Vainshtein mechanism in quasi-dilaton massive gravity · Vainshtein mechanism in quasi-dilaton massive gravity Rampei Kimura RESCEU, The University of Tokyo Mini-workshop “Massive

Vainshtein mechanismin quasi-dilaton massive gravity

Rampei KimuraRESCEU, The University of Tokyo

Mini-workshop “Massive gravity and its cosmological implications”04/09/2013

Collaborator : Gregory Gabadadze (CCPP, New York University)

In progress

dRGT massive gravity

✓ The action

where the potential is given by

(BD) Ghost free massive gravity

de Rham, Gabadadze, Tolley (2011)

Vainshtein screening in quasi-dilaton theory

Rampei Kimura!

Research Center for the Early Universe (RESCEU),The University of Tokyo, Tokyo, 113-0033, Japan

Gregory Gabadadze†

Center for Cosmology and Particle Physics, Department of Physics,New York University, New York, NY 10003, USA

(Dated: April 1, 2013)

Quasi-dilaton theory is the candidate for massive gravity theory, which couples to an addititonalscalar degrees of freedom. Similarly to dRGT massvie gravity theory, there is no BD ghost in thisthoery. In this paper, we show that there is no usual solution, which posses Vainshtein mechanism.Insted, we only have cosmological solution. We clarly show that assymptotically Minkowski solutionhas always ghost in the scalar modes in the decoupling limit of the theory.

I. INTRODUCTION

It is now belived that general relativity is the theory of gravity, which describe solar system scale and it has beentested for a long decade. It seems that there is no contradiction within tests in our solar system scales. However, ifwe extend this theory to ”cosmology”, we still have a number of question that we can not understand yet. One isthe existance of dark matter, and this is now believed as some particle that we have not discovered yet. Nonethelessthis unknow matter could be of the form of some energy or be part of the theory of gravity. Another example is darkenergy, which is responsible for current cosmic accleration of the universe, and this existance has not confirmed yet.This unknow energy constitues 72 percent of the energy in the universe. One possible solution is the cosmologicalconstant, but this model su!ers from the cosmological constant porblem.There might be a chance to explain this cosmic acceleration, for example, modification of gravity or other fluid

that we have not discovered yet. As a candiate of alternative theory of gravity, massive gravity has been recentlyattracted considerable attention. In 1939, Pauli and Fierz found that the ”linearized” massive gravity which doesnot possess ghost. This theory is based on general relativity, and the mass is measured by the di!erence betweenthe fluctuation of the metric and Minkowski metric. However, Boulwer and Deser found that there is always ghostat nonlienar level. Now we have ghost free massive gravity constructed by de Rham, Gabadadze, and Tolley. Thisincludes all the nonliear terms and describe massive spin-2 particle. Now we have some question whether we can addthe additional scalar model in massive gravity, and this has been done by [] by introducing new symmetry, calledquasi-dilaton theory. This model contains massive spin-2 mode, whose number of degree of freedom is five, and onedilaton mode. It is still opened question whether we have Vainshtein mechanism in this thoery.In this paper, we examine the Vainshtein mechanism in quasi-dilaton theory.

II. THEORY

The action for massive gravity can be described by

SMG =M2

Pl

2

!d4x

!"g

"R" m2

4(U2 + !3U3 + !4U4)

#+ Sm[gµ! ,"] (1)

where the potential of the massive graviton is given by

U2 = 2#µ"#$#!%#$Kµ

!K"% = 4

$[K2]" [K]2

%

U3 = #µ"&##!%'#Kµ

!K"%K

&' = "[K]3 + 3[K][K2]" 2[K3]

U4 = #µ"&##!%'$Kµ

!K"%K

&'K

#$ = "[K]4 + 6[K]2[K2]" 3[K2]2 " 8[K][K3] + 6[K4] (2)

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

Vainshtein screening in quasi-dilaton theory

Rampei Kimura!

Research Center for the Early Universe (RESCEU),The University of Tokyo, Tokyo, 113-0033, Japan

Gregory Gabadadze†

Center for Cosmology and Particle Physics, Department of Physics,New York University, New York, NY 10003, USA

(Dated: April 1, 2013)

Quasi-dilaton theory is the candidate for massive gravity theory, which couples to an addititonalscalar degrees of freedom. Similarly to dRGT massvie gravity theory, there is no BD ghost in thisthoery. In this paper, we show that there is no usual solution, which posses Vainshtein mechanism.Insted, we only have cosmological solution. We clarly show that assymptotically Minkowski solutionhas always ghost in the scalar modes in the decoupling limit of the theory.

I. INTRODUCTION

It is now belived that general relativity is the theory of gravity, which describe solar system scale and it has beentested for a long decade. It seems that there is no contradiction within tests in our solar system scales. However, ifwe extend this theory to ”cosmology”, we still have a number of question that we can not understand yet. One isthe existance of dark matter, and this is now believed as some particle that we have not discovered yet. Nonethelessthis unknow matter could be of the form of some energy or be part of the theory of gravity. Another example is darkenergy, which is responsible for current cosmic accleration of the universe, and this existance has not confirmed yet.This unknow energy constitues 72 percent of the energy in the universe. One possible solution is the cosmologicalconstant, but this model su!ers from the cosmological constant porblem.There might be a chance to explain this cosmic acceleration, for example, modification of gravity or other fluid

that we have not discovered yet. As a candiate of alternative theory of gravity, massive gravity has been recentlyattracted considerable attention. In 1939, Pauli and Fierz found that the ”linearized” massive gravity which doesnot possess ghost. This theory is based on general relativity, and the mass is measured by the di!erence betweenthe fluctuation of the metric and Minkowski metric. However, Boulwer and Deser found that there is always ghostat nonlienar level. Now we have ghost free massive gravity constructed by de Rham, Gabadadze, and Tolley. Thisincludes all the nonliear terms and describe massive spin-2 particle. Now we have some question whether we can addthe additional scalar model in massive gravity, and this has been done by [] by introducing new symmetry, calledquasi-dilaton theory. This model contains massive spin-2 mode, whose number of degree of freedom is five, and onedilaton mode. It is still opened question whether we have Vainshtein mechanism in this thoery.In this paper, we examine the Vainshtein mechanism in quasi-dilaton theory.

