Vainshtein mechanism in modiﬁed gravity theories · Vainshtein mechanism in modiﬁed gravity...

Vainshtein mechanismin modified gravity theories

Rampei KimuraHiroshima university

TJ2012 : Japan-Thai workshop in cosmology12/25/2012

Based on : RK, T. Kobayashi, K. Yamamoto PRD 85 024023 (2012) [arXiv : 1111.6749]

+ α

Introduction

(from WMAP website)

Accelerated expansion of the universe

• Implication of cosmological constant?

• Observationally, Λ is fine !

• Cosmological constant problem

　　　　 ~120 orders of magnitude differences

Gµ⌫+⇤gµ⌫ = 8⇡GTµ⌫

Can modification of gravity solve this puzzle ???

Modification of gravity

Gµ⌫ = 8⇡GTµ⌫? ?Gµ⌫ ⇡ 8⇡GTµ⌫

Horizon scaleSolar system scale

(Cosmic acceleration)(Large scale structure)(Earth, Sun)

In order to explain an accelerated expansion of the universe, ONLY long range modification is needed

Modification of gravity Example： Brans-Dicke theory

Inconsistent with long range and short range

S =

Zd

4x

p�g

'R� !BD

'

(r')2�

• Self-accelerating solution

2.2 The most general second-order scalar-tensor theory 15

a di!erent form than (2.6), and later it was rediscovered by De!ayet et al. [64] as a generalizationof the galileon. The equivalence of the two expressions is shown by the authors of Ref. [65]. In thisthesis, we employ the galileon-like expression (2.6) since it is probably more useful than its originalform when discussing the Vainshtein mechanism. The gravitational and scalar-field equations can befound in the Appendix of Ref. [65].

2.2.1 Brans-Dicke theory

The prototype Brans-Dicke theory [14] corresponds to the choice of the following arbitrary functions,

K =!BD

"X, G4 = ", G3 = G5 = 0, (2.8)

where !BD is called the Brans-Dicke parameter, which is dimensionless parameter. In this model,the parametrized post-Newtonian parameter # is given by

#(!BD) =1 + !BD

2 + !BD. (2.9)

Since solar-system test indicates that |1 ! #| < 2.3 " 10!5 [66], the Brans-Dicke parameter mustsatisfy !BD > 4"104. On the other hand, !BD < !3/2 is required in order to explain the acceleratedexpansion of the universe [67]. Thus the Brans-Dicke theory is not consistent with the solar systemexperiment and the accelerated expansion of the universe. However, if we add the potential termV (") in K(", X) and choose an appropriate potential form, then this local constraints can be avoidedthrough Chameleon mechanism [15–17].

2.2.2 f(R) theories

The Lagrangian of f(R) theories [18–21] is given by

L =M2

Pl

2f(R), (2.10)

which corresponds to the choice,

K = !M2

Pl

2(Rf,R ! f), G4 =

MPl

2", G3 = G5 = 0, " = MPlf,R, (2.11)

where " is a scalar degree of freedom and f,R = $f/$R.

2.2.3 Covariant galileon

The covariant galileon (2.4) corresponds to

K = c1"! c2X, G3 =c3M3

X, G4 =M2

Pl

2! c4

M6X2, G5 =

c5M9

X2, (2.12)

where ci are dimensionless model parameters and M is the constant, which has the dimension ofmass.

• Solar system constraint

2.2 The most general second-order scalar-tensor theory 15

a di!erent form than (2.6), and later it was rediscovered by De!ayet et al. [64] as a generalizationof the galileon. The equivalence of the two expressions is shown by the authors of Ref. [65]. In thisthesis, we employ the galileon-like expression (2.6) since it is probably more useful than its originalform when discussing the Vainshtein mechanism. The gravitational and scalar-field equations can befound in the Appendix of Ref. [65].

2.2.1 Brans-Dicke theory

The prototype Brans-Dicke theory [14] corresponds to the choice of the following arbitrary functions,

K =!BD

"X, G4 = ", G3 = G5 = 0, (2.8)

where !BD is called the Brans-Dicke parameter, which is dimensionless parameter. In this model,the parametrized post-Newtonian parameter # is given by

#(!BD) =1 + !BD

2 + !BD. (2.9)

Since solar-system test indicates that |1 ! #| < 2.3 " 10!5 [66], the Brans-Dicke parameter mustsatisfy !BD > 4"104. On the other hand, !BD < !3/2 is required in order to explain the acceleratedexpansion of the universe [67]. Thus the Brans-Dicke theory is not consistent with the solar systemexperiment and the accelerated expansion of the universe. However, if we add the potential termV (") in K(", X) and choose an appropriate potential form, then this local constraints can be avoidedthrough Chameleon mechanism [15–17].

2.2.2 f(R) theories

The Lagrangian of f(R) theories [18–21] is given by

L =M2

Pl

2f(R), (2.10)

which corresponds to the choice,

K = !M2

Pl

2(Rf,R ! f), G4 =

MPl

2", G3 = G5 = 0, " = MPlf,R, (2.11)

where " is a scalar degree of freedom and f,R = $f/$R.

2.2.3 Covariant galileon

The covariant galileon (2.4) corresponds to

K = c1"! c2X, G3 =c3M3

X, G4 =M2

Pl

2! c4

M6X2, G5 =

c5M9

X2, (2.12)

where ci are dimensionless model parameters and M is the constant, which has the dimension ofmass.

Brans & Dicke (1961)

How can modification of gravity reduce General relativity at short distances??

Galileon theory✓ Galileon

L � (@')2 ⇤'

second derivative with respect to space-time

✓ Galileon term contains the second derivative term, but ...

No higher-order derivative terms in EOM !!

(Nicolis et al. ’09)

coupling between scalar and curvature

EOM � (⇤')2 � (rµr⌫')2 �Rµ⌫rµ'r⌫'

Vainshtein mechanism✓ Galileon example (Deffayet et al. ’10)

Horizon scaleSolar system scale

“Nonlinear” “Linear”

Vainshtein radiusrV ⇠ (rsr

2c )

1/3

F� ⇠ GM

r2

✓r

rV

◆3/2

⌧ GM

r2

φ is effectively weakly coupled to matter

F� ⇠ GM

r2

φ is strongly coupled to matter

Inside the Vainshtein radius, general relativity can be recovered

self-accelerating solutionrc ⇠ O(H�1

0 )

L� = �1

2(@�)2 � r2c

2MPl(@�)2⇤�

Horndeski theory

K-essence term L2 � (@�)2, V (�)

Non-minimal derivative coupling

L5 � Gµ⌫rµ�r⌫�

(Germani et al. 2011; Gubitosi, Linder 2011)

Cubic galileon term

L3 � (@�)2⇤�

L2 = K(�, X)

L3 = �G3(�, X)⇤�

L4 = G4(�, X)R+G4,X [(⇤�)2 � (rµr⌫�)(rµr⌫�)]

L5 = G5(�, X)Gµ⌫(rµr⌫�)

� 1

6G5,X

(⇤�)3 � 3(⇤�)(rµr⌫�) (rµr⌫�)

+ 2(rµr↵�)(r↵r��)(r�rµ�)

�

X = �(@�)2/2, GiX = @Gi/@X

✓ Horndeski found the most general Lagrangian whose EOM is second-order differential equation for φ and gμν (also known as Generalized galileon)

Horndeski, Int. J. Theor. Phys. 10,363 (1974) , Deffayet, Gao, Steer (2011)

Einstein-Hilbert term

L4 � (M2Pl/2)R

Question :

Does Vainshtein mechanism work in Horndeski theory ??

