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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

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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, 02th October 2015

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 1 / 33

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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

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  • 1. Introduction
  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 3 / 33

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Scalar-tensor theories of gravity

Extra degree(s) of freedom φA additionally to gµν

Appear in low-energy limit of string theories Kaluza-Klein like models Braneworld scenarios

Historically: time-space dependent GNewton

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

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The end of stellar evolution

Nuclear fusion above iron: energy consuming Stars with MZAMS 8 M⊙ explode as SN → NS, BH

  • U. Sperhake (DAMTP, University of Cambridge)

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Core-collapse scenario (0th-order)

Ni-Fe core reaches Chandrasekhar mass → Collapse EOS stiffens at ρ ρnuc → Bounce Outgoing shock, re-invigorated by νe → Outer layers blast away

  • U. Sperhake (DAMTP, University of Cambridge)

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  • 2. Formalism
  • U. Sperhake (DAMTP, University of Cambridge)

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Notation

ϕA Scalar field(s) γAB Target space metric γA

BC

Target space Christoffel symbols gµν Physical spacetime metric ¯ gµν Spacetime metric in the Einstein frame ds2 Physical line element d¯ s2 Line element in the Einstein frame a(ϕA)2 Conformal factor: gµν = a2(ϕA)¯ gµν ∂A ≡

∂ ∂ϕA

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

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Action and equations

  • cf. Damour & Esposito-Farese CQG 9, 2093

S = c4 4π ¯ G dx4 c

  • −¯

g ¯ R 4 − 1 2 ¯ gµνγAB(∂µϕA)(∂νϕB) + W(ϕA)

  • +Sm[ψm, a2(ϕA)¯

gµν] Energy momentum tensor: T µν =

2

√

−¯ g δSm(ψm,gµν δgµν

¯ T µν = a6T µν ⇒ ¯ Rµν = 2γAB(∂µϕA)(∂νϕB) + 2W(ϕA)¯ gµν + 8π ¯

G c4

¯ Tµν − 1

2 ¯

T ¯ µν

  • ¯

ϕA = −γA

BC ¯

gµν(∂µϕB)(∂νϕC) − 4π ¯

G c4 γAB 1 a(∂Ba)¯

T + γAB∂BW ¯ ∇ν ¯ T µν = 1

a(∂Aa)¯

T ¯ ∇µϕA

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 9 / 33

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Spherically symmetric stars

Line element ds2 = −α2dt2+X 2dr 2+a2r 2dΩ2 , d¯ s2 = −¯ α2dt2+ ¯ X 2dr 2+r 2dΩ2 Auxiliary variables ¯ m ≡ r

2

  • 1 − a2

X 2

  • ,

¯ Φ ≡ ln α

a

  • ,

ηA = ∂rϕA

X

, ψA = ∂tϕA

α

, Ξ = γAB(ηAηB + ψAψB) Matter Tαβ = (ǫ + ρ + p)uαuβ + Pgαβ uα =

1

√

1−v2

1

α, v X , 0, 0

  • Jα = ρuα

“baryonic flow” satisfies ∇µJµ = 0

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 10 / 33

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Spherically symmetric stars

Equation of state P = KρΓ , ǫ =

P ρ(Γ−1)

We typically use: Γ = 2.34 , K = 1187 (c = M⊙ = 1) EOS1 in Novak gr-qc/9707041 Flux conservative variables ¯ D =

a3ρX

√

1−v2

¯ Sr = a4[ρ(1+ǫ)+P]v

1−v2

¯ τ = a4[ρ(1+ǫ)+P]

1−v2

− a4P − ¯ D

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 11 / 33

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The equations: Metric and scalar field

∂rΦ = X 2

a2

  • ¯

m r 2 + 4πr

¯ Sr v + a4P

  • + a2r

2 Ξ

  • ,

∂r ¯ m = 4πr 2(¯ τ + ¯ D) + a2r 2

2 Ξ ,

∂tφA = αψA , ∂tηA = −ηA ∂tX

X + α X

  • ∂rψA + ψA ∂rα

α

  • ,

∂tψA = α

X

  • ∂rηA + 2

r ηA + ηA ∂rα α

  • − ψA ∂tX

X − αγA BC(ψBψC − ηBηC)

−4πα

  • ¯

τ − ¯ Sr v + ¯ D − 3a4P

  • γAB ∂Ba

a2

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 12 / 33

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The equations: Matter variables

