Transcript Numerical Relativity & Gravitational waves

Numerical Relativity
&
Gravitational waves
M. Shibata (U. Tokyo)
I.
Introduction
II. Status
III. Latest results
IV. Summary
I. Introduction
• Detection of gravitational waves is done by
matched filtering (in general)
 Theoretical templates are necessary
• For coalescing binaries & pulsars
 We have post-Newtonian analytic solutions
BUT, for most of other sources
(SN, Merger of 2NS, 2BH, etc),
it is not possible to compute gravitational
waveforms in analytical manner
 Numerical simulation in full GR is
the most promising approach
Goal of our work
• To understand dynamics of general relativistic
dynamical phenomena (merger, collapse)
• To predict gravitational waveforms carrying out
fully GR hydrodynamic simulations
• In particular, we are interested in
* Merger of binary neutron stars (3D)
* Instability of rapidly rotating neutron stars (3D)
* Stellar collapse to a NS/BH (axisymmetric)
* Accretion induced collapse of a NS to a BH
(axisymmetric)
II. Necessary elements for GR
simulations
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Einstein evolution equations solver
Gauge conditions (coordinate condition)
GR Hydrodynamic equations solvers
Realistic initial conditions in GR
Horizon finder
Gravitational wave extraction techniques
Powerful supercomputer
Special techniques for handling BHs.
Status
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Einstein evolution equations solver
OK
Gauge conditions (coordinate condition) OK
GR Hydrodynamic equations solvers
OK
Realistic initial conditions in GR
OK
OK
Horizon finder
Gravitational wave extraction techniques
~OK
Powerful supercomputer NAOJ, VPP5000
Special techniques for handling BHs.
To be developed
Simulations are feasible for
merger of 2NS to BH, stellar collapse to NS/BH
III. Latest Results:
Merger of binary neutron stars
Setting at present
• Adiabatic EOS with various adiabatic constants
P=(G-1)re
(extensible for other EOSs)
• Initial conditions with realistic irrotational velocity
fields (by Uryu, Gourgoulhon, Taniguchi)
• Arbitrary mass ratios (we choose 1:1 & 1:0.9)
• Typical grid numbers (500, 500, 250) with which
L ~ gravitational wavelength &
Grid spacing ~ 0.2M
Low mass merger : Massive Neutron star is formed
Elliptical object.
Evolve as a
result of
gravitational
wave emission
subsequently.
Lifetime ~ 1sec
Formed Massive NS is differentially rotating
Angular
velocity
Kepler angular
Velocity for
Rigidly rotating case
Disk mass for equal mass merger
Negligible for merger of equal mass.
r = 6M.
Mass for r > 6M
~ 0%
Mass for r > 3M
~ 0.1%
Apparent
horizon
Disk mass for unequal mass merger
Merger of unequal mass; Mass ratio is ~ 0.9.
r = 6M.
Mass for r > 6M
~ 6%
Almost
BH
r = 3M.
Mass of r > 3M
~ 7.5%
Disk mass ~ 0.1 Solar_mass
Products of mergers
Equal – mass cases
・ Low mass cases
Formation of short-lived massive neutron stars
of non-axisymmetric oscillation.
(Lifetime would be ~1 sec due to GW by
quasi-stationary oscillations of NS; talk later)
・ High mass cases
Direct formation of Black holes
with negligible disk mass
Unequal – mass cases (mass ratio ~ 90%)
・ Likely to form disk of mass ~ several percents
==> BH(NS) + Disk
Gravitational waveforms along z axis
GW associated
with normal modes
of formed NS
BH-QNM would appear
crash
BH-QNM would appear
crash
~ 2 msec
IIIB Axisymmetric simulations:
Collapses to BH & NS
• Axisymmetric simulations in the Cartesian coordinate
system are feasible (no coordinate singularities)
=> Longterm, stable and accurate simulations
are feasible
• Arbitrary EOS (parametric EOS by Mueller)
• Initial conditions with arbitrary rotational law
• Typical grid numbers (2500, 2500)
• High-resolution shock-capturing hydro code
Example
• Parametric EOS (Following Mueller et al., K. Sato…)
P = PPolytrope  PThermal
PThermal = ( G Thermal - 1) re Thermal
 K1 r G1
PPolytrope = 
G2
K
r
 2
e Thermal = e - e Polytrope
4
G1 ~
3
G2  2
r  r Nuc 

r  r Nuc 
G Thermal = 1.5
Initial condition: Rotating stars with
G =4/3 & r ~ 1.e10 g/cc
Collapse of a rigidly rotating star
with central density ~ 1e10 g/cc to NS
Density
at r = 0
At t = 0,
T/W = 9.e-3
r (r=0) = 1.e10
M = 1.49 Solar
J/M^2 = 1.14
Lapse
at r = 0
Animation
is started here.
Qualitatively the same as Type I of Dimmelmeier et al (02).
Gravitational waveforms
h
2
sin 
Due to quasiradial
oscillation of
protoneutron stars
Time
Characteristic frequency ＝ several １００Ｈｚ
IV Summary
• Hydrodynamic simulations in GR are feasible
for a wide variety of problems both in 3D
and 2D (many simulations are the first ones
in the world)
• Next a couple of years :
Continue simulations for many parameters in
particular for merger of binary neutron stars
and stellar collapse to a NS/BH.
 To make Catalogue for gravitational waveforms
• More computers produce more outputs (2D)
 Appreciate very much for providing Grant !
 Hopefully, we would like to get for next a
couple of years
Review of the cartoon method
Y
Needless
3 points
Solve equations only at y = 0
The same point
In axisymmetric space.
X
・ Use Cartesian coordinates : No coordinate singularity
・ Impose axisymmetric boundary condition at y=+,-Dy
・ Total grid number = N＊3＊N for (x, y, z)