Formulation✓ Newtonian gauge and perturbations on cosmological background

Neglect

EOM �⇢

”mass terms”, ”time derivative terms”,

✓L(t)2 @2✏

◆n

,

✓L(t) @✏

◆m

. . .

�

✏ = ,�, and Q ⌧ 1

Picking up the terms like @2✏, (@2✏)2, (@2✏)3, (@2✏)4, �

2

variety of solutions of the key equation (50). In AppendixC, we present the Fourier transform of the perturbationequations.

II. COSMOLOGY IN THE MOST GENERALSCALAR-TENSOR THEORY

We consider a theory whose action is given by

S =

!d4x

!"g (LGG + Lm) , (1)

where

LGG = K(!, X)"G3(!, X)!!

+G4(!, X)R+G4X

"(!!)2 " (#µ#!!)

2#

+G5(!, X)Gµ!#µ#!!" 1

6G5X

"(!!)3

"3!!(#µ#!!)2 + 2(#µ#!!)

3#,(2)

with four arbitrary functions, K,G3, G4, and G5, of !and X := "("!)2/2. Here GiX stands for "Gi/"X, andhereafter we will use such a notation without stating so.The Lagrangian LGG is a mixture of the gravitationaland scalar-field portions, as the Ricci scalar R and theEinstein tensor Gµ! are included. Note in particular thatif a constant piece is present in G4 then it gives rise to theEinstein-Hilbert term. We assume that matter, describedby Lm, is minimally coupled to gravity.

The Lagrangian (2) gives the most general scalar-tensor theory with second-order field equations in fourdimensions. The most general theory was constructed forthe first time by Horndeski [7] in a di!erent form than (2),and later it was rediscovered by De!ayet et al. [9] as ageneralization of the Galileon. The equivalence of the twoexpressions is shown by the authors of Ref. [13]. In thispaper, we employ the Galileon-like expression (2) sinceit is probably more useful than its original form whendiscussing the Vainshtein mechanism. The gravitationaland scalar-field equations can be found in the Appendixof Ref. [13].

We now replicate the cosmological background equa-tions in the theory (1) [13, 14]. For ! = !(t) and thebackground metric ds2 = "dt2 + a2(t)dx2, the gravita-tional field equations are

E = "#m, (3)

P = 0, (4)

where

E := 2XKX "K + 6X!HG3X " 2XG3"

"6H2G4 + 24H2X(G4X +XG4XX)

"12HX!G4"X " 6H!G4"

+2H3X! (5G5X + 2XG5XX)

"6H2X (3G5" + 2XG5"X) , (5)

P := K " 2X$G3" + !G3X

%+ 2

$3H2 + 2H

%G4

"12H2XG4X " 4HXG4X

"8HXG4X " 8HXXG4XX + 2$!+ 2H!

%G4"

+4XG4"" + 4X$!" 2H!

%G4"X

"2X$2H3!+ 2HH!+ 3H2!

%G5X

"4H2X2!G5XX + 4HX$X "HX

%G5"X

+2"2 (HX)˙+ 3H2X

#G5" + 4HX!G5"", (6)

and #m is the (nonrelativistic) matter energy density,while the scalar-field equation of motion is

S := J + 3HJ " P" = 0, (7)

where

J := !KX + 6HXG3X " 2!G3"

+6H2! (G4X + 2XG4XX)" 12HXG4"X

+2H3X (3G5X + 2XG5XX)

"6H2! (G5" +XG5"X) , (8)

P" := K" " 2X$G3"" + !G3"X

%

+6$2H2 + H

%G4" + 6H

$X + 2HX

%G4"X

"6H2XG5"" + 2H3X!G5"X . (9)

An overdot denotes di!erentiation with respect to t andH = a/a.

III. PERTURBATIONS WITH RELEVANTNONLINEARITIES

We work in the Newtonian gauge, in which the per-turbed metric is written as

ds2 = "(1 + 2")dt2 + a2(1" 2#)dx2, (10)

with the perturbed scalar field and matter energy density,

! $ !(t) + $!(t,x), (11)

#m $ #m(t)[1 + $(t,x)]. (12)

It will be convenient to use

Q := H$!

!, (13)

✓ In field equations

2

variety of solutions of the key equation (50). In AppendixC, we present the Fourier transform of the perturbationequations.

II. COSMOLOGY IN THE MOST GENERALSCALAR-TENSOR THEORY

We consider a theory whose action is given by

S =

!d4x

!"g (LGG + Lm) , (1)

where

LGG = K(!, X)"G3(!, X)!!

+G4(!, X)R+G4X

"(!!)2 " (#µ#!!)

2#

+G5(!, X)Gµ!#µ#!!" 1

6G5X

"(!!)3

"3!!(#µ#!!)2 + 2(#µ#!!)

3#,(2)

with four arbitrary functions, K,G3, G4, and G5, of !and X := "("!)2/2. Here GiX stands for "Gi/"X, andhereafter we will use such a notation without stating so.The Lagrangian LGG is a mixture of the gravitationaland scalar-field portions, as the Ricci scalar R and theEinstein tensor Gµ! are included. Note in particular thatif a constant piece is present in G4 then it gives rise to theEinstein-Hilbert term. We assume that matter, describedby Lm, is minimally coupled to gravity.

The Lagrangian (2) gives the most general scalar-tensor theory with second-order field equations in fourdimensions. The most general theory was constructed forthe first time by Horndeski [7] in a di!erent form than (2),and later it was rediscovered by De!ayet et al. [9] as ageneralization of the Galileon. The equivalence of the twoexpressions is shown by the authors of Ref. [13]. In thispaper, we employ the Galileon-like expression (2) sinceit is probably more useful than its original form whendiscussing the Vainshtein mechanism. The gravitationaland scalar-field equations can be found in the Appendixof Ref. [13].

