Vainshtein mechanism in Horndeski’s general scalar-tensor theory (and in massive gravity)

Tsutomu KobayashiRikkyo University

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Based on work withTastuya Narikawa (Osaka), Ryo Saito (YITP), Daisuke Yamauchi (RESCEU)arXiv:1302.2311

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Mini-workshop "Massive gravity and its cosmological implications" @IPMU

Talk plan

Introduction & Motivation

Horndeski’s most general scalar-tensor theory

Static, spherically symmetric, weak gravitational field

Application

Summary

IntroductionMystery of dark energy

Modified gravity as an alternative to dark energy?

Modification would persist down to small length scales…

Need screening mechanism in the vicinity of matter

Basic idea

Extra d.o.f is effectively weakly coupled to matter

––– Vainshtein mechanism

Basic idea

Vainshtein (1972)

Example

L =1

8�G

��1

2(��)2 � r2

c

3(��)2��

�+ �T µ

µ

�( : dimensionless)

Cubic Galileon non-minimally coupled to matter :

Key non-linearity

r2c�� �� 1can be large even if

� � rg

r� 1, r2

c�� � r2crg

r3� 1 r � (r2

crg)1/3for

Equation of motionStatic, spherically symmetric, non-relativistic source: T µ

µ = ��

1r2

�(r2��)� +

4r2c

3�r(��)2

���

= 8�G�

Quadratic algebraic equation for ��

�� =3r

8r2c

��1 +

�1 +

163

r2crg

r3

�

Vainshtein radius

rV = (r2crg)1/3

�� � rg

r2� ��

�� � rg

r2

�r

rV

�3/2

� ��

r � rV

r � rV

for

for

unscreened

screened

�( : gravitational potential)

Suppose rc = 3 Gpc

rV � 1 Mpc

rV � 100 pc M = M�

M = 1014 M�

for the Sun ( )

for a galaxy cluster ( )

Motivation

Study the Vainshtein mechanism in the most general scalar-tensor theory, clarifying the conditions under which a screened solution is realized

Offer a basic tool to test general scalar-tensor type gravity

Horndeski’s most general scalar-tensor theory

Galileon

Unique scalar-field theory in 4D flat spacetime having

Galilean shift symmetry

2nd-order equation of motion

�� � + bµxµ + c

Vainshtein mechanism operates

Nicolis, Rattazzi,Trincherini (2009)

L = c1� + c2(��)2 + c3(��)2�� + c4(��)2�(��)2 � (�µ���)2

�

+c5(��)2�(��)3 � 3��(�µ���)2 + 2(�µ���)3

�

Burrage, Seery (2010); ………

Generalized Galileon

L = K(�, X)�G3(�, X)��

+G4(�, X)R + G4X

�(��)2 � (�µ���)2

�

+G5(�, X)Gµ��µ���� 16G5X

�(��)3

�3 (��) (�µ���)2 + 2 (�µ���)3�

Include gravity

2nd-order equation of motion both for and

Forget about any symmetry…

gµ� �

G4X :=�G4

�X

Deffayet, Gao, Steer, Zahariade (2011)

X = �12(��)2

Horndeski’s theory

LH = ��⇥⇤µ⌅⇧

�⇥1⇥µ⇥�⇤R ⌅⇧

⇥⇤ +23⇥1X⇥µ⇥�⇤⇥⌅⇥⇥⇤⇥⇧⇥⇤⇤

+⇥3⇥�⇤⇥µ⇤R ⌅⇧⇥⇤ + 2⇥3X⇥�⇤⇥µ⇤⇥⌅⇥⇥⇤⇥⇧⇥⇤⇤

�

+��⇥µ⌅

�(F + 2W )R µ⌅

�⇥ + 2FX⇥µ⇥�⇤⇥⌅⇥⇥⇤ + 2⇥8⇥�⇤⇥µ⇤⇥⌅⇥⇥⇤�

�6 (F⌃ + 2W⌃ �X⇥8) �⇤ + ⇥9

The most general scalar-tensor theory with second-order field equations

Horndeski (1974); Rediscovered by Charmousis et al. (2011)

The generalized Galileon is equivalent to Horndeski’s theory

TK, Yamaguchi, Yokoyama (2011)

Static, spherically symmetric, weak gravitational fieldNarikawa, TK, Yamauchi, Saito, 1302.2311

