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Boson Star collisions in GR

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Boson Starcollisions in GR

ERE 2006

Palma de Mallorca, 6 September 2006

Carlos Palenzuela,

I.Olabarrieta, L.Lehner & S.Liebling

- 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

- 1) model to study the 2 body interaction in GR
- 2) candidates for the dark matter
- 3) study other issues, like wave extraction, gauges, …

- Lagrangian of a complex scalar field in a curved background (natural units G=c=1)

L = - R/(16 π) + [gabaφ* 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

- 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 = [aφbφ* + bφ aφ* – gab (cφ cφ* + m2 |φ|2) ]/2

gaba bφ = m2φ

- 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,…)

- 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

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

- 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

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)

- 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

L=30

R=13

φ0(0)=0.01

ω = 0.976

M=0.361

φ0(0)=0.01

ω = 0.976

M=0.361

- Superposition of two BS with the same mass

|φ|2 (plane z=0) gxx (plane z=0)

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

- Superposition of two BS with different mass

|φ|2 (plane z=0) gxx (plane z=0)

- 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)

|φ|2 (plane z=0) gxx (plane z=0)

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