Mapping between black-hole perturbation theory and numerical relativity: gravitational-wave energy flux

We investigate the $\alpha$-$\beta$ mapping, as previously introduced by Islam et al.~\cite{Islam:2022laz}, which relates numerical relativity (NR) and adiabatic point-particle black hole perturbation theory (BHPT) waveforms in the comparable mass regime for quasi-circular, non-spinning binary black holes. This mapping involves scaling the amplitude of individual modes with different values of $\alpha$ and the time (and therefore the phase) with a single parameter, $\beta$. In this paper, we demonstrate that this scaling, both in terms of time and orbital frequencies, also extends to the overall gravitational-wave energy flux. This means that we can find a single $\alpha_{\mathcal{F}}$ that scales the BHPT flux and a single $\beta_{\mathcal{F}}$ (which matches the value of $\beta$) that scales the BHPT time such a way that it aligns with NR flux evolution. We then explore the connection between the scaling parameter $\alpha_{\mathcal{F}}$ ($\beta_{\mathcal{F}}$) and the missing finite size correction for the secondary black hole within the BHPT framework.