We now replicate the cosmological background equa-tions in the theory (1) [13, 14]. For ! = !(t) and thebackground metric ds2 = "dt2 + a2(t)dx2, the gravita-tional field equations are

E = "#m, (3)

P = 0, (4)

where

E := 2XKX "K + 6X!HG3X " 2XG3"

"6H2G4 + 24H2X(G4X +XG4XX)

"12HX!G4"X " 6H!G4"

+2H3X! (5G5X + 2XG5XX)

"6H2X (3G5" + 2XG5"X) , (5)

P := K " 2X$G3" + !G3X

%+ 2

$3H2 + 2H

%G4

"12H2XG4X " 4HXG4X

"8HXG4X " 8HXXG4XX + 2$!+ 2H!

%G4"

+4XG4"" + 4X$!" 2H!

%G4"X

"2X$2H3!+ 2HH!+ 3H2!

%G5X

"4H2X2!G5XX + 4HX$X "HX

%G5"X

+2"2 (HX)˙+ 3H2X

#G5" + 4HX!G5"", (6)

and #m is the (nonrelativistic) matter energy density,while the scalar-field equation of motion is

S := J + 3HJ " P" = 0, (7)

where

J := !KX + 6HXG3X " 2!G3"

+6H2! (G4X + 2XG4XX)" 12HXG4"X

+2H3X (3G5X + 2XG5XX)

"6H2! (G5" +XG5"X) , (8)

P" := K" " 2X$G3"" + !G3"X

%

+6$2H2 + H

%G4" + 6H

$X + 2HX

%G4"X

"6H2XG5"" + 2H3X!G5"X . (9)

An overdot denotes di!erentiation with respect to t andH = a/a.

III. PERTURBATIONS WITH RELEVANTNONLINEARITIES

We work in the Newtonian gauge, in which the per-turbed metric is written as

ds2 = "(1 + 2")dt2 + a2(1" 2#)dx2, (10)

with the perturbed scalar field and matter energy density,

! $ !(t) + $!(t,x), (11)

#m $ #m(t)[1 + $(t,x)]. (12)

It will be convenient to use

Q := H$!

!, (13)

Q ⌘ (H/�)��

Quasi-static approximation

@t

⌧ @x

higher-order terms

L(t) ⇠ O(H�1)

Field equations

r2 (FT � GT��A1Q)

=B1

2a2H2Q(2) +

B3

a2H2

�r2�r2Q� @i@j�@

i@jQ�

A0r2Q�A1r2 �A2r2�+B0

a2H2Q(2) � B1

a2H2

�r2 r2Q� @i@j @

i@jQ�

� B2

a2H2

�r2�r2Q� @i@j�@

i@jQ�� B3

a2H2

�r2�r2 � @i@j�@

i@j �

� C0

a4H4Q(3) � C1

a4H4U (3) = 0

GTr2 =a2

2⇢m� �A2r2Q

� B2

2a2H2Q(2) � B3

a2H2

�r2 r2Q� @i@j @i@jQ

�� C1

3a4H4Q(3)

✓ Traceless part of the Einstein equations

✓ 00 component of the Einstein equation

✓Scalar field equation

Q(2) ⌘�r2Q

�2 � (@i@jQ)2

Q(3) ⌘�r2Q

�3 � 3r2Q (@i@jQ)2 + 2 (@i@jQ)3

U (3) ⌘ Q(2)r2�� 2r2Q@i@jQ@i@j�+ 2@i@jQ@j@kQ@k@i�

Ai, Bi, and Ci are functions of K, G3, G4, G5

Coefficients

9

Acknowledgments

This work was supported in part by JSPS Grant-in-Aidfor Research Activity Start-up No. 22840011 (T.K.), theGrant-in-Aid for Scientific Research No. 21540270 and

No. 21244033 and JSPS Core-to-Core Program “Inter-national Research Network for Dark Energy”. R.K. ac-knowledges support by a research assistant program ofHiroshima University.

Appendix A: Coe!cients in the field equations

Here we summarize the definitions of the coe!cients in the field equations:

FT := 2!G4 !X

"!G5X +G5!

#$, (A1)

GT := 2!G4 ! 2XG4X !X

"H!G5X !G5!

#$, (A2)

" := !!XG3X + 2HG4 ! 8HXG4X ! 8HX2G4XX + !G4! + 2X!G4!X

!H2!%5XG5X + 2X2G5XX

&+ 2HX (3G5! + 2XG5!X) , (A3)

A0 :="

H2+

"

H+ FT ! 2GT ! 2

GT

H! E + P

2H2, (A4)

A1 :=1

H

dGT

dt+ GT ! FT , (A5)

A2 := GT ! "

H, (A6)

B0 :=X

H

'!G3X + 3

"X + 2HX

#G4XX + 2XXG4XXX ! 3!G4!X + 2!XG4!XX

+"H +H2

#!G5X + !

!2HX +

"H +H2

#X$G5XX +H!XXG5XXX ! 2

"X + 2HX

#G5!X

!!XG5!!X !X"X ! 2HX

#G5!XX

(, (A7)

B1 := 2X!G4X + ! (G5X +XG5XX)!G5! +XG5!X

$, (A8)

B2 := !2X"G4X + 2XG4XX +H!G5X +H!XG5XX !G5! !XG5!X

#, (A9)

B3 := H!XG5X , (A10)

C0 := 2X2G4XX +2X2

3

"2!G5XX + !XG5XXX ! 2G5!X +XG5!XX

#, (A11)

C1 := H!X (G5X +XG5XX) . (A12)

Appendix B: Intermediate regime for G5X = 0

Here we would like to point out that there could be an interesting intermediate regime where the quadratic termin Eq. (50) comes into play so that we have the solution

Q! " ±H

)C"2C2

µ

r. (B1)

This regime can be found in the range

*C#C22

µ

H2

+1/3

# r #*C"C2C21

µ

H2

+1/3

(Case I),

,C2#

C"C2µ

H2

-1/3

# r #*C"C2C21

µ

H2

+1/3

(Case II). (B2)

All coefficients are determined by the background

Spherical symmetric caseαi, βi, and γi are functions of K, G3, G4, G5

where

c2h 0

r� �0

r� ↵1

Q0

r=

�1

H2

✓Q0

r

◆2

+ 2�3

H2

�0

r

Q0

r

0

r+ ↵2

Q0

r=

1

8⇡GT

�M(t, r)

r3� �2

H2

✓Q0

r

◆2

� 2�3

H2

0

r

Q0

r� 2

3

�1H4

✓Q0

r

◆3

↵0Q0

r� ↵1

0

r� ↵2

�0

r= 2

� �0

H2

✓Q0

r

◆2

+�1

H2

0

r

Q0

r+

�2

H2

�0

r

Q0

r+

�3

H2

�0

r

0

r

+�0H4

✓Q0

r

◆3

+�1H4

�0

r

✓Q0

r

◆2�

✓ EOM for gravity and scalar field can be integrated once,

c2h ⌘ FT /GT (Propagation speed of the gravitational waves)