BackgroundStart with the most general scalar-tensor theory

Minkowski background

ds2 = �µ�dxµdx� , � = �0 = const, X = 0

(Require for the theory to admit Minkowski background)K(�0, 0) = 0, K�(�0, 0) = 0

L = K(�, X)�G3(�, X)�� + G4(�, X)R + G4X

�(��)2 � (�µ���)2

�

+G5(�, X)Gµ��µ���� 16G5X

�(��)3 � 3��(�µ���)2 + 2(�µ���)3

�

Approximations

ds2 = �[1 + 2�(r)]dt2 + [1� 2�(r)]dx2

� = �0 + �(r)

Static, spherically symmetric perturbations produced by non-relativistic matter

T tt = ��(r)

Perturbations are small, but non-linear terms can be as large as linear terms(���)n ���

� � rg

rr2c (���)2 � ��� for r � (rgr

2c )1/3

Do not neglect

(� 1)

Gravitational field equationsTime-time component:

Space-space component:

G4(r2��)�

r2�G4�

(r2��)�

2r2� (G4X �G5�)

[r(��)2]�

2r2+ G5X

[(��)3]�

6r2=

�

4

Background quantities G5X = G5X(�0, 0), · · ·

2G4

�r2 (�� � ��)

��

r2� 2G4�

(r2��)�

r2� (G4X �G5�)

[r(��)2]�

r2= 0

(l.h.s.) =1r2

ddr

(· · · )

Scalar field equation

(KX � 2G3�)(r2��)�

r2� 2(G3X � 3G4�X)

[r(��)2]�

r2

+2G4�[r2(2�� �)�]�

r2+ 4(G4X �G5�)

[r��(�� � ��)]�

r2

+2�

G4XX �23G5�X

�[(��)3]�

r2+ 2G5X

[(��)2��]�

r2

= �K��� Neglect “mass term”(Scalar field is screened if it is sufficiently massive)

1r2

ddr

(· · · ) = 0

Three equations are integrated once to givealgebraic equations for ��,��,��

Introduce two mass scales and six dimensionless parameters

G4 =M2

Pl

2,

G4� = MPl�,

KX � 2G3� = �,

�G3X + 3G4�X =µ

�3,

G4X �G5� =MPl

�3�,

G4XX �23G5�X =

�

�6,

G5X = �3MPl

�6�

(MPl,�)

(one is redundant)

�� MPl

r2c

Previous example:

Useful dimensionless quantities:

x(r) :=1�3

��

r, A(r) :=

1MPl�3

M(r)8�r3

Enclosed mass

Master equations:

P (x,A) := �A(r) +��

2+ 3�2

�x + [µ + 6�� � 3�A(r)]x2

+�� + 2�2 + 4��

�x3 � 3�2x5

= 0

MPl

�3

��

r= ��x + �x3 + A(r),

MPl

�3

��

r= �x + �x2 + �x3 + A(r),

and

Problem reduces to solving quintic equation

x(r) = x[A(r)]

Vainshtein radius

1 2 3 4

1

2

3

4

r

A(r) (Concrete form depends on density profile)

for A(r) � 1

x(r) � 1Non-linearity sets in, ,

rV

A(rV ) = 1 Scre

ened

sol

utio

n

Linear approximation is good

Solution we are looking for

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

A

»x»

A� 1 � r � rV A� 1 � r � rV

|x(A)|

Ax� 0

Asymptotically flat

Vainshtein mechanism operates� � � � �GR

Outer region

Inner region

Outer solution

Linear regime: P (x,A) � �A(r) +��

2+ 3�2

�x

� x � xf := � 2�A(r)� + 6�2

� 1

Other solutions (if they exist) do not correspond to asymptotically flat spacetime

� + 6�2 > 0Stable if (Kinetic term for small fluctuations has right sign)

Inner solutionP (x,A) := �A(r) +

��

2+ 3�2

�x + [µ + 6�� � 3�A(r)]x2

+�� + 2�2 + 4��

�x3 � 3�2x5

= 0Structure foris different depending onwhether or

A� 1

� = 0 � �= 0

� �= 0

� = 0

P (x,A) is cubic –– consider separately

P (x,A) � �A� 3�Ax2 � 3�2x5 for A� 1

Inner solution forP (x,A) � �A� 3�Ax2 � 3�2x5

� �= 0

�� < 0

�� > 0

x3 � �A

���

r� A2/3,

��

r� A1/3 Not GR

x3 � �A

�

X

Xand x � x± := ±

��

3�� � � � �GR

O

Matching inner and outer solutions

-0.4 -0.3 -0.2 -0.1 0.0

-2

0

2

4

6

8

x

PHx,AL

(Consider for simplicity the case )� > 0

�A� 1

�A� 1

A

inner sol.

