Maximum black-hole spin from quasicircular binary mergers

Michael Kesden, Guglielmo Lockhart, and E. Sterl Phinney

Phys. Rev. D 82, 124045 – Published 20 December 2010

Abstract

Black holes of mass $M$ must have a spin angular momentum $S$ below the Kerr limit ( $χ \equiv S / M^{2} \leq 1$ ), but whether astrophysical black holes can attain this limiting spin depends on their accretion history. Gas accretion from a thin disk limits the black-hole spin to $χ_{gas} ≲ 0.9980 \pm 0.0002$ , as electromagnetic radiation from this disk with retrograde angular momentum is preferentially absorbed by the black hole. Extrapolation of numerical-relativity simulations of equal-mass binary black-hole mergers to maximum initial spins suggests these mergers yield a maximum spin $χ_{eq} ≲ 0.95$ . Here we show that for smaller mass ratios $q \equiv m / M ≪ 1$ , the superradiant extraction of angular momentum from the larger black hole imposes a fundamental limit $χ_{\lim} ≲ 0.9979 \pm 0.0001$ on the final black-hole spin even in the test-particle limit ( $q \to 0$ ) of binary black-hole mergers. The nearly equal values of $χ_{gas}$ and $χ_{\lim}$ imply that measurement of supermassive black-hole spins cannot distinguish a black hole built by gas accretion from one assembled by the gravitational inspiral of a disk of compact stellar remnants. We also show how superradiant scattering alters the mass and spin predicted by models derived from extrapolating test-particle mergers to finite mass ratios.

Received 29 October 2010

DOI:https://doi.org/10.1103/PhysRevD.82.124045

Authors & Affiliations

Michael Kesden^*

Center for Cosmology and Particle Physics, New York University, Department of Physics, 4 Washington Pl., New York, New York 10003, USA

Guglielmo Lockhart and E. Sterl Phinney^†

California Institute of Technology, MC 350-17, 1216 E. California Blvd., Pasadena, California 91125, USA

^*mhk10@nyu.edu
^†esp@tapir.caltech.edu

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Vol. 82, Iss. 12 — 15 December 2010

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Images

Figure 1
Top panel: The dimensionless energy $E_{SR}$ (thin line) and angular momentum $J_{SR}$ (thick line) extracted from a black hole with the limiting spin $χ_{\lim} = 0.9979$ as the test particle inspirals from infinity to radius $r$ . Bottom panel: The change in final spin per unit test-particle mass $\partial χ / \partial q$ . The dotted horizontal lines show the spin increase $\partial χ_{ISCO} / \partial q$ when the test particle falls from the ISCO into a black hole of spin $χ = 0.996$ (green), 0.997 (blue), $χ_{\lim}$ (black), and 0.999 (red). The curves $\partial χ_{SR} / \partial q$ show the spin decrease as superradiant scattering extracts energy from black holes with spins of the corresponding color. Only at $χ = χ_{\lim}$ do the lines and curves intersect at $r_{ISCO}$ , indicating that the spin remains unchanged by the merger.Reuse & Permissions
Figure 2
The same quantities $E_{SR}$ , $J_{SR}$ , and $\partial χ / \partial q$ presented in Fig. 1 as functions of the orbital radius $r$ . We zoom into the region near the ISCO to better display how these quantities evolve near merger. The colors and styles of the curves are the same as those in Fig. 1.Reuse & Permissions
Figure 3
The spin $χ$ of a black hole initially of mass $M_{0}$ after binary mergers with a mass $Δ M$ of test particles on quasicircular equatorial orbits. The black (solid) curves show the predictions of our model for different initial spins, while the blue (dashed) curves show the predicted spins in the absence of scattering. Spins in our model asymptote to $χ_{\lim} ≃ 0.9979$ as shown by the horizontal dotted line, while without scattering black holes can spin all the way up to the Kerr limit.Reuse & Permissions
Figure 4
The radiative efficiency $ε = 1 - E_{ISCO} (χ)$ of the accreting black holes shown in Fig. 3. In the Bardeen model (dashed blue curves), the black holes spin up to the Kerr limit $χ = 1$ after accreting a finite mass $Δ M$ . The radiative efficiency of a maximally spinning black hole is $ε = 1 - 1 / \sqrt{3} = 0.423$ . In our model (solid black curves), which includes the superradiant scattering of gravitational waves, the black hole spins asymptotically approach $χ_{\lim}$ . A black hole with this spin has the substantially lower efficiency $ε_{\lim} = 0.320$ .Reuse & Permissions
Figure 5
The change in final spin per unit test-particle mass $\partial χ_{f} / \partial q$ as a function of the initial spin $χ_{i}$ . The solid black curve shows the predictions of this paper, with spin down ( $\partial χ_{f} / \partial q \leq 0$ , horizontal dotted line) possible for spins $χ_{i} \geq χ_{\lim}$ (vertical dotted line). The short-dashed blue curve shows how this result changes when superradiant scattering is neglected as in the models of Bardeen [4], HB [23], and K [25]. The long-dashed red curve gives the BKL [24] test-particle prediction, while the short dash-dotted green and long dash-dotted magenta curves are the predictions of the AEI [31] and FAU [32] fitting formulas.Reuse & Permissions

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