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Gravitational wave detection and numerical relativity
曹周键中国科学院数学与系统科学研究院
2015-9-8
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)
10107
10104
GR = Newton Theory + terms (v) + terms (v^2) + ……
10107
6107
10107
6102
Einstein and GW1915, general relativity
1916-2, based on post-Newtonian approximation, claimed “there are no gravitational waves analogous to light waves”
1916-10, based on linear approximation found monopole radiation. 1918, corrected it to quadruple radiation
1936, showed that GW does not exist
Theory of GW1936-1962, debate
1962, Bondi convinced people the existence of GW
Theory of GWBondi’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 GW1969, Weber claimed the detection of GW. But people doubt it
1978, Hulse and Taylor confirmed the quadruple energy balance, implied the existence of GW
2015-2020, 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
RXJ1914.4+2456
Improve SNR
Einstein’s equation
TG 8•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
G T
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, 121103 (2001), 100 t.u.• B. Brugmann et al, PRL 92, 211101 (2004) 150 t.u.• F. Pretorius, PRL 95, 121101 (2005); M. Campanelli et al,
PRL 96, 111101 (2006); J. Baker et al, PRL 96, 111102 (2006), stably!!
• Penn State group, CQG 24, S33 (2007)• Jena group (Brugmann), PRD 76, 104015 (2007); PRD 77,
024027 (2008)• AEI group, PRL 99, 041102 (2007)• Tokyo group, PRD 78, 064054 (2008)• Our group, PRD 78, 124011 (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, 124011 (2008)]
new scheme
Different formalism admits different stability
new scheme
Our modification can reduce numerical noise
[Yo, Lin and Cao, PRD 86, 064027 (2012)]
Different formalism admits different accuracy
Our modification can improve the spin accuracy more than 7 times
[Yo, Cao, Lin and Pan, PRD 92, 024034 (2015)]
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Typically requiring ten of thousands floating point operations per grid point !!!
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Evolution PDE system of Einstein’s equationEinstein 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 (1000-10000)
• Computationally expensive on every grid point (less grid points, much more levels)
01.0~x
Mesh refinement
Cao, Yo, and Yu, 2007 Cao, Yo, and Yu, 2008
Example only, usually 12-16 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
• 2006-2009, AMR infrastructure• 2007-2008, DAGH + Einstein solver, work together with
NCKU• 2009-2012, AMR infrastructure + Einstein solver + GW
calculator + other tools (independent)• 2013-2014, 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 + 1024 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) 104029
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)