On the approximate relation between black-hole perturbation theory and numerical relativity

We investigate the interplay between numerical relativity (NR) and adiabatic point-particle black hole perturbation theory (ppBHPT) in the comparable mass regime for quasi-circular non-spinning binary black holes. Specifically, we reassess the $\alpha$-$\beta$ scaling technique, previously introduced by Islam et al, as a means to effectively match ppBHPT waveforms to NR waveforms within this regime. In particular, $\alpha$ rescales the amplitude and $\beta$ rescales the time (and hence the phase). Utilizing publicly available long NR data (\texttt{SXS:BBH:2265}~\cite{sxs_collaboration_2019}) for a mass ratio of $1:3$, encompassing the final $\sim 65$ orbital cycles of the binary evolution, we examine the range of applicability of such scalings. We observe that the scaling technique remains effective even during the earlier stages of the inspiral. Additionally, we provide commentary on the temporal evolution of the $\alpha$ and $\beta$ parameters and discuss whether they can be approximated as constant values. Consequently, we derive the $\alpha$-$\beta$ scaling as a function of orbital frequencies and demonstrate that it is equivalent to a frequency-dependent correction. We further provide a brief comparison between post-Newtonian (PN) waveforms and the rescaled ppBHPT waveform at a mass ratio of $q=3$ and comment on their regime of validity. Finally, we explore the possibility of using PN theory to obtain the $\alpha$-$\beta$ calibration parameters and still provide a rescaled ppBHPT waveform that matches NR.