II. THEORY

The action for massive gravity can be described by

SMG =M2

Pl

2

!d4x

!"g

"R" m2

4(U2 + !3U3 + !4U4)

#+ Sm[gµ! ,"] (1)

where the potential of the massive graviton is given by

U2 = 2#µ"#$#!%#$Kµ

!K"% = 4

$[K2]" [K]2

%

U3 = #µ"&##!%'#Kµ

!K"%K

&' = "[K]3 + 3[K][K2]" 2[K3]

U4 = #µ"&##!%'$Kµ

!K"%K

&'K

#$ = "[K]4 + 6[K]2[K2]" 3[K2]2 " 8[K][K3] + 6[K4] (2)

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

2

and the new nonlinear tensor is defiend by

Kµ! = !µ! !

!"abgµ"#"$a#!$b (3)

Here $a is called Stuckelberg field, which is responsible for restoring general covariance of the theory. Fixing gauge,corresponding to $a = xa, reduces to original theory of massive gravity. Here we have five degrees of freedom, whichcan be decomposed into two massless graviton, two massless vector, and one massless scalar. It has been shown thatthe this theory is free of BD ghost in the full theory done by Hassan and Rosen.Now we impose the the new global symmetry

% " % ! &MPl, $a " e"$a (4)

and define the new tensor

Kµ! = !µ! ! e#/MPl

!"abgµ"#"$a#!$b (5)

It is very clear that this tensor is invariant under the new global symmetry. Then the action for the new action isgiven by

S =M2

Pl

2

"d4x

#!g

#R! '

M2Pl

gµ!#µ%#!% ! m2

4(U2 + &3U3 + &4U4)

$+ S! + Sm[gµ! ,(] (6)

Here we added the kineti term of the new scalar called dilaton. We also can add the action, which satisfies the grobalsymmetry,

S! = M2Plm

2&5

"d4x

#!g e4#/MPl

!det (gµ"#"$a#!$a) (7)

In usual massive gravity, this term disappears on the boudary, however, in this theory, we can have this kind like termthanks to the new additional degrees of freedom.

III. DECOUPLING LIMIT

Now we want to focus on the nonlinear e!ect of the new additional degrees of freedom. It is very convenient to usethe framework called decoupling limit, which allow us to extract the scalar mode of the graviton. In usual, the vectormode decoupled from the tensor mode in the linear level, therefore, we can safely ignore vector mode. In quasi-dilatonthoery, we can also extract another scalar mode, dilaton, by taking the decoupling limit. In order to to so, we expandStuckelberg field as

$a = !aµxµ ! "aµ#µ)/MPlm

2 (8)

Then we take the follwoing limit

MPl " $, m " 0, " = (MPlm2)1/3 = fixed,

Tµ!

MPl= fixed (9)

The action in the decoupling limit is

LDL = !1

4hµ!E"$

µ! h"$ ! '

2#µ%#µ% ! hµ!

#1

4*µ*!#! &

4"3*µ*!##! +

2"6*µ*!###

$

+ %%4&5"

3 + ,0**#+,1"3**##+

,2"6**###+

,3"9**####

&+

1

MPlhµ!Tµ! (10)

where

& = !3

4&3 ! 1, + = !1

8&3 !

1

2&4, ,0 =

1

2! 2

3&5,

,1 =3

8&3 !

1

2! &5, ,2 =

1

2&4 !

3

8&3 !

2

3&5, ,3 = !1

2&4 !

1

6&5. (11)

and

Hassan and Rosen (2011)

Quasi-dilaton theory

2


Kµ! = !µ! !

!"abgµ"#"$a#!$b (3)


% " % ! &MPl, $a " e"$a (4)


Kµ! = !µ! ! e#/MPl

!"abgµ"#"$a#!$b (5)


S =M2

Pl

2

"d4x

#!g

#R! '

M2Pl

gµ!#µ%#!% ! m2

4(U2 + &3U3 + &4U4)

$+ S! + Sm[gµ! ,(] (6)


S! = M2Plm

2&5

"d4x

#!g e4#/MPl

!det (gµ"#"$a#!$a) (7)





2 (8)


MPl " $, m " 0, " = (MPlm2)1/3 = fixed,

Tµ!

MPl= fixed (9)


LDL = !1

4hµ!E"$

µ! h"$ ! '

2#µ%#µ% ! hµ!

#1

4*µ*!#! &

4"3*µ*!##! +

2"6*µ*!###

$

+ %%4&5"

3 + ,0**#+,1"3**##+

,2"6**###+

,3"9**####

&+

1

MPlhµ!Tµ! (10)

where

& = !3

4&3 ! 1, + = !1

8&3 !

1

2&4, ,0 =

1

2! 2

3&5,

,1 =3

8&3 !

1

2! &5, ,2 =

1

2&4 !

3

8&3 !

2

3&5, ,3 = !1

2&4 !

1

6&5. (11)

✓ Symmetry

2


Kµ! = !µ! !

!"abgµ"#"$a#!$b (3)


% " % ! &MPl, $a " e"$a (4)


Kµ! = !µ! ! e#/MPl

!"abgµ"#"$a#!$b (5)


S =M2

Pl

2

"d4x

#!g

#R! '

M2Pl

gµ!#µ%#!% ! m2

4(U2 + &3U3 + &4U4)

$+ S! + Sm[gµ! ,(] (6)


S! = M2Plm

2&5

"d4x

#!g e4#/MPl

!det (gµ"#"$a#!$a) (7)





2 (8)


MPl " $, m " 0, " = (MPlm2)1/3 = fixed,

Tµ!

MPl= fixed (9)


LDL = !1

4hµ!E"$

µ! h"$ ! '

2#µ%#µ% ! hµ!

#1

4*µ*!#! &

4"3*µ*!##! +

2"6*µ*!###

$

+ %%4&5"

3 + ,0**#+,1"3**##+

,2"6**###+

,3"9**####

&+

1

MPlhµ!Tµ! (10)

where

& = !3

4&3 ! 1, + = !1

8&3 !

1

2&4, ,0 =

1

2! 2

3&5,

,1 =3

8&3 !

1

2! &5, ,2 =

1

2&4 !

3

8&3 !

