1 Testing Modified gravity and gravitational waves Takahiro Tanaka (Dept. of Phys./YITP Kyoto university) Gravitational waves

1

Testing Modified gravity and gravitational waves

Takahiro Tanaka(Dept. of Phys./YITP Kyoto university)

BH重力波Gravitational

waves

2

Motivation for modified gravity1) Incompleteness of General relativity

2) Dark energy problem

GR is non-renormalizabileSingularity formation after gravitational collapse

3) To test General relativityGR has been repeatedly tested since its first proposal.The precision of the test is getting higher and higher.

⇒ Do we need to understand what kind of modification is theoretically possible before experimental test?

Yes, especially in the era of gravitational wave observation!

3

Gravitation wave

detectors

eLISA(NGO)⇒DECIGO/BBO LIGO⇒adv LIGO

TAMA300,CLIO　⇒ KAGRA

4

(Moore, Cole, Berry http://www.ast.cam.ac.uk/~rhc26/sources/)

5

Inspiraling-coalescing binaries

• Inspiral phase (large separation)

Merging phase Numerical relativity

Ringing tail - quasi-normal oscillation of BH

for precision test of general relativity

Clean system: ～ point particlesInternal structure of stars is not so important

(Cutler et al, PRL 70 2984(1993))

for detection for parameter extraction(direction, mass, spin,…)

Accurate theoretical prediction of waveform is possible.

EOS of nuclear matter Electromagnetic counterpart

6

Theoretical prediction of GW waveform

fiefAfh 6/7

Waveform in Fourier space

uuftf cc 16

4

11

331

743

9

201

128

32 3/23/5

MM

DA

L

,,20

1 52536/5

3

3vOfMu

for quasi-circular inspiral

1PN 1.5PN

Standard post Newtonian approximation~ (v/c)expansion

4PN=(v/c)8 computation is ready(Blanchet, Living Rev.Rel.17:2Damour et al. Phys.Rev. D89 (2014) 064058)

7

• Precise determination of orbital parameters

• Mapping of the strong gravity region of BH spacetime

1 cycle phase difference is detectable

GR is correct in strong gravity regime?Many cycles of gravitational waves

from an inspiraling binary

uuuf 16

9

55

756

37151

128

3 3/23/23/5M

10

Scalar-tensor gravity

BD

ss

64

5 221 For binaries composed of similar NSs, 12

21 ≪ss

3vOfMu

Typical modification of GR

aaaBD mdRgxdS

,

,14

16

1

0ln/ln Gms aa BD

BDG23

24

scalar charge:

G-dependence of the gravitational binding energy

Dipole radiation ＝－ 1 PN frequency dependence

often discussed in the context of test by GWs

11

Spontaneous scalarization

,

,4

16

1 BDRgxdS

T 8EOM

As two NS get closer, “spontaneous scalarization” may happen. Sudden change of structure and starting scalar wave emission.

,,

4

2

1

16

1Rgxd

Effective potential for a star with radius R.

8T

2/R2

8T

2/R2

larger radiussmaller radius

More general model

is canonically normalized

12

Scalar-tensor gravity (conti)Current constraint on dipole

radiation: BD ＞ 2.4×104 J1141-6545 (NS(young pulsar)-WD ) (Bhat et al. arXiv:0804.0956)

LISA 1.4M◎NS+1000M◎BH: BD > 5×103

DECIGO1.4M◎NS+10M◎BH ： BD > 8×107

　　 collecting 104events at cosmological distances 　　

Constraint from future observations:

200SNR at 40Mpc corresponding to

(Yagi & TT, arXiv:0908.3283)

Einstein dilaton Gauss-Bonnet, Chern-Simons gravity

Scalar-tensor theory BH no hair

Turu-turu

RR

Rgxd

GS

GB

N*

4

RRRR *

VgxdGN

22

1 24

NS can have a scalar hair

• For constnat , these higher curvature terms are topological invariant. Hence, no effect on EOM. • Higher derivative becomes effective only in strong field.

RRRRRRGB 42

×(higher curvature)

Hairy BH - bold NS

• By contrast, BH solutions in EDGB and CS have scalar monopole and dipole, respectively.

