Core collapse in scalar-tensor theory of gravity U. Sperhake DAMTP - PowerPoint PPT Presentation

Core collapse in scalar-tensor theory of gravity U. Sperhake DAMTP , University of Cambridge M. Horbatsch, H. Silva, D. Gerosa, P . Pani, R. Berti, L. Gualtieri, US arXiv:1505.07462 D. Gerosa, C. Ott, US work in preparation III Amazonian Symposium on Physics, V NRHEP Meeting Belem, 02 th October 2015 U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 1 / 33

Overview Introduction Formalism Neutron stars in bi-STT Core collapse in STT Conclusions U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 2 / 33

1. Introduction U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 3 / 33

Scalar-tensor theories of gravity Extra degree(s) of freedom φ A additionally to g µν Appear in low-energy limit of string theories Kaluza-Klein like models Braneworld scenarios Historically: time-space dependent G Newton Jordan ’59, Fierz ’56, Brans & Dicke ’61 Candidate for explaining the dark sector in cosmology, inflation Many alternative theories can be formulated as ST theories No-hair theorems for BHs ⇒ matter sources often more sensitive to ST effects E.g. spontaneous scalarization Damour & Esposito-Farese ’93 U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 4 / 33

The end of stellar evolution Nuclear fusion above iron: energy consuming Stars with M ZAMS � 8 M ⊙ explode as SN → NS, BH U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 5 / 33

Core-collapse scenario (0 th -order) Ni-Fe core reaches Chandrasekhar mass → Collapse EOS stiffens at ρ � ρ nuc → Bounce Outgoing shock, re-invigorated by ν e → Outer layers blast away U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 6 / 33

2. Formalism U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 7 / 33

Notation ϕ A Scalar field(s) γ AB Target space metric γ A Target space Christoffel symbols BC g µν Physical spacetime metric ¯ g µν Spacetime metric in the Einstein frame ds 2 Physical line element s 2 d ¯ Line element in the Einstein frame a ( ϕ A ) 2 Conformal factor: g µν = a 2 ( ϕ A )¯ g µν ∂ ∂ A ≡ ∂ϕ A In general: bar → Einstein frame variable no bar → Jordan frame variable U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 8 / 33

Action and equations cf. Damour & Esposito-Farese CQG 9 , 2093 � dx 4 � ¯ c 4 R 4 − 1 � � g µν γ AB ( ∂ µ ϕ A )( ∂ ν ϕ B ) + W ( ϕ A ) − ¯ 2 ¯ S = g 4 π ¯ c G + S m [ ψ m , a 2 ( ϕ A )¯ g µν ] T µν = δ S m ( ψ m , g µν 2 √ Energy momentum tensor: δ g µν − ¯ g T µν = a 6 T µν ¯ g µν + 8 π ¯ ¯ � ¯ 2 ¯ R µν = 2 γ AB ( ∂ µ ϕ A )( ∂ ν ϕ B ) + 2 W ( ϕ A )¯ G T µν − 1 � ⇒ T ¯ µν c 4 � ϕ A = − γ A g µν ( ∂ µ ϕ B )( ∂ ν ϕ C ) − 4 π ¯ a ( ∂ B a )¯ ¯ c 4 γ AB 1 G BC ¯ T + γ AB ∂ B W T µν = 1 ∇ ν ¯ a ( ∂ A a )¯ ¯ T ¯ ∇ µ ϕ A U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 9 / 33

Spherically symmetric stars Line element ds 2 = − α 2 dt 2 + X 2 dr 2 + a 2 r 2 d Ω 2 , s 2 = − ¯ α 2 dt 2 + ¯ X 2 dr 2 + r 2 d Ω 2 d ¯ Auxiliary variables � α � � 1 − a 2 m ≡ r ¯ ¯ � , Φ ≡ ln , 2 X 2 a η A = ∂ r ϕ A ψ A = ∂ t ϕ A Ξ = γ AB ( η A η B + ψ A ψ B ) , , X α Matter T αβ = ( ǫ + ρ + p ) u α u β + Pg αβ � 1 u α = 1 v √ � α , X , 0 , 0 1 − v 2 J α = ρ u α “baryonic flow” satisfies ∇ µ J µ = 0 U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 10 / 33

Spherically symmetric stars Equation of state P = K ρ Γ , P ǫ = ρ (Γ − 1 ) We typically use: Γ = 2 . 34 , K = 1187 ( c = M ⊙ = 1 ) EOS1 in Novak gr-qc/9707041 Flux conservative variables a 3 ρ X ¯ √ D = 1 − v 2 S r = a 4 [ ρ ( 1 + ǫ )+ P ] v ¯ 1 − v 2 τ = a 4 [ ρ ( 1 + ǫ )+ P ] − a 4 P − ¯ ¯ D 1 − v 2 U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 11 / 33

The equations: Metric and scalar field � S r v + a 4 P � ∂ r Φ = X 2 ¯ � ¯ + a 2 r m � r 2 + 4 π r 2 Ξ , a 2 τ + ¯ D ) + a 2 r 2 ∂ r ¯ m = 4 π r 2 (¯ 2 Ξ , ∂ t φ A = αψ A , ∂ t η A = − η A ∂ t X ∂ r ψ A + ψ A ∂ r α X + α � � , X α ∂ t ψ A = α ∂ r η A + 2 r η A + η A ∂ r α BC ( ψ B ψ C − η B η C ) − ψ A ∂ t X X − αγ A � � X α S r v + ¯ τ − ¯ γ AB ∂ B a D − 3 a 4 P � � − 4 πα ¯ a 2 U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 12 / 33

