Detectability Scaling Laws for Environmental Phase Modulation in Gravitational-Wave Signals

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 analyze trajectory-level 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 Lambda = Delta phi x SNR. The ROC-AUC follows an approximately sigmoid transition in this parameter. Moderate distortions are detectable even 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.