D-Branes and The Disformal Dark Sector - Danielle Wills and Tomi Koivisto

Cape Town, 11.03.2013

African Ins7tute for Mathema7cal Sciences

D-branes and the disformal dark sector

Danielle Wills and Tomi Koivisto

Institute for Theoretical Astrophysics University of Oslo Centre for Particle Theory Durham University

•  For simplicity, let us take the rela3on to be given by a scalar Φ •  It can be argued that the most general consistent rela3on then has the form •  C ≠ 1 is very well known and extensively studied. We’ll focus on D ≠ 0. The outline: -‐ What is it good for?

1) Mo7va7ons, Phenomenology: the screening -‐ Where does it come from? 2) The DBI string scenario -‐ How to detect it? 3) Cosmology: background expansion, Large-‐scale structure -‐ So what? An outlook and conclusion

On the relation between physical and gravitational geometry

Coupled three-form dark energy

Tomi S. Koivisto

⇤

Institute for Theoretical Astrophysics, University of Oslo

Consider an f(R) theory as

gµ⌫ = �gµ⌫ (1)

gµ⌫ = Cgµ⌫ +Dvµv⌫ (2)

gµ⌫ = f,Rgµ⌫ + f,QRµ⌫ (3)

G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) = D0e��(��0)

V (�) = V0e��

S =

Zd

4x

p�g

R

16⇡G

+

p�gL (matter, gµ⌫)

�(6)

⇤Electronic address: [email protected]


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) = D0e��(��0)

V (�) = V0e��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)


[Bekenstein, Phys.Rev. D48 (1993) ]

Frames of gravity

Brans-‐Dicke theory, e.g. f(R):

( )L R Vφ φ= + ( ) )()(21~~ 2 ϕϕϕ mLURL ++∂−=


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)


•  The generalisa3on of conformal mapping •  Is contained in any modified gravity* beyond f(R) •  and in any scalar-‐tensor theory beyond Brans-‐Dicke

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Screening Modifications of Gravity through Disformally Coupled Fields

Tomi S. Koivisto1, David F. Mota1, Miguel Zumalacarregui1,2

1Institute for Theoretical Astrophysics, University of Oslo, N-0315 Oslo, Norway2Institut de Ciencies del Cosmos, Universitat de Barcelona IEEC-UB, Marti i Franques 1, E-08028 Barcelona, Spain

It is shown that extensions to General Relativity, which introduce a strongly coupled scalar field,can be viable if the interaction has a non-conformal form. Such disformal coupling depends uponthe gradients of the scalar field. Thus, if the field is locally static and smooth, the coupling becomesinvisible in the Solar System: this is the disformal screening mechanism. A cosmological model isconsidered where the disformal coupling triggers the onset of accelerated expansion after a scalingmatter era, giving a good fit to a wide range of background observational data. Moreover, theinteraction leaves signatures in the formation of large-scale structure that can be used to probe suchcouplings.

PACS numbers: 95.36.+x, 04.50.Kd, 98.80.-k

In the standard !CDM model of cosmology, the uni-verse at the present day appears to be extremely finetuned. The energy scale of the ! component is about10!30 times the most naive expectations of theory, andthis ! has begun dominating the universal energy budgetat a redshift which, compared to the redshift to the for-mation of simplest elements, is a fraction of about 10!11.In attempts to avoid these fine tuning problems, ! is of-ten generalized to a dynamical scalar field, whose timeevolution could more naturally result in the observed en-ergy density today [1].High energy physics generically predicts an interaction

between scalar degree of freedom and other forms of mat-ter, which in turn could help to explain why the fieldbecomes dynamically important at the present epoch.There are myriad variations of such models, but in allof them the coupling can be e"ectively described by afield-dependent mass of the dark matter particle. ThoseYukawa-type couplings can be motivated by a conformalrelation to scalar-tensor theories, which includes also thef(R) class of modified gravity [2].However, for any other type of gravity modification,