2

3&5, ,3 = !1

2&4 !

1

6&5. (11)

✓ Define the new tensor

K satisfies the symmetry

massive graviton + scalar (DOF=6)

✓ The action

D’Amico, Gabadadze, Hui, Pirtskhalava (2012)

S =M

2Pl

2

Zd

4x

p�g

R� !

M

2Pl

g

µ⌫@µ�@⌫� � m

2

4(U2 + ↵3U3 + ↵4U4)

�+ Sm[gµ⌫ , ]

Decoupling limit in QMG

✓ Decoupling limit

2


Kµ! = !µ! !

!"abgµ"#"$a#!$b (3)


% " % ! &MPl, $a " e"$a (4)


Kµ! = !µ! ! e#/MPl

!"abgµ"#"$a#!$b (5)


S =M2

Pl

2

"d4x

#!g

#R! '

M2Pl

gµ!#µ%#!% ! m2

4(U2 + &3U3 + &4U4)

$+ S! + Sm[gµ! ,(] (6)


S! = M2Plm

2&5

"d4x

#!g e4#/MPl

!det (gµ"#"$a#!$a) (7)





2 (8)


MPl " $, m " 0, " = (MPlm2)1/3 = fixed,

Tµ!

MPl= fixed (9)


LDL = !1

4hµ!E"$

µ! h"$ ! '

2#µ%#µ% ! hµ!

#1

4*µ*!#! &

4"3*µ*!##! +

2"6*µ*!###

$

+ %%4&5"

3 + ,0**#+,1"3**##+

,2"6**###+

,3"9**####

&+

1

MPlhµ!Tµ! (10)

where

& = !3

4&3 ! 1, + = !1

8&3 !

1

2&4, ,0 =

1

2! 2

3&5,

,1 =3

8&3 !

1

2! &5, ,2 =

1

2&4 !

3

8&3 !

2

3&5, ,3 = !1

2&4 !

1

6&5. (11)

2


Kµ! = !µ! !

!"abgµ"#"$a#!$b (3)


% " % ! &MPl, $a " e"$a (4)


Kµ! = !µ! ! e#/MPl

!"abgµ"#"$a#!$b (5)


S =M2

Pl

2

"d4x

#!g

#R! '

M2Pl

gµ!#µ%#!% ! m2

4(U2 + &3U3 + &4U4)

$+ S! + Sm[gµ! ,(] (6)


S! = M2Plm

2&5

"d4x

#!g e4#/MPl

!det (gµ"#"$a#!$a) (7)





2 (8)


MPl " $, m " 0, " = (MPlm2)1/3 = fixed,

Tµ!

MPl= fixed (9)


LDL = !1

4hµ!E"$

µ! h"$ ! '

2#µ%#µ% ! hµ!

#1

4*µ*!#! &

4"3*µ*!##! +

2"6*µ*!###

$

+ %%4&5"

3 + ,0**#+,1"3**##+

,2"6**###+

,3"9**####

&+

1

MPlhµ!Tµ! (10)

where

& = !3

4&3 ! 1, + = !1

8&3 !

1

2&4, ,0 =

1

2! 2

3&5,

,1 =3

8&3 !

1

2! &5, ,2 =

1

2&4 !

3

8&3 !

2

3&5, ,3 = !1

2&4 !

1

6&5. (11)

✓ The action

↵, �, and �i = functions of ↵3, and ↵4

⇧µ⌫ = @µ@⌫⇡

D’Amico, Gabadadze, Hui, Pirtskhalava (2012)

gµ⌫ = ⌘µ⌫ + hµ⌫

LDL = �1

4hµ⌫E↵�

µ⌫ h↵� � !

2@µ�@µ� � hµ⌫

1

4"µ"⌫⇧� ↵

4⇤3"µ"⌫⇧⇧� �

2⇤6"µ"⌫⇧⇧⇧

�

+ �h�0""⇧+

�1⇤3

""⇧⇧+�2⇤6

""⇧⇧⇧+�3⇤9

""⇧⇧⇧⇧i+

1

MPlhµ⌫Tµ⌫

""⇧ ⌘ "µ↵��"⌫↵��@µ@⌫⇡

"µ"⌫⇧ ⌘ " ↵��µ " �

⌫ ��@↵@�⇡

Galileon terms

Decoupling limit in QMG✓ Transformation

✓ The action with β=0

3

Consider the following transformation,

hµ! ! hµ! + !"µ! +#

!3$µ!$!! (12)

Then the action in the decoupling limit is given by

LDL = "1

4hµ!E"#

µ! h"# " %

2$µ&$µ& " 1

8!

!''"" 2#

!3''""+

1

!6

"#2 +

(

2

#''"""" (

2!9''""""

$

+(

2!6hµ!'µ'!"""+ &

%4#5!

3 + )0''"+)1!3''""+

)2!6''"""+

)3!9''""""

&

+1

MPlhµ!Tµ! +

1

MPl!T +

#

MPl!3$µ!$!!T

µ! (13)

As you can see from this action, this is exactly bi-galileon thoery, which has mixing term of ! field and & field.Throughout this paper, we consider the case ( = 0, for simplicity. Equation of motion for !,

1

4''"" 3#

4!3''""+

#2

2!6''"""" )0''#" 2)1

!3''#"" 3)2

!6''#""" 4)3

!9''#""" (14)

=1

MPlT " 2#

MPl!3"µ!T

µ! (15)

and equation of & is given by

"%6''#+ 4#5!

3 + )0''"+)1!3''""+

)2!6''"""+

)3!9''"""" = 0 (16)

IV. SPHERICALLY SYMMETRIC CASE

In this paper, we use the following time-dependent ansatz,

!(t, x) ! a

2t2 + !(r) (17)

&(t, x) ! b

2t2 + &(r) (18)

where a and b are constants and a = b = 0 gives time-independent spherical symmetric case.After substituting these ansatz we have equation of motion for !

3a

2" 6)0b+

1

r2

!"3

2r2!! " 3#

!3

"ar2!! " r!!2

#+#2

!6

"3ar!!2 " !!3

#(19)

+ 6)0r2&! " 4)1

!3

"br2!! + ar2&! " 2r&!!!

