Numerical relativity simulation with Microphysics

1. Numerical relativity simulation with Microphysics National Astronomical Observatory of Japan Yuichiro Sekiguchi Masaru Shibata (YITP) Kenta Kiuchi (YITP) Koutaro Kyutoku (YITP) Keisuke Taniguchi (Tokyo)

2. Introduction • Exploring phenomena in strong, dynamical gravity • Black hole (BH) formation, Merger of compact objects, Collapse of massive star, etc. • Gravitational waveforms • Test of GR in strong gravity • High energy astrophysical phenomena • Gamma-ray bursts (GRBs), Supernovae etc • Theoretical study ⇔ Observation • Einstein equations : nonlinear partial differential equation • Numerical simulation will be unique approach to the problem ⇒Numerical Relativity

3. Targets of Numerical Relativity • Collapse of massive stellar core, NS/BH-NS/BH merger • Gravitational waveform • EOS of dense matter • High energy astrophysical phenomena

4. What is required to explore the phenomena and what is the problems ?

5. GR effects Chandrasekhar 1964, 1965

6. GR and EOS Van Riper (1988) ApJ 326, 235 Shock velocity @ 300 km (1000km/s) Incompressibility K(sym) (MeV)

7. GR and weak rates Takahara & Sato (1984) PTP 72, 978 Shock energy @ bounce (1052 erg) Log (Shock energy @ ejection)

8. Collapse of massive star • Dense (hot) matter region • 　　　　　　　　⇒ neutrinos drive the thermal / chemical evolution of the core • Neutrinos and weak interaction must be included • Strong dependence of weak rates on temperature ⇒ a finite temperature EOS is required • Currently, Shen EOS and LS EOS available • 　　　　　　　　⇒ β-equilibrium may be achieved • Very different two timescales • Numerically, very ‘ stiff ’ source terms appear • Generally, implicit schemes are necessary • In my study, sophisticated GR leakage scheme is adopted to solve in an explicit manner

9. Merger of NS-NS/BH • Late inspiral phase : NS is ‘ cold ‘ • kBT/ EF << 1 : NS will be described well with zero temperature EOS (Cold EOS) • Extension to finite temperature • Meger phase : Compression, shock heating • kBT/ EF ~ O(0.1) : a finite temperature EOS is required • Currently, Shen EOS and LS EOS • Prompt BH formation and small disk • Effects of finite temperature may be miner (Cold EOS may be sufficient) • HMNS formation or massive disk formation • Shock heating and neutrino emission, etc. are important (finite temperature EOS required)

10. Problems in NR ① • Cauthy problem in GR is constrained system • There are constraint equations (similar to no-monopole condition and gauss’low in EM) • Everything is in terms of energy-momentum tensor • All equations of source field are obtained from • One can not add any source terms to the system • If added, constraint violations will lead to termination of simulations • Neutrino energy momentum should be considered

11. Argument quantities Evolved quantities Problems in NR ② • Existence ofut(Lorentz factor) • There is a procedure to solve nonlinear equations forut • Total energy (depends on ut) is evolved • There is a procedure to recover T or (P) from the evolved total energy • The above two procedure couples in a complex manner Nonlinear eq. with EOS table search e depends on ut Nonlinear eq. with EOS table search

12. Problems in NR ② • Due to these complexity, solving the equation implicitly is very hard in NR • Iteration includes two loops : no guarantee for convergence • Explicit scheme is required • A resolution : GR leakage scheme • Utilizing the fact that ‘ leakage timescale ’ is much longer than the weak timescale • Approximate treatment of neutrino cooling based on ‘ leakage time scale ’

13. GR leakage scheme (hydro) • Basic equation : • Energy-momentum tensor of neutrinos : • ‘Trapped neutrino’ and ‘Streaming neutrino’ parts • Trapped neutrino part is included into Fluid part • The equation to be solved Only leakage timescale appears

14. GR leakage scheme (Lepton conservation) • Source terms: • local rates (electron capture, pair processes : weak timescale) • leak out of trapped neutrinos to be streaming neutrinos (leakage timescale) • Problem : • How to treat the local rates • andβ-equilibrium

15. GR leakage scheme (Lepton conservation) • In the hot matter region, weak timescale becomes too short and the source term becomes too large • We introduce some limiters to the source terms • Assumption: Ynu’s cannot exceed the corresponding values at β-equilibrium • First, trial evolution of total lepton fraction Yl • Note that the source term is in leakage timescale • Under the assumption of β-equilibrium, Ynubeta’s are calculated. These provide the limiters • Second, evolution of lepton fractions • If the local rates are below the limiters, we simply evolve them • On the other hand, if the local rates exceeds the limiters, the values at β-equilibrium are adopted

