Black-hole binary inspiral and merger in scalar-tensor theory of gravity U. Sperhake DAMTP , University of Cambridge General Relativity Seminar, DAMTP , University of Cambridge 24 th January 2014 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 1 / 27
Overview Joined work with E. Berti, V. Cardoso, L. Gualtieri, M. Horbatsch Berti et al. 2013 (PRD 87 ) Introduction, motivation Analytic results Numerical framework Numerical results Conclusions and outlook U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 2 / 27
1. Introduction, motivation U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 3 / 27
Motivation Goal: BHs in ST theory with non-trivial dynamics Time varying BCs (e.g. Cosmology) ⇒ induce scalar charge of BHs Non-uniform scalar field due to galactic matter ≈ non-asymptotically flat BCs Super massive boson stars ⇒ scalar field gradients Scalar field modifications of GR Brans-Dicke Bergmann-Wagoner ω ( φ ) , V ( φ ) Multiple scalar fields Here: single scalar field, vacuum U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 4 / 27
Theoretical framework g J Jordan frame: Physical metric αβ � − g J � � � F ( φ ) R J − 8 π GZ ( φ ) g µν d 4 x Action S = J ∂ µ φ ∂ ν φ − U ( φ ) 16 π G GWs → 3 degs. of freedom Matter couples to g J αβ Conformal metric g αβ = F ( φ ) g J Einstein frame: αβ � 1 / 2 F ′ ( φ ) 2 � 3 � F ( φ ) 2 + 8 π GZ ( φ ) ϕ ( φ ) = d φ 2 F ( φ ) 1 � [ R − g µν ∂ µ ϕ ∂ ν ϕ − W ( ϕ )] √− gd 4 x Action S = 16 π G U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 5 / 27
Einstein vs. Jordan frame Pro Einstein Minimally coupled scalar field ⇒ numerics straightforward F , Z not explicitly present in evolutions ⇒ Evolve whole class of theories at once Pro Jordan Strongly hyperbolic formulation also available Salgado 2005 (CQG 23 ), Salgado et al. 2008 (PRD 77 ) Matter couples to evolved metric g J αβ Here: Einstein frame more suitable U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 6 / 27
GWs in the Einstein and Jordan frames Einstein frame evolution eqs. G αβ = ∂ α ϕ ∂ β ϕ − 1 2 g αβ g µν ∂ µ ϕ ∂ ν ϕ � ϕ = 0 g J g J αβ + δ g J αβ = ¯ g αβ = ¯ Perturbations g αβ + δ g αβ αβ φ = ¯ φ + δφ ϕ = ¯ ϕ + δϕ � � αβ F ′ (¯ δ g J 1 g J δ g αβ − ¯ αβ = φ ) δφ F (¯ φ ) � − 1 / 2 F ′ (¯ φ ) 2 + 8 π GZ (¯ � φ ) 2 φ ) 3 δφ = δϕ F (¯ F (¯ 2 φ ) Newman-Penrose scalar: Ψ 4 = ¨ h + − i ¨ h × Jordan version Ψ J 4 from Ψ 4 , ϕ : see Barausse et al. 2012 (PRD 87 ) U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 7 / 27
2. Analytic solutions U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 8 / 27
Single BH solutions to the linearized equations Equations: R αβ = 0 , � ϕ = 0 i.e. solve Laplace eq. on BH background Schwarzschild in isotropic coordinates � 4 [ d ˜ r − M ) 2 ds 2 = ( 2 ˜ r + M ) 2 dt 2 + r 2 + ˜ � 1 + M r 2 d Ω 2 ] 2 ˜ ( 2 ˜ r � � 1 + M 2 ˜ ⇒ . . . ⇒ ϕ = 2 πσ r cos θ ≈ 2 πσ z 4 ˜ r 2 asymptotically: constant gradient in z dir. Kerr BH; cf. Press 1972 (ApJ 175 ) � z r cos γ + x � ϕ = 2 πσ ( r − M ) r f a sin γ , f a = f a ( M , a , r ) γ = angle between BH spin and z axis U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 9 / 27
Contour plots of ϕ U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 10 / 27