The equations: Matter variables ∂ t ¯ D = ¯ D + a � r 2 α � r 2 ∂ r = s ¯ D , Dv , aX f ¯ f ¯ D S r + 1 S r v + a 4 P , ∂ t ¯ S r = ¯ r 2 α � � r 2 ∂ r X f ¯ = s ¯ S r , f ¯ S r S r − ¯ τ = ¯ τ + 1 r 2 α � � ∂ t ¯ r 2 ∂ r X f ¯ = s ¯ τ , f ¯ D v . τ Flux conservative form! The source terms s ¯ D , s ¯ S r , s ¯ contain no derivatives. τ Suitable for high-resolution shock-capturing methods extension of GR1D O’Connor & Ott 0912:2393 [astro-ph] U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 13 / 33

The static limit → TOV models, initial data All time derivatives vanish � ¯ D , ¯ S r , ¯ � � � Relation τ ↔ ρ, ǫ, v trivial as v = 0 � α, X , P , ϕ A , η A � Gives system of 5 ODEs for Boundary conditions X = 1 , ρ = ρ c , η A = 0 At r = 0 : At r = r S : P = 0 ϕ A = 0 At r → ∞ : (wlog) U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 14 / 33

3. Neutron stars in multi-ST theories U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 15 / 33

Specifying the theory Target geometry: maximally symmetric � 1 + ( ϕ 1 ) 2 +( ϕ 2 ) 2 � r 2 > 0 γ AB = δ AB spherical: 4 r 2 hyperbolic: r 2 < 0 r 2 → ∞ flat: Conformal factor log a = 2 α 0 ϕ 1 − 2 α 1 ϕ 2 + 1 2 ( β 0 + β 1 )( ϕ 1 ) 2 + 1 2 ( β 0 − β 1 )( ϕ 2 ) 2 Complex scalar field: ϕ = ϕ 1 + i ϕ 2 Free parameters: α 0 , α 1 , β 0 , β 1 , r U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 16 / 33

Case 1: α 0 = α 1 = β 1 = 0, β 0 = − 5 O ( 2 ) symmetry: Invariance under rotation in ϕ 1 , ϕ 2 plane Spherical (hyperbolic) target geometry ⇒ scalarization strengthened (weakened) 0.15 0.15 0.10 0.10 0.05 0.05 Im[ ψ 0 ] Im[ ψ 0 ] GR GR 0.00 0.00 −0.05 −0.05 −0.10 1 / 픯 =0 . 2 −0.10 1 / 픯 = − 0 . 2 i 1 / 픯 =1 . 0 1 / 픯 = − 1 . 0 i 1 / 픯 =2 . 0 1 / 픯 = − 2 . 0 i −0.15 −0.15 1 / 픯 =2 . 5 1 / 픯 = − 2 . 5 i −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 Re[ ψ 0 ] Re[ ψ 0 ] U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 17 / 33

Case 2: α 0 = α 1 = 0, β 0 = − 5 , β 1 � = 0 No bi-scalarized solutions! “Circle” → “Cross” r = 5 β 0 = − 5 . 0 , β 1 =0 . 0 0.10 β 0 = − 5 . 0 , β 1 =0 . 01 β 0 = − 5 . 0 , β 1 = − 0 . 01 0.05 Im[ ψ 0 ] GR 0.00 −0.05 −0.10 −0.10 −0.05 0.00 0.05 0.10 Re[ ψ 0 ] U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 18 / 33

Case 2: Scalarization for β 0 ± β 1 � − 4 . 35 Spontaneous scalarization for single-STT if β � − 4 . 35 Damour & Esposito-Farese ’93 Here: log a = 2 α 0 ϕ 1 − 2 α 1 ϕ 2 + 1 2 ( β 0 + β 1 )( ϕ 1 ) 2 + 1 2 ( β 0 − β 1 )( ϕ 2 ) 2 → Like single-STT with β → β 0 ± β 1 Re[ ψ ] Im[ ψ ] M B / M O M B / M O . . 2 2 0.76 0.76 1.5 1.5 0.74 0.74 0.72 0.72 1 1 -9.4 -9.3 -9.2 -9.1 -9 -9.4 -9.3 -9.2 -9.1 -9 -9.5 -9.5 1 / r = 0, β 1 = 0 1 / r = 0, β 1 = 0 1 / r = 0, β 1 = -5 1 / r = 0, β 1 = -5 1 / r = 0, β 1 = +5 1 / r = 0, β 1 = +5 0.5 0.5 1 / r = 2, β 1 = 0 1 / r = 2, β 1 = 0 1 / r = 2, β 1 = -5 1 / r = 2, β 1 = -5 1 / r = 2, β 1 = +5 1 / r = 2, β 1 = +5 0 0 -14 -12 -10 -8 -4 -14 -12 -10 -8 -4 -6 -6 β 0 + β 1 β 0 - β 1 U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 19 / 33

Case 3: ( α 0 , α 1 ) � = 0, β 0 = − 5 α ≡ | ( α 0 , α 1 ) | constrained; But phase not! α � = 0 facilitates bi-scalar solutions | α | = 0.001, 1 / r = 0 Im[ ψ 0 ] 0.2 0 β 1 =0.09 β 1 =0.24 β 1 =0.16 -0.2 β 1 =0.01 0.2 0 β 1 =0.39 β 1 =0.46 β 1 =0.54 β 1 =0.31 -0.2 0.2 0 β 1 =0.61 β 1 =0.69 β 1 =0.76 β 1 =0.99 -0.2 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 Re [ ψ 0 ] U. Sperhake (DAMTP, University of Cambridge) Core collapse in scalar-tensor theory of gravity 02/10/2015 20 / 33