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

Technical Summary: Anatomy of Parameter-Estimation Biases in Overlapping Gravitational-Wave Signals: Detector Network

Problem Statement
Next-generation (XG) ground-based gravitational-wave (GW) detectors, such as the Cosmic Explorer and Einstein Telescope, are projected to detect approximately $10^5$ compact binary coalescence (CBC) events annually. Due to the increased sensitivity and longer signal durations (ranging from hours to days), multiple GW signals will inevitably overlap within the data stream. While a global joint parameter estimation (PE) for all signals is computationally prohibitive, extracting a single signal while ignoring others introduces biases in the estimated parameters. Previous studies have analyzed these biases in single-detector scenarios, primarily focusing on intrinsic parameters (masses, spins) and a limited set of extrinsic parameters. However, the dependence of these biases on the full set of extrinsic geometric parameters—specifically sky location ($\hat{n}$), polarization angle ($\psi$), and inclination angle ($\iota$)—within a detector network remains underexplored. It is unclear whether a network of detectors, with their distinct locations and orientations, inherently suppresses these overlapping biases or, conversely, amplifies them.

Methodology
The authors extend the semi-analytical framework established by Wang et al. [1] for single-detector bias analysis to a multi-detector network. The study employs the Fisher-matrix (FM) method, which approximates the posterior distribution as a multivariate Gaussian in the high signal-to-noise ratio (SNR) limit. This approach allows for a (semi-)analytical characterization of biases without the computational cost of full Bayesian inference.

Key methodological components include:

1. Bias Integral ($J$): The authors introduce a complex quantity called the "bias integral," $J_{D\alpha}$, defined for the $D$-th detector and parameter $\alpha$. This integral isolates the contribution of the overlapping signal to the reduced bias.
2. Geometric Decomposition: The signal model in a network is decomposed to explicitly separate geometric effects. The bias integral in a specific detector is related to a reference "averaged" bias integral ($\bar{J}_\alpha$) located at the Earth's center via:
   • Time Delay ($\Delta \tau_D$): Caused by the different geographic locations of detectors, shifting the coalescence time difference variable.
   • Amplitude and Phase Factors ($a_D, \phi_D$): Caused by the different antenna pattern functions and source orientations ($\hat{n}, \psi, \iota$), scaling the amplitude and rotating the phase of the integral.
3. Stationary Phase Approximation (SPA): The evolution of the bias integral with respect to the coalescence time difference ($\Delta t_c$) is analyzed using SPA. This reveals that the bias integral rotates in the complex plane with a frequency determined by the stationary point of the phase evolution, while its modulus varies slowly.
4. Population Simulation: A population-level simulation is conducted using a 3-detector network (LIGO-Hanford, LIGO-Livingston, Virgo) with CE-2 sensitivity. The simulation samples extrinsic parameters (sky location, orientation, coalescence time, and phase) isotropically and uniformly to assess the statistical likelihood of network-induced bias amplification.

Key Contributions

• Extension to Networks: The work generalizes the "anatomy of biases" from single detectors to a detector network, explicitly deriving the relationship between single-detector biases and network biases through geometric angle parameters.
• Bias Integral Formalism: The introduction of the bias integral $J$ provides a unified tool to characterize how extrinsic parameters modulate biases. It demonstrates that network biases result from the constructive or destructive interference of bias integrals from individual detectors in the complex plane.
• Quantification of Network Effects: The study systematically quantifies how detector location and orientation independently and jointly affect the combination of biases.

Results

1. Mechanism of Bias Combination:
   • Location Effect: Different detector locations introduce time delays ($\Delta \tau_D$). For small time separations between signals ($\Delta t_c \lesssim 0.2$ s), these delays cause the bias integrals in different detectors to be out of phase, potentially leading to destructive interference (reduced network bias). However, as $\Delta t_c$ increases, the phase differences diminish, and the integrals tend to align, leading to constructive interference.
   • Orientation Effect: Different detector orientations introduce constant amplitude and phase factors. Unlike the location effect, these do not diminish with increasing $\Delta t_c$.
2. Network vs. Single Detector:
   • Contrary to the intuition that networks always separate signals and reduce biases, the study finds that nearly half ($\gtrsim 40\%$) of overlapping signals result in larger reduced biases in the network compared to a single detector.
   • This amplification occurs because the network increases the total SNR, thereby reducing statistical uncertainty ($\Delta \theta_{stat}$). Since the reduced bias is the ratio of absolute bias to statistical uncertainty, the reduction in the denominator can make the bias more significant even if the absolute bias does not increase proportionally.
   • The "location effect" (time delay) becomes negligible for $\Delta t_c \gtrsim 0.4$ s, meaning the network size (Earth-scale) is insufficient to temporally separate signals effectively for most overlapping cases.
3. Dependence on Time Threshold:
   • The fraction of samples with larger network biases increases with the time threshold used to define "overlapping" signals. For thresholds ranging from 0.1 s to 5 s, the fraction of cases where the network bias exceeds the single-detector bias remains high (40–50%).
   • The orientation effect (pattern functions) dominates the superposition of biases for larger time separations, while the location effect dominates for very small separations.

Significance
The paper argues that the assumption that detector networks inherently mitigate overlapping signal biases is flawed. Instead, networks can amplify these biases for a significant portion of the population due to the interplay between increased SNR and the geometric alignment of bias integrals. This finding highlights a critical risk: neglecting overlapping signals in a network analysis may lead to even larger parameter estimation errors than in single-detector analyses.

The authors conclude that developing effective algorithms for overlapping signals must account for these network-specific geometric effects. The bias integral serves as a diagnostic tool, suggesting that inconsistencies in single-detector parameter estimates or null-stream analyses could serve as criteria for identifying overlapping signals. The work provides a necessary theoretical foundation for future strategies, such as hierarchical subtraction or joint PE, in the era of high-rate GW observations.