Numerical Relativity Simulations on Super-Computers

Transcription

1 FEATURE ARTICLES BULLETIN Numerical Relativity Simulations on Super-Computers YUICHIRO SEKIGUCHI YUKAWA INSTITUTE FOR THEORETICAL PHYSICS 1. INTRODUCTION: WHAT IS NUMERICAL RELATIVITY? Einstein s general theory of relativity contains the rules of gravity described by Einstein s equations, and provides the basis for modern theories of astrophysics and cosmology. The equations are among the most complicated seen in physics. For many years, many efforts have been made to develop techniques to solve them. Despite more than 70 years of intense study, only a handful of special solutions for Einstein s equations which are relevant for astrophysics and cosmology have been yielded. The theory remains largely untested, except in the weakgravitational-field, slow velocity regime. Also, many of the important dynamical scenarios thought to occur in nature have not been studied in detail. With the advent of supercomputers, however, the situation is now drastically changing; it is now possible to tackle these complicated equations numerically and explore astrophysically realistic, high-velocity, strong-field systems. This is the field of numerical relativity. In this article, we will review recent developments in numerical relativity simulations for black hole (BH) formation and emission of gravitational waves (GWs) from binary neutron stars (BNSs), which are among the most mysterious and exciting predictions made by general relativity. BHs with stellar-size mass are expected to form from stellar collapse when a very massive star exhausts its nuclear fuel and then collapses in on itself. This process may lead to a supernova explosion and the formation of a neutron star. But if the progenitor star is massive enough, its stronger self-gravity will pull it into a runaway collapse and a BH may be formed eventually. Details of this scenario are still uncertain and its clarification is one of the targets of numerical relativity. Gravitational waves are curvature perturbations (ripples in spacetime) that propagate with the speed of the light as a wave. Akin to how electromagnetic radiation is produced by an acceleration of charges, gravitational radiation is produced by an acceleration of mass-energy. The first detection of GWs will be achieved using nextgeneration gravitational-wave detectors such as advanced LIGO, advanced VIRGO, and KAGRA in the next decades [1]. For the direct detection of GWs, theoretical templates of waveforms are necessary. Many researchers now engage in this field of study, performing numerical relativity simulations. 2. SUMMARY OF NUMERICAL RELATIVITY Before describing our latest results, let us review the current status of numerical relativity. Figure 1 shows a sche- Fig. 1: A schematic summary of numerical relativity. Due to the recent rigorous efforts, most of the ingredients have been developed and we are now at a point such that a realistic simulation could be performed. 10

2 OCTOBER 2013 VOL. 23 NO. 5 FEATURE ARTICLES Recall that these degrees of freedom are originated in the general covariance and are different from the choice of coordinate system such as Cartesian or spherical polar coordinate. The lapse and shift need to be specified during simulations. Conversely speaking, we can utilize them to constitute a good coordinate system (e.g., no appearance of both physical and coordinate singularities and resolution of the frame dragging in the binarystar simulations). Based on geometric insights and numerical experimentation, there have been several good gauge conditions for treating important astrophysical problems [3]. Fig. 2: A scematic picture of 3+1 decomposition of spacetime. The line element between two slices t and (t+dt) is shown. Here n a is the unit normal to the slice and t a is the time axis. matic summary of numerical relativity simulations. As can be seen, various elements are included in numerical relativity. - No absolute background spacetime: In numerical simulations, we want to evolve physical quantities on a spacelike hypersurface (a slice) at a certain time to those on a future slice. In a Newtonian framework, the time evolution as a matter of course indicates the integration along the absolute time. In general relativity, in contrast, the notion of time and space is not absolute but relative so that the time and spatial axes must be specified during numerical simulations. Moreover, Einstein s equations in their covariant form contain complicated mixed derivatives in time and space. Thus, to solve Einstein s equations, we must decompose the spacetime into time and space (3+1 decomposition or 3+1 formalism) and rewrite Einstein s equations as an initial value problem (or Cauchy problem) [2]. - Coordinate axes must be specified: Associated with the general covariance equipped with general relativity, we have four degrees of freedom in specifying the 3+1 decomposition: one is the lapse function,, which determines the way how the time proceeds (time slicing); and the others are shift vector, a, which controls the spatial direction of the time axis (spatial gauge). Then, the spacetime metric, g ab, is decomposed in terms of lapse, shift and the induced metric on a slice ab (See Fig. 2). - Einstein s equations as a Cauchy problem: In considering the Cauchy problem of Einstein s equations, it is important to note that they are a constrained system, like Maxwell s equations. Maxwell s equations in the covariant form can be decomposed into Gauss s law, the no-monopole condition, Faraday s law, and Ampere s law. Here, Gauss s law and the no-monopole condition involve only spatial derivatives of the fields and hold at each instant of time. They therefore constrain any possible configurations of the fields, and are correspondingly called the constraint conditions. On the other hand, Faraday s law and Ampere s law describe how the fields evolve forward in time and are therefore called the evolution equations. Likewise, Einstein s equations in the covariant form can be decomposed into constraint and evolution equations [4]. - Construction of initial data: Before starting simulations, we must specify gravitational fields on the initial slice that are compatible with the constraint conditions. The constraints are coupled nonlinear elliptic-type equations and it is computationally expensive to solve them numerically. Also, we need astrophysically realistic configuration of the gravitational and matter fields. Construction of such initial data is one of the important issues in numerical relativity [5]. - Reformulating the original 3+1 system: The existence of the constraint conditions poses another problem in numerical simulations. Because it is hard to solve the constraint equations, they are usually solved only at the beginning of simulations. Mathematically, it is guaranteed that the constraint conditions are always satisfied in the later time provided that the evolution equations are solved correctly and the constraint conditions are satisfied initially. Unfortunately, however, it is not possible to solve the equations without numerical error 11

