1. Boson Star collisions in GR ERE 2006 Palma de Mallorca, 6 September 2006 Carlos Palenzuela, I.Olabarrieta, L.Lehner & S.Liebling

2. I. Introduction

3. 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

4. I. 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, …

5. II. The evolution equations

6. II. The EKG evolution system (I) Lagrangian of a complex scalar field in a curved background (natural units G=c=1) L = - R/(16 π) + [g ab + m 2 |/2 ] L = - R/(16 π) + [g ab  a φ*  b φ + m 2 |φ| 2 /2 ] R : Ricci scalar g ab : spacetime metric φ, φ* : scalar field and its conjugate complex m : mass of the scalar field

7. II. EKG evolution system (II) The Einstein-Klein-Gordon equations are obtained by varying the action with respect to g ab and φ - EE with a real stress-energy tensor (quadratic) - KG : covariant wave equation with massive term R ab = 8π (T ab – g ab T/2) T ab = [ + – g ab ( + m 2 |) ]/2 T ab = [  a φ  b φ* +  b φ  a φ* – g ab (  c φ  c φ* + m 2 |φ| 2 ) ]/2 g ab g ab  a  b φ = m 2 φ

8. II. The harmonic formalism 3+1 decomposition to write EE as a evolution system 3+1 decomposition to write EE as a evolution system - EE in the Dedonder-Fock form - EE in the Dedonder-Fock form - harmonic coordinates Γ a = 0 - harmonic coordinates Γ a = 0 □g ab = … □g ab = … Convert the second order system into first order to useConvert the second order system into first order to use numerical methods that ensure stability (RK3, SBP,…) numerical methods that ensure stability (RK3, SBP,…)

9. III. Testing the numerical code

10. III. The numerical code Infrastructure : had - Method of Lines with 3 rd order Runge-Kutta to integrate in time - Finite Difference space discretization satisfying Summation By Parts (2 nd and 4 th order) - Parallelization - Adaptative Mesh Refinement in space and time

11. III. Initial data for the single BS 1) static spherically symmetric spacetime in isotropic 1) static spherically symmetric spacetime in isotropic coordinates coordinates ds 2 = - α 2 dt 2 + Ψ 4 (dr 2 + r 2 dΩ 2 ) ds 2 = - α 2 dt 2 + Ψ 4 (dr 2 + r 2 dΩ 2 ) 2) harmonic time dependence of the complex scalar field 2) harmonic time dependence of the complex scalar field φ = φ 0 (r) e -iωt φ = φ 0 (r) e -iωt 3) maximal slicing condition 3) maximal slicing condition trK = ∂ t trK = 0 trK = ∂ t trK = 0

12. III. Initial data for the single BS(II) Substitute previous ansatzs in EKG Substitute previous ansatzs in EKG  set of ODE’s, can be solved for a given φ 0 (r=0)  set of ODE’s, can be solved for a given φ 0 (r=0)  eigenvalue problem for {ω : α(r), Ψ(r), φ 0 (r)}  eigenvalue problem for {ω : α(r), Ψ(r), φ 0 (r)} - stable configurations for M max ≤ 0.633/m - stable configurations for M max ≤ 0.633/m φ 0 φ 0 g xx g xx

13. III. Evolution of a single BS The frequency and amplitude of the star gives us a good measure of the validity of the code (+ convergence) The frequency and amplitude of the star gives us a good measure of the validity of the code (+ convergence) φ = φ 0 (r) e -iωt φ = φ 0 (r) e -iωt Re(φ) = φ 0 (r) cos(ωt)

14. IV. Head-on collisions of BS

15. IV. The 1+1 BS system Superposition of two single boson stars Superposition of two single boson stars φ T = φ 1 + φ 2 φ T = φ 1 + φ 2 Ψ T = Ψ 1 + Ψ 2 - 1 Ψ T = Ψ 1 + Ψ 2 - 1 α T = α 1 + α 2 - 1 α T = α 1 + α 2 - 1 - satisfies the constraints up to discretization error if the - satisfies the constraints up to discretization error if the BS are far enough BS are far enough

16. IV. The equal mass case Superposition of two BS with the same mass Superposition of two BS with the same mass L=30 φ 0 (0)=0.01 ω = 0.976 M=0.361 R=13 φ 0 (0)=0.01 ω = 0.976 M=0.361

17. IV. The equal mass case |φ| 2 (plane z=0) g xx (plane z=0) |φ| 2 (plane z=0) g xx (plane z=0)

18. IV. The unequal mass case Superposition of two BS with different mass Superposition of two BS with different mass L=30 φ 0 (0)=0.03 ω = 0.933 M=0.542 R=13 φ 0 (0)=0.01 ω = 0.976 M=0.361 R=9

19. IV. The unequal mass case |φ| 2 (plane z=0) g xx (plane z=0) |φ| 2 (plane z=0) g xx (plane z=0)

20. IV. The unequal phase case Superposition of two BS with the same mass but a difference of phase of π Superposition of two BS with the same mass but a difference of phase of π L=30 φ 0 (0)=0.01 ω = 0.976 M=0.361 R=13 φ = φ 0 (r) e -iωt φ = φ 0 (r) e -i(ωt+π) φ 0 (0)=0.01 ω = 0.976 M=0.361

21. IV. The unequal phase case |φ| 2 (plane z=0) g xx (plane z=0) |φ| 2 (plane z=0) g xx (plane z=0)

22. Future work Develop analysis tools (wave extraction, …) Develop analysis tools (wave extraction, …) Analyze and compare the previous cases with BHs Analyze and compare the previous cases with BHs Study the new cases that appear only in BS collisions Study the new cases that appear only in BS collisions