Gravitational radiation from a binary black hole coalescence in Einstein-scalar-Gauss-Bonnet gravity Félix-Louis Julié Johns Hopkins University, Baltimore Groupe Relativité et Objets Compacts Seminar Laboratoire Univers et Théories, Observatoire de Meudon March 5, 2020
Introduction The era of gravitational wave astronomy • GW150914: first observation of a BBH coalescence by LIGO-Virgo • GW170817: first BNS with EM counterparts (multimessenger astronomy) • Since April 2019: third observation run (O3) ongoing… Opportunity of new tests of general relativity and modified gravities , in the strong-field regime of a compact binary merger.
Introduction “Knowing the chirp to hear it”... In general relativity: PN theory, self-force calculations, EOB framework, numerical relativity...
Introduction How to adapt these tools to derive analytical waveforms in modified gravities ? Consider the example of Einstein-scalar-Gauss-Bonnet (ESGB) theories. • Félix-Louis Julié, Emanuele Berti , “Post-Newtonian dynamics and black hole thermodynamics in Einstein-scalar-Gauss-Bonnet gravity,” Phys.Rev. D100 (2019) no.10, 104061 • Marcela Cardenas, Félix-Louis Julié, Nathalie Deruelle , ”Thermodynamics sheds light on black hole dynamics,” Phys. Rev. D97, 12, 124021, 2018. • Félix-Louis Julié , ”Gravitational radiation from compact binary systems in Einstein-Maxwell-dilaton theories,” JCAP 1810, 10, 033, 2018. • Félix-Louis Julié , ”Reducing the two-body problem in scalar-tensor theories to the motion of a test particle: a scalar-tensor e ff ective-one-body approach,” Phys. Rev. D97, 2, 024047, 2018. • Félix-Louis Julié, Nathalie Deruelle , ”Two-body problem in scalar-tensor theories as a deformation of general relativity: an e ff ective-one-body approach,” Phys. Rev. D95, 12, 124054, 2017.
Introduction Einstein-Scalar-Gauss-Bonnet gravity ESGB vacuum action ( G = c = 1) − g ( R − 2 g μν ∂ μ φ ∂ ν φ + α f ( φ ) ℛ 2 GB ) 16 π ∫ d 4 x 1 I ESGB = • Massless scalar field φ • Gauss-Bonnet scalar ℛ 2 GB = R μνρσ R μνρσ − 4 R μν R μν + R 2 • Fundamental coupling with dimensions L 2 and defines the ESGB theory α f ( φ ) ∫ d D x − g ℛ 2 is a boundary term in [Myers 87] • D ⩽ 4 GB Second order field equations g μν R μν =2 ∂ μ φ ∂ ν φ − 4 α ( ) ∇ α ∇ β f ( φ ) P μανβ − 2 P αβ □ φ = − 1 4 α f ′ � ( φ ) ℛ 2 GB with P μνρσ = R μνρσ − 2 g μ [ ρ R σ ] ν + 2 g ν [ ρ R σ ] μ + g μ [ ρ g σ ] ν R
Introduction Hairy black holes in ESGB gravity Analytical solutions in the small Gauss-Bonnet coupling limit α • Einstein-dilaton-Gauss-Bonnet, f ( φ ) = e φ Mignemi-Stewart 93 at 𝒫 ( α 2 ) , Maeda at al. 97 at , Yunes-Stein 11 at 𝒫 ( α ) 𝒫 ( α ) 𝒫 ( α 2 , S 2 ) 𝒫 ( α 2 , S 2 ) 𝒫 ( α 7 , S 5 ) Ayzenberg-Yunes 14 at , Pani et al. 11 at , Maselli et al. 15 at • Shift-symmetric theories, f ( φ ) = φ Sotiriou-Zhou 14 at 𝒫 ( α 2 ) • Generic ESGB theories Julié-Berti 19 at 𝒫 ( α 4 ) Numerical solutions • Einstein-dilaton-Gauss-Bonnet, f ( φ ) = e φ Kanti et al. 95 , Pani-Cardoso 09 , Kleihaus 15 (includes spins) • Shift-symmetric theories, f ( φ ) = φ Delgado et al. 20 (includes spin) • Generic ESGB theories Antoniou et al. 18 f ( φ ) = − e − λφ 2 • Quadratic couplings, f ( φ ) = φ 2 (1 + λφ 2 ) and Doneva-Yazadjiev 17 , Silva et al. 17, Minamitsuji-Ikeda 18, Macedo et al. 19, etc… How to address (analytically) the motion and gravitational radiation of two coalescing ESGB black holes? See also Yagi et al. 12 ; and Witek et al. 19 , Okounkova 20 for a numerical relativity analysis.
