Gravitational wave detection and numerical relativity 曹周键 中国科学院数学与系统科学研究院 中国科学技术大学交叉学科理论研究中心

Content Gravitational wave, its detection and modeling Introduction to NR and AMSS-NCKU code Application to gravitational wave modeling Summary and prospect

GR and its test perihelion advance of mercury (1915, v≈ ) Light bending (1919, v≈ ) Gravitational redshift (1965, v≈ ) Gravitational time delay (1968, v≈ ) Indirect evidence of GW (1978, v≈ ) Gravitational draging (2010, v ≈ ) GW detection (?, v≈1) GR = Newton Theory + terms (v) + terms (v^2) + ……

Einstein and GW 1915, general relativity , based on post-Newtonian approximation, claimed “there are no gravitational waves analogous to light waves” , based on linear approximation found monopole radiation. 1918, corrected it to quadruple radiation 1936, showed that GW does not exist

Theory of GW , debate 1962, Bondi convinced people the existence of GW

Theory of GW Bondi’s boundary condition is an essential assumption in his work For Einstein’s Eq including cosmological constant 1.Bondi’s original boundary condition no GW any more [Ashtekar, Bonga and Kesavan, CQG, 2015] 2.New boundary condition Similar GW behavior to Bondi’s original work [He and Cao, IJMPD, 2015] The behavior of GW in different gravitational theory is different So GW detection is possible to test gravitational theory

Experiment of GW 1969, Weber claimed the detection of GW. But people doubt it 1978, Hulse and Taylor confirmed the quadruple energy balance, implied the existence of GW , AdvLIGO ?

What is GW geodesic deviation Do not need linearization Do not need perturbation

Importance of GW detection This will be an unprecedented direct test of general relativity, especially in the highly dynamical and non-linear strong-field regime Direct evidence for black holes, as well as give valuable information on stellar evolution theory and large scale structure formation and evolution in the universe Information for neutron star and particle physics ……

Importance of GW detection This will be an unprecedented direct test of general relativity, especially in the highly dynamical and non-linear strong-field regime Direct evidence for black holes, as well as give valuable information on stellar evolution theory and large scale structure formation and evolution in the universe Information for neutron star and particle physics …… Gravitational Wave Astronomy

Can we detect this signal?

Data analysis: Matched Filtering Data from detector Theoretical wave form (strongly dynamical spacetime, numerical method)

Data analysis and template

Roughly speaking, a good source model can improve the detection ability 10 to 100 times

Power of GW model RXJ Improve SNR

Einstein’s equation Geometry respect: metric; diffeomorphism invariant PDE respect: second order “hyperbolic” partial differential equation (coordinate dependent) Nonlinearity: is nonlinear functions of metric; depends on metric nonlinearly also Complexity: several thousands of terms

Exact solution Although “Exact Solutions of Einstein’s Field Equations” have near 700 pages, from 1915 till now, we have only two physically interesting solutions  Kerr solution: single rotating star (vacuum).  Friedmann-Robertson-Walker cosmology: homogenous isotropic universe.

Exact solution Although “Exact Solutions of Einstein’s Field Equations” have near 700 pages, from 1915 till now, we have only two physically interesting solutions  Kerr solution: single rotating star (vacuum).  Friedmann-Robertson-Walker cosmology: homogenous isotropic universe. For real atrophysical systems: no symetry at all !!!

Approximate methods Post-Newtonian method: slowly varied spacetime (while strongly dynamical spacetime reduce gravitational wave) Perturbation method: spacetime = known back ground + small field as perturbation (known back ground means we almost know the solution already, linearity approximation)

Approximate methods Post-Newtonian method: slowly varied spacetime (while strongly dynamical spacetime reduce gravitational wave) Perturbation method: spacetime = known back ground + small field as perturbation (known back ground means we almost know the solution already, linearity approximation) Weak GW cases

Numerical methods Numbers and + - * /

Stability problem Hahn and Lindquist, first BBH simulation (1964) Smarr, Eppley, Choptuik, …… P. Anninos, et al, first 3D BBH simulation, PRD 52, 2059 (1995) B. Brugmann, Int. J. Mod. Phys. D 8, 85 (1999), 35 t.u. S. Brandt et al, PRL 85, 5496 (2000), 50 t.u.

