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# Effective potentials and morphological transitions for binary black-hole spin precession

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### Abstract

Binary black holes (BBHs) on quasicircular orbits are fully characterized by their total mass $$M$$, mass ratio $$q$$, spins $$\mathbf{S}_1$$ and $$\mathbf{S}_2$$, and orbital angular momentum $$\mathbf{L}$$. When the binary separation $$r \gg GM/c^2$$, the precession timescale is much shorter than the radiation-reaction time on which $$L = |\mathbf{L}|$$ decreases due to gravitational-wave (GW) emission. We use conservation of the total angular momentum $$\mathbf{J} = \mathbf{L} + \mathbf{S}_1 + \mathbf{S}_2$$ (with magnitude $$J$$) and the projected effective spin $$\xi \equiv M^{-2} [(1+q) \mathbf{S}_1 + (1+q^{-1})\mathbf{S}_2] \cdot \hat{\mathbf{L}}$$ on the precession time to derive an effective potential for BBH spin precession. This effective potential allows us to solve the orbit-averaged spin-precession equations analytically for arbitrary mass ratios and spins. These solutions are quasiperiodic functions of time: after a period $$\tau(L, J, \xi)$$ the angular momenta return to their initial relative orientations and precess about $$\mathbf{J}$$ by an angle $$\alpha(L, J, \xi)$$. We classify BBH spin precession into three distinct morphologies between which BBHs can transition during their inspiral. Our new solutions constitute fundamental progress in our understanding of BBH spin precession and also have important astrophysical applications. We derive a precession-averaged evolution equation $$dJ/dL$$ that can be numerically integrated on the radiation-reaction time, allowing us to statistically track BBH spins from formation to merger far more efficiently than was possible with previous orbit-averaged precession equations. This will greatly help us predict the signatures of BBH formation in the GWs emitted near merger and the distributions of final spins and gravitational recoils. The solutions may also help efforts to model and interpret GWs from generic BBH mergers.

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### Relativistic Suppression of Black Hole Recoils

(2010)
Numerical-relativity simulations indicate that the black hole produced in a binary merger can recoil with a velocity up to v_max ~ 4,000 km/s with respect to the center of mass of the initial binary. This challenges the paradigm that most galaxies form through hierarchical mergers, yet retain supermassive black holes at their centers despite having escape velocities much less than v_max. Interaction with a circumbinary disk can align the binary black hole spins with their orbital angular momentum, reducing the recoil velocity of the final black hole produced in the subsequent merger. However, the effectiveness of this alignment depends on highly uncertain accretion flows near the binary black holes. In this Letter, we show that if the spin S_1 of the more massive binary black hole is even partially aligned with the orbital angular momentum L, relativistic spin precession on sub-parsec scales can align the binary black hole spins with each other. This alignment significantly reduces the recoil velocity even in the absence of gas. For example, if the angle between S_1 and L at large separations is 10 degrees while the second spin S_2 is isotropically distributed, the spin alignment discussed in this paper reduces the median recoil from 864 km/s to 273 km/s for maximally spinning black holes with a mass ratio of 9/11. This reduction will greatly increase the fraction of galaxies retaining their supermassive black holes.
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### Author and article information

###### Journal
2014-11-03
###### Article
1411.0674