"" 2R□

• NS in EDGB and CS do not have any scalar charge.

"" 23 RxdQ ""1 24 RxdT

topological invariant, which vanishes on topologically trivial spacetime.

EDGB ： monopole charge 　　 dipole radiation (-1PN order)

CS ： dipole charge 　　　 2PN order corrections

(Yagi, Stein, Yunes, Tanaka (2012))

Observational bounds• EDGB Cassini 　 cm103.1 122/1 EDGB (Amendola, Charmousis, Davis (2007))

• CS Gravity Probe B, LAGEOS 　(Ali-Haimound, Chen (2011))

cm10132/1 CS

cm104 52/1 EDGB

Future Ground-based GW observation SNR=20, 6Msol ＋ 12Msol

Low mass X-ray binary, A0620-00cm109.1 52/1 EDGB (Yagi, arXiv:1204.4524)

cm10 762/1 CS

Future Ground-based GW observation with favorable spin alignment: 100Mpc, a ～ 0.4M (This must be corrected…)

(Yagi, Stein, Yunes, TT, arXiv:1110.5950)

(Yagi, Yunes, TT, arXiv:1208.5102)

Pulsar : ideal clock

Test of GR by pulsar binaries

（ J.M. Weisberg, Nice and J.H. Taylor, arXiv:1011.0718)

Periastron advance due to GW emission

16

PSR B1913+16Hulse-Taylor binary dPorb/dt2.423×10-12

Test of GW generation

Agreement with GR

prediction

002.0997.0 GR

obs

P

P

Is there any possibility that gravitons disappear during its propagation over a cosmological distance?

We know that GWs are emitted from binaries.

But, then what can be a big surprise when we first detect GWs?

Just fast propagation of GWs can be realized in Lorentz violating models such as Einstein Æther theory.

Chern-Simons Modified Gravity

Right-handed and left-handed gravitational waves are amplified/decreased differently during propagation, depending on the frequencies. hhh iRL

2

1,

RRgxdS 4

8

Vgxd 22

24

(Yunes & Spergel, arXiv:0810.5541)

The origin of this effect is clear in the effective action. (Flanagan & Kamionkowski, arXiv:1208.4871)

,

22

,,

223

22 1

4 kk AARLA p

Ap hkh

Makdd

mS

The time variation of this factor affects the amplitude of GWs.

,

22

,,

223

2

14 kk AA

RLA pA

p hkhM

akddm

S

But the model has a ghost for large , and the variation of GW amplitude is significant only for marginally large .

(Ali-Haimoud, (2011)

: J0737-3039(double pulsar, periastron precession)

Current constraint on the evolution of the background scalar field :

16 Hz10

In other words, 1 modes are in the strong coupling regime,

which are outside the validity of effective field theory.

obs

2

emit

2,

emit,

obs 11pp

RLRL

MM

hh

Bi-gravity(De Felice, Nakamura, TT arXiv:1304.3920)

Bi-gravity

Both massive and massless gravitons exist. → oscillation-like phenomena? First question is whether or not we can construct a viable cosmological model.

22

,

16

~~

16 G

matter

G M

gLRgRg

M

L

Massive gravity0h□ 02 hm□

Simple graviton mass term is theoretically inconsistent → ghost, instability, etc.

22

1) Ghost-free bigravity model exists.

2) It has a FLRW background very similar to the GR case at low energy.

3) The non-linear mechanism seems to work to pass the solar system constraints. (Vainshtein mechanism)4) Two graviton eigen modes are superposition of two metric perturbations, which are mass eigen states at low frequencies and g and g themselves at high frequencies.

~

5) Graviton oscillations occur only at around the crossover frequency, but there is some chance for observation.

Ghost free bi-gravity

,10 V ,11 V ,2212 V

nn Tr kj

ikij gg ~

~

4

022 22

~~

2 n G

matternn

G M

LVc

gRgRg

M

L

~

(Hassan, Rosen (2012))

When g is fixed, de Rham-Gabadadze-Tolley massive gravity.

Even if g is promoted to a dynamical field, the model remains to be free from ghost.

only 5 possible terms including 2 cosmological constants.