3 FEATURE ARTICLES BULLETIN in numerical simulations. Then, violation of constraints increases with time and simulations break down eventually in the original version of formulation [4]. There are several reformulations of the 3+1 decomposition that have proven to be successful in a number of numerical simulations. The most popular and successful one is the BSSN (Baumgarte, Shapiro, Shibata Nakamura) formulation [6]. In the BSSN formulation, decompositions of geometrical variables and new auxiliary variables which is essentially the spatial derivative of ab are introduced. The equations are rewritten using these new variables and the constraint equations. Then the hyperbolic property of the original equations is improved by suppressing the violation of constraints, thus making it possible to perform long-term simulations stably [7]. - Treatment of BH: BH interiors contain curvature singularities which cause simulations to terminate prematurely. Special care therefore needs to be taken to avoid encountering such a singularity. This problem was one of the biggest problems in numerical relativity until 2005, when the first successful simulation of binary BH mergers was reported by Frans Pretorius [8]. He used BH excision in which an interior region of a BH is excised from the computational region [9]. Another breakthrough came with the so-called moving puncture method without excision, first suggested independently by Baker et al. [10] and Campanelli et al. [11] in Implemented with appropriate gauge conditions, this moving puncture method works very well; evolving spacetime with BHs, one of the biggest problems until 2006, is now not very hard to do. - Matter fields: Besides gravity, the description of various forms of matter is essential because most relativistic astrophysical systems involve matter sources: stars, accretion flows, jets, and so on. Relativistic hydrodynamic (HD) matter is particularly important in many astrophysical systems. Magnetic fields also play an important role. In many astrophysical applications, the gas is highly ionized and a very excellent conductor of current. Then the ideal magneto-hydrodynamics (MHD) should be applied to treat them. When photons and neutrinos are involved, we must include a radiation stress-energy tensor, which is determined by solving equations of radiation-hydrodynamics (RadHD: hydrodynamic equations coupled with a radiation transfer equation). In solving the hydrodynamic evolution equations, we need an equation of state (EOS) in order to close the system. Ideally, EOS should be obtained from microphysical theories such as nuclear or hadron physics. However, in the early epoch, some particularly simple EOS, like polytrope and ideal gas, were widely used in numerical relativity groups. Only very recently simulations with EOS based on the microphysical theories have begun. In many astrophysical phenomena shocks are formed in general. Classical finite difference schemes present deficiencies in dealing with the shock discontinuities and we need so-called high-resolution shock-capturing (HRSC) schemes. Basic issues in developing HRSC schemes for general relativistic hydrodynamics and magnetohydrodynamics have been settled now [12]. Also, cuttingedge studies [13] in solving radiation transfer equations for neutrinos enabled us to perform general relativistic radiation-hydrodynamic simulations with detailed microphysics such as weak interactions and nuclear-theory based finite temperature EOS. 3. WE NEED SUPER-COMPUTERS In the following, we briefly review our first results of numerical relativity simulations for BH formation [14] and BNS merger [15], which are performed incorporating both a finite-temperature EOS [16] and neutrino cooling [17]. For BH formation, we need to resolve the BH horizon size (~km) while at the same time the much bigger whole stellar core (~10000km) should also be taken into account. In the BNS merger, similarly, two different scales, the size of a neutron star (~10km) and the wavelength of GWs (~500km) are treated. Consequently, numerical relativity simulations including the whole of the ingredients summarized in Fig. 1 are computationally very expensive and we need supercomputers. Actually, all four known forces in nature play important roles in numerical relativity simulations and our numerical code is very complicated. Elaborate tuning of such a complicated code to draw out a good performance of super-computer is now getting more important. Our simulations were done using a number of supercomputers in the Yukawa Institute for Theoretical Physics (HITACHI SR16000, NEC SX-9), the Center for Computational Astrophysics of NAOJ (Cray XT4, Cray XC30, and NEC SX-9), and the University of Tokyo (Fujitsu FX10), and K computer at the RIKEN. 12