#" 6)2

!6

"br!!2 + 2ar&!!! " &!!!2

#" 8)3

!9

"b!!3 + 3a&!!!2

#$(20)

= " 1

MPl(1 + 2#a)* (21)

and equation of motion of & field

"%b+ 4#5!3 + 6)0a+

1

r2

!%r2&! " 6)0r

2!! +4)1!3

"ar2!! " r!!2

#+

2)2!6

"3ar!!2 " !!3

#+

8)3!9

a!!3$= 0 (22)

(23)

A. !5 = 0 and a = b = 0

Defining the new variables

+ # !!

!3r, +$ # &!

!3r, r" #

"M

M2Plm

2

#1/3

, (24)

LDL = �1

4hµ⌫E↵�

µ⌫ h↵� � !

2@µ�@µ� � 1

8⇡

""⇧� 2↵

⇤3""⇧⇧+

↵2

⇤6""⇧⇧⇧

�

+ �h�0""⇧+

�1⇤3

""⇧⇧+�2⇤6

""⇧⇧⇧+�3⇤9

""⇧⇧⇧⇧i

+1

MPlhµ⌫Tµ⌫ +

1

MPl⇡T +

↵

MPl⇤3@µ⇡@⌫⇡T

µ⌫

where

Bi-galileon interaction term

2


Kµ! = !µ! !

!"abgµ"#"$a#!$b (3)


% " % ! &MPl, $a " e"$a (4)


Kµ! = !µ! ! e#/MPl

!"abgµ"#"$a#!$b (5)


S =M2

Pl

2

"d4x

#!g

#R! '

M2Pl

gµ!#µ%#!% ! m2

4(U2 + &3U3 + &4U4)

$+ S! + Sm[gµ! ,(] (6)


S! = M2Plm

2&5

"d4x

#!g e4#/MPl

!det (gµ"#"$a#!$a) (7)





2 (8)


MPl " $, m " 0, " = (MPlm2)1/3 = fixed,

Tµ!

MPl= fixed (9)


LDL = !1

4hµ!E"$

µ! h"$ ! '

2#µ%#µ% ! hµ!

#1

4*µ*!#! &

4"3*µ*!##! +

2"6*µ*!###

$

+ %%4&5"

3 + ,0**#+,1"3**##+

,2"6**###+

,3"9**####

&+

1

MPlhµ!Tµ! (10)

where

& = !3

4&3 ! 1, + = !1

8&3 !

1

2&4, ,0 =

1

2! 2

3&5,

,1 =3

8&3 !

1

2! &5, ,2 =

1

2&4 !

3

8&3 !

2

3&5, ,3 = !1

2&4 !

1

6&5. (11)

Spherically symmetric case✓ Master equation for λ

4

then after integrating equation of motion for ! field gives

3"! 6#"2 + 2#2"3 ! 6"! ! 16$1"!"! 12$2"!"2 = 2

!r!r

"3(25)

and equation of motion for % field

"! =1

&(3"+ 4$1"

2 + 2$2"3) (26)

Eliminating "! gives the master equation of "

3

2

#1! 6

&

$"!

#3#+

36$1&

$"2 +

##2 ! 32$21 + 24$2

&

$"3 ! 40

$1$2&

"4 ! 12$22&

"5 =!r!r

"3(27)

When r " r!, then possible solutions are obtained by solving P (") = 0, where P (") is defined by left hand side ofEq. (?). We call this solution as " = 0,"1,2,3,4. " = 0 corresponds to asymptotic Minkoski solution, which is given by

" # 2&

3(& ! 6)

!r!r

"3(28)

for r " r!.When r $ r!, the highest nonlinear term dominates, so there are two solutions depending on the sign of &, which

are

" # ±#

3|&|16(1 + #)2

$1/5 !r!r

"3/5(29)

negative " corresponds to positive &, and positive " corresponds to negative &. These solutions are only solutions nomatter what the solution ouside the Vainshtein radius. However, negative & leads to ghost of % field (see Appendix).Therefore, we disregard this solution and only consider positive &, which corresponds to negative " solution.It is interesting to see the e!ective energy density contributed from the scalar fieldis. For the constant " solutions

correspoding to constant " which are "1,2,3,4, the e!ective energy density is

' = MPlG00 # !3"(1! #")"3MPl (30)

and the e!ective pressure is

p =MPl

3Gi

i # !"(!2 + #")"3MPl (31)

Since we are focusing on negative " solution, we have positive energy and negative pressure if # > 0.For assymptotic Minkowski solution, the e!ective energy density is

' # !4#&2"3MPl

3(w ! 6)2

!r!r

"6(32)


p # 4#&2"3MPl

9(w ! 6)2

!r!r

"6(33)

In this case, the e!ective equation of state is w = !1/3. In order to have positive energy density we require # < 0 forthis solution. However, we need positive # to avoid ghost inside distributed matter because of disformal coupling of! field to matter. Therefore, there is no stable assymptotic Minkowski solution.Now we want to see what is going to happen in perturbations. We summerized the quadratic action for the scalar

fields. The leading term of time component of the kinetic term in the Vainshtein region is

L(2)DL # ! 34/5

10% 21/5(1 + #)(10 + 7#)

&

#&

(1 + #)2

$4/5 !r!r

"12/5((t))

2 (34)

Thus as long as # is positive, we always have ghost and there is no stable solution for any solutions.

4


3"! 6#"2 + 2#2"3 ! 6"! ! 16$1"!"! 12$2"!"2 = 2

!r!r

"3(25)


"! =1

&(3"+ 4$1"

2 + 2$2"3) (26)


3

2

#1! 6

&

$"!

#3#+

36$1&

$"2 +

##2 ! 32$21 + 24$2

&

$"3 ! 40

$1$2&

"4 ! 12$22&

"5 =!r!r

"3(27)


" # 2&

3(& ! 6)

!r!r

"3(28)


are

" # ±#

3|&|16(1 + #)2

$1/5 !r!r

"3/5(29)



' = MPlG00 # !3"(1! #")"3MPl (30)


p =MPl

3Gi

i # !"(!2 + #")"3MPl (31)


' # !4#&2"3MPl

3(w ! 6)2

!r!r

"6(32)


p # 4#&2"3MPl

9(w ! 6)2

!r!r

"6(33)



L(2)DL # ! 34/5

10% 21/5(1 + #)(10 + 7#)

&

#&

(1 + #)2

$4/5 !r!r

"12/5((t))

2 (34)


3

Consider the following transformation,

hµ! ! hµ! + !"µ! +#

!3$µ!$!! (12)

Then the action in the decoupling limit is given by

LDL = "1

4hµ!E"#

µ! h"# " %

2$µ&$µ& " 1

8!

