Boson Star collisions in GR

1. Boson Starcollisions 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) + [gabaφ* bφ + m2 |φ|2 /2 ] • R : Ricci scalar • gab : 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 gab and φ • - EE with a real stress-energy tensor (quadratic) • - KG : covariant wave equation with massive term Rab = 8π (Tab – gab T/2) Tab = [aφbφ* + bφ aφ* – gab (cφ cφ* + m2 |φ|2) ]/2 gaba bφ = m2φ

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

9. III. Testing the numerical code

10. III. 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

11. III. Initial data for the single BS 1) static spherically symmetric spacetime in isotropic coordinates ds2 = - α2 dt2 + Ψ4 (dr2 + r2 dΩ2) 2) harmonic time dependence of the complex scalar field φ = φ0(r) e-iωt 3) maximal slicing condition trK = ∂t trK = 0

12. III. Initial data for the single BS(II) • Substitute previous ansatzs in EKG •  set of ODE’s, can be solved for a given φ0(r=0) •  eigenvalue problem for {ω : α(r), Ψ(r), φ0(r)} • - stable configurations for Mmax ≤ 0.633/m φ0 gxx

13. III. Evolution of a single BS φ = φ0(r) e-iωt Re(φ) = φ0(r) cos(ωt) • The frequency and amplitude of the star gives us a good measure of the validity of the code (+ convergence)

14. IV. Head-on collisions of BS

15. IV. The 1+1 BS system • Superposition of two single boson stars • φT = φ1 + φ2 • ΨT = Ψ1 + Ψ2 - 1 • αT = α1 + α2 - 1 • - satisfies the constraints up to discretization error if the • BS are far enough

16. L=30 R=13 φ0(0)=0.01 ω = 0.976 M=0.361 φ0(0)=0.01 ω = 0.976 M=0.361 IV. The equal mass case • Superposition of two BS with the same mass

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

18. L=30 R=13 R=9 φ0(0)=0.03 ω = 0.933 M=0.542 φ0(0)=0.01 ω = 0.976 M=0.361 IV. The unequal mass case • Superposition of two BS with different mass

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

20. IV. The unequal phase case • Superposition of two BS with the same mass but a difference of phase of π L=30 R=13 φ0(0)=0.01 ω = 0.976 M=0.361 φ0(0)=0.01 ω = 0.976 M=0.361 φ = φ0(r) e-iωt φ = φ0(r) e-i(ωt+π)

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

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