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About  Yukawa  Ins@tute  for  Theore@cal  Physics

 Visi@ng  programs:        Visi@ng  professors,          Suppor@ng  stay  during  sabba@cal  period,        Atoms,        GCOE/YITP  Visitors,        BIEP,        Post  docs  (JSPS,  YITP)  

Graviton  Oscilla.on  in  a  viable  bigravity  model  

 Takahiro  Tanaka  (YITP)  with    

Antonio  De  Felice  and  Takashi  Nakamura

3  

Gravita@on  wave  detectors  

eLISA(NGO)  ⇒DECIGO/BBO   LIGO⇒adv  LIGO  

TAMA300,CLIO  　⇒KAGRA  

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  

4

PSR  B1913+16  Hulse-­‐Taylor  binary        dPorb/dt=-2.423×10-­‐12  

Test  of  GW  genera@on  

Agreement  with  GR  predic@on  

002.0997.0 ±=GR

obs

PP

Is  there  possibility  that  graviton  disappear  during  its  propaga@on  over  cosmological  distance?  

We  know  that  GWs  are  emiced  from  binaries.  

What  is  the  possible  big  surprise  when  we  directly  detect  GWs?  

Braneworld

7  

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 .

µx

Brane  

??

y

Minkowski Bulk

0=y

σ

( ) ( )( )∫∫ ++= mattLRMgxdRgxdMS 424

44535

crMM 2/2435 =

Dvali-­‐Gabadadze-­‐Porra@  model  (2000)

very  different  from  the  other  braneworld  

models  

Cri@cal  length  scale  22

24

35 1:1:

rrrrM

rM

c

=

8  

Gravitons  are  trapped  to  the  brane  but  not  completely.  

µx

Brane  

??

y Minkowski

Bulk 0=y

σ

( ) ( )[ ] ( )JyMyM δφδ =+ 424

35 □□

crMM 2/2435 =

Source  term  5D  scalar  toy  model:  

( ) ( ) ( )JypMypM y~~~ 22

4223

5 δφδφ −=+∂−

( )µ

µφφ xipey~=

JpMM y~~~2 22

435 −=+∂− φφ

( )∫−ε

εequationdy

[ ] JpMpM ~~2 224

35 −=+ φ

pye−= 0~~φφ

Solu@on  in  the  bulk  is  given  by

2

~~prp

J

c+∝φ

9  

2

~~prp

J

c+∝φ

23 11

rkekd ikr ∝∝ ∫φ

( ) ( )ωδδω ∝∝ ∫∫ ⋅− xexdedtJ xkiti 3~Sta@c  pointlike  source  on  the  brane

large  scale (small k)

rkekd ikr 11

23 ∝∝ ∫φsmall  scale (small k)

five  dimensional  behavior

four  dimensional  behavior

Aher  propaga@on  over  cosmological  distance,  GWs  may  escape  into  the  bulk?  

     

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+≈

−≈

−+−=

−+−=

−+−

−−

−

−

∞

∞−−

−

∫

∫

∫

∫

∫

2

2

4/

~

~ 4/

22122

22122

22122

3

212

22

12

2

22

2

2

sin2

c

ri

c

i

C

uri

C

c

i

uri

Cc

ikrc

i

c

rire

r

ur

re

iuduee

r

rur

eiu

dueer

krkdkke

ir

krkdkkkr

r

ekrk

kd

πωπ

ωω

π

ωω

π

ωω

π

ωω

π

ωω

ω

πω

π

ω

kx

ｘ ｘ

k

ｘ ｘ

k

C

u

ｘ

C ～

k=ω+iu/r r≪ωrc

2

" Chern-­‐Simons  Modified  Gravity

(Ali-Haimoud, (2011) : J0737-3039(double pulsar)

Right-­‐handed  and  leh-­‐handed  gravita@onal  waves  are  magnified  differently  during  propaga@on,  depending  on  the  frequencies.

Current  constraint  on  the  evolu@on  of  the  background  scalar  field  θ :

( ) 16Hz10 −<θα

( ) ( ) ( )( )×+ += hhh iRL

21,

( ) ( )

emit

,,obs 1 θαω ±≈ RLRL hh

∫ −⊃ RRgxdS *4

4θ

α

Moreover,   1≈θωα  modes  are  in  the  strong  coupling  regime.      outside  the  validity  range  of  EFT.

"  Bi-­‐gravity

Both  massive  and  massless  gravitons  exist.  　→ν    oscilla@on-­‐like  phenomena?    

First  ques@on  is  whether  or  not  we  can  construct  a  viable  cosmological  model.    

( ) ++−

+−

= φκππ

,16

~~

16gL

GRg

GRg

L matterNN

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

∑=

+−+−

=4

016 nmatternn

N

LVcgGRg

Lπ,10 =V ,11 τ=V ,2

212 ττ −=V [ ]nn Tr γτ ≡ kj

ikij gg ~≡γ

~

No  gauge  degrees  of  freedom,  but  by  introducing  a  field   ++→ µνµνµν π ;2gg

µνµνµν ;2Λ+→ ggΛ−→ππ

 becomes  a  gauge  symmetry.  

