Averaging the average: Morphology transitions in spin precession of black-hole binaries

Transcription

1 Averaging the average: Morphology transitions in spin precession of black-hole binaries U. Sperhake DAMTP, University of Cambridge M. Kesden, D. Gerosa, R. O Shaughnessy, E. Berti VII Black Holes Workshop Aveiro University, 18 th December 2014 U. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 1 / binar 13

2 Overview Introduction The model The morphologies Precession averaged inspiral Conclusions. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 2 / binar 13

3 The 2-body problem Newtonian Point masses Kepler orbits General Relativity Dissipative GWs Black holes Spins Additional parameters GW observations: LIGO, VIRGO,.... Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 3 / binar 13

4 Spin precessing BH binaries Inspiral due to GW emission Precession of spins S 1, S 2, orbital angular momentum L Orbital motion U. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 4 / binar 13

5 Timescale Adiabatic inspiral at 3.5PN; spin-spin coupling at 2PN order Zero eccentricity Timescales Radiation reaction: t RR r 4 Precession: t pre r 5/2 Orbital motion: t orb r 3/2 t orb t pre t RR Usual PN dynamics: orbit averaged t orb t t pre Here: precession averaged t pre t t RR U. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 5 / binar 13

6 Parametrizing BBHs Zero eccentricity 9 parameters: S 1, S 2, L Align z axis (e.g. with L or J) 7 Rotation about z axis 6 S 1, S 2 conserved 4 L r 1/2 is a measure for the separation, i.e. time 3 U. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 6 / binar 13

7 Option 1: Orientation of spin vectors θ 1,2 = (S 1,2, L) φ = (S 1,, S 2, ) Advantage: Easy to visualize Drawback: All vary on t pre Conserved or averaged variables better! U. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 7 / binar 13

8 Option 2: Variables adapted to timescales ξ M 2[ (1 + q)s 1 + (1 + q 1 )S 2 ] ˆL conserved! J conserved on t pre S ϕ varies on t pre U. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 8 / binar 13

9 The precession cycle S min = max { J L, S 1 S 2 } S max = min { J + L, S 1 + S 2 } Constraint on S, ϕ: ξ(s, ϕ) = { (J 2 L 2 S 2 ) [ S 2 (1 + q) 2 (S 2 1 S2 2 )(1 q2 ) ] (1 q 2 )A cos ϕ } /(4qM 2 S 2 L) A = A(J, L, S 1, S 2, S) 0 cos ϕ = 1 effective potentials ξ ± (S) : ξ ξ ξ + A = 0 if S = S min or S = S max ξ ± (S) form closed loop. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 9 / binar 13

10 The precession cycle for L > S 1 + S 2 φ = 0 or π on ξ ± φ jumps 0 π if cos θ i = ±1 ξ = ξ max : φ = π resonance Schnittman 04 ξ = ξ min : φ = 0 resonance Librating φ 0 Circulating Librating φ π. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 10 / binar 13

11 The morphologies Over t RR, a binary can change morphology Empirically: Only from circulating to librating!. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 11 / binar 13

12 Evolution of J Note: S parametrizes the precession Spin precession eqs. ds dt = 3(1 q2 ) 2q S 1 S 2 S (η 2 M 3 ) 3 L 5 ( 1 ηm2 ξ L ) sin θ 1 sin θ 2 sin φ Except for a new resonant case α = 2πn, this gives dj [ dl = 1 2JL J 2 + L 2 2 ] S+ S 2 ds τ S ds/dt with τ 2 S + S ds ds/dt Evolve J numerically but with t pre t t RR. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 12 / binar 13

13 Conclusions Use hierarchy of time scales: t orb t pre t RR Use convenient variables: ξ, J, S BBHs precession represented in S ξ diagram φ = 0, π on edge of allowed configurations 3 morphologies: circulating, librating about φ = 0, π BBHs undergo phase transitions over t RR Precession averaged inspiral faster algorithm for evolving J U. Sperhake (DAMTP, University of Cambridge) Averaging the average: Morphology transitions in spin precession 18/12/2014 of black-hole 13 / binar 13