Introduction of Numerical Relativity

Masaru Shibata YITP, Kyoto University

Content

I.  Introduction (long): background II.  Outline of numerical relativity III.  Latest progress

What is numerical relativity ?

Gµν = 8πGc4Tµν

∇µTνµ = 0

∇µGνµ = 0( )

Spacetime dynamics GR Matter dynamics

4

Furthermore in Numerical Relativity

∇µTνµ = 0

∇µ ρuµ( ) = 0∇µ ρuµYl( ) =Ql∇µF

µν = −4π jν

∇[µFνλ ] = 0

pα∂α f + !pα ∂f∂pα

= S

+EOS

$

%

&&&&&&&&&&&&&

'

(

)))))))))))))

Relativistic object = high-density, high-temperature All 4 forces in nature often come into play and hence many eqs. have to be solved

Composition evolution

Maxwell’s eq

Radiation transfer

Continuity equation

Major roles in physics & astrophysics

•  Predicting gravitational waveforms from promising gravitational-wave sources

•  Predicting electromagnetic signals from promising gravitational-wave sources

•  Describing high-energy phenomena such as core-collapse supernovae & GRB

•  Predicting new phenomena in GR •  AdS/CFT, high-dimensional gravity ….

Major roles in physics & astrophysics

•  Predicting gravitational waveforms from promising gravitational-wave sources

•  Predicting electromagnetic signals from promising gravitational-wave sources

•  Describing high-energy phenomena such as core-collapse supernovae & GRB

•  Predicting new phenomena in GR •  AdS/CFT, high-dimensional gravity ….

Currently most important subjects

Gravitational wave detectors LIGO: 2015/9 ~

VIRGO: 2016 ~?

KAGRA: 2018~ ?

Planned sensitivity of advanced LIGO arXiv:1304.0670v

~10Hz—several kHz

Impact of gravitational-wave detection

•  First direct detection of gravitational waves •  First direct observation of black-hole formation •  First detection of coalescing binary black holes •  Revealing the central engine of short GRB •  Constraining the neutron-star equation of state •  Revealing the origin of heavy elements by rapid

neutron capture processes (such gold, silver ..) •  Examining the robustness of general relativity

and hopefully discovering violation of GR

GW    detec(on    will    provide    a    new    tool    for      exploring    a    variety    of    unsolved    issues  !    

Detecting gravitational waves needs accurate theoretical prediction

-2e-21

-1e-21

0

1e-21

2e-21

2.96 2.98 3 3.02 3.04 3.06 3.08 3.1

h

t (s)

Expected noise curve of KAGRA + hypothetical signal

Detecting gravitational waves needs accurate theoretical prediction

-2e-21

-1e-21

0

1e-21

2e-21

2.96 2.98 3 3.02 3.04 3.06 3.08 3.1

h

t (s)

SNR~10 event

Expected sensitivity of adv LIGO

NS-NS@100Mpc：Average: SNR~10

arXiv:1304.0670v

Detecting gravitational waves needs accurate theoretical prediction

-2e-21

-1e-21

0

1e-21

2e-21

2.96 2.98 3 3.02 3.04 3.06 3.08 3.1

h

t (s)

SNR~10 event

Detection will be achieved only by taking cross correlation with theoretical waveforms

Numerical relativity is necessary for deriving accurate waveforms

Many aspects of binary neutron stars 0. 7 coalescing NS-NS binaries have been observed 1.  Potential laboratory for studying high-density

nuclear matter poorly understood 2.  Promising progenitors of short-hard GRBs 3.  Possible origins of r-process heavy elements

?

Gravitational-wave + Electromagnetic obs will contribute to solving all these issues

II Outline of ���numerical relativity

• Formulation • Slicing

16

Einstein’s equation = hyperbolic equations

Gµν = Rµν −12gµνR = 8π G

c4Tµν

Gµν : Einstein tensor

Rµν : Ricci tensor

"#$

%$

de-Donder Gauge: ∂α −ggαβ( ) = 0

⇒ Gµν = −ggαβ∂α∂β −ggµν( )+O ∂gµν( )2(

)*+,-

Einstein’s eq=Wave equation

e.g., Landau-Lifshitz, “Classical fields of theory”

Essence of numerical relativity

•  Einstein’s equation = hyperbolic equation à The basic equations are similar to massless scalar-field equation

