Bayesian Calibration of Gravitational-Wave Detectors Using Null Streams Without Waveform Assumptions

Abstract

We introduce a Bayesian null-stream method to constrain calibration errors in closed-geometry gravitational-wave (GW) detector networks. Unlike prior methods requiring electromagnetic counterparts or waveform models, this method uses sky-independent null streams to calibrate the detectors with any GW signals, independent of general relativity or waveform assumptions. We show a proof-of-concept study to demonstrate the feasibility of the method. We discuss prospects for next-generation detectors like Einstein Telescope, Cosmic Explorer, and LISA, where enhanced calibration accuracy will advance low-frequency GW science.

Figures (5)

• Figure 1: The plot shows the variation of the ratio of the posterior to prior generalized standard deviation of calibration parameters with increasing network SNR of the signal. The generalized standard deviation is defined as the determinant of the covariance matrix raised to $1 / (2N)$, where $N$ is the number of parameters. A ratio below $1$ indicates improved constraints, with $y = 1$ marking no improvement. As network SNR increases from 20 to 300, with the corresponding null stream SNR leakage shown at the top, the ratio decreases approximately linearly, demonstrating tighter constraints.
• Figure 2: The plot shows the ratio of the posterior to prior generalized standard deviation of the calibration parameters as a function of the number of signals (1--3). The generalized standard deviation is defined as the determinant of the covariance matrix raised to $1 / (2N)$, where $N$ is the number of parameters. A ratio below $1$ indicates improved constraints, with $y = 1$ marking no improvement. The single-signal case has a null stream SNR of 26.3, with signals in the 2- and 3-signal cases scaled to maintain this null stream SNR. Increasing the number of signals from one to three tightens the posterior, breaking polarization degeneracies, consistent with the theoretical expectations.
• Figure 3: The plot shows the posterior distributions for the amplitude ratio $(1 + \delta A_{j}) / (1 + \delta A_{1})$ and phase difference $\delta \phi_{j} - \delta \phi_{1}$ of calibration errors for the $j$th ET detector compared to the first, at 32 Hz, where $\delta A_{j}$ and $\delta \phi_{j}$ are the amplitude and phase errors, respectively. The top panel compares the second detector to the first, and the bottom panel compares the third to the first. Contours represent 90% credible regions. As the number of signals increases from one to three (null stream SNR fixed at 26.3), the posteriors tighten significantly, converging to the true values.
• Figure S1: The calibration errors applied to the mock data for the three detectors. The top panel shows the amplitude errors, while the bottom panel shows the phase errors.
• Figure S2: The plot shows the posterior distributions for the amplitude ratio $(1 + \delta A_{j}) / (1 + \delta A_{1})$ and phase difference $\delta \phi_{j} - \delta \phi_{1}$ of calibration errors for the $j$th detector compared to the first, across the frequency range 8--512 Hz, where $\delta A_{j}$ and $\delta \phi_{j}$ are the amplitude and phase errors, respectively. The left panel compares the second detector (ET2) to the first (ET1), and the right panel compares the third (ET3) to the first (ET1). The shaded regions represent 90% credible intervals. The results are from the three-signal case with a null stream of 26.3. Across the frequency range, the posteriors are significantly tighter than the priors, accurately recovering the true values (dashed lines).