FLRW background

2222 dxdttads

22222~ dxdttctbsd

046 122

3 baacbccc

ab

branch 1 branch 2

branch 1 ： Pathological: Strong coupling Unstable for the homogeneous anisotropic mode.

(Comelli, Crisostomi, Nesti, Pilo (2012))

branch 2 ： Healthy

Generic homogeneous isotropic metrics

Branch 2 background

ab is algebraically determined as a function of

→ c for → 0

ff

F 2

~log

A very simple relation holds

We consider only the branches with F > 0, F’< 0.

33

2210 663:log ccccf 0

~ 22

ff

M G 44

33

221 24186:log

~ ccccf

We further focus on low energy regime.

required for the absence of Higuchi

ghost (Yamashita and TT)

Branch 2 background

a

a

c

1

1

natural tuning to coincident light cones (c=1) at low energies ( → 0)!

22

31

GM

Pc

effective energy density due to

mass term

We expand with respect to = c

33 22 f

MH

G

cc

f

22 1

1

222

13 Gc MH

Effective gravitational coupling is weaker because of

the dilution to the hidden sector.

Effective graviton

mass

vDVZ discontinuity

27

In GR, this coefficient is 1/2

TgTg

3

11□current bound <10-5

Solar system constraint: basics

To cure this discontinuity we go beyond the linear perturbation (Vainshtein)

NG 22

Schematically

gN

N

r

r

Gr

Gr 3

2

2

cmcm 531310 101010 Mpc3001

Correction to the Newton potential

22222 drdredteds vuvu

rer R~

cfC

Erasing , Rvu and ~,~

2

22

22

G

mji Muu

Cu

Gravitational potential around a star in the limit c→1Spherically symmetric static configuration:

Then, the Vainshtein radius

222~~2~~22 ~~~ drrdedtesd vuvuc

, which can be tuned to be extremely large. 3/1

2

g

V

Crr

can be made very large, even if -1 << 300Mpc .

Mpc3001 CSolar system constraint:

2~G

m

Mv

v is excited as in GR.

2

2~

3 GMH

Erasing u, v and R

2

22

22 ~~

~~

G

mji Muu

Cu

Excitation of the metric perturbation on the hidden sector:

The metric perturbations are almost conformally related with each other: 222~ dssd cNon-linear terms of u (or equivalently u) play the role of the source of gravity.

2~~

G

m

Mv

v is also excited like v. ~

~

u is also suppressed like u. ~

0~2 hhmhh g

Gravitational wave propagation

0~~~

2

22 hh

cmhch

c

g

Short wavelength approximation ：Hmk g

(Comelli, Crisostomi, Pilo (2012))

ffcf

mg

6

1

32

C ≠ １ is important.

12:

c

2

222 1

:

gm

mass term is important. Eigenmodes are

hhc

~2 ,~hh

Eigenmodes are

hh~

, modified dispersion relation

due to the effect of mass modified dispersion relation due to different light cone

kc

k

At the GW generation, both and are equally excited.

hhc

~2 ,~hh hh

~,

hh~

kc

Only the first mode is excited

Only the first mode is detected

We can detect only h.Only modes with k ～ kc picks up the non-trivial dispersion relation of the second mode.

X

X

If the effect appears ubiquitously, such models would be already ruled out by other observations.

k

Interference between two modes.

Graviton oscillations

SummaryGravitational wave observations open up a new window for modified gravity.

Even the radical idea of graviton oscillations is not immediately denied. We may find something similar to the case of solar neutrino experiment in near future. Although space GW antenna is advantageous for the gravity test in many respects, more that can be tested by KAGRA will be remaining to be uncovered.

Branch 2 background

branch 2 ：

33

2210 663: ccccmass

a

a

c

1

1

Natural Tuning to c=1 for → 0.

