The final spin from the coalescence of aligned-spin black-hole binaries

TL;DR

The paper addresses the challenge of predicting the final spin $a_{ m fin}$ of black-hole binaries by focusing on aligned-spin, unequal-mass systems, reducing the parameter space to a 2D problem in $(a,\nu)$. It introduces a simple analytic fit for $a_{ m fin}$ as a polynomial in $a$ and $\nu$, with coefficients constrained by the extreme-mass-ratio limit and calibrated to numerical-relativity data; the model achieves accuracy at the few-percent level. Key results include explicit fitted coefficients, a Schwarzschild boundary curve defined by $a_{ m fin}(a,\nu)=0$, and insights into spin-up/down behavior, plus an indication that the approach extends to unequal initial spins via a natural generalization. This compact, physically grounded description provides a practical tool for modeling merger remnants and informing gravitational-wave templates and astrophysical interpretations of BH mergers.

Abstract

Determining the final spin of a black-hole (BH) binary is a question of key importance in astrophysics. Modelling this quantity in general is made difficult by the fact that it depends on the 7-dimensional space of parameters characterizing the two initial black holes. However, in special cases, when symmetries can be exploited, the description can become simpler. For black-hole binaries with unequal masses but with equal spins which are aligned with the orbital angular momentum, we show that the use of recent simulations and basic but exact constraints derived from the extreme mass-ratio limit allow to model this quantity with a simple analytic expression. Despite the simple dependence, the expression models very accurately all of the available estimates, with errors of a couple of percent at most. We also discuss how to use the fit to predict when a Schwarzschild BH is produced by the merger of two spinning BHs, when the total angular momentum of the spacetime ``flips'' sign, or under what conditions the final BH is ``spun-up'' by the merger. Finally, suggest an extension of the fit to include unequal-spin binaries, thus potentially providing a complete description of the final spin from the coalescence of generic black-hole binaries with spins aligned to the orbital angular momentum.

Figures (5)

• Figure 1: Global dependence of the final spin on the symmetric mass ratio and on the initial spins as predicted by expression (\ref{['eq:4']}). Squares refer to numerical estimates while circles to the EMRL constraints.
• Figure 2: Upper panel: Comparison of the numerical data with the 2D fit through (\ref{['eq:4']}) in the case of equal-mass binaries, ($\nu = 1/4$). Empty circles indicate the AEI data Rezzolla_etal:2007a, stars the FAU-Jena data Marronetti_etal:2007], a long-dashed line the BKL, and a short-dashed one the fit. Lower panel: residuals between the different estimates and the fit.
• Figure 3: Upper panel: Comparison of the numerical data with the 2D fit through (\ref{['eq:4']}) in the case of nonspinning binaries. Empty circles indicate the Jena data Berti_etal:2007, stars the Goddard data Buonanno_etal:2007a], a long-dashed line the quadratic EOB fit Damour_Nagar:2007a and a short-dashed line our 2D fit. Lower panel: residuals between the different estimates and the 2D fit.
• Figure 4: Upper panel: Set of initial spins and mass ratios leading to a final Schwarzschild BH: i.e., $a_{\rm fin}(a,\nu)=0$. The two curves refer to the BKL estimate (long dashed) and to the 2D fit (short dashed), respectively. Indicated with a star is a numerical example leading to $a_{\rm fin} =0.005$. Lower panel: Comparison between the BKL prediction (symbols) and the 2D fit (solid, dashed and long-dashed lines) near the EMRL. Different curves refer to different values of $\nu$ and the match is complete for $\nu=0$.
• Figure 5: Critical values of the initial spin and mass ratio leading to a final BH having the same spin as the initial ones i.e., $a_{\rm fin}(a,\nu)=a$. A magnification is shown in the inset, where the dashed/non-dashed region refers to binaries spun-down/up by the merger.