Numerical methods GW detection will be earlier than Numerical simulation of black hole collisions Kip Thorne, In 2000

Brief history of Stability problem J. Baker et al, PRL 87, (2001), 100 t.u. B. Brugmann et al, PRL 92, (2004) 150 t.u. F. Pretorius, PRL 95, (2005); M. Campanelli et al, PRL 96, (2006); J. Baker et al, PRL 96, (2006), stably!! Penn State group, CQG 24, S33 (2007) Jena group (Brugmann), PRD 76, (2007); PRD 77, (2008) AEI group, PRL 99, (2007) Tokyo group, PRD 78, (2008) Our group, PRD 78, (2008)

Formalism problem (gauge) Reality, solvable Num tech, coding Gauge, finite distance Numerical Relativity

Formalism problem

Our modification is more stable [Cao, Yo, and Yu, PRD 78, (2008)] new scheme Different formalism admits different stability

new scheme Our modification can reduce numerical noise [Yo, Lin and Cao, PRD 86, (2012)] Different formalism admits different accuracy Our modification can improve the spin accuracy more than 7 times [Yo, Cao, Lin and Pan, PRD 92, (2015)]

Evolution PDE system of Einstein’s equation Einstein summation convention Covariant derivative operator Ricci tensor and trace free notation Typically requiring ten of thousands floating point operations per grid point !!!

Evolution PDE system of Einstein’s equation Einstein summation convention Covariant derivative operator Ricci tensor and trace free notation Typically requiring ten of thousands floating point operations per grid point !!! Face to so massive computational request, Solvable?

Parallized Mesh refinement Several scales involved black hole (1)  separation of black holes (10) wave length of gravitational wave (50) asymptotic region ( ) Computationally expensive on every grid point (less grid points, much more levels)

Mesh refinement Cao, Yo, and Yu, 2007 Cao, Yo, and Yu, 2008 Example only, usually levels 3x64x64x64 3x128x128x64 Take the advantage of spacetime symmetry

Boundary treatment Real physical system, no boundary (non possible for numerics) Compactify --- energy piles up Artificial boundary (how to set BD condition) Radiative boundary condition [Shibata and Nakamura PRD ‘95] Fortunately, it is STABLE! but produce extra error!

Constraint preserving BD Smooth BD required by theory Reduce phase error 10 times Hilditch, Bernuzzi, Thierfelder, Cao, Tichy and Brugeman (2013)

NR code on the world

AMSS-NCKU code , AMR infrastructure , DAGH + Einstein solver, work together with NCKU , AMR infrastructure + Einstein solver + GW calculator + other tools (independent) , add GPU supporting, work with THU In 2009, Jena NR group named our code AMSS-NCKU In 2013, Einstein Toolkit leader gave us the pronunciation

AMSS-NCKU code 标准 BSSN 、非 GPU 部分已获得计 算机软件著作权

Parallel Scaling behavior 13x128x128x64, strong scaling test Cao, 2010 (MPI, OpenMP) Weak scaling of Einstein Toolkit Loffler’s talk, 2009

Test of AMSS-NCKU GPU code Titan: top 1 super computer around the world (now Tianhe 2) 1024x16 cores GPUs, Du Zhihui, 2013 The only GPU numerical relativity code to date

Structure of AMSS-NCKU GPU code Two groups MPI processes, one for cpu and one for gpu MPI + OpenMP + CUDA

Application of AMSS-NCKU code

Horizon corresponds to black hole

BBH source model EOB: phenomenological model, Sun Baosan and Pan Yi, 2013 NR: AMSS-NCKU simulation result, Cao, 2013

Different GW behavior between GR and f(R) Cao, Pablo, and Li, PRD 87 (2012) BBH merge faster in f(R), More complicated GW waveform show up in f(R)

Summary and Prospect GW detection is hard but important to science and theoretical model is criticaly important to the detection AMSS-NCKU NR code has been well developed for GW source modeling AMSS-NCKU code is portable to other astrophysical research including hydrodynamics and EM, which is needed by the GW source modeling of AdvLIGO (multi-messenger)