Black-hole binary inspiral and merger in scalar-tensor theory of gravity

Transcription

1 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 24/01/2014 gravity 1 / 27

2 Overview Joined work with E. Berti, V. Cardoso, L. Gualtieri, M. Horbatsch Berti et al (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 24/01/2014 gravity 2 / 27

3 1. Introduction, motivation U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of 24/01/2014 gravity 3 / 27

4 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 Multiple scalar fields ω(φ), V (φ) Here: single scalar field, vacuum U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of 24/01/2014 gravity 4 / 27

5 Theoretical framework Jordan frame: Physical metric gαβ J g Action S = d 4 J [ x 16πG GWs 3 degs. of freedom Matter couples to g J αβ F(φ)R J 8πGZ (φ)g µν J ] µφ ν φ U(φ) Einstein frame: Conformal metric g αβ = F(φ)gαβ J [ 3 F (φ) 2 ] 1/2 8πGZ (φ) ϕ(φ) = dφ + 2 F (φ) 2 F(φ) Action S = 1 [R g µν µ ϕ ν ϕ W (ϕ)] gd 4 x 16πG U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of 24/01/2014 gravity 5 / 27

6 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 (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 24/01/2014 gravity 6 / 27

7 GWs in the Einstein and Jordan frames Einstein frame evolution eqs. G αβ = α ϕ β ϕ 1 2 g αβg µν µ ϕ ν ϕ ϕ = 0 Perturbations g J αβ = ḡj αβ + δgj αβ g αβ = ḡ αβ + δg αβ φ = φ + δφ δg J αβ = 1 F ( φ) δφ = [ δg αβ ḡ J αβ F ( φ)δφ [ 3 F ( φ) F ( φ) 2 ] 1/2 8πGZ ( φ) F ( φ) δϕ ϕ = ϕ + δϕ ] Newman-Penrose scalar: Ψ 4 = ḧ+ iḧ Jordan version Ψ J 4 from Ψ 4, ϕ: see Barausse et al (PRD 87) U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of 24/01/2014 gravity 7 / 27

8 2. Analytic solutions U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory of 24/01/2014 gravity 8 / 27

9 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 ds 2 = (2 r M)2 dt 2 + ( 1 + M ) 4 (2 r+m) 2 2 r [d r 2 + r 2 dω 2 ] ( )... ϕ = 2πσ 1 + M2 r cos θ 2πσz 4 r 2 asymptotically: constant gradient in z dir. Kerr BH; cf. Press 1972 (ApJ 175) ϕ = 2πσ(r M) [ z r cos γ + x 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 24/01/2014 gravity 9 / 27

10 Contour plots of ϕ U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 10 / 27

11 Boundary conditions and multipolar expansion of ϕ Outgoing radiation condition at large r ϕ = ϕ ext + Φ(t r,θ,φ) r r (rϕ) + t (rϕ) = 4πσr cos θ Multipolar expansion of Φ Φ(t r, θ, φ) = M + n i Ḋ i ni n j Q ij +... n r/r M D i Q ij Monopole Dipole Quadrupole U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 11 / 27

12 Scalar radiation from BH binaries Scalar field background: ϕ ext = 2πσr 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 Ωt)] [ ϕ = 2πσr sin θ sin φ 1 + ] f m e im(φ Ωt) m ϕ lm [e i(m+1)ωt + e i(m 1)Ωt] Monopole: Oscillation with Ω Dipole: Oscillation with 2Ω Confirmed by more elaborate calculation U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 12 / 27

13 3. Numerical framework U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 13 / 27

14 Evolution system 3+1 formalism with BSSN Baumgarte & Shapiro 1998 (PRD 59), Shibata & Nakamura 1995 (PRD 52) Matter variables: ϕ, ( t L β )ϕ = 2αK ϕ 3+1 Matter sources 8πG ρ = 2Kϕ iϕ i ϕ 8πG j i = 2K ϕ i ϕ 8πG S ij = i ϕ j ϕ 1 2 γ ij m ϕ m ϕ + 2γ ij Kϕ 2 8πG S = 1 2 m ϕ m ϕ + 6Kϕ 2 Straightforward to add to Lean code Moving punctures Campanelli et al.2005, Baker et al 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 theory24/01/2014 of gravity 14 / 27

15 Initial data Scalar field: Initialize as Error: σ 2, M 2 /4 r 2 Brief transient at early times ϕ = 2πσz BHs: Spectral solver Ansorg et al (PRD 70) Limits on σ Scalar field energy ( ϕ) 2 σ 2 const Total scalar energy M σ 2 R 3 Horizon if M/R σ 2 R 2 1 σ < R 1 = O(10 3 M 1 BH ) Conservative choice: M BH σ = U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 15 / 27

16 4. Numerical results U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 16 / 27

17 Schwarzschild BH: Num. vs. lin. solution Mσ = 10 5 ϕ 10,lin = 4π 3 2πσ(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 theory24/01/2014 of gravity 17 / 27

18 Schwarzschild BH: σ dependence r ex = 50 M: Signs of collapse of scalar field for Mσ = 10 4 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 18 / 27

19 Schwarzschild BH: Scalar multipoles, Mσ = 10 5 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 19 / 27

20 Schwarzschild BH: Scalar multipoles, Mσ = 10 4 U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 20 / 27

21 BH binary: Animation of r t ϕ U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 21 / 27

22 BH binary: Gravitational waves, Mσ = 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 theory24/01/2014 of gravity 22 / 27

23 BH binary: Gravitational waves, Mσ = 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 theory24/01/2014 of gravity 23 / 27

24 BH binary: Scalar dipole radiation, Mσ = r ex = M U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 24 / 27

25 BH binary: Scalar dipole radiation, Mσ = Dipole oscillates at 2Ω orb as expected U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 25 / 27

26 Features of the radiation Ringdown of a/m = BH GWs: Mω 11 lin = i, Mω 11 num = i Scal.: Mω 11 lin = i, Mω 11 num = i Drift in ϕ 11 EFT calculation predicts some drift Contribution from BH kick expected but not large enough Frame dragging: order of magnitude ok, but r dependence not Injection of scalar field energy through BCs Probably: Combination of all effects. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 26 / 27

27 Conclusions and outlook Numerical simulations of BHs in ST Theory work very well! Einstein frame Simulate whole class of theories at once Single BHs: Excellent agreement with linearized calculations Large Mσ induces collapse of scalar field Our Mσ values expected for dark matter models Large Mσ may still be possible: e.g. boson stars... Scalar radiation: Monopole oscillates at Ω orb Dipole oscillates at 2Ω orb Scalar gradients circumvent the no-hair theorem U. Sperhake (DAMTP, University of Cambridge) Black-hole binary inspiral and merger in scalar-tensor theory24/01/2014 of gravity 27 / 27