!''"" 2#

!3''""+

1

!6

"#2 +

(

2

#''"""" (

2!9''""""

$

+(

2!6hµ!'µ'!"""+ &

%4#5!

3 + )0''"+)1!3''""+

)2!6''"""+

)3!9''""""

&

+1

MPlhµ!Tµ! +

1

MPl!T +

#

MPl!3$µ!$!!T

µ! (13)

As you can see from this action, this is exactly bi-galileon thoery, which has mixing term of ! field and & field.Throughout this paper, we consider the case ( = 0, for simplicity. Equation of motion for !,

1

4''"" 3#

4!3''""+

#2

2!6''"""" )0''#" 2)1

!3''#"" 3)2

!6''#""" 4)3

!9''#""" (14)

=1

MPlT " 2#

MPl!3"µ!T

µ! (15)

and equation of & is given by

"%6''#+ 4#5!

3 + )0''"+)1!3''""+

)2!6''"""+

)3!9''"""" = 0 (16)

IV. SPHERICALLY SYMMETRIC CASE

In this paper, we use the following time-dependent ansatz,

!(t, x) ! a

2t2 + !(r) (17)

&(t, x) ! b

2t2 + &(r) (18)

where a and b are constants and a = b = 0 gives time-independent spherical symmetric case.After substituting these ansatz we have equation of motion for !

3a

2" 6)0b+

1

r2

!"3

2r2!! " 3#

!3

"ar2!! " r!!2

#+#2

!6

"3ar!!2 " !!3

#(19)

+ 6)0r2&! " 4)1

!3

"br2!! + ar2&! " 2r&!!!

#" 6)2

!6

"br!!2 + 2ar&!!! " &!!!2

#" 8)3

!9

"b!!3 + 3a&!!!2

#$(20)

= " 1

MPl(1 + 2#a)* (21)

and equation of motion of & field

"%b+ 4#5!3 + 6)0a+

1

r2

!%r2&! " 6)0r

2!! +4)1!3

"ar2!! " r!!2

#+

2)2!6

"3ar!!2 " !!3

#+

8)3!9

a!!3$= 0 (22)

(23)

A. !5 = 0 and a = b = 0

Defining the new variables

+ # !!

!3r, +$ # &!

!3r, r" #

"M

M2Plm

2

#1/3

, (24)

where

✓ Solution inside Vainshtein radius

π is suppressed by usual gravity → Newton gravity

� ' �✓

3!

16(1 + ↵)2

◆1/5 ⇣r⇤r

⌘3/5

Negative λ (repulsive force)

⇡0 / r2/5

Spherically symmetric case✓ Solution outside Vainshtein radius

4


3"! 6#"2 + 2#2"3 ! 6"! ! 16$1"!"! 12$2"!"2 = 2

!r!r

"3(25)


"! =1

&(3"+ 4$1"

2 + 2$2"3) (26)


3

2

#1! 6

&

$"!

#3#+

36$1&

$"2 +

##2 ! 32$21 + 24$2

&

$"3 ! 40

$1$2&

"4 ! 12$22&

"5 =!r!r

"3(27)


" # 2&

3(& ! 6)

!r!r

"3(28)


are

" # ±#

3|&|16(1 + #)2

$1/5 !r!r

"3/5(29)



' = MPlG00 # !3"(1! #")"3MPl (30)


p =MPl

3Gi

i # !"(!2 + #")"3MPl (31)


' # !4#&2"3MPl

3(w ! 6)2

!r!r

"6(32)


p # 4#&2"3MPl

9(w ! 6)2

!r!r

"6(33)



L(2)DL # ! 34/5

10% 21/5(1 + #)(10 + 7#)

&

#&

(1 + #)2

$4/5 !r!r

"12/5((t))

2 (34)


0

� = 0 and 4 constants

4


3"! 6#"2 + 2#2"3 ! 6"! ! 16$1"!"! 12$2"!"2 = 2

!r!r

"3(25)


"! =1

&(3"+ 4$1"

2 + 2$2"3) (26)


3

2

#1! 6

&

$"!

#3#+

36$1&

$"2 +

##2 ! 32$21 + 24$2

&

$"3 ! 40

$1$2&

"4 ! 12$22&

"5 =!r!r

"3(27)


" # 2&

3(& ! 6)

!r!r

"3(28)


are

" # ±#

3|&|16(1 + #)2

$1/5 !r!r

"3/5(29)



' = MPlG00 # !3"(1! #")"3MPl (30)


p =MPl

3Gi

i # !"(!2 + #")"3MPl (31)


' # !4#&2"3MPl

3(w ! 6)2

!r!r

"6(32)


p # 4#&2"3MPl

9(w ! 6)2

!r!r

"6(33)



L(2)DL # ! 34/5

10% 21/5(1 + #)(10 + 7#)

&

#&

(1 + #)2

$4/5 !r!r

"12/5((t))

2 (34)


✓ λ=0 solution (Asymptotically flat solution)

✓ λ=constant solutions (Asymptotic cosmological solutions)

� ' �1, �2, �3, �4

h00 / hij / r2

⇡0 / 1

r2

⇡0 / r

vDVZ solution

Ghost ??✓ Quadratic Lagrangian

6

Appendix A: Perturbations

We perturbe the scalar and dilaton,

! ! !(r) + "(t, x) (A1)

# ! "(r) + $(t, x) (A2)

Then, the quadratic Lagrangian for the scalar parts is

L(2)DL = A1(%t")

2 "A2(%r")2 "A3(%!")