Case 1 :

Case 2 :

Case 3 :

G4X = 0, G5 = 0

5

Equation (39) can easily be solved to give

Q!

r=

H2

B

!"1 +

2BCµH2r3

! 1

#. (41)

At short distances, r3 " r3" := BCµ/H2, one finds

Q! # H

B

"2BCµr

. (42)

In order for this solution to be real, we require

BC > 0 $ G3X (XG3X +G4!) > 0. (43)

In this regime, the metric potentials are given by

!! # GN!M

r2! H("1 + "2)

B

"2BCGN!M

r, (44)

"! # GN!M

r2! H"2

B

"2BCGN!M

r, (45)

where

8#GN :=1

2G4. (46)

If BC % O(1), the typical length scale r" % (µ/H2)1/3

can be estimated using Eq.(46) as

$ µ

H2

%1/3# 120

&H0

H

'2/3& !M

M#

'1/3

pc, (47)

where H0 = 70km/s/Mpc.Note that in this case the Friedmann equation can be

written as

3H2 = 8#Gcos ($m + $!) , (48)

where Gcos = 1/16#G4 = GN and $! = 2XKX ! K +6X%HG3X!2XG3!!6H%G4!. Thus, the e#ective grav-itational coupling governing short-distance gravity is thesame as the one in the Friedmann equation.The Vainshtein mechanism successfully screens the ef-

fect of the fluctuation !%, so that the two metric po-tentials coincide and exhibit the Newtonian behavior atleading order. However, GN is time-dependent since it isa function of the time-dependent field %(t), which meansthat the Vainshtein mechanism cannot suppress the timevariation of GN in a cosmological background. This factwas first noticed in Ref. [20]. We will discuss this pointfurther in the next subsection.

B. G5X = 0

Let us consider a class of models with G5X = 0. Inthis case, we see that &3 = '1 = 0. For the other nonzerocoe$cients we have the following relations:

c2h = 1 + 2&1 ( &= 1), &1 + &2 + 2'0 = 0. (49)

With G5X = 0, the problem reduces to solving a cu-bic equation one can handle. Indeed, using Eqs. (28)and (29) to remove !! and "! from Eq. (30), we obtainthe cubic equation for Q!:

(Q!)3 + C2H2r(Q!)2 +

&C12H4r2 !H2C"

µ

r

'Q!

!H4C#µ2

= 0, (50)

where r-independent coe$cients are defined as

C# :="1 + c2h"2

2&1&2 + c2h&22 ! '0

,

C" :=&1 + c2h&2

2&1&2 + c2h&22 ! '0

,

C1 :="0 + (2"1 + c2h"2)"2

2&1&2 + c2h&22 ! '0

,

C2 :=2&0 + 3

("1&2 + "2&1 + c2h"2&2

)

2 (2&1&2 + c2h&22 ! '0)

. (51)

We note the expressions for "! and !! in terms of Q!:

!! = c2hµ

r2! ("1 + c2h"2)Q

! ! (&1 + c2h&2)Q!2

H2r,

"! =µ

r2! "2Q

! ! &2Q!2

H2r. (52)

Linearizing Eq. (50) at r3 ' (C"/C1)µ/H2, one obtainsthe solution (35) as expected. This will be matched toone of the following three solutions at short distances:

Q! # +H

"C"

µ

r, !H

"C"

µ

r, !C#

C"H2r

2. (53)

If simply C# % C" % C1 % C2 = O(1),2 the two regimesare connected at around r % r", where

r" :=

&C2#

C21C"

µ

H2

'1/3

for Q! # ±H

"C"

µ

r,

r" :=

&!C"C1

µ

H2

'1/3

for Q! # !C#C"

H2r

2. (54)

If C" > 0 and C1 > 0, the solution with the bound-ary condition (35) at large r can be matched either toQ! # +H

*C"µ/r or to Q! # !H

*C"µ/r. We call

this situation Case I. This is possible for (C2, C#) in theshaded region in Fig. 1. For C# > 0 (respectively C# < 0),the short-distance solution is given by +H

*C"µ/r (re-

spectively !H*

C"µ/r). Outside this region one cannot

2 This is probably the most natural case if one considers a modelthat accounts for the present cosmic acceleration and the coe!-cients are evaluated at present time, because in that case thereis only one typical length scale L = H!1

0 .

G5X 6= 0

aQ02 + bQ0 + c = 0

aQ03 + bQ02 + cQ0 + d = 0

aQ06 + bQ05 + cQ04 + dQ03 + eQ02 + f Q0 + g = 0

Linear solution✓ For sufficiently large r, (linearlize ψ, Φ, and φ)

�0 =1

8⇡GT

c2h↵0 � ↵21

↵0 + (2↵1 + c2h↵2)↵2

�M

r2

0 =1

8⇡GT

↵0 + ↵1↵2

↵0 + (2↵1 + c2h↵2)↵2

�M

r2

Q0 =1

8⇡GT

↵1 + c2h↵2

↵0 + (2↵1 + c2h↵2)↵2

�M

r2

In general, the gravitational coupling Geff in the Poisson equation is different from Newton’s constant on cosmological scales

(time-dependent)effective gravitational coupling G(Linear)

e↵ ( 6= GN )

Case 1:

7

As in the case of the previous subsection, GN is in gen-eral time-dependent, as it is a function of time-dependent!(t) and X = !2(t)/2. We thus illustrate how the Vain-shtein mechanism fails to suppress the time-variation ofGN in a cosmological background within the context ofsome generic scalar-tensor theories minimally coupled tomatter. The claim was originally suggested using theEinstein frame action in Ref. [20]. Here we explicitlygive the concrete formula with which one can evaluatethe time-variation of GN for a given model.If GN happens to vary very slowly, we can say that

usual Newtonian gravity is reproduced in the vicinity ofthe source. However, in general, one expects that GN

varies on cosmological time scales. The time-variation|G/G| is constrained from Lunar Laser Ranging experi-ments to be |GN/GN | < 0.02H0 [21].At this stage it is interesting to look at the background

evolution for the models with G5X = 0. The (modified)Friedmann equation (3) can be written as

3H2 = 8"Gcos (#m + #!) , (61)

where the gravitational coupling in the Friedmann equa-tion read o! from the above exactly coincides with theexpression for GN ,

Gcos = GN , (62)

and

#! := 2XKX !K ! 2XG3!

+6H!X!G3X ! 2X!G4!X ! !G4!

".