•  Metric is smooth à We can use well-known numerical methods like finite-differencing methods based on Taylor expansion or other analytic expansions •  We have to solve the hyperbolic equation

for unknown spacetime manifold

−∂t2 +Δ( )φ = S

18

General dynamical spacetime

At each time step, time and space slice have to be chosen appropriately

Time

Unknown manifold

19

Schwarzschild spacetime

t is the time coordinate.

t is a radial coordinate

( )1

2 2 2 2 2 2 2 22

2 21 sinGM GMds c dt dr r d dr rc

θ θ ϕ−

⎛ ⎞ ⎛ ⎞= − − + − + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Time has to be always “time” for dynamical evolution à Use ADM (3+1) formalism

t = const

Coordinate singularity

20

2 Draw timelike unit normal wrt spacelike surface, and choose lapse

1  Choose a spacelike surface & give space-metric

ADM formalism (3+1 formalism)

4 Evolve γij & Kij

t n

Time

Space

t

t + Δt iβ

αΔt

3  Choose time axes at each point

γ ij , Kij( )Kij ~ !γ ij : Talk later

nn = -1

21

The problem is not so simple: Linearized Einstein’s equation can tell us

Linearized Einstein equations with gtt=−1 & gti = 0

gij = γ ij = δij + hij ; |hij |≪1

Evolution eq. : ""hij = Δhij − hik ,kj − hjk ,ki + hkk ,ij

This causes a serious problem: Not only propagation (physical) mode but also non-propagation (unphysical) mode appear due to tiny numerical error Need reformulation

22

Analysis Decomposition:

hij = Aδij +C,ij + 2B(i , j ) + hijTT

A, C : scalar, Bi : vector, hijTT : tensor

(Definition: Bi ,i = 0, hij , jTT = hii

TT = 0)

⇒ Evolution eqs. :

!!hijTT = Δhij

TT

!!A = ΔA!!B(i , j ) = 0!!C = A

#

$

%%

&

%%

Wave equation

local equation

23

Pathological Solution

Hamiltonian constraint: ΔA = 0However, if constraint is violated and A ≠ 0 initially,

A = Ylm!!flm(r − t)+ !!g(r + t)

rl ,m∑

⇒C = Ylmf (r − t)+ g(r + t)

r+C1t +C2∑

Constraint violation is serious

Small error in A results in serious error in C

24

One prescription: BSSN formalism Essence •  Erase the pathological modes

Define new variablesFi := hij , jΦ := hii

"#$

%$

and rewrite basic eq. as!!hij = Δhij − Fi , j − Fj ,i +Φ,ij

Evolution eqs. for Fi and Φ ?

Consider them as independent variables

Wave equations

25

Reformulation using constraint equations

Momentum constraint: !hij , j − !hjj ,i = 0

⇒ !Fi − !Φ,i = 0 : Evolution eq for FiTrace of !!hij = Δhij − Fi , j − Fj ,i +Φ,ij

⇒ !!Φ = 2ΔΦ− 2Fi ,iHamiltonian constraint: ΔΦ− Fi ,i = 0

⇒ !!Φ = 0 ⇒ !Φ = 0 ⇒ !Fi = 0

⇒ !!hij = Δhij

The problem disappears !!

Decouple from hij

26

Reformulation increasing the variables and using constraints

Standard ADM: γ ij , Kij( ) ⇓BSSN formulation:

γ = det γ ij( ), !γ ij = γ −1/3γ ij , Γi = −γ , jij

K = Kkk , !Aij = γ

−1/3 Kij −13γ ijK

$

%&

'

()

12  Variables 12 Equations with 4 constraints

17 Variables 17 Equations with 9 constraints

27

Appropriate slicing is the key to avoid black-hole singularity

t Singularity

Or, effectively excised

Singularity

crash

Bad slicing

Singularity is not avoided: This is bad slicing

28

Singularity avoiding

29

Freeze: good ! But, …

30

Well-known possible good slice

Drawback: development of coordinate singularity

Maximal slice: K = 0 = !K ⇒Δα =αS γ ij ,Kij ,Tµν#$

%&

Estbrook et al. PRD 1973

dl2 = γ ijdxidx j : dl↑⇔ γ ij ↑

Singularity avoiding & excision is needed Freeze: good !

