Scaling Laws for Template-Free Detection of Environmental Phase Modulation in Gravitational-Wave Signals

TL;DR

This paper addresses the detectability of environmental phase modulation in gravitational-wave signals caused by hierarchical triple systems, using a template-free approach. It introduces LOSA as a smooth time warp that yields a cumulative phase distortion $\Delta\phi_{env}$, and analyzes detectability via a single-sample, trajectory-based statistic derived from time-frequency centroids computed with the continuous wavelet transform. A key finding is a universal one-parameter scaling, $\Lambda = \Delta\phi_{env} \times \mathrm{SNR}$, that governs ROC-AUC through a sigmoid transition, implying that observability is set by the balance between deformation strength and signal power. The results inform practical environmental searches in current and future detectors, suggesting a lightweight diagnostic to flag potential LOSA effects and guiding extensions to more realistic waveforms and acceleration profiles.

Abstract

Environmental effects such as hierarchical triple motion can introduce cumulative phase modulation in gravitational-wave signals through time-dependent line-of-sight acceleration. Whether such smooth time-warp distortions are observable depends jointly on deformation strength and signal-to-noise ratio (SNR), yet this relationship has not been quantified in a template-free framework. We study the detectability of these distortions using time-frequency representations derived from the continuous wavelet transform. Instead of reconstruction error alone, we examine trajectory-based statistics, in particular the evolution of the power-weighted frequency centroid. We find that environmental modulation can be detected using a single-sample statistic referenced to an isolated-binary distribution, without requiring matched templates. Across a grid of cumulative phase distortions and SNR, detection performance collapses onto a single scaling parameter defined as the product of phase distortion and SNR. The ROC-AUC follows a sigmoid transition in this parameter. Moderate distortions are detectable at low SNR, whereas smaller distortions require higher SNR. These results indicate that smooth environmental phase modulation is not generically absorbed by intrinsic waveform variability; instead, detectability is governed by a simple scaling between cumulative phase distortion and signal strength.

Figures (5)

• Figure 1: Environmental phase modulation for $\Delta\phi = 3\,\mathrm{rad}$ (noise-free). (a) Zoomed strain overlay showing the isolated chirp $h_{\mathrm{iso}}(t)$ (blue) and the LOSA-deformed signal $h_{\mathrm{LOSA}}(t)$ (red) during the final $\sim 0.5\,\mathrm{s}$ prior to peak strain. The dashed curve shows the strain difference $\Delta h(t) = h_{\mathrm{LOSA}} - h_{\mathrm{iso}}$. (b) Time--frequency centroid trajectory and its deviation, which will be defined formally in Sec. \ref{['sec:detection_statistic']}.
• Figure 2: Environmental phase modulation applied to a post-Newtonian (TaylorT4) inspiral waveform with $\Delta\phi_{\mathrm{env}} = 3\,\mathrm{rad}$ (noise-free). (a) Zoomed strain overlay during the final $\sim 0.4\,\mathrm{s}$ before merger. The LOSA-deformed signal (red) exhibits a cumulative phase slip relative to the isolated waveform (blue); the dashed curve shows the strain difference $\Delta h(t)$. (b) Corresponding time--frequency centroid trajectory $f_c(t)$ (top) and its deviation $\Delta f_c(t)$ (bottom). The deformation appears as a coherent modification of the centroid evolution, consistent with a smooth time reparameterization rather than a constant phase offset.
• Figure 3: AUROC as a function of cumulative phase distortion $\Delta\phi_{\mathrm{env}}$ and SNR. Detectability increases with both deformation strength and signal amplitude.
• Figure 4: AUROC as a function of the composite parameter $\Lambda = \Delta\phi_{\mathrm{env}} \times \mathrm{SNR}$. Points from all $(\mathrm{SNR}, \Delta\phi)$ combinations collapse onto a single curve, demonstrating one-parameter scaling of environmental detectability. The red curve shows a sigmoid fit to the collapsed data. Vertical lines indicate characteristic threshold values corresponding to AUROC = 0.8 and 0.95.
• Figure 5: Time evolution of the centroid shift $\Delta f_c(t)$ for representative cumulative phase distortions. Larger $\Delta\phi_{\mathrm{env}}$ produces systematically larger centroid deviations, while preserving the characteristic smooth temporal structure.