The situation here is the same as what we have seenin the previous subsection. We refer a constraint inRef. [22], obtained by translating the Big Bang nucle-osynthesis (BBN) bound on extra relativistic degrees offreedom,

####1!GN |BBN

GN |now

#### ! 0.1, (63)

where GN |BBN (respectively, GN |now) is evaluated at thetime of BBN (respectively, today).Having thus seen that the Newtonian behavior is re-

produced with time-dependent GN , let us then evaluateleading order corrections to the potentials. In this casewe need to keep the subleading term in Q!:

Q! " ±H

$C"

µ

r+

H2(C# ! 2C"C2)r4C"

. (64)

From this we obtain the corrections "#! = #! !GN$M/r2 and "$! = $! !GN$M/r2 as

"#! = #H

%%2 +

&2

2C"(C# ! 2C"C2)

&$2GN$M

r, (65)

"$! = #H

%%1 + c2h%2 +

&1 + c2h&2

2C"(C# ! 2C"C2)

&

$$

2GN$M

r.(66)

For the solution Q! " !(C#/C")H2r/2, we find

#! " c2h8"GT

$M

r2+O(r), $! " 1

8"GT

$M

r2+O(r), (67)

implying that the parameterized post-Newtonian param-eter ' is given by ' = 1/c2h. Therefore, c

2h is tightly con-

strained from solar-system tests in this case: |1 ! '| <2.3$ 10"5 [23].

When the coe%cients C#, C" , C1, and C2 have hi-erarchies in their values, we find a variety of solutionsto Eq. (50) on an intermediate scale between the linearregime at large r and the small r limit of Case I or CaseII. The details are summarized in Appendix B, whichcould be potentially confronted with observations.

C. G5X != 0

Let us finally discuss the most general case where allthe coe%cients in Eqs. (28)–(30) are nonzero. Althoughone can still eliminate #! and $! to get an equationsolely in terms of Q!, the resulting equation will be asextic equation. This hinders us to analyze a variety ofpossible solutions in detail. However, for &3 %= 0 and'1 %= 0 one can show that there is no solution such that#! " $! & 1/r2 on su%ciently small scales. To showthis, one substitutes #! " $! & 1/r2 to Eq. (28). Thesecond term in the right hand side can be compensatedby the other provided that Q! & r or Q! & 1/r2. IfQ! & r, one cannot find a term that compensates theforth term in the right hand side of Eq. (30) which be-haves as #!$!/r2 & 1/r6. If Q! & 1/r2, then one cannotfind a term that compensates the last term in the righthand side of Eq. (29). Thus, there is no consistent so-lution with #! " $! & 1/r2 on su%ciently small scales.This implies that the typical length scale associated withB3 and C1 must be as small as O(100 µm) [24], though itis uncertain whether or not we can have the Newtonianbehavior of the potentials on intermediate scales.

V. APPLICATIONS

A. Evolution of density perturbations

Since it is assumed that matter is minimally coupledto !, no modification is made for the energy conservationequation and the Euler equation for matter. Therefore,the nonlinear evolution equation for $ is given by

$ + 2H $ ! 4

3

$2

1 + $= (1 + $)

'2

a2#. (68)

However, as we have seen in the previous section, therelation between # and $ is modified. One may tacklethe nonlinear equations using the perturbative approach(e.g., [25]). We derive the Fourier transform of the non-linear equations in Appendix C.

Time-dependence of GN can be tested by BBN

Friedmann equation

7





3H2 = 8"Gcos (#m + #!) , (61)


Gcos = GN , (62)

and

#! := 2XKX !K ! 2XG3!

+6H!X!G3X ! 2X!G4!X ! !G4!

".


####1!GN |BBN

GN |now

#### ! 0.1, (63)



Q! " ±H

$C"

µ

r+

H2(C# ! 2C"C2)r4C"

. (64)


"#! = #H

%%2 +

&2

2C"(C# ! 2C"C2)

&$2GN$M

r, (65)

"$! = #H

%%1 + c2h%2 +

&1 + c2h&2

2C"(C# ! 2C"C2)

&

$$

2GN$M

r.(66)


#! " c2h8"GT

$M

r2+O(r), $! " 1

8"GT

$M

r2+O(r), (67)


2h is tightly con-



C. G5X != 0


V. APPLICATIONS



$ + 2H $ ! 4

3

$2

1 + $= (1 + $)

'2

a2#. (68)


(Uzan 2011)

7





3H2 = 8"Gcos (#m + #!) , (61)


Gcos = GN , (62)

and

#! := 2XKX !K ! 2XG3!

+6H!X!G3X ! 2X!G4!X ! !G4!

".


####1!GN |BBN

GN |now

#### ! 0.1, (63)



Q! " ±H

$C"

µ

r+

H2(C# ! 2C"C2)r4C"

. (64)


"#! = #H

%%2 +

&2

2C"(C# ! 2C"C2)

&$2GN$M

r, (65)

"$! = #H

%%1 + c2h%2 +

&1 + c2h&2

2C"(C# ! 2C"C2)

&

$$

2GN$M

r.(66)


#! " c2h8"GT

$M

r2+O(r), $! " 1

8"GT

$M

r2+O(r), (67)


2h is tightly con-



C. G5X != 0


V. APPLICATIONS



$ + 2H $ ! 4

3

$2

1 + $= (1 + $)

'2

a2#. (68)


G4X = 0, G5 = 0

✓ Kinetic gravity braiding with non-minimal coupling (Deffayet et al. 2010)

4

IV. SPHERICALLY SYMMETRICCONFIGURATIONS

We now want to consider a spherically symmetric over-density on a cosmological background. For this purposeit is convenient to use the coordinate r = a(t)

!!ijxixj .

We are primarily interested in scales much smaller thanthe horizon radius, rH ! 1. Under this circumstancethe background metric may be written as ds2 " #dt2 +dr2 + r2d!2, where d!2 is the line element of the unit2-sphere.

