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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 24th 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

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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

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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

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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

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Theoretical framework

Jordan frame: Physical metric gJ

αβ

Action S =

d4x
−gJ

16πG

F(φ)RJ − 8πGZ(φ)gµν

J ∂µφ ∂νφ − U(φ)

GWs → 3 degs. of freedom

Matter couples to gJ

αβ

Einstein frame: Conformal metric gαβ = F(φ)gJ

αβ

ϕ(φ) =

dφ

3 2 F ′(φ)2 F(φ)2 + 8πGZ(φ) F(φ) 1/2 Action S = 1 16πG

[R − gµν∂µϕ ∂νϕ − W(ϕ)] √−gd4x
U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 5 / 27

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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 gJ

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

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GWs in the Einstein and Jordan frames

Einstein frame evolution eqs. Gαβ = ∂αϕ ∂βϕ − 1

2gαβgµν∂µϕ ∂νϕ

ϕ = 0 Perturbations gJ

αβ = ¯

gJ

αβ + δgJ αβ

gαβ = ¯ gαβ + δgαβ φ = ¯ φ + δφ ϕ = ¯ ϕ + δϕ δgJ

αβ = 1 F(¯ φ)

δgαβ − ¯

gJ

αβF ′(¯

φ)δφ

δφ =
3

2 F ′(¯ φ)2 F(¯ φ)2 + 8πGZ(¯ φ) F(¯ φ)

−1/2 δϕ Newman-Penrose scalar: Ψ4 = ¨ h+ − i¨ h× Jordan version ΨJ

4 from Ψ4, 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

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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

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Single BH solutions to the linearized equations

Equations: Rαβ = 0 , ϕ = 0 i.e. solve Laplace eq. on BH background Schwarzschild in isotropic coordinates ds2 = (2˜

r−M)2 (2˜ r+M)2 dt2 +

1 + M

2˜ r

4 [d˜ r 2 + ˜ r 2dΩ2] ⇒ . . . ⇒ ϕ = 2πσ

1 + M2

4˜ r 2

˜

r cos θ ≈ 2πσz asymptotically: constant gradient in z dir. Kerr BH;

cf. Press 1972 (ApJ 175)

ϕ = 2πσ(r − M) z

r cos γ + x r fa sin γ

,

fa = fa(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

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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

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Boundary conditions and multipolar expansion of ϕ

Outgoing radiation condition at large r ϕ = ϕext + Φ(t−r,θ,φ)

r

⇒ ∂r(rϕ) + ∂t(rϕ) = 4πσr cos θ Multipolar expansion of Φ Φ(t − r, θ, φ) = M + ni ˙ Di + 1

2ninj ¨

Qij + . . .

n ≡

r/r M Monopole Di Dipole Qij Quadrupole

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 11 / 27

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Scalar radiation from BH binaries

Scalar field background: ϕext = 2πσr sin θ sin φ Orbital plane yz ⇒ θ relative to x axis Consider rotating source with frequency Ω ⇒ Modulation in ϕ = ϕext[1 + f(φ − Ωt)] ⇒ ϕ = 2πσr sin θ sin φ

1 +
m

fmeim(φ−Ωt)

⇒ ϕlm ∼
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 theory of gravity 24/01/2014 12 / 27

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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

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Evolution system

“3+1” formalism with BSSN

Baumgarte & Shapiro 1998 (PRD 59) , Shibata & Nakamura 1995 (PRD 52)

Matter variables: ϕ , (∂t − Lβ)ϕ = −2αKϕ “3+1” Matter sources 8πG ρ = 2K 2

ϕ + 1 2∂iϕ ∂iϕ

8πG ji = 2Kϕ∂iϕ 8πG Sij = ∂iϕ ∂jϕ − 1

2γij∂mϕ ∂mϕ + 2γijK 2 ϕ

8πG S = − 1

2∂mϕ ∂mϕ + 6K 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

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Initial data

Scalar field: Initialize as ϕ = 2πσz Error: σ2, M2/4˜ r 2 ⇒ Brief transient at early times BHs: Spectral solver Ansorg et al. 2004 (PRD 70) Limits on σ

Scalar field energy ∼ (∇ϕ)2 ∼ σ2 ∼ const Total scalar energy M ∼ σ2R3 Horizon if M/R ∼ σ2R2 ∼ 1 ⇒ σ < R−1 = O(10−3 M−1

BH )

Conservative choice: MBHσ = 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

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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

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Schwarzschild BH: Num. vs. lin. solution Mσ = 10−5

ϕ10,lin =

4π

3 2πσ(r − M)

rex = 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

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Schwarzschild BH: σ dependence

rex = 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 theory of gravity 24/01/2014 18 / 27

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Schwarzschild BH: Scalar multipoles, Mσ = 10−5

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 19 / 27

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Schwarzschild BH: Scalar multipoles, Mσ = 10−4

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 20 / 27

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BH binary: Animation of r∂tϕ

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 21 / 27

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BH binary: Gravitational waves, Mσ = 0

q = 1/3, S = 0, yz plane: Multipoles of Ψ4

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 22 / 27

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BH binary: Gravitational waves, Mσ = 2 × 10−7

q = 1/3, S = 0, yz plane: Multipoles of Ψ4

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 23 / 27

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BH binary: Scalar dipole radiation, Mσ = 2 × 10−7

rex = 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

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BH binary: Scalar dipole radiation, Mσ = 2 × 10−7

Dipole oscillates at 2Ωorb 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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Features of the radiation

Ringdown of a/M = 0.543 BH

GWs: Mω11 lin = 0.476 − 0.0849i , Mω11 num = 0.48 − 0.081i Scal.: Mω11 lin = 0.351 − 0.0936i , Mω11 num = 0.36 − 0.070i

Drift in ϕ11

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

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 26 / 27

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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 Ωorb Dipole oscillates at 2Ωorb

Scalar gradients circumvent the no-hair theorem

U. Sperhake (DAMTP, University of Cambridge)

Black-hole binary inspiral and merger in scalar-tensor theory of gravity 24/01/2014 27 / 27