2 + B1(%t$)2 " B2(%r$)

2 " B3(%!$)2 (A3)

+C1%t"%t$ " C2%r"%r$ " C3%!"%!$ (A4)

where

A1 =3

4" 1

#3

!3

2&

"!!! + 2

!!

r

#+ 2'1

""!! + 2

"!

r

#$(A5)

+1

#6

!3

2&2

"!!2

r2+ 2

!!!!!

r

#" 6'2

""!!!

r2+

"!!!!

r+

"!!!!

r

#$(A6)

" 12'3#9

""!!!!2

r2+ 2

"!!!!!!

r2

#(A7)

A2 =3

4+

3

2&a+ 2'1b"

1

#3

!3(&+ &2a" 2'2b)

!!

r+ 2(2'1 " 3'2a)

"!

r

$(A8)

+1

#6

!3

2

%&2 + 8'3b

& !!2

r2" 6('2 " 4'3a)

"!!!

r2

$(A9)

A3 =3

4+

3

2&a+ 2'1b"

1

#3

!3

2(&+ &2a" 2'2b)

"!!! +

!!

r

#+ (2'1 " 3'2a)

""!! +

"!

r

#$(A10)

+1

#6

!3

2

%&2 + 8'3b

& !!!!!

r" 3('2 " 4'3a)

!!"!! +"!!!!

r

$(A11)

B1 =(

2(A12)

B2 =(

2(A13)

B3 =(

2(A14)

C1 = "6'0 "4'1#3

"!!! + 2

!!

r

#" 6

'2#6

"!!2

r2+ 2

!!!!!

r

#" 24'3

#9

!!2!!!

r2(A15)

C2 = "6'0 + 4'1b"8'1 " 6'2(a+ b)

#3

!!

r" 6'2 " 8'3(2a+ b)

#6

!!2

r2(A16)

C3 = "6'0 + 4'1b"4'1 " 3'2(a+ b)

#3

"!!! +

!!

r

#" 6'2 " 8'3(2a+ b)

#6

!!!!!

r(A17)

(A18)

Diagonalizing the time derivative terms gives

L(2)DL # A1(%t")

2 + B1(%t$)2 + C1(%t")(%t$) (A19)

= A1

"%t"+

C12A1

%t$

#2

+

"B1 "

C21

4A1

#(%t$)

2 (A20)

Thus we require

A1 > 0, B1 "C21

4A1> 0 (A21)

Thus as one can see from the coe$ents, the minimum condition of the parameter ( for avoiding ghost is ( > 0.

6



! ! !(r) + "(t, x) (A1)

# ! "(r) + $(t, x) (A2)


L(2)DL = A1(%t")

2 "A2(%r")2 "A3(%!")

2 + B1(%t$)2 " B2(%r$)

2 " B3(%!$)2 (A3)

+C1%t"%t$ " C2%r"%r$ " C3%!"%!$ (A4)

where

A1 =3

4" 1

#3

!3

2&

"!!! + 2

!!

r

#+ 2'1

""!! + 2

"!

r

#$(A5)

+1

#6

!3

2&2

"!!2

r2+ 2

!!!!!

r

#" 6'2

""!!!

r2+

"!!!!

r+

"!!!!

r

#$(A6)

" 12'3#9

""!!!!2

r2+ 2

"!!!!!!

r2

#(A7)

A2 =3

4+

3

2&a+ 2'1b"

1

#3

!3(&+ &2a" 2'2b)

!!

r+ 2(2'1 " 3'2a)

"!

r

$(A8)

+1

#6

!3

2

%&2 + 8'3b

& !!2

r2" 6('2 " 4'3a)

"!!!

r2

$(A9)

A3 =3

4+

3

2&a+ 2'1b"

1

#3

!3

2(&+ &2a" 2'2b)

"!!! +

!!

r

#+ (2'1 " 3'2a)

""!! +

"!

r

#$(A10)

+1

#6

!3

2

%&2 + 8'3b

& !!!!!

r" 3('2 " 4'3a)

!!"!! +"!!!!

r

$(A11)

B1 =(

2(A12)

B2 =(

2(A13)

B3 =(

2(A14)

C1 = "6'0 "4'1#3

"!!! + 2

!!

r

#" 6

'2#6

"!!2

r2+ 2

!!!!!

r

#" 24'3

#9

!!2!!!

r2(A15)

C2 = "6'0 + 4'1b"8'1 " 6'2(a+ b)

#3

!!

r" 6'2 " 8'3(2a+ b)

#6

!!2

r2(A16)

C3 = "6'0 + 4'1b"4'1 " 3'2(a+ b)

#3

"!!! +

!!

r

#" 6'2 " 8'3(2a+ b)

#6

!!!!!

r(A17)

(A18)


L(2)DL # A1(%t")

2 + B1(%t$)2 + C1(%t")(%t$) (A19)

= A1

"%t"+

C12A1

%t$

#2

+

"B1 "

C21

4A1

#(%t$)

2 (A20)

Thus we require

A1 > 0, B1 "C21

4A1> 0 (A21)


6



! ! !(r) + "(t, x) (A1)

# ! "(r) + $(t, x) (A2)


L(2)DL = A1(%t")

2 "A2(%r")2 "A3(%!")

2 + B1(%t$)2 " B2(%r$)

2 " B3(%!$)2 (A3)

+C1%t"%t$ " C2%r"%r$ " C3%!"%!$ (A4)

where

A1 =3

4" 1

#3

!3

2&

"!!! + 2

!!

r

#+ 2'1

""!! + 2

"!

r

#$(A5)

+1

#6

!3

2&2

"!!2

r2+ 2

!!!!!

r

#" 6'2

""!!!

r2+

"!!!!

r+

"!!!!

r

#$(A6)

" 12'3#9

""!!!!2

r2+ 2

"!!!!!!

r2

#(A7)

A2 =3

4+

3

2&a+ 2'1b"

1

#3

!3(&+ &2a" 2'2b)

!!

r+ 2(2'1 " 3'2a)

"!

r

$(A8)

+1

#6

!3

2

%&2 + 8'3b

& !!2

r2" 6('2 " 4'3a)

"!!!

r2

$(A9)

A3 =3

4+

3

2&a+ 2'1b"

1

#3

!3

2(&+ &2a" 2'2b)

"!!! +

!!

r

#+ (2'1 " 3'2a)

""!! +

"!

r

#$(A10)

+1

#6

!3

2

%&2 + 8'3b

& !!!!!