The spherical symmetry allows us to write

a!2$2Q = r!2(r2Q")",

a!4"$2"$2Q# "i"j""

i"jQ#

= 2r!2(r""Q")",

a!6U (3) = 2r!2$""(Q")2

%",

where a prime denotes di#erentiation with respect to r.The gravitational field equations and scalar-field equa-tion of motion can then be integrated once, leading to

c2h$"

r# ""

r# #1

Q"

r=

$1

H2

&Q"

r

'2

+ 2$3

H2

""

r

Q"

r, (28)

$"

r+ #2

Q"

r=

1

8%GT

!M(t, r)

r3# $2

H2

&Q"

r

'2

# 2$3

H2

$"

r

Q"

r# 2

3

&1H4

&Q"

r

'3

, (29)

#0Q"

r# #1

$"

r# #2

""

r= 2

(# $0

H2

&Q"

r

'2

+$1

H2

$"

r

Q"

r+

$2

H2

""

r

Q"

r+

$3

H2

""

r

$"

r+

&0H4

&Q"

r

'3

+&1H4

""

r

&Q"

r

'2),

(30)

where we defined

!M(t, r) = 4%'m(t)

* r

!(t, r) r"2dr", (31)

c2h := FT /GT , and dimensionless coe%cients

#i(t) :=Ai

GT, $i(t) :=

Bi

GT, &i(t) :=

Ci

GT. (32)

Note that ch is the propagation speed of gravitationalwaves which may be di#erent from 1 in general [13]. Notealso that in deriving Eqs. (28)–(30) we have set the in-tegration constants to be zero, requiring that "" = $" =Q" = 0 is a solution if !M = 0.For su%ciently large r, we may neglect all the nonlinear

terms in the above equations. The solution to the linearequations is given by

"" =c2h#0 # #2

1

#0 + (2#1 + c2h#2)#2

µ

r2, (33)

$" =#0 + #1#2

#0 + (2#1 + c2h#2)#2

µ

r2, (34)

Q" =#1 + c2h#2

#0 + (2#1 + c2h#2)#2

µ

r2. (35)

where we defined µ := !M/8%GT . In this regime, theparameterized post-Newtonian parameter & is given by

& =#0 + #1#2

c2h#0 # #21

, (36)

which in general di#ers from unity.

A. G4X = 0, G5 = 0

A simple example for which nonlinear terms can op-erate is the model with G4X = 0 = G5 and G3X %= 0,i.e.,

L = G4(()R+K((, X)#G3((, X)!(. (37)

In this case, we have $1 = $2 = $3 = &0 = &1 = 0 andc2h = 1. We also have the relation

$0 =#1

2+ #2 ( %= 0). (38)

The Lagrangian (37) corresponds to a nonminimallycoupled version of kinetic gravity braiding [15], and hasbeen studied extensively in the context of inflation [16]and dark energy/modified gravity [17–19]. Of course thenonminimal coupling can be undone by performing a con-formal transformation, but in the present analysis wehave no particular reason to do so. Note that even inthe case of G4 =const the scalar ( is coupled to the cur-vature at the level of the field equation, which signals“braiding.”

Using Eqs. (28) and (29), Eq. (30) reduces to aquadratic equation

BH2

&Q"

r

'2

+2Q"

r= 2C µ

r3, (39)

where

B :=4$0

#0 + 2#1#2 + #22

, C :=#1 + #2

#0 + 2#1#2 + #22

. (40)

4

IV. SPHERICALLY SYMMETRICCONFIGURATIONS

We now want to consider a spherically symmetric over-density on a cosmological background. For this purposeit is convenient to use the coordinate r = a(t)

!!ijxixj .

We are primarily interested in scales much smaller thanthe horizon radius, rH ! 1. Under this circumstancethe background metric may be written as ds2 " #dt2 +dr2 + r2d!2, where d!2 is the line element of the unit2-sphere.

The spherical symmetry allows us to write

a!2$2Q = r!2(r2Q")",

a!4"$2"$2Q# "i"j""

i"jQ#

= 2r!2(r""Q")",

a!6U (3) = 2r!2$""(Q")2

%",

where a prime denotes di#erentiation with respect to r.The gravitational field equations and scalar-field equa-tion of motion can then be integrated once, leading to

c2h$"

r# ""

r# #1

Q"

r=

$1

H2

&Q"

r

'2

+ 2$3

H2

""

r

Q"

r, (28)

$"

r+ #2

Q"

r=

1

8%GT

!M(t, r)

r3# $2

H2

&Q"

r

'2

# 2$3

H2

$"

r

Q"

r# 2

3

&1H4

&Q"

r

'3

, (29)

#0Q"

r# #1

$"

r# #2

""

r= 2

(# $0

H2

&Q"

r

'2

+$1

H2

$"

r

Q"

r+

$2

H2

""

r

Q"

r+

$3

H2

""

r

$"

r+

&0H4

&Q"

r

'3

+&1H4

""

r

&Q"

r

'2),

(30)

where we defined

!M(t, r) = 4%'m(t)

* r

!(t, r) r"2dr", (31)

c2h := FT /GT , and dimensionless coe%cients

#i(t) :=Ai

GT, $i(t) :=

Bi

GT, &i(t) :=

Ci

GT. (32)

Note that ch is the propagation speed of gravitationalwaves which may be di#erent from 1 in general [13]. Notealso that in deriving Eqs. (28)–(30) we have set the in-tegration constants to be zero, requiring that "" = $" =Q" = 0 is a solution if !M = 0.For su%ciently large r, we may neglect all the nonlinear

terms in the above equations. The solution to the linearequations is given by

"" =c2h#0 # #2

1

#0 + (2#1 + c2h#2)#2

µ

r2, (33)

$" =#0 + #1#2

#0 + (2#1 + c2h#2)#2

µ

r2, (34)

Q" =#1 + c2h#2

#0 + (2#1 + c2h#2)#2

µ

r2. (35)

where we defined µ := !M/8%GT . In this regime, theparameterized post-Newtonian parameter & is given by

& =#0 + #1#2

c2h#0 # #21

, (36)

which in general di#ers from unity.

A. G4X = 0, G5 = 0

A simple example for which nonlinear terms can op-erate is the model with G4X = 0 = G5 and G3X %= 0,i.e.,

L = G4(()R+K((, X)#G3((, X)!(. (37)

In this case, we have $1 = $2 = $3 = &0 = &1 = 0 andc2h = 1. We also have the relation

$0 =#1

2+ #2 ( %= 0). (38)

The Lagrangian (37) corresponds to a nonminimallycoupled version of kinetic gravity braiding [15], and hasbeen studied extensively in the context of inflation [16]and dark energy/modified gravity [17–19]. Of course thenonminimal coupling can be undone by performing a con-formal transformation, but in the present analysis wehave no particular reason to do so. Note that even inthe case of G4 =const the scalar ( is coupled to the cur-vature at the level of the field equation, which signals“braiding.”

