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Numerical Relativity & Gravitational waves

Numerical Relativity & Gravitational waves. M. Shibata (U. Tokyo). Introduction Status Latest results Summary. I. Introduction. Detection of gravitational waves is done by matched filtering (in general) Theoretical templates are necessary

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Numerical Relativity & Gravitational waves

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

  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 • 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. OK OK OK OK OK ~OK To be developed Simulations are feasible for merger of 2NS to BH, stellar collapse to NS/BH

  6. 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

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

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

  9. 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

  10. Disk mass for unequal mass merger Merger of unequal mass; Mass ratio is ~ 0.9. r = 6M. Mass for r > 6M ~ 6% r = 3M. Mass of r > 3M ~ 7.5% Almost BH Disk mass ~ 0.1 Solar_mass

  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. 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

  13. 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

  14. Example • Parametric EOS(Following Mueller et al., K. Sato…) Initial condition: Rotating stars with G =4/3 & r ~ 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 (r=0) = 1.e10 M = 1.49 Solar J/M^2 = 1.14 Density at r = 0 Lapse at r = 0 Animation is started here. Qualitatively the same as Type I of Dimmelmeier et al (02).

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

  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 Y Needless Solve equations only at y = 0 The same point In axisymmetric space. 3 points X ・　Use Cartesian coordinates : No coordinate singularity ・　Impose axisymmetric boundary condition at y=+,-Dy ・　Total grid number = N＊3＊N for (x, y, z)

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