∂t ¯ D + a

r 2 ∂r

  • r 2 α

aX f¯ D

  • = s¯

D ,

f¯

D = ¯

Dv , ∂t ¯ Sr + 1

r 2 ∂r

  • r 2 α

X f¯ Sr

  • = s¯

Sr ,

f¯

Sr = ¯

Sr v + a4P , ∂t ¯ τ + 1

r2 ∂r

  • r 2 α

X f¯ τ

  • = s¯

τ ,

f¯

τ = ¯

Sr − ¯ D v . Flux conservative form! The source terms s¯

D, s¯ Sr , s¯ τ

contain no derivatives. 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

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The static limit → TOV models, initial data

All time derivatives vanish Relation ¯ D, ¯ Sr, ¯ τ

  • ↔
  • ρ, ǫ, v
  • trivial as v = 0

Gives system of 5 ODEs for

  • α, X, P, ϕA, ηA

Boundary conditions At r = 0 : X = 1, ρ = ρc, ηA = 0 At r = rS : P = 0 At r → ∞ : ϕA = 0 (wlog)

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 14 / 33

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  • 3. Neutron stars in multi-ST

theories

  • U. Sperhake (DAMTP, University of Cambridge)

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Specifying the theory

Target geometry: maximally symmetric γAB = δAB

  • 1 + (ϕ1)2+(ϕ2)2

4r2

  • spherical:

r2 > 0 hyperbolic: r2 < 0 flat: r2 → ∞ Conformal factor log a = 2α0ϕ1 − 2α1ϕ2 + 1

2(β0 + β1)(ϕ1)2 + 1 2(β0 − β1)(ϕ2)2

Complex scalar field: ϕ = ϕ1 + iϕ2 Free parameters: α0, α1, β0, β1, r

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 16 / 33

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Case 1: α0 = α1 = β1 = 0, β0 = −5

O(2) symmetry: Invariance under rotation in ϕ1, ϕ2 plane Spherical (hyperbolic) target geometry ⇒ scalarization strengthened (weakened)

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

Re[ψ0 ]

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

Im[ψ0]

GR

1/픯 =0.2 1/픯 =1.0 1/픯 =2.0 1/픯 =2.5

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

Re[ψ0 ]

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

Im[ψ0]

GR

1/픯 =−0.2i 1/픯 =−1.0i 1/픯 =−2.0i 1/픯 =−2.5i

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 17 / 33

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Case 2: α0 = α1 = 0, β0 = −5, β1 = 0

No bi-scalarized solutions! “Circle” → “Cross” r = 5

−0.10 −0.05 0.00 0.05 0.10

Re[ψ0 ]

−0.10 −0.05 0.00 0.05 0.10

Im[ψ0 ]

GR

β0 =−5.0, β1 =0.0 β0 =−5.0, β1 =0.01 β0 =−5.0, β1 =−0.01

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 18 / 33

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Case 2: Scalarization for β0 ± β1 −4.35

Spontaneous scalarization for single-STT if β −4.35

Damour & Esposito-Farese ’93

Here: log a = 2α0ϕ1 − 2α1ϕ2 + 1

2(β0 + β1)(ϕ1)2 + 1 2(β0 − β1)(ϕ2)2

→ Like single-STT with β → β0 ± β1

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4

β0 + β1

0.5 1 1.5 2

MB / MO .

1 / r = 0, β1 = 0 1 / r = 0, β1 = -5 1 / r = 0, β1 = +5 1 / r = 2, β1 = 0 1 / r = 2, β1 = -5 1 / r = 2, β1 = +5

Re[ψ]

  • 9.5
  • 9.4
  • 9.3
  • 9.2
  • 9.1
  • 9

0.72 0.74 0.76

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4

β0 - β1

0.5 1 1.5 2

MB / MO .

1 / r = 0, β1 = 0 1 / r = 0, β1 = -5 1 / r = 0, β1 = +5 1 / r = 2, β1 = 0 1 / r = 2, β1 = -5 1 / r = 2, β1 = +5

Im[ψ]

  • 9.5
  • 9.4
  • 9.3
  • 9.2
  • 9.1
  • 9

0.72 0.74 0.76

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 19 / 33

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Case 3: (α0, α1) = 0, β0 = −5

α ≡ |(α0, α1)| constrained; But phase not! α = 0 facilitates bi-scalar solutions

  • 0.2

0.2

Im[ψ0]

  • 0.2

0.2

  • 0.2

0.2

  • 0.2

0.2

  • 0.2

0.2

Re[ψ

0]