r" 3('2 " 4'3a)

!!"!! +"!!!!

r

$(A11)

B1 =(

2(A12)

B2 =(

2(A13)

B3 =(

2(A14)

C1 = "6'0 "4'1#3

"!!! + 2

!!

r

#" 6

'2#6

"!!2

r2+ 2

!!!!!

r

#" 24'3

#9

!!2!!!

r2(A15)

C2 = "6'0 + 4'1b"8'1 " 6'2(a+ b)

#3

!!

r" 6'2 " 8'3(2a+ b)

#6

!!2

r2(A16)

C3 = "6'0 + 4'1b"4'1 " 3'2(a+ b)

#3

"!!! +

!!

r

#" 6'2 " 8'3(2a+ b)

#6

!!!!!

r(A17)

(A18)


L(2)DL # A1(%t")

2 + B1(%t$)2 + C1(%t")(%t$) (A19)

= A1

"%t"+

C12A1

%t$

#2

+

"B1 "

C21

4A1

#(%t$)

2 (A20)

Thus we require

A1 > 0, B1 "C21

4A1> 0 (A21)


where

Ghost ??

✓ Quadratic Lagrangian (Inside the source + Vainshtein radius)

✓ The static clump of dust of constant density

Acknowledgments

We would like to thank Claudia de Rham, Rachel A. Rosen, Andrew Tolley andArkady Vainshtein for helpful comments on the manuscript, and Rampei Kimuraand Mehrdad Mirbabayi for discussions. LB is supported by funds provided bythe University of Pennsylvania, GG is supported by NSF grant PHY-0758032 andNASA grant NNX12AF86G S06.

Appendix A

In order to study the stability, we split the scalar degree of freedom into the back-ground ! and the fluctuation ! as follows

" = !+ !. (A-I)

As a result, to the second order in perturbations, the Lagrangian (3) becomes5

L! =

!"!3

2+ 3

#

"33

!!+3

2

#2

"63

#($"$#!)

2 ! (!!)2$%

%µ$

+

"!3

#

"33

$µ$$!+ 3#2

"63

(!$"$µ!$"$$!+!!$µ$$!)

%&

"$µ!$$! +1

Mpl!T +

#

Mpl"33

$µ!$$!Tµ$ . (A-II)

For the static clump of dust of constant density, that is for Tµ$ = &'0µ'0$((R! r), we

obtain the following kinetic term (to the leading order) inside the source'#

&

Mpl"33

+ 9

(#1/3r!R

)2*($t!)

2. (A-III)

Here, the second term in brackets comes from the quartic Galileon term after we havesubstituted the classical solution within Vainshtein region (9). After the replacement& # M/R3, (A-III) leads to the conclusion that in # < 0 parameter space the scalarperturbations have ghost-like kinetic term within the source of radius R $ #1/3r!.The latter condition can be translated on the language of the source density as

#& > Mpl"33. (A-IV)

For phenomenologically interesting value of ""13 % 1000 km the above bound be-

comes #& > 10"29 g/cm3.

5Here we give only the scalar part of the Lagrangian, since the tensor mode is completelydecoupled and propagates according to the linearized Einstein-Hilbert action.

12

L(2)DL �

"↵

⇢

MPl⇤3� 34/5

10⇥ 21/5(1 + ↵)(10 + 7↵)

!

✓!

(1 + ↵)2

◆4/5 ⇣r⇤R

⌘12/5#(@t�)

2

from galileon and bi-galileon

R ⌧ ⇠�5/3r⇤ ⇠ ⌘ (1 + ↵)(10 + 7↵)

↵!

✓!

(1 + ↵)2

◆4/5

↵ > 0

✓ Quadratic Lagrangian (Outside the source, but inside Vainshtein radius)

L(2)DL � � 34/5

10⇥ 21/5(1 + ↵)(10 + 7↵)

!

✓!

(1 + ↵)2

◆4/5 ⇣r⇤r

⌘12/5(@t�)

2

Negative sign of the kinetic term

6



! ! !(r) + "(t, x) (A1)

# ! "(r) + $(t, x) (A2)


L(2)DL = A1(%t")

2 "A2(%r")2 "A3(%!")

2 + B1(%t$)2 " B2(%r$)

2 " B3(%!$)2 (A3)

+C1%t"%t$ " C2%r"%r$ " C3%!"%!$ (A4)

where

A1 =3

4" 1

#3

!3

2&

"!!! + 2

!!

r

#+ 2'1

""!! + 2

"!

r

#$(A5)

+1

#6

!3

2&2

"!!2

r2+ 2

!!!!!

r

#" 6'2

""!!!

r2+

"!!!!

r+

"!!!!

r

#$(A6)

" 12'3#9

""!!!!2

r2+ 2

"!!!!!!

r2

#(A7)

A2 =3

4+

3

2&a+ 2'1b"

1

#3

!3(&+ &2a" 2'2b)

!!

r+ 2(2'1 " 3'2a)

"!

r

$(A8)

+1

#6

!3

2

%&2 + 8'3b

& !!2

r2" 6('2 " 4'3a)

"!!!

r2

$(A9)

A3 =3

4+

3

2&a+ 2'1b"

1

#3

!3

2(&+ &2a" 2'2b)

"!!! +

!!

r

#+ (2'1 " 3'2a)

""!! +

"!

r

#$(A10)

+1

#6

!3

2

%&2 + 8'3b

& !!!!!

r" 3('2 " 4'3a)

!!"!! +"!!!!

r

$(A11)

B1 =(

2(A12)

B2 =(

2(A13)

B3 =(

2(A14)

C1 = "6'0 "4'1#3

"!!! + 2

!!

r

#" 6

'2#6

"!!2

r2+ 2

!!!!!

r

#" 24'3

#9

!!2!!!

r2(A15)

C2 = "6'0 + 4'1b"8'1 " 6'2(a+ b)

#3

!!

r" 6'2 " 8'3(2a+ b)

#6

!!2

r2(A16)

C3 = "6'0 + 4'1b"4'1 " 3'2(a+ b)

#3

"!!! +

!!

r

#" 6'2 " 8'3(2a+ b)

#6

!!!!!

r(A17)

(A18)


L(2)DL # A1(%t")

2 + B1(%t$)2 + C1(%t")(%t$) (A19)

= A1

"%t"+

C12A1

%t$

#2

+

"B1 "

C21

4A1

#(%t$)

2 (A20)

Thus we require

A1 > 0, B1 "C21

4A1> 0 (A21)


For

↵

MPl⇤3@µ⇡@⌫⇡T

µ⌫ = ↵⇣r⇤R

⌘3

✓ Quadratic action for π and σ

6



! ! !(r) + "(t, x) (A1)

# ! "(r) + $(t, x) (A2)


L(2)DL = A1(%t")

2 "A2(%r")2 "A3(%!")

