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07 Nov 2003 Gravitational Wave Phenomenology Workshop 1
Numerical Relativity
Deirdre Shoemaker Cornell University
The role of numerical relativity in gravitational-wave astrophysics
Status of Binary Black Hole (BBH) simulations
In the absence of a complete solution to the BBH problem, what can we do today?– Introduce approximations
• Sequences of quasi-circular orbits • Lazarus approach to BBH mergers
– Extract info from single BH evolutions• The spectrum of highly distorted black-holes
Open Questions
07 Nov 2003 Gravitational Wave Phenomenology Workshop 2
Numerical Relativity: Modeling gravitational wave sources
Full NR:• Compact object binaries• Supernovae ?
Less NR:• Extreme mass ratio binaries• NS oscillations
07 Nov 2003 Gravitational Wave Phenomenology Workshop 3
Today’s black-hole source simulations
1. Single BH evolutions:
• Static
• Boosted
• Wobbling
• Distorted
2. BBH Mergers
3. BBH < 30% orbit
07 Nov 2003 Gravitational Wave Phenomenology Workshop 4
Quasi-equilibrium orbits
BHb MME 2
0,
BHMJb
l
E
Binding Energy:
• We have the numerical infrastructure to solve elliptic equations (i.e. initial data)
• Quasi-adiabatic phase of the BBH system ~ sequences of quasi-circular initial data orbits
• Identification of the “innermost stable circular orbit”
Effective Potential Method(Cook & Baumgarte)
bE
07 Nov 2003 Gravitational Wave Phenomenology Workshop 5
Thin-Sandwich Method (Meudon)
Data Satisfy: Circular orbits found from:
0
0
K
g
t
ijt KomarADM MM
07 Nov 2003 Gravitational Wave Phenomenology Workshop 6
• Goal: to resurrect a death non-linear calculation
• How: On the 3rd day, match it to a perturbative calculation
• Lazarus Project:
• Start with ISCO data, merger takes place before code dies, continue with Teukolsky evolution.
• This approach is a natural extension of close limit approximation (Price & Pullin)
Non-linear matching to perturbative regime
07 Nov 2003 Gravitational Wave Phenomenology Workshop 7
07 Nov 2003 Gravitational Wave Phenomenology Workshop 8
07 Nov 2003 Gravitational Wave Phenomenology Workshop 9
Lazarus
Equal mass spinning black holes (~2.25%) Baker et al (2003)
07 Nov 2003 Gravitational Wave Phenomenology Workshop 10
Generating modes in black-hole spacetimes
What modes are generated if you hit a black hole hard enough?
• Study highly distorted black-holes
• End result of the merger of two black holes
• Prediction of mode content impacts data analysis algorithms
• In the linear regime, the modes are well known
• In the non-linear regime, it is possible to study mode coupling
• Allen et. al (1998)
• Papadopoulos (2002)
• Zlochower et. al (2003)
• DS et. al (2003)
07 Nov 2003 Gravitational Wave Phenomenology Workshop 11
Distorted black holes
Using Characteristic codes: mode-mode coupling for Schwarzschild DBHs.
• Papadopoulos(2002)
• Zlochower et. al (2003)
07 Nov 2003 Gravitational Wave Phenomenology Workshop 12
Distorted black holes in Cauchy formulism
MrAH 2 Mr 15 MrO 25
D. Shoemaker, H. Pfeiffer, L. Kidder, and S. Teukolsky
log(A) vs evolution time FFT(A) vs period
Amplitude of the l=2,m=0 mode for A(t=0) = 0.01 and A(t=0) = 0.1
07 Nov 2003 Gravitational Wave Phenomenology Workshop 13
Numerical Relativity
• We need more approaches that involve approximations.
• Lower dimension calculations (i.e. axisymmetry)
• Impact of numerical accuracy on data analysis tools.
• More astrophysical input in initial data
• And of course the usual
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