1. Current status of numerical relativity Gravitational waves from coalescing compact binaries Masaru Shibata (Yukawa Institute, Kyoto University)

2. Initial LIGO Adv LIGO, LCGT… Prediction only by Num. Rela h=h(f)f Merge Frequency f (Hz) GW spectrum from compact binaries Assume BH=10M sun NS=1.4M sun Chirp

3. We need to solve Einstein equation Equation of motion (Usually, hydro eqs.) Maxwell equation, Radiation hydro, Elementary physics, … Non linear equations: NR is the unique method

4. Needed implementations 1. Einstein’s evolution equations solver 2. GR Hydrodynamic equations solver 3. Gauge conditions (coordinate conditions) 4. Realistic initial conditions 5. Gravitational wave extraction techniques 6. Apparent horizon (Event horizon) finder 7. Special techniques for handling BHs 8. Micro physics (EOS, neutrino processes, B-field, radiation transfer …) 9. Powerful supercomputers or AMR

5. Present status 1. Einstein’s evolution equations solver 2. GR Hydrodynamic equations solver 3. Gauge conditions (coordinate conditions) 4. Realistic initial conditions 5. Gravitational wave extraction techniques 6. Apparent horizon (Event horizon) finder 7. Special techniques for handling BHs 8. Micro physics (EOS, neutrino processes, B-field, radiation transfer …) 9. Powerful supercomputers or AMR ○○○○○○○△○○○○○○○○△○ Not yet

6. Before 2005 1. Einstein’s evolution equations solver 2. GR Hydrodynamic equations solver 3. Gauge conditions (coordinate conditions) 4. Realistic initial conditions 5. Gravitational wave extraction techniques 6. Apparent horizon (Event horizon) finder 7. Special techniques for handling BHs 8. Micro physics (EOS, neutrino processes, B-field, radiation transfer …) 9. Powerful supercomputers or AMR ○○○○○○××△○○○○○○××△

7. Summary of current status Simulation for BH spacetime (BH-BH, BH- NS, collapse to BH) is now feasible. Simulation for NS-NS with a variety of equations of state is in progress. Adaptive mesh refinement (AMR) enables to perform a longterm simulation for inspiral. MHD effects, finite temperature EOS, neutrino cooling, etc, start being incorporated ⇒ Application to relativistic astrophysics (but still in a primitive manner)

8. § BH-BH: Status Simulation from late inspiral through merger phases is feasible: Evolve ~10 orbits accurately by several groups For nonspining BHs, an excellent analytical modeling (i.e. Taylor-T4 formula) has been found for orbital evolution and gravitational waveforms For the spinning BH-BH, several works exist, but still a large parameter space is left; good modeling has not been done yet.

9. Gravitational waves from BBH merger By F. Pretorius QNM BH ringing Inspiral waveform

10. Universal Fourier spectrum f  7/6 inspiral f  2/3 merger Buonanno, Cook, & Pretorius, PRD75 (2007) e  ringdown h(f)h(f) (15+15M sun )

11. advLIGO, LCGT 1st LIGO Frequency (Hz) Current level Assume BH=10 M sun h=h(f)f Larger mass Detection is possible now

12. High-precision computation by Cornell-Caltech group Nonspining Equal-mass 15 orbits

13. Excellent agreement with Taylor T4 formula

14. § NS-NS: Status Late inspiral phase by AMR: It is possible to follow >~5 orbits before merger with nuclear-theory based EOS  Will clarify the dependence of GWs on EOSs at the onset of merger Merger phase: It is feasible to follow evolution to a stationary state of BH/NS. BUT, still, with simple EOS/microphysics  More detailed modeling is left for the future work

15. 1.5M sun Merger to BH Akmal-Pandharipande-Ravenhall EOS Kiuchi et al. (2009) 1.4M sun Merger to NS

16. 1.5M sun Merger to BH Akmal-Pandharipande-Ravenhall EOS Kiuchi et al.

17. Merger to BH+disk 1.3M sun 1.6M sun

18. Merger to NS 1.4M sun

19. Gravitational waveform for black-hole formation case Inspiral Merger Ring down

20. Universal spectrum for BH formation Inspiral h eff ~ f n n~-1/6 Damp BH QNM Merger Different from BBH Bump Damp

21. Gravitational waveform for hyper- massive NS formation case Inspiral Merger QPO

22. Spectrum for two EOSs advLIGO EOS-dependence of f cut

23. Assume BH ： 10M sun 、 NS: 1.4M sun Frequency f (Hz) f  Spectrum BH form Initial LIGO advLIGO h HMNS form ~ 3kHz ~ 6 kHz

24. § Status of BH-NS It is possible to follow several orbits. Significant difference between tidal- disruption and no-disruption waveforms  Merger waveforms depend significantly on the neutron star radius Still in an early stage; simulations have been performed with simple EOSs  Next task: Survey for waveforms using a wide variety of EOSs (on going)

25. Inspiral: (M/R) NS =0.145,  =2 polytrope M BH /M NS =2 M BH /M NS =5 ~5 orbits ~7.5 orbits

26. (M/R) NS =0.145, M BH /M NS =2

27. (M/R) NS =0.145, M BH /M NS =4 (& >4) No disk

28. Gravitational waveforms (M/R) NS =0.145, M BH /M NS =2 Dotted curve = 3 PN fit inspiral disruption Quick shutdown

29. Gravitational waveforms (M/R) NS =0.145, M BH /M NS =5 Dotted curve = 3 PN fit ringdown

30. (M/R) NS =0.178 M BH /M NS =3 (M/R) NS =0.145 Clear ringdown Not very clear No disruption Mass shedding

31. Typical spectrum (M/R) NS =0.145, M BH /M NS =3 Inspiral Damp ∝ exp[-(f/f cut ) n ] BH-QNM Hz

32. Spectrum (M/R) NS =0.145, 0.160, 0.178, M BH /M NS =3 No-disruption  Spectrum extends to high-frequency Ringdown frequency Inspiral Damp

33. Relation between Compactness (C) and mass ratio (Q) C For small mass ratio, strong dependence of f cut on NS compactness QNM frequency

34. Summary Accurate GR simulation can be performed. Many simulations are ongoing for many groups not only for BH-BH, but also for NS-NS and BH-NS. In 3—5 years, a variety of theoretical waveforms will be derived.  These may be used for deciding design for next-generation detectors

35. With spin: Q=2, C=0.145, 0.160, 0.178 S>0 shifts lower f Cyan = with spin a=0.5

36. Relation between Compactness (C) and mass ratio (Q) C spin

37. Spectrum (M/R) NS =0.145, M BH /M NS =2–5 No-disruption  Spectrum extends to high-frequency ∝ exp[-(f/f cut ) n ] Ringdown