Using Eqs. (28) and (29), Eq. (30) reduces to aquadratic equation

BH2

&Q"

r

'2

+2Q"

r= 2C µ

r3, (39)

where

B :=4$0

#0 + 2#1#2 + #22

, C :=#1 + #2

#0 + 2#1#2 + #22

. (40)

Propagation speed of the gravitational waves

Newton’s constant GN(t)

PPN parameter � ⌘ 0/�0 = 1

Gravitational potentials

0 = �0 ' 1

16⇡G4(t)

�M

r2

✓ Inside the Vainshtein radius rV

Q0 ' H

B

r2BCµr

⌧ GM

r2

✓ Scalar field solution

Q0

r=

H2

B

r1 +

⇣rVr

⌘3� 1

!

rV =

✓2BCµH2

◆1/3Vainshtein radius

rV~100 pc for sun

c2h = 1 + 2�1 6= 1

Propagation speed of the gravitational waves

✓ Lagrangian

B. G5X = 0

L =K(�, X)�G3(�, X)⇤�

+G4(�, X)R+G4X

⇥(⇤�)2 � (rµr⌫�)

2⇤

+G5(�)Gµ⌫rµr⌫�

does not depend on the kinetic term X

✓ Scalar field equations

(Q0)3 + C2H2r(Q0)2 +

✓C12H4r2 �H2C�

µ

r

◆Q0 � H4C↵µ

2= 0

C1, C2, Cα, and Cβ are functions of K, G3, G4, G5 µ ⌘ �M

8⇡GT

✓ 3 possible solutions at short distance can be matched to the linear solution

Q0 ' ±H

rC�

µ

r, �C↵

C�H2r

2

Case 2:

✓Solution II


✓Solution I

7





3H2 = 8"Gcos (#m + #!) , (61)


Gcos = GN , (62)

and

#! := 2XKX !K ! 2XG3!

+6H!X!G3X ! 2X!G4!X ! !G4!

".


####1!GN |BBN

GN |now

#### ! 0.1, (63)



Q! " ±H

$C"

µ

r+

H2(C# ! 2C"C2)r4C"

. (64)


"#! = #H

%%2 +

&2

2C"(C# ! 2C"C2)

&$2GN$M

r, (65)

"$! = #H

%%1 + c2h%2 +

&1 + c2h&2

2C"(C# ! 2C"C2)

&

$$

2GN$M

r.(66)


#! " c2h8"GT

$M

r2+O(r), $! " 1

8"GT

$M

r2+O(r), (67)


2h is tightly con-



C. G5X != 0


V. APPLICATIONS



$ + 2H $ ! 4

3

$2

1 + $= (1 + $)

'2

a2#. (68)


Time-dependence of GN can be tested by BBN

Friedmann equation

7





3H2 = 8"Gcos (#m + #!) , (61)


Gcos = GN , (62)

and

#! := 2XKX !K ! 2XG3!

+6H!X!G3X ! 2X!G4!X ! !G4!

".


####1!GN |BBN

GN |now

#### ! 0.1, (63)



Q! " ±H

$C"

µ

r+

H2(C# ! 2C"C2)r4C"

. (64)


"#! = #H

%%2 +

&2

2C"(C# ! 2C"C2)

&$2GN$M

r, (65)

"$! = #H

%%1 + c2h%2 +

&1 + c2h&2

2C"(C# ! 2C"C2)

&

$$

2GN$M

r.(66)


#! " c2h8"GT

$M

r2+O(r), $! " 1

8"GT

$M

r2+O(r), (67)


2h is tightly con-



C. G5X != 0


V. APPLICATIONS



$ + 2H $ ! 4

3

$2

1 + $= (1 + $)

'2

a2#. (68)


Q0 ' ±H

rC�

µ

r

Q0 ' �C↵C�

H2r

2

0 = �0 ' C (t)

8⇡GT (t)

�M

r2


PPN parameter � ⌘ 0/�0 = 1


�0 ' c2h 0 ' c2h

8⇡GT

�M

r2


PPN parameter

� ⌘ 0

�0 =1

c2h( 6= 1) propagation speed of

gravitational waves

Solar-system tests (Will 2005)

explicitly give the concrete formula with which one canevaluate the time variation of GN for a given model.

If GN happens to vary very slowly, we can say that theusual Newtonian gravity is reproduced in the vicinity of thesource. However, in general, one expects that GN varies oncosmological time scales. The time variation j _G=Gj isconstrained from lunar laser ranging experiments to bej _GN=GNj< 0:02H0 [21].

At this stage it is interesting to look at the backgroundevolution for the models with G5X ! 0. The (modified)Friedmann equation (3) can be written as

3H2 ! 8!Gcos""m # "#$; (61)

where the gravitational coupling in the Friedmann equationread off from the above exactly coincides with the expres-sion for GN ,

Gcos ! GN; (62)

and

"# :! 2XKX % K % 2XG3#

# 6H"X _#G3X % 2X _#G4#X % _#G4#$:

The situation here is the same as what we have seen in theprevious subsection. We refer to a constraint in Ref. [22],obtained by translating the big bang nucleosynthesis(BBN) bound on extra relativistic degrees of freedom, as

!!!!!!!!1%GNjBBNGNjnow

!!!!!!!!& 0:1; (63)

where GNjBBN (respectively, GNjnow) is evaluated at thetime of BBN (respectively, today).

Having thus seen that the Newtonian behavior is repro-duced with time-dependent GN , let us then evaluate lead-ing order corrections to the potentials. In this case we needto keep the subleading term in Q0:

Q0 ’ &H

"""""""""""C$

%

r

r#H2"C& % 2C$C2$r

4C$: (64)

From this we obtain the corrections !"0 ! "0 %GN'M=r2 and !#0 ! #0 %GN'M=r2 as

!"0 ! 'H#&2 #

$2

2C$"C& % 2C$C2$

$ """""""""""""""""2GN'M

r

s; (65)

!#0 ! 'H#&1 # c2h&2

# $1 # c2h$2

2C$"C& % 2C$C2$

$ """""""""""""""""2GN'M

r

s: (66)

For the solution Q0 ’ %"C&=C$$H2r=2, we find

"0 ’ c2h8!GT

'M

r2#O"r$; #0 ’ 1

8!GT

'M

r2#O"r$; (67)

implying that the parametrized post-Newtonian parame-ter ( is given by ( ! 1=c2h. Therefore, c2h is tightlyconstrained from solar-system tests in this case:j1% (j< 2:3( 10%5 [23].When the coefficients C&, C$, C1, and C2 have hierar-

chies in their values, we find a variety of solutions toEq. (50) on an intermediate scale between the linear regimeat large r and the small r limit of Case I or Case II. Thedetails are summarized in Appendix B, which could bepotentially confronted with observations.

C. G5X ! 0

Let us finally discuss the most general case where all thecoefficients in Eqs. (28)–(30) are nonzero. Although onecan still eliminate "0 and #0 to get an equation solely interms of Q0, the resulting equation will be a sextic equa-tion. This hinders us from analyzing a variety of possiblesolutions in detail. However, for $3 ! 0 and (1 ! 0 onecan show that there is no solution such that "0 ’ #0 )1=r2 on sufficiently small scales. To show this, one sub-stitutes"0 ’ #0 ) 1=r2 to Eq. (28). The second term in theright-hand side can be compensated by the other providedthat Q0 ) r or Q0 ) 1=r2. If Q0 ) r, one cannot find a termthat compensates the forth term in the right-hand side ofEq. (30) which behaves as "0#0=r2 ) 1=r6. If Q0 ) 1=r2,then one cannot find a term that compensates the last termin the right-hand side of Eq. (29). Thus, there is no con-sistent solution with "0 ’ #0 ) 1=r2 on sufficiently smallscales. This implies that the typical length scale associatedwith B3 and C1 must be as small as O"100 %m$ [24],though it is uncertain whether or not we can have theNewtonian behavior of the potentials on intermediatescales.

