Numerical Relativity&Gravitational wavesIntroductionStatusLatest resultsSummaryM. Shibata (U. Tokyo)
I. IntroductionDetection 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 workTo 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 solverGauge conditions (coordinate condition)GR Hydrodynamic equations solversRealistic initial conditions in GRHorizon finderGravitational wave extraction techniquesPowerful supercomputerSpecial techniques for handling BHs.
StatusOKOKOKOKOK~OKTo be developedSimulations are feasible for merger of 2NS to BH, stellar collapse to NS/BHEinstein evolution equations solverGauge conditions (coordinate condition)GR Hydrodynamic equations solversRealistic initial conditions in GRHorizon finderGravitational wave extraction techniquesPowerful supercomputer NAOJ, VPP5000Special techniques for handling BHs.
III. Latest Results: Merger of binary neutron starsAdiabatic 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
Setting at present
Low mass merger : Massive Neutron star is formedElliptical object.
Evolve as a result of gravitationalwave emissionsubsequently.
Lifetime ~ 1sec
Kepler angular Velocity for Rigidly rotating caseFormed Massive NS is differentially rotating Angularvelocity
Disk mass for equal mass mergerr = 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 mergerr = 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_massAlmostBH
Products of mergersEqual 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 massUnequal mass cases (mass ratio ~ 90%) Likely to form disk of mass ~ several percents ==> BH(NS) + Disk
BH-QNM would appearBH-QNM would appearGW associatedwith normal modesof formed NScrashcrash~ 2 msecGravitational waveforms along z axis
IIIB Axisymmetric simulations:Collapses to BH & NSAxisymmetric simulations in the Cartesian coordinate system are feasible (no coordinate singularities) => Longterm, stable and accurate simulations are feasibleArbitrary EOS (parametric EOS by Mueller) Initial conditions with arbitrary rotational lawTypical grid numbers (2500, 2500)High-resolution shock-capturing hydro code
ExampleParametric EOS(Following Mueller et al., K. Sato)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 NSAt t = 0, T/W = 9.e-3r (r=0) = 1.e10M = 1.49 SolarJ/M^2 = 1.14
Animationis started here.Densityat r = 0Lapseat r = 0Qualitatively the same as Type I of Dimmelmeier et al (02).
Gravitational waveformsTimeCharacteristic frequency several Due to quasiradialoscillation ofprotoneutron stars
IV SummaryHydrodynamic 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 waveformsMore 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 methodXYUse Cartesian coordinates : No coordinate singularityImpose axisymmetric boundary condition at y=+,-DyTotal grid number = N3N for (x, y, z)NeedlessThe same point In axisymmetric space.3 pointsSolve equations only at y = 0
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