0624

618

36 3

324

23

12

01

c

cc

cc

cc

c

221

31

Gcc

c

M

Pc

046 122

3 baacbccc ab branch 1 branch 2

becomes a function of

effective energy density due to mass term

→ c for → 0

22

3 G

mass

MH

d

d mass

222

13 Gc MH

Effective gravitational coupling is

weaker because of the dilution to the hidden sector.

ab

Healthy branch

Gradient instability

Higuchi ghost

JCAP1406 (2014) 037 De Felice, Gumrukcuoglu, Mukohyama, Tanahashi and TT

Stability of linear perturbation

22222 drdredteds vuvu

rer R~

log

log

d

dA

Erasing , and truncating at the second order Rvu and ~,~

2

22

22

G

mji Muu

Au

Gravitational potential around a star

Spherically symmetric static configuration:

Then, the Vainshtein radius

222~~2~~22 ~~~ drrdedtesd vuvuc

can be tuned to be extremely large. 3/1

2

g

V

Arr

can be made very large, even if -1 << 300Mpc .

Mpc3001 ASolar system constraint:

2~~

G

m

Mvv

Both v and v are excited as in GR. 2

2~

3 GMH

~

(PTEP2014 043E01 De Felice, Nakamura and TT)

ab

Healthy branch


Meaning of A >>1

0~2 hhmhh g

Gravitational wave propagation

0~~~

2

22 hh

cmhch

c

g

Short wavelength approximation ：Hmk g


c ≠ １ is important.

12:

c

2

222 1

:

gm

mass term is important. Eigenmodes are

hh~

,~2hh c

Eigenmodes are

hh~

, modified dispersion relation due to the

effect of mass

modified dispersion relation due to different

light cone

kc

k

3321

2 612 cccccmg

2

1

2/12

08.0100

1Hz100

pcc

Gravitational wave propagation over a long distance D

22

22

2,1 1

1211

22

1

2xxx

x

cDD

k

Phase shift due to the modified dispersion relation:

log10x

=0.2

=1

=100

2 1 1 2

1.5

1.0

0.5

0.5

1.0

1~log10

cD

2

2 12

c

x

02131 cHDcD

becomes O(1) after propagation over the horizon distance

2

1


hh~

,

hh~

kc

Only the first

mode is excited

Only the first

mode is detected

We can detect only h.Only modes with k ～ kc pick up the non-trivial dispersion relation of the second

mode.

X

X

If the effect appears ubiquitously, the model would be already ruled out.

k

Interference between two modes


hh~

hh c

~2

fifififi GRGR efBefBfAfh 2121

Gravitational wave oscillations=0.2 =1 =100 B1

B2

At high frequencies only the first mode is

observed.

2

2 12

c

x

2 1 1 2

4

2

2

4

log10x

B

At low frequencies only the first mode is

excited.

Detectable range of parameters by KAGRA, assuming NS-NS binary

at 200Mpc.

(Phys. Rev. D91 (2015) 062007 Narikawa, Tagoshi, TT, Kanda and Nakamura)


hh~

,

hh~

kc

Only the first

mode is excited

Only the first

mode is detected

We detect both and .

All modes with k > kc pick up the non-trivial dispersion relation of the second mode.

X

X k

Interference between two modes


hh~

hh c

~2

If Vainshtein mechanism works for GW detectors…

hh~

～

X

For detection by aLIGO, aVirgo and KAGRA:

Window is narrow but open.

Mpc3001 CSolar system constraint:0.1pc1

No Vainshtein effect in the inter-galactic space: Mpc30001 C

GMCm 2MeV10 Bare mass:

ab

22plMm

Cosmological

solutiondisappe

ars


4MeV10

Can Bigravity with large A be naturally realized as a low energy effective theory?

Higher dimensional model?!KK graviton spectrumOnly first two modes remain at low energy

2ds2~sd

222~ dssd cIf the internal space is stabilized

h~

h

0h

1h

1c

Matter on right brane couples to h.

However, pinched throat configuration looks quite unstable…

44

Induced gravity on the branes

• Induced gravity terms play the role of potential well.

• Lowest KK graviton mass • KK graviton mass x

Bra

ne

??

y

MinkowskiBulk

mattmatt LRMgxdLRMgxdRgxdMS 24

424

4535

crMM 2/24

35

Dvali-Gabadadze-Porrati model (2000)

Critical length scale

Bra

ne

yrc 1

ym 1

cry is required to reproduce bigravity.

than in this model.

• To construct a viable model,

the radion (=brane separation) must be stabilized. • Radions can be made as heavy as KK gravitons.• However, the energy density cannot be made large.