outer sol.x � xf

x�

|x|LargeNon-linear regime Linear regime

|x|Small

0 2 4 6 8-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

A

x

x� x�

x � xf � 0

x(A)

A� 1A� 1

Outer region Inner region

A = 1 � r = rV

Profile of x

Conditions for smooth matching

-0.4 -0.3 -0.2 -0.1 0.0

-2

0

2

4

6

8

x

PHx,AL

x�

has a single root in for anyP (x,A) = 0 (x�, 0) A > 0

x � xf

Otherwise...

-0.4 -0.3 -0.2 -0.1 0.0-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

x

PHx,AL

3 roots for A

Ex� x�

0.0 0.1 0.2 0.3 0.4 0.5

-0.4

-0.3

-0.2

-0.1

0.0

A

x

x� x�

x � xf � 0

Case I

-0.4 -0.3 -0.2 -0.1 0.0

-2

0

2

4

6

8

x

PHx,AL

No local extrema in (x�, 0)

P (x�, A) = P (x�) < 0

x� A�

�P (x�, A�)�x

= 0,�2P (x�, A�)

�x2= 0

and do not exist satisfying

Case II

-0.8 -0.6 -0.4 -0.2 0.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

x

PHx,AL

A�

�P (x�, A�)�x

= 0,�2P (x�, A�)

�x2= 0

P (x�, A�) < 0

Local maximum never exceeds P=0

Otherwise...

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

0.0

0.2

0.4

0.6

0.8

x

PHx,AL

P (x�, A�) > 0

Inner solution for � = 0

P (x,A)� �A +��

2+ 3�2

�x + (µ + 6��) x2 +

�� + 2�2

�x3

Solution for A� 1

x3 � x3i := � �A

� + 2�2(< 0)

(required from stability)

� � � � �GR

For this inner solution Vainshtein mechanism operates

Matching inner and outer solutions

-2.0 -1.5 -1.0 -0.5 0.5

-1

1

2

3

4

5

6

-1.5 -1.0 -0.5

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

y = ���

2+ 3�2

�x� (µ + 6��) x2 �

�� + 2�2

�x3

y = �A

A� 1

A� 1

inner sol.

outer sol.

x � xi

x � xf � 0

X

O Condition for smooth matching:

no local extrema in x<0

µ + 6�� < 0

�� + 2�2

� �� + 6�2

�� 2

3(µ + 6��)2

µ + 6�� � 0

Local extrema are in x>0 if

No local extrema if

ExampleDecoupling limit of massive gravity de Rham, Gabadadze, Tolley (2011)

de Rham, Heisenberg (2011)

L = �12hµ�E��

µ� h�� + hµ��X(1)

µ� +a2

�3X(2)

µ� +a3

�6X(3)

µ�

�+

12MPl

hµ�Tµ�

Helicity-2 mode Interactions with helicity-0 mode

MPl ��, m� 0

�3 = MPlm2 fixed

“Covariantization”

hµ�X(1)µ� �� �R

hµ�X(2)µ� �� �Gµ��µ����

hµ�X(3)µ� ��

�2Rµ��� + · · ·

��µ���������

X(1)µ� := �µ������ gµ�

X(2)µ� := · · ·

�� G4 �MPl�

�� G4 � X

�� G5 � �3X

in Horndeski’s language

K = 0 = G3

G4 =M2

Pl

2+ MPl� +

MPl

�3�X

G5 = �3MPl

�6�X

(� = a2, � = a3)

Decoupling limit of massive gravity = 2-parameter subclass of Horndeski’s theory

Smooth matching of asymptotically flat and Vainshtein solutions is possible for:

-4 -2 0 2 4-4

-2

0

2

4

6

8

�

�

� =

�5 +

�13

24

��2

� = 0

Smooth matching is possible

Previous results [Sjors and Mortsell (2011); Sbisa et al. (2012)] are correctly reproduced

Application

Application: Gravitational lensingLensing convergence can be computed for any density profile and for any scalar-tensor theory from

�(� + �) =�3

MPl

1r2

ddr

�r2(�x2 + 2�x3 + 2A)

�

Interesting signature in cluster lensing?

