Detecting Unmodeled, Source-dependent Signals in Gravitational Waves with SCoRe

New physics and systematic errors can lead to deviations between the models used to analyze gravitational wave data and the actual signal. Such deviations will generally be correlated between detectors and manifest differently across the gravitational wave source parameter space. The previously introduced \ttt{SCoRe} framework uses these features to distinguish these deviations from noise and extract physical information from their source-dependent variation. In this work, we further analyze the hierarchical component of the method -- we include the expected dependence of the deviations on the source parameters into the inference process, obtaining more physically informative results. As a specific example, we study a deviation that scales as a power law of the mass scale of black hole binaries -- as, for example, in Effective Field Theory of gravity. We show how the signal-to-noise ratio of the cross-correlated residual power can be used to recover the power-law index. We demonstrate how both the distribution in source and deviation strength determine which region of source parameter space influences the inference most. Finally, we forecast the constraint on the power law index for a network of two Cosmic Explorer-like detectors with a year of observation period.