(JCAP 1406 (2014) 004 Yamashita and TT)

Why? We know that self-accelerating branch has a ghost.

2log Kranr cc

Normal branch condition:

DHK 4n n

DH 4 cannot be made larger cr1

• Despite the fore-mentioned limitation, it would be interesting to see how bigravity derives from brane setup without fine tuning of coupling constants.

• We neglect radion stabilization, for simplicity.• Thus, we consider a system of bigravity with radion

(Yamashita and TT, in preparation)Deriving bigravity without fine tuning

Gradient expansion

with the scaling assumptions:

Ky K :bulk extrinsic curvaturey :Brane separation

∂ lrcy

Krc

We solve the bulk equations of motion for given boundary metrics

NN

Rgl

KKKKKN y

142

12

Rl

KKK 2

2 12

Substituting back the obtained bulk solution into the action, we obtain at the quadratic order in perturbation

This looks very complicated but can be recast into the form of bigravity + radion, which is coupled to the averaged metric:

24242

662

HRgxdHRgxdM

S pl

XH

HXgggxd 2

4*

22242 41

4

3□

gH

hhHHH

gX

2341

3

2 22

1

24*

22

2*

□□with

24242

662

HRgxdHRgxdM

S pl

2242 44

3Hgggxd □

aHH /16/cKr

hhHg 2

4*

33

22

4

3□

RR

Why do we have this attractor behavior, c→ 1 and → c, at low energies?

DGP 2-brane model?!

Only first two modes remain at low energy

2ds 2~sd

222~ dssd c

h~h

0h

1h

1c

～～

??

d

KK graviton mass spectrum

~1/d 2

potential wells due to induced gravity

terms

d→ 0 identical light cone

RgxdRgxd~~& 44

Gravitational wave propagation over a long distance D

22

22

2,1 1

1211

22

1

2xxx

x

cDD

k

Phase shift due to the modified dispersion relation:

log10x

=0.2

=1

=100

2 1 1 2

1.5

1.0

0.5

0.5

1.0

1~log10

cD

2

2 12

c

x

02131 cHDcD

becomes O(1) after propagation over the horizon distance

2

1

1) At the time of generation of GWs from coalescing binaries, both h and h are equally excited.


Gravitational wave oscillations

=0.2 =1 =100 B1

B2

At high frequencies only the first mode is

observed.

~

2

2 12

c

x

2 1 1 2

4

2

2

4

log10x

B

At low frequencies only the first mode is

excited.

2) When we detect GWs, we sense h only.


=0.2 =1

=100 B1

B2

Graviton oscillations occur only around the frequency

4

1

2/12

02

2

08.0100

1Hz100

16

pcHc

c

GW

2 1 1 2

4

2

2

4

log10x

B

0213 cHD Phase shift is as small as ?

1x

log10x =0.2

=1 =100

2 1 1 2

1.5

1.0

0.5

0.5

1.01~log10

cD

No, x << 1 when the GWs are propagating the inter-galactic low density region.

SummaryGravitational wave observations give us a new probe to modified gravity.

Even graviton oscillations are not immediately denied, and hence we may find something similar to the case of solar neutrino experiment in near future. Although space GW antenna is advantageous for the gravity test in many respects, we should be able to find more that can be tested by KAGRA.

53

(Yunes & Pretorius (2009))

Parametorized post-Einstein fiefAfh 6/7

fAufA GRi

ai

i

1

i

biGW

iuff

pulsarconstraint

ba

pulsarconstraint

Constraints from BH-BH merger

Constraints from BH-BH merger

12MOBH-6MOBH and 18MOBH-6MOBH mergers

Better constraint than pulsar timing for ai>0 or bi>-5/3.

GW waveform for Quasi-circular orbits

corresponding to Newtonian order

(Cornish, Sampson, Yunes, Pretorius. (2011))

Einstein Æther

• The Lorentz violating effects should be suppressed.

• At the lowest order in the weak field approximation, there is no correction to the metric if U // u(≡the four momentum of the star).

gUUcccggcM 321

with

UUMRgxdS 4

16

1

1UU

two constraints among the four coefficients

Compact self-gravitating bodies can have significant scalar charge due to the strong gravity effect.

