Functional dependence of merger and ringdown parameters on normalized incidence angle

Characterize the analytical trends of the merger-phase sin‑Gaussian parameters and ringdown quasi‑normal mode parameters of the Weyl scalar Ψ4 with respect to the normalized incidence angle θ/θ0 for dynamical capture of two equal‑mass non‑spinning black holes, resolving the current absence of clear functional dependencies due to oscillatory behavior in the fitted quantities.

Background

The waveform model uses a sin‑Gaussian description for both the close hyperbolic encounter and merger portions of Ψ4, and a damped quasi‑normal mode for ringdown. While the CHE parameters exhibit well‑behaved trends versus θ/θ0, the merger and ringdown parameters do not show clear functional relationships.

In the merger phase, the authors report no clear trends across scenarios, likely due to clustered parameter magnitudes and the approximate division between merger and ringdown. In the ringdown phase, fitted parameters display mild oscillations and no analytical trends, complicating efforts to build a compact phenomenological model that parametrizes these emissions in terms of θ/θ0.