4 OCTOBER 2013 VOL. 23 NO. 5 FEATURE ARTICLES Fig. 3: Density contour (log of g/cm3) and velocity fields in x-z plane at selected time slices of a BH-accretion disk system formed after the collapse of rapidly rotating massive stellar core. Due to the formation of a torus-shaped shock, falling materials are accumulated into the central neutron star via oblique shock and dissipate their energy there, which enhances the energy conversion (top-left). As a result, pressure gradient in the polar region of the neutron star increases and an outflow is driven (top-right). The neutron star eventually collapses to a BH and an accretion disk is formed around it (bottom-left). The resulting system is not stationary but highly dynamical (bottom-right). 4. BH FORMATION AND GAMMA-RAY BURSTS While BH formation is on its own very interesting subject in the field, recognition of its importance has increased due to recent observations which have indicated that the formation of a BH is likely to be associated with the central engines of gamma-ray bursts (GRBs). Although progenitors of GRBs have not been fully clarified yet, there is accumulating observational evidence that GRBs, which are of long-duration and soft in spectra, are associated with collapse of massive stars [18]. The observational association between long GRBs and supernovae has provided strong support to a scenario, the so-called collapsar model, which claims that long GRBs originate in the collapse of a massive, rapidly rotating star which forms a BH and a disk[19]. Among the characteristic properties of GRBs, their huge energetics and rapid time variability should be noted. Any model of a GRB central engine therefore should explain these properties. Huge gravitational binding energy released in the collapse is stored as thermal energy in the accretion disk and is carried away by neutrinos. Then pair annihilation of neutrinos could supply sufficient energy to make GRBs. Also, huge rotational energy of the BH could be 13

5 FEATURE ARTICLES BULLETIN point of view. It is more than just sophistication; it brings in much richer phenomena. The formation of a torus-shaped shock acts as an accumulator of the matter into the central region by bending the flow line via the oblique shock. Then, the kinetic energy is more efficiently converted to thermal energy. This mechanism is advantageous to explain the huge energetics of GRBs. Fig. 4: Contour of entropy per baryon (k B) in x-z plane at a selected time slice. Strong velocity shears are developed at the interface of the thick torus and accretion flows. extracted and used to induce GRBs, if strong magnetic fields of order gauss are present. To investigate the above scenarios, we performed numerical relativity simulations taking into account the detailed microphysics [14]. The latest simulations adopting a realistic progenitor star model [20] clarified that BH formation proceeds in a highly dynamical manner accompanying torus-shaped shock formation, convection, Kelvin- Helmholtz instability and outflows (Fig. 4). All these features were not observed in the simulation performed in 2005 [21] in which any of microphysical processes were taken into account. This fact shows the importance of performing realistic simulations even in a qualitative The convective activities are activated because the energy balance is not satisfied: locally dissipated energy increases due to the above mass accumulation mechanism, while cooling by advection onto the BH decreases due to the rapid rotation so that additional cooling mechanisms such as convection and outflows are inevitable. Kelvin- Helmholtz instability occurs because there are strong velocity shears at the boundary of the torus and infalling flows (Fig. 4). As a result of these fluid instabilities, neutrino luminosities show violent time variability (Fig. 5). Thus, the convection and Kelvin-Helmholtz instability naturally introduce time variability to the system, which may by associated with the observed variability of GRB light curves. 5. BINARY NEUTRON STAR (BNS) MERGERS Coalescence of BNSs is one of the most promising sources for next-generation kilo-meter-size gravitational-wave detectors, and also a possible candidate for the progenitor of short-hard GRBs [22]. Observations of GWs will provide us unique information regarding the interiors of Fig. 5: Time evolution of neutrino luminosities before (left panel) and after (right panel) the BH formation. Note the difference of the vertical scale in two panels: luminosities after the BH formation is in log scale. 14