Dipole radiation.

U is not coupled to matter field

directly.

55

Scalar-tensor gravity (conti)

Current constraint on dipole radiation: BD ＞ 2.4×104 J1141-6545 (NS(young pulsar)-WD )

(Bhat et al. arXiv:0804.0956)

LISA 1.4M◎NS+1000M◎BH: BD > 5×103

Decigo1.4M◎NS+10M◎BH ： BD > 8×107

　　 collecting 104events at cosmological distances 　

　


200SNR at 40Mpc corresponding to

(Yagi & TT, arXiv:0908.3283)

31 ccc

c

c

The case of Einstein Æther ⇒

(Yagi et al. arXiv:1311.7144)

2adD

Graviton mass effect

56

uuf g 16

3

128

9

55

756

37151

128

3 3/23/5M

Simple addition of mass to graviton

f

DfcfDtf

gphase 2

22

222

2

2

11

21

f

mkfc

gphase

phase velocity of massive graviton

Phase shift depending on frequencies

2

2

gg

DM

3vOfMu

LISA 107M◎BH+106M◎BH at3Gpc: graviton compton wavelength g > 4kpc


(Yagi & TT, arXiv:0908.3283)

57

Prediction of the event rate for binary NS mergers

(Faulkner et al ApJ 618 L119 (2005))

double pulsar

NS-WD

total coalescence time

GW)( ci Time to spin-down to the current spin velocity + time to elapse before coalescence

　　　 the volume in which we can detect an observed binary NS when it is placed there.

event rate per Milky way galaxy

i

gal

iiV

VR

max

iVmax

0.4 ～ 400yr-1 for advLIGO/Virgo (Abadie et al. 2010)

>1.5yr-1 for advanced detector network (Yonetoku et al. 1402.5463)If short -ray bursts are binary NS mergers,

58

Gravitation wave

detectors

eLISA(NGO)⇒DECIGO/BBO LIGO⇒adv LIGO

TAMA300,CLIO　⇒ KAGRA

年15 16 17 18 19 20 21

iKAGRA bKAGRA

adv LIGO

adv Virgo

22

LIGO India

LISA pathfinder eLISA

27 28 29 30

DECIGODECIGO pathfinder Pre DECIGO

59

Ghost free bi-gravityWhen g is fixed, de Rham-Gabadadze-Tolley massive gravity.

4

016 nmatternn

N

LVcgG

RgL

,10 V ,11 V ,2

212 V n

n Tr kjiki

j gg ~

~

No gauge degrees of freedom, but by introducing a field ;2gg

;2 gg

becomes a gauge symmetry.

Fixing the gauge by =0 ⇒ original theoryImposing condition on g⇒ becomes dynamical

, we consider flat metric + perturbation:

nn Tr kj

ikij gg ~

If , its variation gives higher derivative terms.

,~ g

;;;;2 g

,

; g

To avoid higher derivatives of in the EOM,

Setting

,

;,

;L

,

,

,

,10 V ,11 V ,2212 V

In other words:

10 (metric components) – 4 (constraints) ＝ 6

Since massive spin 2 field has 5 components, one scalar remains, which becomes a ghost (kinetic term with wrong sign).

If constraints do not completely fix the Lagrange multipliers, g0, their consistency relation gives an additional condition. As a result, the residual scalar degree of freedom disappears.

(Hassan, Rosen (2011))

0~

2 22 hhmahhaHh g

EOM of Gravitational waves

0~~~

22~

2222 hh

cmahchccaHh g

Short wavelength approximation ： Hmk g

3321

2 612 cccccmg


0~11 22

22222

2222

h

h

mkcm

mmk

gg

gg

hhh gg

~sincos1

Eigen modedecomposition

22

2222

2,1 1

1211

2xxxk

Two propagation speeds are not same for c≠1. [≠–oscillation]

hhh gg

~cossin2

2

2 12

c

x

2

222 1

gm

c

ccg

x

2

112cot

22


=0.2 =1

=100 B1

B2

Graviton oscillations occur only around the frequency

4

1

2/12

02

2

08.0100

1Hz100

16

pcHc

c

GW

2 1 1 2

4

2

2

4

log10x

B

0213 cHD Phase shift is as small as ?

