$, $\psi$, and $\iota$. This effect multiplies the orange arrow $\bar{J}(\Delta t_c + \Delta \tau_D)$ by an amplitude factor $a_D^{(1)}a_D^{(2)}/\bar{a}^2$ and rotates it by $\Delta \varphi_D$, leading to the ultimate bias integral in the $D$-th detector, marked in black.
• Figure 2: Evolution of the bias integral $\bar{J}_\alpha$ in the complex plane, as a function of the coalescence time difference $\Delta t_c$ for three parameters $\cal M$, $\eta$, and $d_L$. The red line marks $\bar{J}_\alpha$ where $\Delta t_c\in[-0.1\,{\rm s},0\,{\rm s}]$, while the remaining part is for $\Delta t_c\in[0\,{\rm s},1\,{\rm s}]$ and plotted with color bar. Though $\bar{J}_\alpha$ is dimensional (the reciprocal of $\theta^\alpha$) and has different scales for different parameters, here we are concerned only about its dependence on $\Delta t_c$. Note that here and below in similar figures, we report the bias integral in arbitrary unit because only their relative magnitudes are important in the context. We also create an animated version of this figure, which shows the evolution of $\bar{J}_\alpha$ with $\Delta t_c$ more clearly supplemental_material.
• Figure 3: Bias integrals in the individual detectors (blue, yellow, and red) and the whole network (green) for the parameter $\cal M$, only considering the time delay in different detectors and ignoring their different pattern functions (by adopting the angle-averaged waveform). Two sources are located at opposite ends of the H1$-$L1 line, and the time delays of the three detectors from the Earth's center are given in the bottom right. In the left panel, we show the bias integrals in the complex plane at the maximum point of $\Re J^{\rm net}_\alpha$. The black arrow represents the averaged bias integral $\bar{J}_\alpha$, while the gray line shows the evolution track of $\bar{J}_\alpha$ as a function of the coalescence time difference $\Delta t_c$. The right panel shows the real part of the bias integral as a function of $\Delta t_c$. The maximum point of $\Re J^{\rm net}_\alpha$ is marked with a magenta line for reference. We also create an animated version of this figure, which shows the evolution of $\bar{J}_\alpha$ and $J_{D\alpha}$ with $\Delta t_c$ and their addition more clearly supplemental_material.
• Figure 4: The real part of the bias integral as a function of the coalescence time difference $\Delta t_c$ for the parameter $\cal M$, where only the time delay effect is considered. From top to bottom, two sources are located at opposite ends of the H1$-$L1 line (same as Fig. \ref{['fig:J time delay']}), V1$-$H1 line, and V1$-$L1 line, respectively. The colors for individual detectors and the network are kept the same as in Fig. \ref{['fig:J time delay']}.
• Figure 5: Same as Fig. \ref{['fig:J time delay']}, but now only considering the different pattern functions of the detectors while ignoring the time delay effect. The additional amplitude and phase factors (calculated from pattern functions) in converting from the averaged bias integral $\bar{J}_\alpha$ to the detector-dependent $J_{D\alpha}$ are marked. We also create an animated version of this figure supplemental_material.