[Paper Review] Black-Hole Binary Remnant Mass and Spin Revisited: A New Phenomenological Formula

C. O. Loustó, Yosef Zlochower|arXiv (Cornell University)|Dec 19, 2013

Pulsars and Gravitational Waves Research1 citations

TL;DR

This paper presents a new phenomenological formula for predicting the final mass and spin of equal-mass black hole binary mergers, leveraging 38 numerical simulations with aligned spins of magnitude 0.8 and varying initial spin-orbit angles. The formula, derived via least-squares fitting to simulation data, reproduces remnant properties within 2.5% relative error by exploiting conserved angular momentum and spin-orbit angle dynamics during inspiral.

ABSTRACT

We perform a set of 38 numerical simulations of equal-mass BH binaries in a configuration where the BH spins in the binary are equal in both magnitude and direction, to study precession effects. We vary the initial direction of the total spin S with respect to the orbital angular momentum L, covering the 2 dimensional space of orientation angles with 38 configurations consisting of 36 configurations distributed in the azimuthal angle phi and polar angle theta, and two configurations on the poles. In all cases, we set the initial dimensionless BH spins to 0.8. We observe that during the late-inspiral stage, the total angular momentum of the system J remains within 5 deg of its original direction, with the largest changes in direction occurring when the spins are nearly counter-aligned with the orbital angular momentum. We also observe that the angle between S and L is nearly conserved during the inspiral phase. These two dynamical properties allow us to propose a new phenomenological formula for the final mass and spin of merged BHs in terms of the individual masses and spins of the progenitor binary at far separations. We determine coefficients of this formula (in the equal-mass limit) using a least-squares fit to the results of this new set of 38 runs, an additional set of five new configurations with spins aligned/counteraligned with the orbital angular momentum, and over 100 recent simulations. We find that our formulas reproduce the remnant mass and spin of these simulations to within a relative error of 2.5%. We discuss the region of validity of this dynamical picture for precessing unequal-mass binaries. Finally, we perform a statistical study to see the consequence of this new formula for distributions of spin-magnitudes and remnant masses with applications to BH-spin distributions and gravitational radiation in cosmological scenarios involving several mergers.

Motivation & Objective

• To develop a predictive model for the final mass and spin of equal-mass black hole binary mergers that accounts for precessing spin configurations.
• To identify conserved dynamical quantities—specifically, the direction of total angular momentum and the angle between total spin and orbital angular momentum—during the late-inspiral phase.
• To calibrate a phenomenological formula using a comprehensive dataset of 38 new simulations, five aligned/counteraligned runs, and over 100 recent simulations.
• To assess the validity of the model for unequal-mass precessing binaries and to explore its implications for cosmological merger populations.

Proposed method

• Performing 38 numerical relativity simulations of equal-mass black hole binaries with dimensionless spin magnitude 0.8 and varying initial spin orientations relative to the orbital angular momentum.
• Distributing configurations across the full 2D angular space (azimuthal and polar angles), including two on the poles, to sample diverse precession dynamics.
• Tracking the evolution of total angular momentum J and the angle between total spin S and orbital angular momentum L to identify conserved quantities.
• Using least-squares fitting to determine coefficients of a phenomenological formula for final mass and spin in terms of initial binary parameters.
• Validating the formula against an additional 105 simulations (38 new runs + 5 aligned/counteraligned + 100 recent results) to assess accuracy.
• Applying the formula to statistical studies of spin-magnitude distributions and remnant masses in cosmological merger scenarios.

Experimental results

Research questions

• RQ1 How do the directions of total angular momentum and the spin-orbit angle evolve during the late-inspiral phase of precessing black hole binaries?
• RQ2 To what extent can the conservation of the total angular momentum direction and the spin-orbit angle angle be leveraged to construct a predictive phenomenological formula?
• RQ3 What is the accuracy of the proposed formula in predicting final black hole mass and spin across diverse initial spin configurations?
• RQ4 How does the model’s validity extend to unequal-mass precessing binaries?
• RQ5 What are the statistical implications of the formula for the distribution of black hole spins and remnant masses in cosmological merger populations?

Key findings

• The direction of the total angular momentum J remains within 5 degrees of its initial value throughout the late-inspiral phase, with the largest deviations occurring when spins are nearly counter-aligned with orbital angular momentum.
• The angle between the total spin vector S and the orbital angular momentum L is nearly conserved during the inspiral, supporting the use of this angle as a key invariant.
• The proposed phenomenological formula reproduces the final mass and spin of black hole mergers with a relative error of less than 2.5% across the tested simulation set.
• The model’s accuracy is validated using a combined dataset of 143 simulations, including 38 new runs and over 100 recent results.
• The formula enables statistical predictions of remnant mass and spin distributions, with implications for understanding spin evolution in cosmological merger scenarios.
• The dynamical picture based on conserved J direction and S-L angle is found to be applicable, with caveats, to unequal-mass precessing binaries.