  • 0.2

0.2

  • 0.2

0.2 β1=0.01 β1=0.09 β1=0.16 β1=0.24 β1=0.54 β1=0.46 β1=0.39 β1=0.31 β1=0.61 β1=0.69 β1=0.76 β1=0.99

|α| = 0.001, 1 / r = 0

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 20 / 33

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Case 3: (α0, α1) = 0, β0 = −5

α ≡ |(α0, α1)| constrained; But phase not! α = 0 facilitates bi-scalar solutions

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 21 / 33

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Case 3: (α0, α1) = 0, β0 = −5

Zoom into upper left panel of above (β1 = 0.01) Fine structure of (weakly) scalarized solutions

  • 0.2
  • 0.1

0.1

Im[ψ0]

  • 0.005

0.005

Re[ψ0] β1 = 0.01, |α| = 0.001, 1 / r = 0

  • 0.0002

0.0002

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 22 / 33

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  • 4. Core collapse in single-ST

theories

  • U. Sperhake (DAMTP, University of Cambridge)

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Core collapse

Massive stars: MZAMS ≈ 8 . . . 100M⊙ Core compressed from ∼ 1500 km to ∼ 15 km ∼ 1010 g/cm3 to ∼> 1015 g/cm3 Released gravitational energy: O(1053) erg ∼ 99 % in neutrinos, ∼ 1051 erg in outgoing shock, explosion Explosion mechanism: still uncertainties... Failed explosion ⇒ BH formation Collapsar possible engine for long-soft GRB

  • U. Sperhake (DAMTP, University of Cambridge)

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Code test: Static NS models

¯ M = 2.4 M⊙, ¯ R = 13.1 km model with α0 = 0, β0 = −6

Novak gr-qc/9707041

Baryon density, metric functions, scalar field

  • FIG. 1. Density (n

˜ ), metric potentials (A and N), and scalar

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 25 / 33

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Code test: NS collapse to BH

Intial model: ¯ R = 11.8 km, ¯ M = 2.07 M⊙ α0 = 0.0025, β0 = −5

Novak gr-qc/9707041

Discrepancy due to sign error in α0 in Novak As α0 → 0, we agree with Novak

  • U. Sperhake (DAMTP, University of Cambridge)

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Code test: Transition from GR to scalarized star

Unstable GR-like model: ¯ R = 13.2 km, ¯ M = 1.378 M⊙ ... migrates to scalarized model: ¯ R = 13.0 km, M = 1.373 M⊙ Here: α0 = 0.01 , β0 = −6

Novak gr-qc/9806022

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 27 / 33

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Core collapse: Hybrid EOS

Model stiffening of EOS through change in polytropic index P = Pcold + Pthermal where Pcold = Polytrope(Γ1, Γ2) matched at ρ = ρnuc Pthermal = (Γth − 1)ρ(ǫ − ǫ0) Before shock formation: ǫ = ǫ0 → Pthermal models non-adiabatic shock flow We use Γ1 = 1.3, Γ2 = 2.5, Γth = 1.25 . . . 1.5

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 28 / 33

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Presupernova model: s12WH2007

From stellar evolution codes up to the onset of core collapse

Woosley & Heger Phys.Rep.442, 269

Solar metalicity ZAMS mass M = 12 M⊙ Mpre−SN Generated with Newtonian gravity Set α0 = 0 to trigger scalar field

  • U. Sperhake (DAMTP, University of Cambridge)

Core collapse in scalar-tensor theory of gravity 02/10/2015 29 / 33

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Core bounce

ρ, ϕ profiles at different t Core bounce ⇒ outgoing shock

  • U. Sperhake (DAMTP, University of Cambridge)

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Wave signal: Varying Γth; α0 = 0.01, β0 = −5

Extra pressure ∝ Γth Small Γth ⇒ more massive NS ⇒ Spontaneous scalarization Similar to WD collapse

Novak & Ibañez astro-ph/9911298

Detectable to ∼ 1 Mpc But depends on α0!!

  • U. Sperhake (DAMTP, University of Cambridge)

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  • 5. Conclusions
  • U. Sperhake (DAMTP, University of Cambridge)

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Conclusions

Formalism for multi-scalar theories similar to single ST For α0 = 0, effectively like ST Bi-scalarized solutions require α0 = 0 Complex structure in (ϕ1

c, ϕ2 c) plane

Core collapse in spherical symmetry Code tested successfully; identified few typos in literature Core bounce dynamics so far similar to GR Scalar waveforms strongly dependent on EOS Detectability ∝ α0

  • U. Sperhake (DAMTP, University of Cambridge)

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