2 established formulations: •  Apparent horizon boundary condition (Unruh, Pretorius, Caltech-Cornell-CITA) •  Effective excision using special formulation (BSSN-puncture formulation)

Excision

31

III. Latest progress

1)  High-precession calculation for binary black hole inspiral, merger, and ringdown by Caltech-Cornell-CITA group

2)  Constraining equations of state by gravitational waves from binary neutron stars

3)  Mass ejection and nucleosynthesis of binary neutron star merger

1) BH-BH simulations by Caltech-Cornell-CITA group

•  Modified harmonic gauge formulation + apparent horizon excision

•  Numerical calculation by pseudo-spectral method (note: metric is a smooth function in vacuum) à quite accurate & efficient numerical results

•  Spin with S<~0.995M2 ; Mass ratio 1—10 ; ~ 100 orbits in half a year

•  Now, gravitational waveforms for a variety of parameter space are available

174 Waveforms by Mroue et al.

Orbits of BH-BH binary with spin 0.97

35

Lovelace+ 2012

Almost “exact” solution for 25.5 orbits

Modeling by Effective-one-body formalism

Taraccini et al. arXiv1311.2544

2) Constraining the nuclear-matter equation of state

•  The EOS for neutron-star matter is still poorly known

•  Merger of neutron-star binary could provide a great opportunity for constraining it

•  Specifically, gravitational waves will carry the information

1.5

2.0

10 km 12 km 14 km

Mass-radius relation for various EOS

2.5 M

ass (

sola

r)

Radius (km)

1.0

Strong constraint; but not strong enough

2.0

1.5

J0348+0432

Imprint of EOS on late inspiral waveform In a binary system, the tides raised on each NS depend on the deformability of that NS:

Courtesy    J.  Friedman

Stiff EOS = lager radius = large deformability

Soft EOS = small radius = small deformability

φ ~ −GMr

−3Iij

TFnin j

2r3: Iij

TF =O r−3( ) Lai et al. (1994)

-0.15-0.1

-0.05 0

0.05 0.1

0.15

0 10 20 30 40 50 60 70

h D

/ m

0

tret (ms)

APR4ALF2

H4

Latest Numerical Relativity Waveforms Last several phases are different

Hotokezaka et al. 2015

Mass: 1.35-1.35 solar mass

SOFT                              11.1km  MODERATE      12.4km  STIFF                              13.6km

Appreciable difference in phase

10-23

10-22

10-21

1

f hf (

r=10

0 M

pc)

f (kHz)

APR4ALF2

H4aLIGO

BH-BH

Overall spectrum by NR simulations

BH-BHby SpEC

SNR=0.3—1@ D=180Mpc

0.5

4hred − hblue( )

2

Sn f( )df

500Hz

1.2kHz

∫ ≈ 4 @ Deff =180Mpc

or SNR=20

$%&

'&

R=11.1km R=12.4km R=13.6km

3) Mass ejection & EM counterpart

BH

θobs

θjTidal Tail & Disk Wind

Ejecta−ISM Shock

Merger Ejecta

v ~ 0.1−0.3 c

Optical (hours−days)

KilonovaOptical (t ~ 1 day)

Jet−ISM Shock (Afterglow)

GRB(t ~ 0.1−1 s)

Radio (weeks−years)

Radio (years)

Metzger & Berger 2012

Key of ejecta: •  Mass •  Velocity •  neutron richness à opacity

Need to quantify for near-future observation

β-decay

From  Metzger  @  Santa  Brabara

Mass ejection of neutron rich matter à nucleosynthesis by rapid neutron capture à　

β-decay/fission à heat up à UV ~ IR (Li-Paczynski ‘98)

capture decayn βτ τ− −<

β-decay

44

Electron fraction (x-y)

Electron fraction (x-z)

νeνeνothers

High temperature ⇒ γγ→ e− + e+ , n+ e+ → p+νeNeutrino emission ⇒ n+ν→ p+ e−

Sekiguchi et al. (2015)

Ye

Neutrino luminosity

Simulation results for different EOSs

This-time case

A broad distribution of Ye is likely

Sekiguchi et al. 2015

Consistent with solar abundance pattern

Summary

•  Numerical relativity is now a mature field •  Many (astrophysically) accurate simulations are

ongoing à Templates of gravitational waves, light curve of EM counterparts …

•  Numerical relativity will contribute a lot to gravitational-wave projects and to solving unsolved issue in astronomy/astrophysics and nuclear physics

•  Hopefully, our numerical-relativity prediction will be confirmed by near-future observations !!