2 + B1(%t$)2 " B2(%r$)

2 " B3(%!$)2 (A3)

+C1%t"%t$ " C2%r"%r$ " C3%!"%!$ (A4)

where

A1 =3

4" 1

#3

!3

2&

"!!! + 2

!!

r

#+ 2'1

""!! + 2

"!

r

#$(A5)

+1

#6

!3

2&2

"!!2

r2+ 2

!!!!!

r

#" 6'2

""!!!

r2+

"!!!!

r+

"!!!!

r

#$(A6)

" 12'3#9

""!!!!2

r2+ 2

"!!!!!!

r2

#(A7)

A2 =3

4+

3

2&a+ 2'1b"

1

#3

!3(&+ &2a" 2'2b)

!!

r+ 2(2'1 " 3'2a)

"!

r

$(A8)

+1

#6

!3

2

%&2 + 8'3b

& !!2

r2" 6('2 " 4'3a)

"!!!

r2

$(A9)

A3 =3

4+

3

2&a+ 2'1b"

1

#3

!3

2(&+ &2a" 2'2b)

"!!! +

!!

r

#+ (2'1 " 3'2a)

""!! +

"!

r

#$(A10)

+1

#6

!3

2

%&2 + 8'3b

& !!!!!

r" 3('2 " 4'3a)

!!"!! +"!!!!

r

$(A11)

B1 =(

2(A12)

B2 =(

2(A13)

B3 =(

2(A14)

C1 = "6'0 "4'1#3

"!!! + 2

!!

r

#" 6

'2#6

"!!2

r2+ 2

!!!!!

r

#" 24'3

#9

!!2!!!

r2(A15)

C2 = "6'0 + 4'1b"8'1 " 6'2(a+ b)

#3

!!

r" 6'2 " 8'3(2a+ b)

#6

!!2

r2(A16)

C3 = "6'0 + 4'1b"4'1 " 3'2(a+ b)

#3

"!!! +

!!

r

#" 6'2 " 8'3(2a+ b)

#6

!!!!!

r(A17)

(A18)


L(2)DL # A1(%t")

2 + B1(%t$)2 + C1(%t")(%t$) (A19)

= A1

"%t"+

C12A1

%t$

#2

+

"B1 "

C21

4A1

#(%t$)

2 (A20)

Thus we require

A1 > 0, B1 "C21

4A1> 0 (A21)


6



! ! !(r) + "(t, x) (A1)

# ! "(r) + $(t, x) (A2)


L(2)DL = A1(%t")

2 "A2(%r")2 "A3(%!")

2 + B1(%t$)2 " B2(%r$)

2 " B3(%!$)2 (A3)

+C1%t"%t$ " C2%r"%r$ " C3%!"%!$ (A4)

where

A1 =3

4" 1

#3

!3

2&

"!!! + 2

!!

r

#+ 2'1

""!! + 2

"!

r

#$(A5)

+1

#6

!3

2&2

"!!2

r2+ 2

!!!!!

r

#" 6'2

""!!!

r2+

"!!!!

r+

"!!!!

r

#$(A6)

" 12'3#9

""!!!!2

r2+ 2

"!!!!!!

r2

#(A7)

A2 =3

4+

3

2&a+ 2'1b"

1

#3

!3(&+ &2a" 2'2b)

!!

r+ 2(2'1 " 3'2a)

"!

r

$(A8)

+1

#6

!3

2

%&2 + 8'3b

& !!2

r2" 6('2 " 4'3a)

"!!!

r2

$(A9)

A3 =3

4+

3

2&a+ 2'1b"

1

#3

!3

2(&+ &2a" 2'2b)

"!!! +

!!

r

#+ (2'1 " 3'2a)

""!! +

"!

r

#$(A10)

+1

#6

!3

2

%&2 + 8'3b

& !!!!!

r" 3('2 " 4'3a)

!!"!! +"!!!!

r

$(A11)

B1 =(

2(A12)

B2 =(

2(A13)

B3 =(

2(A14)

C1 = "6'0 "4'1#3

"!!! + 2

!!

r

#" 6

'2#6

"!!2

r2+ 2

!!!!!

r

#" 24'3

#9

!!2!!!

r2(A15)

C2 = "6'0 + 4'1b"8'1 " 6'2(a+ b)

#3

!!

r" 6'2 " 8'3(2a+ b)

#6

!!2

r2(A16)

C3 = "6'0 + 4'1b"4'1 " 3'2(a+ b)

#3

"!!! +

!!

r

#" 6'2 " 8'3(2a+ b)

#6

!!!!!

r(A17)

(A18)


L(2)DL # A1(%t")

2 + B1(%t$)2 + C1(%t")(%t$) (A19)

= A1

"%t"+

C12A1

%t$

#2

+

"B1 "

C21

4A1

#(%t$)

2 (A20)

Thus we require

A1 > 0, B1 "C21

4A1> 0 (A21)


✓ Inside Vainshtein radius,

B1 = !/2

C21

4A1/

⇣r⇤r

⌘12/5

One of the scalar field has always ghost !!!

Ghost ??

L(2)DL ' A1(@t�)

2 � C21

4A1(@t )

2

Summary

Quasi-dilaton massive gravity with β=0 (Restricted bi-galileon)

• does not have “healthy” asymptotically flat solution (vDVZ

solution) as well as cosmological solution

• It seems that there is some “healthy” time-dependent solutions (in

progress)

Documents

Vainshtein mechanism in quasi-dilaton massive gravity · Vainshtein mechanism in quasi-dilaton massive gravity Rampei Kimura RESCEU, The University of Tokyo Mini-workshop “Massive