Morphologies and phase transitions in precessing black-hole binaries - PowerPoint PPT Presentation

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 Iberian Gravitational-Wave Meeting 2015 Barcelona, 14 th May 2015 U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 1 / 32

Overview Introduction Time scales The precessional time scale Evolutions on t RR Conclusions U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 2 / 32

1. Introduction U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 3 / 32

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

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

2. Time scales U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 6 / 32

Time scales Adiabatic inspiral at 2 . 5 PN; spin-spin coupling at 2 PN 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) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 7 / 32

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

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

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

Option 2: Variables adapted to timescales J = | J | θ L = ∠ ( L , J ) ϕ ′ = rotation of S 1 , S 2 around S Replace θ L with S J const on t pre S , ϕ ′ vary on t pre U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 11 / 32

Constant of motion Note: S , ϕ ′ , J completely determine the binary evolution on t pre : L = L ( S , ϕ ′ ; J , S 1 , S 2 , L ) , S 1 , 2 = S 1 , 2 ( S , ϕ ′ ; J , S 1 , S 2 , L ) At 2 PN spin-precession, 2.5 PN Radiation Reaction: { ( J 2 − L 2 − S 2 )[ S 2 ( 1 + q 2 ) − ( S 2 ξ ( S , ϕ ′ ) 1 − S 2 2 )( 1 − q 2 )] = − ( 1 − q 2 ) A 1 A 2 A 3 A 4 cos ϕ ′ } / ( 4 qM 2 S 2 L ) A i = A i ( J , L , S , S 1 , S 2 ) ≥ 0 conserved: projected effective spin ⇒ trade ϕ ′ for ξ Constraint on ( S , ϕ ′ ) U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 12 / 32

Summary on t pre Choose binary parameters S 1 , S 2 , ξ, q , M Fix J of the binary Fix its separation r by specifying L ⇒ On the precession timescale t pre , 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

Physically allowed parameter ranges I | cos ϕ ′ | ≤ 1 Recall { ( J 2 − L 2 − S 2 )[ S 2 ( 1 + q 2 ) − ( S 2 ξ ( S , ϕ ′ ) 1 − S 2 2 )( 1 − q 2 )] = − ( 1 − q 2 ) A 1 A 2 A 3 A 4 cos ϕ ′ } / ( 4 qM 2 S 2 L ) Then ξ − ≤ ξ ≤ ξ + with { ( J 2 − L 2 − S 2 )[ S 2 ( 1 + q 2 ) − ( S 2 1 − S 2 2 )( 1 − q 2 )] ξ ± ( S ) = ± ( 1 − q 2 ) A 1 A 2 A 3 A 4 } / ( 4 qM 2 S 2 L ) ϕ ′ = 0 or ϕ ′ = π Note: ξ ( S , ϕ ′ ) = ξ ± ( S ) ⇒ ⇒ L , S 1 , S 2 co-planar U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 14 / 32

Phycially allowed parameter ranges II S min ≤ S ≤ S max , where � � S min = max | J − L | , | S 1 − S 2 | , � � S max = min J + L , S 1 + S 2 S = S min or S = S max ⇒ . . . ⇒ one A i = 0 ⇒ ξ = ξ − = ξ + For any other value of S : all A i > 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

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

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

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

The precession cycle U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 19 / 32

Summary We now have 3 morphologies! Where do they meet? Answer: When S ± = S min or S max Note: At ξ max , ξ min 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

Switching coordinates: ( J , S , ξ ) ↔ ( θ 1 , θ 2 , ∆Φ) One straightforwardly shows (cosine theorem): � J 2 − L 2 − S 2 − 22 qM 2 ξ 1 � cos θ 1 = , 2 ( 1 − q ) S 1 L 1 + q − J 2 − L 2 − S 2 + 22 M 2 ξ � � q cos θ 2 = , 2 ( 1 − q ) S 2 L 1 + q S 2 − S 2 1 − S 2 2 cos θ 12 = , 2 S 1 S 2 cos θ 12 − cos θ 1 cos θ 2 cos ∆Φ = , sin θ 1 sin θ 2 U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 21 / 32

3 morphologies for ∆Φ Recall: ϕ ′ = 0 or ϕ ′ = π sin ϕ ′ = 0 ⇒ ⇒ L , S 1 , S 2 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

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

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 θ 12 − cos θ 1 cos θ 2 ⇒ = ± 1 on loop sin θ 1 sin θ 2 Discontinuous change only possible if sin θ 1 = 0 or sin θ 2 = 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

4. Evolutions on t RR U. Sperhake (DAMTP, University of Cambridge) Morphologies and phase transitions in precessing black-hole binaries 14/05/2015 25 / 32

Traditional orbit averaged PN evolutions Valid for times t ≈ t orb ≪ t pre Ignore precession: L ( t ) , S 1 ( t ) , S 2 ( t ) held fixed � 2 π � d J � = 1 d J d ψ Then we have: d ψ/ dt dt T dt 0 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