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Morphologies and phase transitions in precessing black-hole binaries U. Sperhake DAMTP , University of Cambridge M. Kesden, D. Gerosa, R. OShaughnessy, E. Berti Phys. Rev. Lett. 114 (2015) 081103 + work in preparation Iberian

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Morphologies and phase transitions in precessing black-hole binaries

  • U. Sperhake

DAMTP , University of Cambridge

  • M. Kesden, D. Gerosa, R. O’Shaughnessy, E. Berti
  • Phys. Rev. Lett. 114 (2015) 081103

+ work in preparation

Barcelona, 14th May 2015 Iberian Gravitational-Wave Meeting 2015

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 1 / 32

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Overview

Introduction Time scales The precessional time scale Evolutions on tRR Conclusions

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 2 / 32

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  • 1. Introduction
  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 3 / 32

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The 2-body problem

Newtonian

Point masses Kepler orbits

General Relativity

Dissipative ⇒ GWs Black holes Spins ⇒ Additional parameters

GW observations: LIGO, VIRGO,...

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 4 / 32

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Spin precessing BH binaries

Inspiral due to GW emission Precession of spins S1, S2, orbital angular momentum L Orbital motion

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 5 / 32

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  • 2. Time scales
  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 6 / 32

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Time scales

Adiabatic inspiral at 2.5 PN; spin-spin coupling at 2 PN order Zero eccentricity Timescales

Radiation reaction: tRR ∼ r 4 Precession: tpre ∼ r 5/2 Orbital motion: torb ∼ r 3/2

torb ≪ tpre ≪ tRR Usual PN dynamics: orbit averaged torb ≪ t ≪ tpre Here: precession averaged tpre ≪ t ≪ tRR

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 7 / 32

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  • 3. The precessional time scale
  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 8 / 32

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Parametrizing BBHs

Zero eccentricity ⇒ 9 parameters: S1, S2, L Align z axis (e.g. with L or J) → 7 Rotation about z axis → 6 S1, S2 conserved → 4 L ∼ r 1/2 is a measure for the separation, i.e. time → 3

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 9 / 32

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Option 1: Orientation of spin vectors

θ1,2 = ∠(S1,2, L) ∆φ = ∠(S1,⊥, S2,⊥) Advantage: Easy to visualize Drawback: All vary on tpre Conserved or averaged variables better!

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 10 / 32

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Option 2: Variables adapted to timescales

J = |J| θL = ∠(L, J) ϕ′ = rotation of S1, S2 around S Replace θL with S J const on tpre S, ϕ′ vary on tpre

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 11 / 32

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Constant of motion

Note: S, ϕ′, J completely determine the binary evolution on tpre: L = L(S, ϕ′; J, S1, S2, L) , S1,2 = S1,2(S, ϕ′; J, S1, S2, L) At 2 PN spin-precession, 2.5 PN Radiation Reaction: ξ(S, ϕ′) = {(J2 − L2 − S2)[S2(1 + q2) − (S2

1 − S2 2)(1 − q2)]

−(1 − q2) A1 A2 A3 A4 cos ϕ′} / (4qM2S2L) Ai = Ai(J, L, S, S1, S2) ≥ 0 conserved: projected effective spin Constraint on (S, ϕ′) ⇒ trade ϕ′ for ξ

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 12 / 32

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Summary on tpre

Choose binary parameters S1, S2, ξ, q, M Fix J of the binary Fix its separation r by specifying L ⇒ On the precession timescale tpre, its evolution is a 1-parameter evolution in S

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 13 / 32

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Physically allowed parameter ranges I

| cos ϕ′| ≤ 1 Recall ξ(S, ϕ′) = {(J2 − L2 − S2)[S2(1 + q2) − (S2

1 − S2 2)(1 − q2)]

−(1 − q2) A1 A2 A3 A4 cos ϕ′} / (4qM2S2L) Then ξ− ≤ ξ ≤ ξ+ with ξ±(S) = {(J2 − L2 − S2)[S2(1 + q2) − (S2

1 − S2 2)(1 − q2)]

±(1 − q2) A1 A2 A3 A4 } / (4qM2S2L) Note: ξ(S, ϕ′) = ξ±(S) ⇒ ϕ′ = 0 or ϕ′ = π ⇒ L, S1, S2 co-planar

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 14 / 32

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Phycially allowed parameter ranges II

Smin ≤ S ≤ Smax , where Smin = max

  • |J − L|, |S1 − S2|
  • ,

Smax = min

  • J + L, S1 + S2
  • S = Smin or S = Smax

⇒ . . . ⇒

  • ne Ai = 0

⇒ ξ = ξ− = ξ+ For any other value of S: all Ai > 0 ⇒ ξ−(S) ≤ ξ(S, ϕ′) ≤ ξ+(S) as cos ϕ′ varies from +1 to −1 ⇒ Closed loop in (S, ξ) plane: allowed configs. inside