Fixing  the  gauge  by  π =0  ⇒ original  theory  Imposing  condi@on  on  gµν ⇒ π    becomes  dynamical

,  we  consider  flat  metric  +  π  perturba@on:    

[ ]nn Tr γτ ≡ kjiki

j gg ~≡γ

If                                                  ,  its  varia@on  gives  higher  deriva@ve  terms.    

,~µνµν η=g

ρνµρµνµνµν πππη ;;;;2 ++→g

µν

µνρν

µνµν πδηγ ,

;+== g

 To  avoid  higher  deriva@ves  of  π    in  the  EOM,    

Senng

νµ

µνππ ,

;,;⊃L

,νβ

µα

αβξζµνξζ γγεε ,ργ

νβ

µα

αβγζµνρζ γγγεε ,σ

δργ

νβ

µα

αβγδµνρσ γγγγεε

,10 =V ,11 τ=V ,2212 ττ −=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  (kine@c  term  with  wrong  sign).  

If  constraints  do  not  completely  fix  the  Lagrange  mul@pliers,  g0µ,  their  consistency  rela@on  gives  an  addi@onal  condi@on.  As  a  result,  the  residual  scalar  degree  of  freedom  disappears.    

(Hassan,  Rosen  (2011))

Ghost  free  bi-­‐gravity

Now  g  is  promoted  to  a  dynamical  field.  Even  in  this  case,  it  was  shown  that  the  model  remains  to  be  free  from  ghost.    

∑=

+−

+−

+−

=4

022 22

~~

2 n G

matternn

G MLVc

gRgRgML

κ,10 =V ,11 τ=V ,2

212 ττ −=V

[ ]nn Tr γτ ≡ kjiki

j gg ~≡γ

~

(Hassan,  Rosen  (2012))

FLRW  background

( )( )2222 dxdttads +−=

( ) ( )( )22222~ dxdttctbsd +−=

( )( )

µνµν δδgSTmass

mass 2=

( ) 0=∇ massTµνµ ( )( ) 046 12

23 =ʹ′−ʹ′++ baacbccc ξξ

ab≡ξ

 branch  1  branch  2  branch  1：  　Pathological:  At  the  linear  perturba@on,  expected  scalar  and  vector  perturba@ons  are  absent.  Strong  coupling?  Unstable  for  the  homogeneous  anisotropic  mode.

(Comelli,  Crisostomi,  Nes@,  Pilo    (2012))

 branch  2：  　Healthy:  All  perturba@on  modes  are  equipped.  

Branch  2  background

 branch  2：

33

2210 663: ξξξρ ccccmass +++=

aa

cʹ′

=ʹ′

− ξξ11

Natural  Tuning  to  c=1  for  ρ  → 0.

062461836 33

242

31

20

1 =+⎟⎠

⎞⎜⎝

⎛ −+⎟⎠

⎞⎜⎝

⎛ −+⎟⎠

⎞⎜⎝

⎛ −+− ξξκ

ξκκκξ

ρ cccccccc

( )( ) 22131

Gcc

c

MPcκξ

κξρ+Γ

+=−

( )( ) 046 122

3 =ʹ′−ʹ′++ baacbccc ξξ ab≡ξ  branch  1  branch  2

 ξ    becomes  a  func@on  of  ρ.

 effec@ve  energy  density  due  to  mass  term

 ξ  → ξc  for  ρ →0.

22

3 G

mass

MH ρρ +

=

ξρdd mass=Γ

( ) 222

13 Gc MH

κξρ

+= Effec@ve  gravita@onal  coupling  is  weaker  

because  of  the  dilu@on  to  the  hidden  sector.

Why  do  we  have  this  acractor  behavior,  c→1  and  ξ→ξc,  at  low  energies?  

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

2ds2~sd

222~ dssd cξ=If  the  internal  space  is  stabilized

h~ h

0h

1h

1=c

Macer  on  right  brane  couples  to  h.

