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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Evolution PDE system of Einstein’s equationEinstein summation convention

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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)