V. APPLICATIONS


Since it is assumed that matter is minimally coupled to#, no modification is made for the energy conservationequation and the Euler equation for matter. Therefore, thenonlinear evolution equation for ' is given by

$'# 2H _'% 4

3

_'2

1# '! "1# '$r

2

a2": (68)

However, as we have seen in the previous section, therelation between " and ' is modified. One may tacklethe nonlinear equations using the perturbative approach(e.g., [25]). We derive the Fourier transform of the non-linear equations in Appendix C.

VAINSHTEIN SCREENING IN A COSMOLOGICAL . . . PHYSICAL REVIEW D 85, 024023 (2012)

024023-7

B. G5X = 0Case 2:

✓ At sufficiently small scales, the inverse square law for the gravitational force

is not a solution anymore !

The Vainshtein mechanism no longer works in the presence of G5X !!

0(r), �0(r) / 1

r2

The coupling between the Einstein tensor and the kinetic term XGμν leads to a strong modification of gravity even at short distances

CASE 3: G5X 6= 0

Current Problems

How can we determine arbitrary functions in Horndeski theory?

• There is no principle to determine arbitrary functions

• Probably, they are determined from the effective theory of a

fundamental theory

• Need to construct such a theory

Massive gravity

✓ The action

U2(K) =1

2!✏µ⌫⇢�✏

↵�⇢�K µ↵ K ⌫

�

U3(K) =1

3!✏µ⌫⇢�✏

↵��K µ↵ K ⌫

� K ⇢�

U4(K) =1

4!✏µ⌫⇢�✏

↵��K µ↵ K ⌫

� K ⇢� K �

�

where

(BD) Ghost free massive gravity

de Rham, Gabadadze, Tolley (2011)

S = M

2Pl

Zd

4x

p�g

R

2+m

2 (U2 + ↵3U3 + ↵4U4) + Lm[gµ⌫ , ]

�

Kµ⌫ = �µ⌫ � (

pg�1f)µ⌫

Massive gravityde Rham, Gabadadze, Tolley (2011)

✓ The action in the decoupling limit (extracting the scalar mode of graviton)

S =

Zd

4

x

�1

2hµ⌫Eµ⌫,↵�

h↵� +5X

n=2

�n

(⇤3

3

)n�2

L(n)galileon

+�X

⇤6

3

h

µ⌫X

(3)

µ⌫

+1

MPhµ⌫T

µ⌫ +1

MP�T +

�T

⇤3

3

MP@µ�@⌫ �T

µ⌫

�

L(2)

galileon

= �1

2(@�)2

L(3)

galileon

= �1

2(@�)2[⇧]

L(4)

galileon

= �1

2(@�)2

�[⇧]2 � [⇧2]

�

L(5)

galileon

= �1

2(@�)2

�[⇧]3 � 3[⇧][⇧2] + 2[⇧3]

�

Horndeski theory includes the decoupling limit of dRGT massive gravity

βi are functions of model parameters αi

Current Problems 2

Massive gravity is a candidate of the galileon effective thoery, but

• homogeneous and isotropic cosmological solution suffers from

ghost instabilities

Gumrukcuoglu, Lin, Mukohyama (2011)Gumrukcuoglu, Lin, Mukohyama (2012)

Bi-gravity theory

Vainshtein mechanism in bigravity theory

Rampei Kimura!

Department of Physical Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan(Dated: December 19, 2012)

I. INTRODUCTION

II. BIGRAVITY

Consider the action

S = M2f

!d4x

"!f

R[f ]

2+M2

g

!d4x

"!g

#R[g]

2+m2 (L2 + !3L3 + !4L4) + Lm[gµ! ,"]

$(1)

where

L2 =1

2

%[K]2 ! [K2]

&

L3 =1

6

%[K]3 ! 3[K][K2] + 2[K3]

&

L4 =1

24

%[K]4 ! 6[K]2[K2] + 3[K2]2 + 8[K][K3]! 6[K4]

&(2)

and Kµ! = #µ! ! (

"g"1f)µ! . Note that there is just one general covariance in this action, and the line element of each

metric is given by ds2g = gµ!dxµdx! and ds2f = fµ!dxµdx! , respectively.Now we introduce the Stuckelberg scalar $a in the metric f ,

fµ!(x) =%$a

%xµ

%$b

%x!fab($), (3)

This allows us to bring two general covariance in the action and the line element for the metric f can be also writtenas ds2f = fabd$ad$b. The unitary gauge choice $a = xa reduces the original bigravity theory.

III. BACKGROUND EVOLUTION

We assume spacially flat FRW metric for the metric g,

ds2g = !N(t)2dt2 + a(t)2#ijdxidxj (4)

and for the metric f

ds2f = !(d$0)2 + b($0)2#ijd$id$j (5)

We choose the following gauge of the Stuckelberg field $0 = f(t) and $i = xi, then the metric f can be written as

fµ! = Diag

'!f(t)2, b[f(t)]2#ij

((6)

Here an overdot denotes di!erentiation with respect to t. In this case,

Kµ! = Diag

'1! f

N,

)1! b

a

*#ij

((7)

!Email: rampei"at"theo.phys.sci.hiroshima-u.ac.jp

Bigravity = massive graviton + massless graviton

S = M

2f

Zd

4x

p�f

R[f ]

2

+M

2g

Zd

4x

p�g

R[g]

2+m

2 (U2 + ↵3U3 + ↵4U4) + Lm[gµ⌫ , ]

�

✓ The action

where

There is a viable cosmological (self-accelerating) solution without ghost-instabilities

Strauss et al. (2011)

Hassan and Rosen (2011)

Decoupling limit of massive gravity

• Scalar mode of graviton can be extracted from Stuckelberg field

• Explicit relation with galileon theory

Decoupling limit of bi-gravity theory

• UNKNOWN...

• Does really Vainshtein mechanism operate ??

• How can we extract the scalar mode of massive graviton ??

Bi-gravity theory

fµ⌫ = ⌘ab@µYa@⌫Y

b

SummaryVainshtein screening successfully operates in Horndeski theory, but

• Newton’s constant G=G(t)

• constrained from PPN and BBN

• inverse-square law can not be reproduced at small scales if G5X≠0

Bigravity theory

• Can we extract the scalar mode of massive graviton ?

• What about Vainshtein mechanism ?

I’m still working ...

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