�(�) =(�S � �L)�L

�S

� �

0dZ

�a2L

(� + �)

Convergence:

� x�(r)

5

IV. DECOUPLING LIMIT OF MASSIVEGRAVITY

Let us confirm that the conditions for smooth match-ing indeed reproduce the previous result obtained in thecontext of massive gravity [15, 16]. To do so, we startwith finding out the concrete form of K,G3, G4, G5 cor-responding to the decoupling limit of massive gravity.The correspondence can be seen more clearly if we moveto the covariantized version of the decoupling limit La-grangian, i.e., the “proxy theory” proposed in Ref. [24].It turns out that the proxy theory corresponds to

K = 0 = G3, G4 =M2

Pl

2+MPl!+

MPl

!3"X,

G5 = !3MPl

!6#X. (38)

In massive gravity, the strong coupling scale ! is givenby m2!3 = MPl, where m is the graviton mass.Since the proxy theory contains the Riemann dual ten-

sor while the Lagrangian of the generalized Galileon not,one may wonder how the former is included in the latter.Actually, G5 " X corresponds to the term containingthe Riemann dual tensor in the proxy theory. The easi-est way to verify this is to compare the field equations ofthe two theories.From Eq. (38) one finds

$ = µ = % = 0, & = 1, " #= 0, # #= 0, (39)

so that the parameter space collapses to a two-dimensional space. The inner solution x! exists onlyfor # > 0 and is given by x! = !1/

$3#. Let us define

' :=$#/". Then, the condition (31) reads

P (x!, A) =2

3

x!'2

!1! 3

$3' + 6'2

"< 0. (40)

Solving the equation (xP (x", A") ! x"(2xP (x", A") = 0,which does not in fact depend on A", one finds

x" =1$5

x!|'|

#1 + 2'2 !

$1 + 4'2 ! 11'4

%1/2&1/2. (41)

This exists if

|'| %

'2 +

$15

11& 0.73. (42)

The equation P (x,A) = 0 has three roots in (x!, 0) forsome interval of A if

P (x", A") > 0 ' 0 < ' <

'5 +

$13

24& 0.6. (43)

Therefore, smooth matching of the asymptotically flatsolution and the Vainshtein solution is possible providedthat

" < 0 or

$#

"(

'5 +

$13

24. (44)

FIG. 1: The profile of x as a function of the radial coordi-nate r. The curves are plotted for (!,") = (0.5, 0.3) (dot-ted red), (0.8, 0.34) (dot-dashed green), and (0.985, 0.375)(dashed blue), respectively. As a halo density profile weadopt the NFW model with Mvir = 1.28 ! 1015M!/h andcvir = 12.8.

Thus, we have confirmed that the previous result [15, 16]is reproduced.2

V. GRAVITATIONAL LENSING IN MODIFIEDGRAVITY

In this section, we are going to relate our sphericallysymmetric solution to gravitational lensing observations.To do so, it is instructive to begin with seeing the typicalbehavior of the Vainshtein solution in massive gravity,adopting the Navarro-Frenk-White (NFW) halo densityprofile [25, 26] for the source )(r) := !T t

t . (See Ap-pendix B for the detailed description of halo density pro-files.) Figures 1 and 2 show the profile of x and its deriva-tive, respectively, as a function of the radial coordinate rfor di"erent values of " and #. The fiducial parametersof the NFW model we use are Mvir = 1.28) 1015 M#/hand cvir = 12.8 , which correspond to )s = 5.91)104 )cr,0and rs = 154 kpc/h, respectively. The strong couplingscale is taken to be !3 = (100H0)2MPl = (46.4 km)!3.Then, the Vainshtein radius determined from Eq. (24) isrV = 209 kpc/h. (As the parameters characterizing theprofile we choose to use the virial cluster mass Mvir and

2 Note that our notation is di!erent from those in [15, 16]. Inparticular, !ours = !!Sbisa et al..

6

FIG. 2: The radial derivative of x as a function of r. Param-eters and definitions of curves are the same as in Fig. 1.

the concentration parameter cvir rather than !s and rs.)One can see that x(r) can have a sharp transition fromouter to inner solutions, depending on the parameters ofthe theory, which leads to a peak in x!(r). This occursat around the Vainshtein radius.Having seen the typical behavior of the radial profile

x(r), we now move to investigate how the lensing sig-nal is modified in massive gravity. We assume that thebackground evolution of the universe does not deviatemuch from conventional cosmology and use the !CDMbackground with "m = 0.3 , "! = 0.7 , and h = 0.7.The background metric (7) is understood to define thephysical coordinates at the location of the lens object.The basic quantity in gravitational lensing is the con-

vergence, ", which is expressed in terms of the sum ofthe two metric potentials #+ := (#+$)/2 as

