Boson StarBoson Starcollisions in GRcollisions in GR

ERE 2006ERE 2006

Palma de Mallorca, 6 September 2006Palma de Mallorca, 6 September 2006

Carlos Palenzuela, I.Olabarrieta, L.Lehner & S.Liebling

I. IntroductionI. Introduction

I. What is a Boson Star (BS)?I. What is a Boson Star (BS)?

• Boson Stars: compact bodies composed of a complex massive scalar field, minimally coupled to the gravitational field

- simple evolution equation for the matter it does not tend to develop shocks it does not have an equation of state

I. MotivationI. Motivation

1) model to study the 2 body interaction in GR 2) candidates for the dark matter

3) study other issues, like wave extraction, gauges, …

II. The evolution equationsII. The evolution equations

II. The EKG evolution system (I)II. The EKG evolution system (I) • Lagrangian of a complex scalar field in a curved

background (natural units G=c=1)

LL = - R/(16 = - R/(16 ππ) + ) + [g[gabab aφ* bφ + m + m22 | |φ|2 /2 ]/2 ]

R : Ricci scalar gab : spacetime metric φ, φ* : scalar field and its conjugate complex m : mass of the scalar field

II. EKG evolution system (II)II. EKG evolution system (II) • The Einstein-Klein-Gordon equations are obtained by

varying the action with respect to gab and φ - EE with a real stress-energy tensor (quadratic) - KG : covariant wave equation with massive term

RRabab = = 8 8ππ (T (Tabab – g – gabab T/2) T/2)

TTabab = [ = [aφ bφ* + + bφ aφ* – g – gabab ( (cφ cφ* + m + m22 | |φ|2) ]/2) ]/2

ggabab a b φ = m2 φ

II. The harmonic formalismII. The harmonic formalism

• 3+1 decomposition to write EE as a evolution system3+1 decomposition to write EE as a evolution system - EE in the Dedonder-Fock form- EE in the Dedonder-Fock form - harmonic coordinates - harmonic coordinates ΓΓaa = 0 = 0

□□ggabab = = ……

• Convert the second order system into first order to useConvert the second order system into first order to use numerical methods that ensure stabilitynumerical methods that ensure stability (RK3, SBP,…) (RK3, SBP,…)

III. Testing the numerical codeIII. Testing the numerical code

III. The numerical codeIII. The numerical code

• Infrastructure : had - Method of Lines with 3rd order Runge-Kutta to integrate in time

- Finite Difference space discretization satisfying Summation By Parts (2nd and 4th order)

- Parallelization

- Adaptative Mesh Refinement in space and time

III. Initial data for the single BSIII. Initial data for the single BS

1) static spherically symmetric spacetime in isotropic 1) static spherically symmetric spacetime in isotropic coordinatescoordinates

dsds22 = - = - αα22 dt dt22 + + ΨΨ44 (dr (dr22 + r + r22 d dΩΩ22))

2) harmonic time dependence of the complex scalar field2) harmonic time dependence of the complex scalar field

φφ = = φφ00(r) e(r) e-i-iωωtt

3) maximal slicing condition3) maximal slicing condition

trK = trK = ∂∂tt trK = 0 trK = 0

III. Initial data for the single BS(II)III. Initial data for the single BS(II)• Substitute previous ansatzs in EKGSubstitute previous ansatzs in EKG set of ODE’s, can be solved for a given set of ODE’s, can be solved for a given φφ00(r=0)(r=0)

eigenvalue problem for eigenvalue problem for {{ωω : : αα(r), (r), ΨΨ(r)(r), , φφ00(r)(r)}}

- stable configurations for M- stable configurations for Mmaxmax ≤ 0.633/m ≤ 0.633/m

φφ00

ggxxxx

III. Evolution of a single BSIII. Evolution of a single BS

• The frequency and amplitude of the star gives us a The frequency and amplitude of the star gives us a good measure of the validity of the code (+ convergence)good measure of the validity of the code (+ convergence)

φφ = = φφ00(r) e(r) e-i-iωωtt

Re(Re(φφ) = ) = φφ00(r) cos((r) cos(ωωt)t)

IV. Head-on collisions of BSIV. Head-on collisions of BS

IV. The 1+1 BS system IV. The 1+1 BS system

• Superposition of two single boson starsSuperposition of two single boson stars

φφTT = = φφ11 + + φφ22

ΨΨTT = = ΨΨ11 + + ΨΨ22 - 1 - 1

ααTT = = αα11 + + αα22 - 1 - 1

- satisfies the constraints up to discretization error if the - satisfies the constraints up to discretization error if the BS are far enoughBS are far enough

IV. The equal mass case IV. The equal mass case

• Superposition of two BS with the same massSuperposition of two BS with the same mass

L=30

φ0(0)=0.01ω = 0.976 M=0.361

R=13

φ0(0)=0.01ω = 0.976 M=0.361

IV. The equal mass case IV. The equal mass case

||φφ||22 (plane z=0) g (plane z=0) gxxxx (plane z=0) (plane z=0)

IV. The unequal mass case IV. The unequal mass case

• Superposition of two BS with different massSuperposition of two BS with different mass

L=30

φ0(0)=0.03ω = 0.933 M=0.542

R=13

φ0(0)=0.01ω = 0.976 M=0.361

R=9

IV. The unequal mass case IV. The unequal mass case

||φφ||22 (plane z=0) g (plane z=0) gxxxx (plane z=0) (plane z=0)

IV. The unequal phase caseIV. The unequal phase case• Superposition of two BS with the same mass but a Superposition of two BS with the same mass but a difference of phase of difference of phase of ππ

L=30

φ0(0)=0.01ω = 0.976 M=0.361

R=13

φ = φ0(r) e-iωt φ = φ0(r) e-i(ωt+π)

φ0(0)=0.01ω = 0.976 M=0.361

IV. The unequal phase caseIV. The unequal phase case

||φφ||22 (plane z=0) g (plane z=0) gxxxx (plane z=0) (plane z=0)

Future workFuture work

• Develop analysis tools (wave extraction, …) Develop analysis tools (wave extraction, …) • Analyze and compare the previous cases with BHsAnalyze and compare the previous cases with BHs• Study the new cases that appear only in BS collisionsStudy the new cases that appear only in BS collisions