1. Numerical Relativity & Gravitational waves I.Introduction II.Status III.Latest results IV.Summary M. Shibata (U. Tokyo)

2. 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

3. 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)

4. 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.

5. Status 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.

6. 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

7. Low mass merger : Massive Neutron star is formed Elliptical object. Evolve as a result of gravitational wave emission subsequently. Lifetime ~ 1sec

8. Kepler angular Velocity for Rigidly rotating case Formed Massive NS is differentially rotating Angular velocity

9. 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

10. 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 Almost BH

11. 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

12. BH-QNM would appear GW associated with normal modes of formed NS crash ~ 2 msec Gravitational waveforms along z axis

13. 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

14. Example Parametric EOS (Following Mueller et al., K. Sato…) Initial condition: Rotating stars with  =4/3 &  ~ 1.e10 g/cc

15. 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.e10 M = 1.49 Solar J/M^2 = 1.14 Animation is started here. Density at r = 0 Lapse at r = 0 Qualitatively the same as Type I of Dimmelmeier et al (02).

16. Gravitational waveforms Time Characteristic frequency ＝ several １００Ｈｚ Due to quasiradial oscillation of protoneutron stars

17. 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

18. 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 points Solve equations only at y = 0