6 OCTOBER 2013 VOL. 23 NO. 5 FEATURE ARTICLES Fig. 6: Contours of density (top panels), temperature (middle panel), and anti-electron neutrino emissivity (bottom panels) at a selected time slice after the merger for x-y (left panels) and x-z (right panels) planes. 15

7 FEATURE ARTICLES BULLETIN Fig. 7: Gravitational waves (upper panel) and neutrino luminosities as functions of time. Before the merger, the waveform is characterized by the so-called chirp signal and neutrino luminosities are essentially zero. After the merger, GWs are emitted from the rotating and oscillating hypermassive neutron star. Neutrino luminosities increase after the merger due to the compressional and shock heating. neutron stars, which otherwise cannot be obtained [23]. With these motivations, numerical relativity simulations have been extensively performed in the past decade since the first success was recorded in 2000 [24]. BNSs evolve due to a gravitational radiation reaction and eventually merge. Before the merger sets in, each neutron star is cold. By contrast, after the merger sets in, shocks are generated by hydrodynamic interactions and the maximum temperature increases to ~30 50MeV, and hence, copious neutrinos are emitted (see Fig. 7, which shows the result for an equal-mass binary with individual mass of 1. 5 Msolar). Note that the neutrino emissivity is higher near the polar surface, which is a favorable feature for the merger hypothesis of short-hard GRBs. The neutrino luminosities in the early evolution are quite huge at ergs/s, which seem to be sufficiently large to explain the GRB energetics (lower panel of Fig. 7). It is remarkable that even for the relatively high mass binary, the dynamical outcome after the merger is a hypermassive neutron star, for a stiff EOS [16] adopted here which is compatible with the latest observations of massive neutron stars [25]. The primary reason is that the thermal pressure plays an important role for sustaining the hypermassive neutron star. Accordingly its lifetime would be determined by the time scale of the subsequent neutrino cooling, which is much longer than its dynamical time scale. Gravitational waveforms emitted from the system are shown in Fig. 7 (upper panel). If the BNS merger happens at a relatively short source distance of ~20Mpc or is located in an optimistic direction with a distance of ~50Mpc, such GWs may be detected by advanced gravitational-wave detectors with the signal-to-noise ratio of 5, and the hypermassive neutron star formation will be confirmed. Furthermore, information of the physics of neutron star matter would be extracted from the waveforms and the characteristic frequency[26]. 6. SUMMARY AND FUTURE PROSPECTS We have reviewed the current status of numerical relativity simulations and our latest results of rotating stellar core collapses to BH and BNS mergers, performed incorporating a finite-temperature EOS and neutrino cooling effects. Recently, we have developed a formulation of general relativistic radiation transfer [13] based on a moment formalism. Also, we have succeeded in implementing a code which can solve the neutrino transfer with a detailed microphysics. Using this code, we plan to perform 16

