1. 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

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

3. GR and its test perihelion advance of mercury (1915, v≈ )
GR = Newton Theory + terms (v) + terms (v^2) + ……
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)
GPS v = 2 x 10^(-10)

4. Einstein and GW 1915, general relativity
1916-2, 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

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

6. Theory of GW
Bondi’s boundary condition is an essential assumption in his work
For Einstein’s Eq including cosmological constant
Bondi’s original boundary condition
no GW any more [Ashtekar, Bonga and Kesavan, CQG, 2015]
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

7. Experiment of GW confirmed the quadruple energy balance,
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 ?

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

9. 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 ……

10. 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

12. Can we detect this signal?

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

14. Data analysis and template

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

16. Power of GW model
Improve SNR
RXJ

17. 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

18. 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.
Why not exactly solve it?

19. For real atrophysical systems: no symetry at all !!!
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 !!!
Why not exactly solve it?

20. 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)

21. Weak GW cases 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

22. Numerical methods
Numbers and
+ - * /

23. 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.
1995: National center for supercomputing Applications + Washington university + UIUC

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

25. 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)
1995: National center for supercomputing Applications + Washington university + UIUC

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

27. Formalism problem

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

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

30. 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 per time step
Typically requiring ten of thousands floating point operations per grid point !!!

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

32. 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)

33. Mesh refinement Take the advantage of spacetime symmetry
Example only, usually levels
3x64x64x x128x128x64
Cao, Yo, and Yu, 2007
Cao, Yo, and Yu, 2008

34. 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!

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

36. NR code on the world

37. AMSS-NCKU code 2006-2009, 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

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

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

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

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

42. Application of AMSS-NCKU code

43. Horizon corresponds to black hole

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

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

46. 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)