" =

! !S

0d#

(#S ! #)#

#S%"#+, (45)

with #, #S, and %" being the comoving angular diame-ter distance, the comoving distance between the observerand the source, and the comoving transverse Laplacian,respectively. Using the thin lens approximation, we canrewrite the convergence as

" " (#S ! #L)#L

#S

! !S

0d#%#+, (46)

where #L is the comoving distance between the observerand the lens object and % is the comoving three dimen-sional Laplacian. Let us now introduce a new spatialcoordinate as Z = aL(# ! #L), whose origin is locatedat the center of the lens object. The projected radius is

FIG. 3: The lensing convergence ! as a function of " fordi!erent values of the parameters of the theory. In these plots,the NFW profile is used with Mvir = 1.28 ! 1015 M!/h andcvir = 12.8. Parameters and definitions of the curves are thesame as in Fig. 1. The points with the error bars represent theobservational data for the high-mass cluster A1689 providedby Umetsu et al. [27–29].

written as r" = aL#L$, where $ is the polar angle fromthe axis connecting the observer and the lens object, andaL is the scale factor at the lens object. In terms of these,the convergence (46) can be written as

"($) =2(#S ! #L)#L

#S

! #

0dZ

%

a2L#+(r), (47)

where r ="

r2" + Z2. Using Eqs. (21) and (22), we findthat in Horndeski’s theory

%

a2L#+(r) =

1

r2d

dr

#r2#+(r)

$

=!3

MPl

[%%x2 + 2&x3 + 2A

&r3]!

2r2. (48)

Figure 3 shows the lensing convergence for the NFWprofile with di&erent choices of the parameters of the the-ory. The points with the error bars indicate the observa-tional data for the high-mass cluster A1689 provided byUmetsu et al. [27–29]. An interesting feature observed inFig. 3 is that a dip appears at a particular polar anglecorresponding to the Vainshtein radius. The dip is mostenhanced for the parameters near the boundary of the re-gion in which the smooth matching is possible. Clearly,this is caused by the sharp peak in x!(r) at the Vain-shtein radius, as seen in Fig. 2. We see from Fig. 4 thatthe peak location is certainly determined by the Vain-

�(r) =�s

(r/rs)(1 + r/rs)2NFW profile:

Massive gravity (2-parameter subclass of Horndeski)

x�(r) can be large at transition from screened to unscreened regions

-4 -2 0 2 4-4

-2

0

2

4

6

8

Sharp transition occursfor parameters near boundary

6

FIG. 2: The radial derivative of x as a function of r. Param-eters and definitions of curves are the same as in Fig. 1.

the concentration parameter cvir rather than !s and rs.)One can see that x(r) can have a sharp transition fromouter to inner solutions, depending on the parameters ofthe theory, which leads to a peak in x!(r). This occursat around the Vainshtein radius.Having seen the typical behavior of the radial profile

x(r), we now move to investigate how the lensing sig-nal is modified in massive gravity. We assume that thebackground evolution of the universe does not deviatemuch from conventional cosmology and use the !CDMbackground with "m = 0.3 , "! = 0.7 , and h = 0.7.The background metric (7) is understood to define thephysical coordinates at the location of the lens object.The basic quantity in gravitational lensing is the con-

vergence, ", which is expressed in terms of the sum ofthe two metric potentials #+ := (#+$)/2 as

" =

! !S

0d#

(#S ! #)#

#S%"#+, (45)

with #, #S, and %" being the comoving angular diame-ter distance, the comoving distance between the observerand the source, and the comoving transverse Laplacian,respectively. Using the thin lens approximation, we canrewrite the convergence as

" " (#S ! #L)#L

#S

! !S

0d#%#+, (46)

where #L is the comoving distance between the observerand the lens object and % is the comoving three dimen-sional Laplacian. Let us now introduce a new spatialcoordinate as Z = aL(# ! #L), whose origin is locatedat the center of the lens object. The projected radius is