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 15 / 32

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Effective potential diagram

Note: ϕ′ = 0 on ξ− ; ϕ′ = π on ξ+

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 16 / 32

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The precession cycle

Now consider a binary with fixed ξ: As S varies, binary moves on horizontal line in (S, ξ) plane Turning points: ξ(S) = ξ± ⇒ . . . ⇒ there are 2 solutions for S : S− , S+ Binary precession quantified by evolution of S ∈ [S−, S+]

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 17 / 32

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The precession cycle

What about ϕ′ during a precession cycle? 1) Both turning points on ξ+ ⇒ cos ϕ′ = −1 ⇒ ϕ′ = π ⇒ ϕ′ oscillates around π but never reaches 0 2) Both turning points on ξ− ⇒ cos ϕ′ = +1 ⇒ ϕ′ = 0 ⇒ ϕ′ oscillates around 0 but never reaches π 3) One turning point on ξ−, the other on ξ+ ⇒ ϕ′ circulates all the way from 0 to π and back

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 18 / 32

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The precession cycle

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 19 / 32

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Summary

We now have 3 morphologies! Where do they meet? Answer: When S± = Smin or Smax Note: At ξmax, ξmin we have S = const

Schnittmann’s (2004) spin-orbit resonances

Note: What about q → 1 ? Then ξ+(S) = ξ−(S) ⇒ S fixed by prescribing ξ “Egg” squashed to “line”

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 20 / 32

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Switching coordinates: (J, S, ξ) ↔ (θ1, θ2, ∆Φ)

One straightforwardly shows (cosine theorem): cos θ1 = 1 2(1 − q)S1 J2 − L2 − S2 L − 22qM2ξ 1 + q

  • ,

cos θ2 = q 2(1 − q)S2

  • −J2 − L2 − S2

L + 22M2ξ 1 + q

  • ,

cos θ12 = S2 − S2

1 − S2 2

2S1S2 , cos ∆Φ = cos θ12 − cos θ1 cos θ2 sin θ1 sin θ2 ,

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 21 / 32

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3 morphologies for ∆Φ

Recall: ϕ′ = 0 or ϕ′ = π ⇒ sin ϕ′ = 0 ⇒ L, S1, S2 co-planar ⇒ sin ∆Φ′ = 0 ⇒ ∆Φ = 0 or ∆Φ = π 3 morphologies in ∆Φ: 1) ∆Φ librates around π 2) ∆Φ librates around 0 3) ∆Φ circulates in [0, π] Warning: Although sin ϕ′ = 0 ⇔ sin ∆Φ = 0, ϕ′ = 0 or π and ∆Φ = 0 or π is NOT determined!!

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 22 / 32

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Morphology diagrams

Depending on BH parameters, not all morphologies may be available

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 23 / 32

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When do morphologies change?

Clearly, ∆Φ must change on the ξ± loop! We know: ϕ′ = 0 or π on loop ⇒ ∆Φ = 0 or π on loop ⇒ cos ∆Φ = cos θ12 − cos θ1 cos θ2 sin θ1 sin θ2 = ±1 on loop Discontinuous change only possible if sin θ1 = 0 or sin θ2 = 0 ⇒ one BH spin aligned with L

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 24 / 32

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  • 4. Evolutions on tRR
  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 25 / 32

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Traditional orbit averaged PN evolutions

Valid for times t ≈ torb ≪ tpre Ignore precession: L(t), S1(t), S2(t) held fixed Then we have: dJ dt

  • = 1

T 2π dJ dt dψ dψ/dt Here: T = orbital period ψ = e.g. Kepler’s true anomaly Then handle precession as a quasi-adiabatic process...

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 26 / 32

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Averaging the average

Let X be some quantity Precession average Xpre := 2 τ S+

S−

Xorb dS |dS/dt| . . . ⇒ 1.5 PN angular momentum flux: dJ dL = 1 2LJ

  • J2 + L2 − Spre
  • with

dS dt = −3(1 − q2) 2q S1S2 S (η2M3)3 L5

  • 1 − ηM2ξ

L

  • sin θ1 sin θ2 sin ∆Φ
  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 27 / 32

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Comments

ξ conserved on tRR ⇒ only L, J vary on t > tpre One exception: Special resonance configurations. Set of measure zero in parameter space. But: Possibly interesting physics... (under study) “Only” 1.5PN, but: Comparison with full PN orbit averaged evolutions ⇒ excellent agreement down to at least 50 M separation Precession averaged: Huge computational speed-up!

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 28 / 32

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Example applications I: Phase transition

Evolution of the ∆Φ range during inspiral: circulating → librating

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 29 / 32

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Example applications II: BH binary memory effect

Determine morphology with LIGO/Virgo measurements Color shows θ1, θ2 of BHs at large distances

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 30 / 32

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  • 5. Conclusions
  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 31 / 32

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Conclusions

Use hierarchy of time scales: torb ≪ tpre ≪ tRR 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 tRR Precession averaged inspiral ⇒ faster algorithm for evolving J

  • U. Sperhake (DAMTP, University of Cambridge)

Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 32 / 32