1x

log10x

=0.2 =1

=100

2 1 1 2

1.5

1.0

0.5

0.5

1.01~log10

cD

No, x << 1 when the GWs are propagating the inter-galactic low density region.

Ordinary Vainshtein mechanism is not good enough!

65

Solar system constraint

Ordinary Vainshtain mechanism tells that can be simply neglected on small length scales for .

int2 TTMG G

intT

Then, however, “local effective gravitational coupling ” ≠ “cosmological one ” 221 Gc M

2GM

Here, we do not send , but we only tune the graviton mass to be small:

0int T

ic2

hh~

“local effective gravitational coupling”= 221 Gc M

0int T

In the DGP two-brane model stabilized at a small brane separation, this Vainshtein mechanism can be easily understood.

66

Vainshtein a la brane

Junction conditions:

cG rKgKTMG 2

small separation

dggKTMr Gc ~2

gg ~

Furthermore,

Nearly identical metrics

crgKKG ~~~~~

KK~

TMrrG Gcc21~1

TM Gc2121

2ds 2~sd??

d

22~

GM

Tgg

tightly stabilized brane separation

d should shrink to maintain stabilization at large energies

67

Induced gravity on the brane

• For r<rc , 4-D induced gravity term dominates?

• Extension is infinite, but 4-D GR seems to be recovered for r < rc .

x

Bra

ne

??

y

MinkowskiBulk

0y

mattLRMgxdRgxdMS 424

44535

crMM 2/24

35

Dvali-Gabadadze-Porrati model (2000)

very different from the other

braneworld models

Critical length scale22

24

35 1

:1

:rrrr

M

r

M

c

68

Gravitons are trapped to the brane but not completely.

x

Bra

ne

??

yMinkowski

Bulk0y

JyMyM 424

35 □□

crMM 2/24

35

Source term

5D scalar toy model:

JypMypM y

~~~ 224

2235

xipey

~

JpMM y

~~~2 22

435

equationdy

JpMpM~~

2 224

35

pye 0

~~

Solution in the bulk is given by

2

~~

prp

J

c

69

2

~~

prp

J

c

23 11

rkekd ikr

xexdedtJ xkiti

3~Static pointlike source on the brane

large scale (small k)

rkekd ikr 11

23 small scale (small

k)

five dimensional behavior

four dimensional behavior

After propagation over cosmological distance, GWs may escape into the bulk?

2

2

4/

~

~ 4/

22122

22122

22122

3

21

2

2

21

2

2

22

2

2

sin2

c

ri

c

i

C

uri

C

c

i

uri

Cc

ikr

c

i

c

r

ire

r

u

r

r

e

iu

duee

r

rur

eiu

duee

r

krk

dkke

ir

krk

dkkkr

r

ekrk

kd

kx

ｘｘ

k

ｘｘ

k

C

u

ｘ

C～

kiu/rr≪rc

2

71(taken from Esposito-Farese, gr-qc/0402007)

72(taken from Esposito-Farese, gr-qc/0402007)

73

74

estimated merger event rate

LISA detectable event rate

(Gair et al, CQG 21 S1595 (2004) )

75

Probably clean system

yr101010

105.41

2

1

612

EddM

m

MM

M

◎◎

•Interaction with accretion disk

(Narayan, ApJ, 536, 663 (2000))

df

obs

t

T

f

f

df

obsobsobs t

TfTTfN

Frequency shift due to interaction

Change in number of cycles

,assuming almost spherical accretion (ADAF)

satellite

reldf mG

vt

2

3

log4

obs. period ~1yr

76

Black hole perturbation GTG 8g

21 hhgg BH

v/c can be O(1)

BH重力波

M≫

11 8 GTG h:master equation

Linear perturbation

11 4 TgL

Gravitationalwaves

Regge-Wheeler formalism (Schwarzschild)Teukolsky formalism (Kerr) Mano-Takasugi-Suzuki’s method (systematic PN expansion)

Documents

1 Testing Modified gravity and gravitational waves Takahiro Tanaka (Dept. of Phys./YITP Kyoto university) Gravitational waves