Predicting the Direction of the Final Spin From the Coalescence of Two Black Holes

Abstract

Knowledge of the spin of the black hole resulting from the merger of a generic black-hole binary is of great importance for studying the cosmological evolution of supermassive black holes. Several attempts have been made to model the spin via simple expressions exploiting the results of numerical-relativity simulations. While these expressions are in reasonable agreement with the simulations, they neglect the precession of the binary's orbital plane, and cannot therefore be applied directly-i.e., without evolving the system to small separations using post-Newtonian theory-to binaries with separations larger than a few hundred gravitational radii. While not a problem in principle, this may be impractical if the formulas are employed in cosmological merger trees or N-body simulations, which provide the spins and angular momentum of the two black holes when their separation is of hundreds or thousands of gravitational radii. The formula that we propose is instead built on improved assumptions and gives, for any separation, a very accurate prediction both for the norm of the final spin and for its direction. By comparing with the numerical data, we also show that the final-spin direction is very accurately aligned with the binary's total angular momentum at large separation. Hence, observations of the final-spin direction (e.g., via a jet) can provide information on the binary's orbital plane at large separations and could be relevant, for instance, for studying X-shaped radio sources.

Where angular momentum goes in a precessing black-hole binary

“…Here we propose a simple model of the final remnant BH's intrinsic spin and mass based on the observations that the total spin magnitude is conserved to a very high degree during the simulation and that the direction between the instantaneous orbital angular momentum and the total spin is approximately conserved (see also [30]). Given the assumption that, even for precessing binaries, the component of the total spin along the orbital angular momentum leads to the well-known hangup effect [4], we find that the remnant final spin is given by the hangup spin with the addition of the in-plane component of the initial total spin.…”

“…Thus suggesting [30], for instance, that AGN jets, which are associated with the direction of the spin of the central black hole engine, are good indicators of the direction of the total angular momentum of the progenitor binary system. While the accurate modeling of the remnant mass leads to specific predictions for the spectrum of the cosmological gravitational-waves background radiation that could discern between light and heavy seeds scenarios [36].…”

Mergers of black-hole binaries with aligned spins: Waveform characteristics

“…For the final spin, one model with just enough complexity for our data sets here was given by [65,66] 2 :…”

“…M f,AH and αAH come from the AHFinderDirect code [53,54] and the GSFC spin calculator [67]; quoted uncertainties are a combination of the post-merger variability of the irreducible mass and spin and the difference between the measured mass and spin from the highest and second-highest resolutions. M f,AEI (MAH) and αAEI use the numerically tuned formulas (13) and (11) due to [14,66,68]; quoted uncertainties here are due to uncertainty in the fitting coefficients (12), (14). and (3, 2); other modes never attain 0.1% of the (2,2) power (equivalently, 3% of the (2,2) amplitude).…”

Is Black-Hole Ringdown a Memory of Its Progenitor?

“…There were three sets of simulations: (1) was the same as that for the corresponding non-spinning binary for (q, χ fin ) = (2, 0.62), (3, 0.54), and (4, 0.47), using the final-spin fits in [3,9] and (3) There were a total of 40 configurations, not including additional tests to verify that the results were robust against changes in the number of inspiral orbits. All simulations were performed with the BAM code [10].…”