Up-down instability of binary black holes in numerical relativity

Vijay Varma, Matthew Mould, Davide Gerosa, Mark A. Scheel, Lawrence E. Kidder, and Harald P. Pfeiffer

Phys. Rev. D 103, 064003 – Published 1 March 2021

Abstract

Binary black holes with spins that are aligned with the orbital angular momentum do not precess. However, post-Newtonian calculations predict that “up-down” binaries, in which the spin of the heavier (lighter) black hole is aligned (antialigned) with the orbital angular momentum, are unstable when the spins are slightly perturbed from perfect alignment. This instability provides a possible mechanism for the formation of precessing binaries in environments where sources are preferentially formed with (anti)aligned spins. In this paper, we present the first full numerical relativity simulations capturing this instability. These simulations span $\sim 100$ orbits and $\sim 3 - 5$ precession cycles before merger, making them some of the longest numerical relativity simulations to date. Initialized with a small perturbation of 1°–10°, the instability causes a dramatic growth of the spin misalignments, which can reach $\sim 90 °$ near merger. We show that this leaves a strong imprint on the subdominant modes of the gravitational wave signal, which can potentially be used to distinguish up-down binaries from other sources. Finally, we show that post-Newtonian and effective-one-body approximants are able to reproduce the unstable dynamics of up-down binaries extracted from numerical relativity.

Received 13 December 2020
Accepted 5 February 2021

DOI:https://doi.org/10.1103/PhysRevD.103.064003

Physics Subject Headings (PhySH)

Gravitational waves

Numerical relativity

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Vijay Varma ^1,2,3,*, Matthew Mould ⁴, Davide Gerosa ⁴, Mark A. Scheel³, Lawrence E. Kidder ², and Harald P. Pfeiffer⁵

¹Department of Physics, Cornell University, Ithaca, New York 14853, USA
²Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA
³TAPIR 350-17, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA
⁴School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom
⁵Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany

^*vvarma@cornell.edu

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Vol. 103, Iss. 6 — 15 March 2021

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Images

Figure 1
The up-down instability is demonstrated in NR. We consider binaries with mass ratios $q = 0.9$ and spin magnitudes $χ_{1} = χ_{2} = 0.8$ , while the component spin vectors are initially perturbed from a perfectly aligned-spin configuration by an angle $θ_{pert} = 10 °$ . The purple (orange) arrows represent the spin $χ_{1}$ ( $χ_{2}$ ) of the heavier (lighter) BH near merger. The colored curves trace the evolution of $χ_{1, 2} (t)$ as the binary precesses. Colors darken linearly in time as indicated on the color bars. The spins of the up-up (top-left), down-down (top-right), and down-up (bottom-left) configurations precess stably about $\hat{z}$ , while those of the up-down (bottom-right) configuration are unstable and become largely misaligned. Smaller modulations occur on the shorter orbital timescale. An animated version of this figure is available at http://www.davidegerosa.com/spinprecession.
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Figure 2
Precessional dynamics and its effect on the gravitational waveform. Each subplot shows the four configurations we consider: up-up (blue), down-down (orange), down-up (green) and up-down (red). Each column corresponds to a different initial perturbation ( $θ_{pert}$ , indicated at the top) of the spin directions with respect to a perfectly aligned-spin system. The first (second) row shows the evolution of the spin perturbation angles [ $Δ θ_{1, 2}$ ; cf. Eq. (5)] of the heavier (lighter) BH. The third row shows the evolution of the orbital plane tilt angle [ $θ_{L}$ ; cf. Eq. (6)]. The fourth and fifth rows show the real part of the (2,2) and (2,1) GW modes, respectively. All quantities are shown as a function of the Newtonian separation $r$ [cf. Eq. (3)] and are terminated at the separation where the common horizon is found. The instability development is reflected in a rapid growth of the angles $Δ θ_{1, 2}$ and $θ_{L}$ for binaries near the up-down configuration. This in turn impacts the observed GW signal, in particular the subdominant modes like (2,1), as the orbital precession causes power leakage from the dominant (2,2) mode.
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Figure 3
Comparison of the spin dynamics predicted by PN and EOB approximants against NR. We show the spin perturbation angles $Δ θ_{1, 2}$ as a function of the Newtonian separation $r$ for each of the four configurations: up-up (top-row), down-down (second-row), down-up (third-row) and up-down (bottom-row), with an initial perturbation $θ_{pert} = 5 °$ . The left (right) column shows $Δ θ_{1}$ ( $Δ θ_{2}$ ). For the stable configurations, the inset shows an enlarged version of the same panel. The Schwarzschild ISCO radius ( $r = 6 M$ ) is indicated by a gray marker. Both approximants are able to track the precession modulations in $Δ θ_{1, 2}$ found in NR, including the unstable growth of the up-down instability, but fail to match the smaller spin oscillations occurring on the orbital timescale.
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Figure 4
Comparison of the spin dynamics and gravitational waveform between simulations with two different NR resolutions for the up-down system with an initial perturbation $θ_{pert} = 10 °$ . We show the spin perturbation angle $Δ θ_{1}$ and the real part of the (2,1) mode, as a function of the Newtonian separation $r$ .
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