Numerical Relativity&

Gravitational waves

I. Introduction

II. Status

III. Latest results

IV. Summary

M. Shibata (U. Tokyo)

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

• 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

OK

OK

OK

OK

OK

~OK

To be developed

Simulations are feasible for merger of 2NS to BH, stellar collapse to NS/BH

• 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 NAOJ, VPP5000• Special techniques for handling BHs.

III. Latest Results: Merger of binary neutron stars

• Adiabatic EOS with various adiabatic constants

P (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

Setting at present

Low mass merger : Massive Neutron star is formed

Elliptical object.

Evolve as a result of gravitationalwave emissionsubsequently.

Lifetime ~ 1sec

Kepler angular Velocity for Rigidly rotating case

Formed Massive NS is differentially rotating

Angularvelocity

Disk mass for equal mass merger

r = 6M.Mass for r > 6M~ 0%

Negligible for merger of equal mass.

Mass for r > 3M~ 0.1%

Apparent horizon

Disk mass for unequal mass merger

r = 6M.Mass for r > 6M~ 6%

Merger of unequal mass; Mass ratio is ~ 0.9.

r = 3M.Mass of r > 3M~ 7.5%

Disk mass ~ 0.1 Solar_mass

AlmostBH

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

BH-QNM would appear

BH-QNM would appear

GW associatedwith normal modesof formed NS

crash

crash

~ 2 msec

Gravitational waveforms along z axis

• 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

IIIB Axisymmetric simulations:Collapses to BH & NS

Example

• Parametric EOS 　 (Following Mueller et al., K. Sato…)

Initial condition: Rotating stars with =4/3 & ~ 1.e10 g/cc

Polytrope Thermal

Thermal Thermal Thermal

11 Nuc

Polytrope2

2 Nuc

Thermal Polytrope

1 2 Thermal

1

4 ~ 2 1.5

3

P P P

P

KP

K

Collapse of a rigidly rotating star with central density ~ 1e10 g/cc to NS

At t = 0, T/W = 9.e-3(r=0) = 1.e10M = 1.49 SolarJ/M^2 = 1.14

Animationis started here.

Densityat r = 0

Lapseat r = 0

Qualitatively the same as Type I of Dimmelmeier et al (02).

Gravitational waveforms

2sin

h

Time

Characteristic frequency ＝ several １００Ｈｚ

Due to quasiradialoscillation ofprotoneutron stars

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

X

Y

・　 Use Cartesian coordinates : No coordinate singularity・　 Impose axisymmetric boundary condition at y=+,-y・　 Total grid number = N ＊ 3 ＊ N for (x, y, z)

Needless

The same point In axisymmetric space.

3 po

ints

Solve equations only at y = 0