Boundary conditions and multipolar expansion of ϕ Outgoing radiation condition at large r ϕ = ϕ ext + Φ( t − r ,θ,φ ) r ⇒ ∂ r ( r ϕ ) + ∂ t ( r ϕ ) = 4 πσ r cos θ Multipolar expansion of Φ Φ( t − r , θ, φ ) = M + n i ˙ 2 n i n j ¨ D i + 1 Q ij + . . . n ≡ � � r / r M Monopole D i Dipole Quadrupole Q ij U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 11 / 27
Scalar radiation from BH binaries Scalar field background: ϕ ext = 2 πσ r sin θ sin φ Orbital plane yz θ relative to x axis ⇒ Consider rotating source with frequency Ω ⇒ Modulation in ϕ = ϕ ext [ 1 + f ( φ − Ω t )] � � � f m e im ( φ − Ω t ) ⇒ ϕ = 2 πσ r sin θ sin φ 1 + m � e − i ( m + 1 )Ω t + e − i ( m − 1 )Ω t � ⇒ ϕ lm ∼ Monopole: Oscillation with Ω Dipole: Oscillation with 2 Ω Confirmed by more elaborate calculation U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 12 / 27
3. Numerical framework U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 13 / 27
Evolution system “3+1” formalism with BSSN Baumgarte & Shapiro 1998 (PRD 59 ) , Shibata & Nakamura 1995 (PRD 52 ) Matter variables: ϕ , ( ∂ t − L β ) ϕ = − 2 α K ϕ “3+1” Matter sources 8 π G ρ = 2 K 2 ϕ + 1 2 ∂ i ϕ ∂ i ϕ 8 π G j i = 2 K ϕ ∂ i ϕ 8 π G S ij = ∂ i ϕ ∂ j ϕ − 1 2 γ ij ∂ m ϕ ∂ m ϕ + 2 γ ij K 2 ϕ 8 π G S = − 1 2 ∂ m ϕ ∂ m ϕ + 6 K 2 ϕ Straightforward to add to Lean code Moving punctures Campanelli et al.2005, Baker et al. 2005 Cactus, Carpet, AHFinder Schnetter et al. 2003, Thornburg 1995, 2003 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 14 / 27
Initial data Scalar field: Initialize as ϕ = 2 πσ z Error: σ 2 , M 2 / 4 ˜ r 2 ⇒ Brief transient at early times BHs: Spectral solver Ansorg et al. 2004 (PRD 70) Limits on σ ∼ ( ∇ ϕ ) 2 ∼ σ 2 ∼ const Scalar field energy M ∼ σ 2 R 3 Total scalar energy M / R ∼ σ 2 R 2 ∼ 1 Horizon if σ < R − 1 = O ( 10 − 3 M − 1 ⇒ BH ) Conservative choice: M BH σ = 10 − 7 . . . 10 − 4 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 15 / 27
4. Numerical results U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 16 / 27
Schwarzschild BH: Num. vs. lin. solution M σ = 10 − 5 � 4 π ϕ 10 , lin = 3 2 πσ ( r − M ) r ex = 5 , 10 , 15 , 20 , 30 , 40 , 50 M U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 17 / 27
Schwarzschild BH: σ dependence r ex = 50 M : Signs of collapse of scalar field for M σ = 10 − 4 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 18 / 27
M σ = 10 − 5 Schwarzschild BH: Scalar multipoles, U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 19 / 27
M σ = 10 − 4 Schwarzschild BH: Scalar multipoles, U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 20 / 27
BH binary: Animation of r ∂ t ϕ U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 21 / 27
BH binary: Gravitational waves, M σ = 0 q = 1 / 3 , S = 0 , yz plane: Multipoles of Ψ 4 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 22 / 27
M σ = 2 × 10 − 7 BH binary: Gravitational waves, q = 1 / 3 , S = 0 , yz plane: Multipoles of Ψ 4 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 23 / 27
M σ = 2 × 10 − 7 BH binary: Scalar dipole radiation, r ex = 56 . . . 112 M U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 24 / 27
M σ = 2 × 10 − 7 BH binary: Scalar dipole radiation, Dipole oscillates at 2 Ω orb as expected U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 25 / 27
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