1. ESGB black holes and their thermodynamics 2. The post-newtonian (PN) dynamics of an ESGB black hole binary 3. Beyond the PN approximation: “EOBization” of an ESGB black hole binary 4. Gravitational radiation from an ESGB black hole binary
1. ESGB black holes and their thermodynamics Static, spherically symmetric ESGB black holes Just coordinate system ds 2 = − A ( r ) dt 2 + dr 2 A ( r ) + B ( r ) r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ϵ = α f ′ � ( φ ∞ ) Solve iteratively the field equations around a Schwarzschild spacetime with ≪ 1 m 2 r + ∑ B ( r ) = 1 + ∑ φ ( r ) = φ ∞ + ∑ A ( r ) = 1 − 2 m ϵ i A i ( r ) , ϵ i B i ( r ) , ϵ i φ i ( r ) i i i g μν R μν =2 ∂ μ φ ∂ ν φ − 4 α ( ) ∇ α ∇ β f ( φ ) P μανβ − 2 P αβ □ φ = − 1 4 α f ′ � ( φ ) ℛ 2 GB with ℛ 2 GB = R μνρσ R μνρσ − 4 R μν R μν + R 2 ESGB black hole, at leading order for simplicity: r + 𝒫 ( B = 1 + 𝒫 ( r 3 ) + 𝒫 ( ( 2 2 2 m 2 ) m 2 ) 2 r + m 2 2 r 2 + 2 m 3 m 2 ) α f ′ � α f ′ � φ = φ ∞ + α f ′ � ( φ ∞ ) α f ′ � A = 1 − 2 m m ∞ ∞ ∞ , , m 2 Two integration constants: and , at all orders in the Gauss-Bonnet coupling. m φ ∞
1. ESGB black holes and their thermodynamics Introduction ESGB black hole thermodynamics • Temperature: T = κ κ 2 = − 1 where 2 ( ∇ μ ξ ν ∇ μ ξ ν ) r H is the surface gravity 4 π • Wald entropy: ∂ℒ S w = − 8 π ∫ r H with d θ d ϕ σ ϵ μν ϵ ρσ ϵ μν = n [ μ l ν ] ∂ R μνρσ S w = H in ESGB gravity. + 4 απ f ( φ H ) 4 • Mass as a global charge: r + 𝒫 ( M = m + ∫ D d φ ∞ r 2 ) φ = φ ∞ + D 1 is the scalar “charge” defined as D [Henneaux et al. 02, Cardenas et al. 16, Anabalon-Deruelle-FLJ 16,…] The quantities above are calculated in terms of and . At leading order for simplicity: m φ ∞ T = 8 π m [ 1 + 𝒫 ( + 𝒫 ( [ 1 + 𝒫 ( ] [ 1 + α f ( φ ∞ ) ] m 2 ) ] 2 2 m 2 ) m 2 ) α f ′ � α f ′ � D = α f ′ � ( φ ∞ ) α f ′ � ∞ ∞ ∞ , S w = 4 π m 2 , m 2 2 m The variations of and with respect to and are such that: S w M m φ ∞ T δ S w = δ M
1. ESGB black holes and their thermodynamics 2. The post-newtonian (PN) dynamics of an ESGB black hole binary 3. Beyond the PN approximation: “EOBization” of an ESGB black hole binary 4. Gravitational radiation from an ESGB black hole binary
2. The post-newtonian (PN) dynamics of an ESGB black hole binary “Skeletonizing” an ESGB black hole [in GR: Mathisson 1931, Infeld 1950,...] − g ( R − 2 g μν ∂ μ φ ∂ ν φ + α f ( φ ) ℛ 2 GB ) + I A 16 π ∫ d 4 x 1 I ESGB = pp Generic ansatz for compact bodies A ] = − ∫ m A ( φ ) ds A pp [ g μν , φ , x μ I A − g μν dx μ with A dx ν . ds A = A is a function of the local value of to encompass the e ff ect of the background scalar • m A ( φ ) φ field on the equilibrium of a body [Eardley 75, Damour-Esposito-Farèse 92]. • Strong equivalence principle violation Question: How to derive for an ESGB black hole? m A ( φ ) Answer: by identifying the BH's fields to those sourced by the particle.
2. The post-newtonian (PN) dynamics of an ESGB black hole binary Comparing the asymptotic expansions of the fields R μν = 2 ∂ μ φ ∂ ν φ − 4 α ( P μανβ − 1 2 g μν P αβ ) ∇ α ∇ β f ( φ ) + 8 π ( T A ) μν − 1 2 g μν T A δ (3) ( x − x A ( t )) □ φ = − 1 GB + 4 π ds A dm A 4 α f ′ � ( φ ) ℛ 2 dt d φ − g dx μ A = m A ( φ ) δ (3) ( x − x A ( t )) dx ν A A with T μν dt dt dx β dx α A A gg αβ dt dt Fields of particle A in its rest frame, x i Fields of the ESGB black hole A = 0 ) + 𝒫 ( r ) + 𝒫 ( g μν = η μν + δ μν ( g μν = η μν + δ μν ( r 2 ) r 2 ) 2 m A ( φ ∞ ) 1 2 m 1 r ˜ ˜ ˜ ˜ d φ ( φ ∞ ) + 𝒫 ( r + 𝒫 ( r 2 ) r 2 ) dm A φ = φ ∞ − 1 1 φ = φ ∞ + D 1 r ˜ ˜ ˜ ˜
2. The post-newtonian (PN) dynamics of an ESGB black hole binary Matching • the identification yields Matching conditions m A ( φ ∞ ) = m m ′ � A ( φ ∞ ) = − D • For an ESGB black hole with “secondary hair” , yields a first order di ff erential equation. D = D ( m , φ ∞ ) At leading order, for simplicity: 1 + 𝒫 ( A ) dm A d φ + α f ′ � ( φ ) α f ′ � = 0 m 2 2 m A ( φ ) • Its resolution involves a unique integration constant . μ A
2. The post-newtonian (PN) dynamics of an ESGB black hole binary The sensitivity of a hairy ESGB black hole A ] = − ∫ m A ( φ ) ds A pp [ g μν , φ , x μ I A • In an arbitrary ESGB theory , BHs are described by a unique constant parameter: m A ( φ ) = μ A ( 1 − α f ( φ ) + ⋯ ) S w where μ A = M irr = 2 μ 2 4 π A • Recall: ESGB first law of thermodynamics: T δ S w = δ M where . δ M = δ m + D δφ ∞ Matching conditions (a) and (b) ⇒ δ M = 0 (a) m A ( φ ∞ ) = m As a consequence, δ S w = 0 (b) m ′ � A ( φ ∞ ) = − D When varies slowly, the black hole readjusts its equilibrium configuration, i.e. , φ ∞ m in keeping its Wald entropy fixed.
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