FIG. 3: The lensing convergence ! as a function of " fordi!erent values of the parameters of the theory. In these plots,the NFW profile is used with Mvir = 1.28 ! 1015 M!/h andcvir = 12.8. Parameters and definitions of the curves are thesame as in Fig. 1. The points with the error bars represent theobservational data for the high-mass cluster A1689 providedby Umetsu et al. [27–29].

written as r" = aL#L$, where $ is the polar angle fromthe axis connecting the observer and the lens object, andaL is the scale factor at the lens object. In terms of these,the convergence (46) can be written as

"($) =2(#S ! #L)#L

#S

! #

0dZ

%

a2L#+(r), (47)

where r ="

r2" + Z2. Using Eqs. (21) and (22), we findthat in Horndeski’s theory

%

a2L#+(r) =

1

r2d

dr

#r2#+(r)

$

=!3

MPl

[%%x2 + 2&x3 + 2A

&r3]!

2r2. (48)

Figure 3 shows the lensing convergence for the NFWprofile with di&erent choices of the parameters of the the-ory. The points with the error bars indicate the observa-tional data for the high-mass cluster A1689 provided byUmetsu et al. [27–29]. An interesting feature observed inFig. 3 is that a dip appears at a particular polar anglecorresponding to the Vainshtein radius. The dip is mostenhanced for the parameters near the boundary of the re-gion in which the smooth matching is possible. Clearly,this is caused by the sharp peak in x!(r) at the Vain-shtein radius, as seen in Fig. 2. We see from Fig. 4 thatthe peak location is certainly determined by the Vain-

Dip in convergence power spectrum

(if we are lucky enough…?)

7

FIG. 4: The convergence ! as a function of " for di!er-ent strong coupling scales. The curves correspond to "3 =(150H0)

2MPl (dotted red), "3 = (100H0)2MPl (dashed blue),

"3 = (50H0)2MPl (dot-dashed green), and "CDM (black

solid), respectively. We take # = 0.985 and $ = 0.375.

shtein scale. From Fig. 4 we also find that the depth ofthe dip increases as ! decreases.In Figs. 5 and 6 we compare di"erent assumptions on

the halo density profile. Two representative profiles areconsidered here: the generalized NFW (gNFW) [30–32]and the Einasto [33–36] profiles. We see that a dip ap-pears at a characteristic polar angle in the gNFW and theEinasto profiles as well. The depth of the dip is enhancedfor larger !s and larger #.The appearance of a dip is expected to be a generic

feature of scalar-tensor theories exhibiting the Vainshteinmechanism, because the essential structure of the masteralgebraic equation (23) in general cases are the same as inmassive gravity. This helps us put constraints on scalar-tensor modification of gravity through the observationsof cluster lensing.The decoupling limit of massive gravity constitutes a

subclass with two free parameters in Horndeski’s theory,which motivated us to use it for an illustrative purpose.We would like to point out here that there are severalcaveats to be aware of when putting observational con-straints on massive gravity based on our analysis. First,there will be some corrections to the decoupling limitbecause m and 1/MPl are not exactly zero in reality.However, the corrections are small enough in the regionoutside the Schwarzschild radius rg of a lens object andinside the Compton length of the graviton [15]. Thus, thedecoupling limit can be used safely at around the Vain-shtein radius, which is relevant to our purpose, unless thegraviton mass is so large that the Schwarzschild radius

FIG. 5: The convergence ! as a function of " for the gNFWprofile with Mvir = 1.28!1028 M!/h, and %l = 3. The curvescorrespond to %s = 0.5 (dotted red) and %s = 1.5 (dot-dashedgreen), respectively. For comparison, the convergence for theNFW profile (%s = 1) (with the same theory parameters)is shown by the blue dashed line. We take # = 0.985 and$ = 0.375.

FIG. 6: The convergence ! as a function of " for the Einastoprofile with Mvir = 1.28 ! 1028 M!/h and r"2 = 154 kpc/h.The curves correspond to # = 0.1 (dotted red) and # = 0.3(dot-dashed green), respectively. We take # = 0.985 and$ = 0.375.

Dip is not a consequence of the specific choice of the density profile

Summary

Static, spherically symmetric, weak gravitational field sourced by non-relativistic matter in Horndeski’s most general scalar-tensor theory

The problem reduces to solving a quintic algebraic equation

Conditions under which a screened solution is realized are clarified

Interesting applications such as testing gravity with cluster lensing

Cosmological background? ––– Sixth-order algebraic equation with time-dependent coefficients… [Kimura, TK, Yamamoto (2012)]

Application to other cosmological probes?

Thank you!