" vDVZ discontinuity

20  

In  GR,  this  coefficient  is  1/2  

⎟⎠

⎞⎜⎝

⎛ −∝ − TgTg µνµνµνδ311□ current bound  <10-­‐5  

Solar  system  constraint:  basics  

To  cure  this  discon@nuity  •   Make  the  extra  helicity  0  mode  massive                                                                                                                            (Chameleon)  •   Go  beyond  the  linear  perturba@on  (Vainshtein)  

( ) ρδµδ NG=Φ∂∂+ΦΔ − 22Schema@cally

( ) ggN rrrrrG 32/ µρµδ ≈≈ΦΦ

( ) ( )cmcm 531310 101010 µ≥− Mpc3001 ≥−µ

mm1.01 ≥−µ

( )22222 Ω++−= +− drdredteds vuvu

rer R=~

( )( )ξloglog

ddC Γ

∝

Erasing                                            ,   Rvu and ~,~

( ) ( ) ( )( ) 222

22

G

mji MuuCu ρ

µµ ≈∂∂−Δ−−Δ

Gravita.onal  poten.al  around  a  star  in  the  limit  c→1

Spherically  symmetric  sta@c  configura@on:

Then,  the  Vainshtein  radius

( )[ ]222~~2~~22 ~~~ Ω++−= +− drrdedtesd vuvucξ

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

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛≈

µg

V

Crr

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

Mpc3001 ≥−µCSolar  system  constraint:

2~G

m

Mv ρ≈Δ v  is  excited  as  in  GR.  

22 ~3 GM

H ρ=

Erasing  u, v  and  R  

( ) ( ) ( )( ) 222

22 ~~

~~

G

mji MuuCu ρ

µµ ≈∂∂−Δ−−Δ

Excita@on  of  the  metric  perturba@on  on  the  hidden  sector:

The  metric  perturba@ons  are  almost  conformally  related  with  each  other: 222~ dssd cξ≈Non-­‐linear  terms  of  u  (or  equivalently  u)  play  the  role  of  the  source  of  gravity.  

2~~

G

m

Mv ρ≈Δ

v  is  also  excited  like  v.       ~

~

u  is  also  suppressed  like  u.       ~

( ) 0~2 22 =−+Δ−ʹ′ʹ′+ʹ′ʹ′ hhmahhaHh g

EOM  of  Gravita@onal  waves

( ) ( ) 0~~~22~2

222 =−−Δ−ʹ′ʹ′−ʹ′++ʹ′ʹ′ hhcmahchccaHh g κξξξ

Short  wavelength  approxima@on： Hmk g >>>>

( )( )33212 612 ξξξ cccccmg +++=

(Comelli,  Crisostomi,  Pilo    (2012))

0~11 22

22222

2222

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

++−−

−++−

h

h

mkcm

mmk

gg

gg

κξω

κξ

ω

( )hhh ggA~sincos ξκθθ +=

Eigen  mode  decomposi@on

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+

−++−= 2

2

2222

2,1 11211

2xxxk

κξκξµ

ω

Two  propaga@on  speeds  are  not  same  for  c≠1.    [≠ν –oscilla@on]

( )hhh gg~cossin2 ξκθθ +−=

( )2

2 12µ

ω −≡

cx

2

222 1κξκξ

µ+

= gm

( ) ( )c

ccg

xξκ

κξκξθ

2112cot

22 −++=

Gravita@onal  wave  propaga@on  over  a  long  distance  D

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

+

−++

−−=

Δ−≡Φ 2

2

22

2,1 11211

221

2xxx

xcDDk

κξκξµ

ωδ

Phase  shih  due  to  the  modified  dispersion  rela@on:

log10x

κ =0.2

κ =1

κ =100

- 2 - 1 1 2

- 1.5

- 1.0

- 0.5

0.5

1.0 1~log10

−

Φ

cDµδ ( )

2

2 12µ

ω −≡

cx

( ) 02131 Ω+≈− cHDcD κξµ

 becomes  O(1)  aher  propaga@on  over  the  horizon  distance

δ Φ2

δ Φ1

1)  At  the  @me  of  genera@on  of  GWs  from  coalescing  binaries,      both  h  and  h  are  equally  excited.    

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]fifififi GRGR efBefBfAfh 2121

Φ+ΦΦ+Φ +∝ δδ

Gravita.onal  wave  oscilla.ons

κξ 2 =0.2   κξ 2 =1   κξ 2 =100   B1  

B2  

At  high  frequencies  only  the  first  mode  is  

observed.

~

( )2

2 12µ

ω −≡

cx

- 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

Φ+ΦΦ+Φ +∝ δδ

κξ 2 =0.2   κξ 2 =1   κξ 2 =100   B1  

B2  

Graviton  oscilla@ons  occur  only    around    the  frequency    

( ) ( )

4

1

2/12

02

2

08.01001Hz100

16 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

Ω+≈ −

−

pcHc

c

GWµκξ

κξ

µω

- 2 - 1 1 2

- 4

- 2

2

4

log10x

B

( ) 0213 Ω+ cHD κξPhase  shih  is  as  small  as                                                              ?  

1≈x

log10x κ =0.2

κ =1 κ =100

- 2 - 1 1 2

- 1.5

- 1.0 - 0.5

0.5 1.0

1~log10−

Φ

cDµδ

No,  x << 1  when  the  GWs  are  propaga@ng    the  inter-­‐galac@c  low  density  region.  

Summary Gravita@onal  wave  observa@ons  give  us  a  new  probe  to  modified  gravity.

Even  graviton  oscilla@ons  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.