8 OCTOBER 2013 VOL. 23 NO. 5 FEATURE ARTICLES simulations of the stellar core collapse to BH and BNS and BH-neutron star mergers. Fruitful and novel results will be reported in the near future. ACKNOWLEDGEMENTS This work was supported by a Grant-in-Aid for Scientific Research ( , , , ) and the HPCI Strategic Program of Japanese MEXT. References [1] J. Abadie et al., Nucl. Insturm. Methods Phys. Res. Sect. A 624 (2010) 223; T. Accadia et al., Classical and Quantum Gravity 28 (2011) ; K. Kuroda et al. Classical and Quantum Gravity 27 (2010) [2] e.g., Y. Chouet-Bruhat General Relativity and the Einstein Equations (Oxford University Press, 2009) [3] e.g., E. Gourgoulhon 3+1 Formalism in General Relativity: Bases of Numerical Relativity (Springer, 2012); T. W. Baungarte and S. L. Shapiro, Numerical Relativity: Solving Einstein s Equations on the Computer (Cambridge University Press, 2010) [4] R. Arnowitt, S. Deser, and C. W. Misner The dynamics of general relativity, in Gravitation : An Introduction to Current Research, ed. L. Witten (Wiley, 1962); J. K. York, Kinematics and Dynamics of General Relativity, in Sources of Gravitational Radiation, ed. L. Smarr, (Cambridge University Press, 1979). [5] e.g., G. Cook, Living Reviews in Relativity 3 (2000) 5; E. Gourgoulhon. Journal of Physics: Conference Series 91 (2007) [6] M. Shibata and T. Nakamura, Physical Review D 52 (1995) 5428; T. W. Baumgarte and S. L. Shapiro, Physical Review D 59 (1999) [7] e.g., M. Alcubierre, Introduction to 3+1 Numerical Relativity (Oxford University Press, 2008) [8] F. Pretoruis, Physical Review Letters 95 (2005) ; Classical and Quantum Gravity 22 (2005) 425 [9] Unruh, unpublished (1984) ; J. Thornburg, Classical and Quantum Gravity 4 (1987) [10] Baker et al., Physical Review Letters 96 (2006) [11] Campanelli et al., Physical Review Letters 96 (2006) [12] e.g., J. A. Font, Living Reviews in Relativity 11 (2008) 7. [13] e.g., M. Shibata et al. Progress of Theoretical Physics 125 (2011) 1255; M. Shibata and Y. Sekiguchi, Progress of Theoretical Physics 127 (2012) 535 [14] Y. Sekiguchi and M. Shibata, Astrophysical Journal 737 (2011) 6; Y. Sekiguchi et al., Progress of Theoretical and Experimental Physics 01 (2012) A304. [15] Y. Sekiguchi et al., Physical Review Letters 107 (2011) ; 107 (2011) [16] H. Shen et al., Nuclear Physics A 637 (1998) 435; Astrophysical Journal Supplement 197 (2011) 20. [17] Y. Sekiguchi, Progress of Theoretical Physics 124 (2010) 331; Classical and Quantum Gravity 27 (2010) 331. [18] e.g., S. E. Woosley and J. S. Bloom, Anural Review of Astronomy and Astrophysics 44 (2006) 507 [19] S. E. Woosley, Astrophysical Journal 405 (1993) 273; A. I. MacFadyen and S. E. Woosley, Astrophysical Journal 524 (1999) 262 [20] H. Umeda and K. Nomoto, Astrophysical Journal 673 (2008) [21] Y. Sekiguchi and M. Shibata, Physical Review D 71 (2005) [22] R. B. P. Narayan and T. Piran, Astrophysical Journal Letters 395 (1992) 83: E. Nakar, Physics Report 442 (2007) 166. [23] J. S. Reed et al., Physical Review D 79 (2009) ; Kiuchi et al., Physical Review Letters 104 (2010); [24] M. Shibata and K. Uryu, Physical Review D 61 (2000) [25] P. Demorest et al. Nature 467 (2010) 1081; J. Antoniadis et al. Science 340 (2013) 448. [26] A. Bauswein and H. T. Janka, Physical Review Letters 108 (2012) Yuichiro Sekiguchi received his PhD degree from the University of Tokyo in From 2008 to 2010, he worked at National Astronomical Observatory of Japan as a postdoctoral researcher. His research interests center on general relativistic physics and astrophysics using computer simulations. He was awarded the Young Scientist Award of the Physical Society of Japan in 2013 due to his contributions in this field. He is now a research assistant professor at the Yukawa Institute for Theoretical Physics. 17