Final spin from the coalescence of two black holes

We provide a compact analytic formula to compute the spin of the black hole produced by the coalescence of two black holes following a quasicircular inspiral. Without additional fits than those already available for binaries with aligned or antialigned spins, but with a minimal set of assumptions, we derive an expression that can model generic initial spin configurations and mass ratios, thus covering all of the 7-dimensional space of parameters. A comparison with simulations already shows very accurate agreements with all of the numerical data available to date, but we also suggest a number of ways in which our predictions can be further improved.

Figure 1

Left panel: rescaled residual for aligned binaries. The circles refer to equal-mass, equal-spin binaries presented in Refs. 11, 12, 13, 20, 21, 22, triangles to equal-mass, unequal-spin binaries presented in Refs. 11, 21, and squares to unequal-mass, equal-spin binaries presented in Refs. 13, 20, 21, 22. Here and in the right panel the “binary order number” is just a dummy index labelling the different configurations. Right panel: the top part reports with asterisks the final spins computed for misaligned binaries. Hexagons refer to data from Ref. 8 (labeled “RIT”), squares to the data Table  (labeled “AEI”), circles to data from Ref. 23 (labeled “FAU”), and triangles to data from Ref. 24 (labeled “PSU-UTA”). Note that these latter data points refer to the aligned component $a_{\mathrm{fin}}^{\parallel }$ since this is the only component available from Ref. 24. The bottom part of this panel shows instead the rescaled residuals for these misaligned binaries.

Figure 2

Using the same data (and convention for the symbols) as in the right panel of Fig. 1, we here report the angle between the final spin vector and the initial orbital angular momentum ${\theta }_{\mathrm{fin}}$. Shown instead with asterisks and circles are the values predicted for the numerical data (as taken from Refs. 8, 23, 24 and from Table ) by our analytic fit (asterisks) and by the point-particle approach suggested in Ref. 5 (circles).