1 testing modified gravity and gravitational waves takahiro tanaka (dept. of phys./yitp kyoto...
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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
Gradient instability
Meaning of A >>1
0~2 hhmhh g
Gravitational wave propagation
0~~~
2
22 hh
cmhch
c
g
Short wavelength approximation :Hmk g
(Comelli, Crisostomi, Pilo (2012))
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
At the GW generation, both and are equally excited.
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
Graviton oscillations
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)
At the GW generation, both and are equally excited.
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
Graviton oscillations
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
Gradient instability
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.
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.
2) When we detect GWs, we sense h only.
fifififi GRGR efBefBfAfh 2121
=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
Constraint from future observations:
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
Constraint from future observations:
(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
(Comelli, Crisostomi, Pilo (2012))
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
fifififi GRGR efBefBfAfh 2121
=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)