General Covariant evolution formalisms

General Covariant evolution formalisms

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

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. 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 equations II. The EKG evolution system (I)

Lagrangian of a complex scalar field in a curved background (natural units G=c=1) L = - R/(16 ) + [gab a* b + m2 || | 2 /2 ] R : Ricci scalar gab : spacetime metric , * : scalar field and its conjugate complex m : mass of the scalar field

II. EKG evolution system (II) The Einstein-Klein-Gordon equations are obtained by varying the action with respect to gab and - EE with a real stress-energy tensor (quadratic) - KG : covariant wave equation with massive term Rab = 8 (Tab gab T/2)

Tab = [ [ a b* + b a* gab ( ( c c* + m2 || | 2) ]/2 gab a b = m2 II. The harmonic formalism 3+1 decomposition to write EE as a evolution system

- EE in the Dedonder-Fock form - harmonic coordinates a = 0 gab = Convert the second order system into first order to use numerical methods that ensure stability (RK3, SBP,)

III. Testing the numerical code 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 III. Initial data for the single BS 1) static spherically symmetric spacetime in isotropic coordinates ds2 = - 2 dt2 + 4 (dr2 + r2 d2) 2) harmonic time dependence of the complex scalar field

= 0(r) e-it 3) maximal slicing condition trK = t trK = 0 III. Initial data for the single BS(II) Substitute previous ansatzs in EKG set of ODEs, can be solved for a given 0(r=0) eigenvalue problem for { : (r), (r), 0(r)} - stable configurations for Mmax 0.633/m

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

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

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

0(0)=0.01 = 0.976 M=0.361 IV. The equal mass case ||2 (plane z=0) gxx (plane z=0)

IV. The unequal mass case Superposition of two BS with different mass L=30 R=9 0(0)=0.03 = 0.933 M=0.542

R=13 0(0)=0.01 = 0.976 M=0.361 IV. The unequal mass case ||2 (plane z=0)

gxx (plane z=0) IV. The unequal phase case Superposition of two BS with the same mass but a difference of phase of L=30 0(0)=0.01 = 0.976 M=0.361

= 0(r) e-it R=13 0(0)=0.01 = 0.976 M=0.361 = 0(r) e-i(t+)

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

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