16. GR leakage scheme (Lepton conservation) • Important issues : • Use the EOS table with arguments (ρ,Yl, T) • In this case, only one dimensional search is required • Otherwise two dimensional search (Yl, e) ⇒ (Ye, T) required, which in general may be convergent

17. Summary of microphysics • EOS: Tabulated EOS can be used • Currently Shen EOS + electrons + radiation • Weak rates • Electron capture:FFN1985, rate on NSE back ground • e±annihilation: Cooperstein et al. 1985, Itoh et al. 1996 • plasmon decay: Ruffert et al. 1996, Itoh et al. 1996 • Bremsstrahlung: Burrows et al. 2006, Itoh et al. 1996 • Neutrino leakage • Opacity based on Burrows et al. 2006 • (n, p, A) scattering • Including correction such as ion-ion correlation • (n, p, A) absorption

18. GR leakage works well • Neutrino luminosities consistent with result by 1D GR radiation hydro (Liebendoefer et al. 04) • Collapse of 15 Msun model by WHW02 • Besides convection induced modulation in luminosities • Neutrino luminosities in BNS merger and GRB will be estimated Results by Sekiguchi (2009) Liebendoerfer et al. (2004)

19. GR leakage works well • Results consistent with Liebendorfer et al. 2004

20. Convective activities

22. Applications : PopIII core collapse

23. BH Time Profile γ線 0 10 20 30 40 50 [s] High energy astrophysics: GRB central engines: BH+DiskStellar core collapse NS-NS/BH merger Jet Disk 1051erg/s<Liso<1054erg/s Most violent explosion in the universe

24. Hot disk Gamma-ray burst by neutrino pair annhilation

25. PopIII core collapse • BH formation with microphysics • black hole excision technique for hydrodynamics & microphysics • puncture evolution for geometry • Initial condition • Simplified model (S = Ye = const core) • S=7kB, 8kB; Ye=0.5 Ye entropy per baryon ( kB ) density log( g/cm3 )

26. Collapse dynamics : Weak bounce • Do not directly collapse to BH • Weak bounce • At bounce • ρ ~ 1013 g/cm3 • subnuclear ! • T ~ 18 MeV • Ye ~ 0.2

27. Bounce due to gas pressure • He → 2p + 2n • Gas pressure (Γ=5/3) increase • Indeed Γth >4/3 • Gas pressure dominates at ρ~1013g/cm3, T~18 MeV • EOS becomes stiffer ⇒ weak bounce

28. Collapse dynamics : Disk formation Neutrino emission rate [erg/cm3/s]

29. Collapse dynamics : Disk formation Neutrino emission rate [erg/cm3/s]

30. Final state of the simulation • Neutrino torus is formed • Density along the rotational axis > 108 g/cm3 • Higher for the formation of GRB fireball via ν-annihilation density

31. Final state of the simulation • Some fluctuation can be seen in Ye • Heavy elements are completely dissociated Ye

32. Neutrino emission Neutrino emission from the torus AH formation

33. Expected neutrino pair annihilation Neutrino luminosity ~ 1054 erg/s Average energy ~ 20-30MeV According to the results by Setiawan et al. pair annihilation luminosity of >1052 erg/s is expected Setiawan et al. (2005) To estimate the pair annihilation rates more accurately, Ray-tracing calculations are planned (Harikae, Sekiguchi, Takiwaki, Kotake)

34. ~300km Neutrino interaction is important The results in which first order correction to the neutron / proton magnetic moment is considered

35. Evolution of BH mass • Assuming Kerr BH geometry • BH mass = 6~7 Msolar • Rotational energy = MBH – Mirr～ 1054 erg • If strong magnetic field exists, the rotational energy can be extracted • Mass accretion rates is still large as > several Msolar/s

36. Summary • Effects of GR cannot be ignored • In NR, to treat weak interactions such as electron capture and neutrino cooling is difficult • We developed GR leakage scheme in which these can be treated approximately • GR leakage scheme works well • We applied the GR leakage code to collapse of PopIII core • Neutrino luminosity is sufficient to produce the GRB fireball by neutrino pair annihilation

37. Very preliminary result (just started) • Simulations are ongoing with electron capture and GR neutrino leakage • Some room for improvement in EOS construction, atmosphere treatments, etc • If you have good EOS, let us use it ! Density profile