the relation between the matter and gravitational metricwill be non-conformal. This can also be motivated e.g.in a DBI type scenario where matter is allowed to enterthe additional dimensions [3]. When given by a scalarfield !, the disformal relation can be parametrized as

gµ! = C(!)gµ! +D(!)!,µ!,! , (1)

where commas denote partial derivatives. Consideringthe most general physical case, Bekenstein [4] argued thatboth functions C and D may also depend upon ("!)2,but we will focus on the simpler case here. Previous ap-plications of such a relation to cosmology include vary-ing speed of light theories [5], inflation [6], dark energy[7, 8], gravitational alternatives to [9, 10] and extensionsof [11] dark matter. The disformal generalization of cou-pled quintessence here introduced is a simple set-up thatis useful to study generic features of the relation (1) indi"erent scenarios.

The disformal matter coupling. Our aim is to explorethe novel features from the disformal coupling in the min-imal setting where gravity is Einstein’s and the scalarfield is coupled to a single matter species. The two met-rics enter into the action for gravity and the coupledscalar-matter system in mutually exclusive sectors

S =

!

d4x

"

!"g

#

1

16#GR+ L"

$

+!"gLm

%

, (2)

where the matter Lagrangian is constructed using gµ!from Eq. (1). The stress energy tensors definition

T µ!" #

2!"g

$ (!"gL")

$gµ!, T µ!

m #2

!"g

$&!

"gLm

'

$gµ!,

ensures that Einstein field equations have the usual formGµ! = 8#GT µ!. However, the covariant conservation ofenergy momentum does not hold for the coupled compo-nents separately. Instead, we obtain that

$µTµ!m # "Q! ,! , (3)

where Q =C"

2CTm "$#

#

D

C!,µT

µ#m

$

+D"

2C!,µ!,!T

µ!m

The coupling will then generically involve second deriva-tives, which entail the distortion of causal structure.For a point particle, and taking into account the cor-

rect weight of the delta function, we have!"gLm = "#m

(

"gµ!xµx!$(4)(x" x(%)) . (4)

From the point of view of the physical frame, the propertime the particle experiences is dilated by the conformalfactor C. In addition, the disformal factor D gives adirection-dependent e"ect proportional to the projectionof the four-velocity along the gradient of the field:

˙x2 # gµ! x

µx! = Cx2 +D(x · "!)2 . (5)

Extremising the proper time of the particle along its pathshows that it follows the disformal geodesics:

xµ + $µ$% x

$x% = 0 , (6)

* The generic ghost problem of higher deriva3ve theories may be avoided in nonlocal gravity that may further be asympto(cally free! [Biswas, TK, Mazumdar: PRL (2012)].

•  A modified gravity with in the 1st order formalism [1]:

è Γ is the Levi-‐Civita connec3on of •  Horndeski scalar-‐tensor theory, “covariant galileon”:

is the E-‐H theory for [2]

An example of both

1. [TK: PRD (2007)] 2. [Zumalacarregui, TK, Mota: PRD (2013, to appear)]


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)




DBI Galileons in the Einstein Frame: Local Gravity and Cosmology

Miguel Zumalacarregui1,2, Tomi S. Koivisto2, and David F. Mota21 Instituto de Fısica Teorica IFT-UAM-CSIC, Universidad Autonoma de Madrid,

C/ Nicolas Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain and2 Institute for Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway

It is shown that a disformally coupled theory in which the gravitational sector has the Einstein-Hilbert form is equivalent to a quartic DBI Galileon Lagrangian, possessing non-linear higher deriva-tive interactions, and hence allowing for the Vainshtein e↵ect. This Einstein Frame descriptionconsiderably simplifies the dynamical equations and highlights the role of the di↵erent terms. Thestudy of highly dense, non-relativistic environments within this description unravels the existence ofa disformal screening mechanism, which represents an alternative way to investigate the Vainshteine↵ect. Disformal couplings to matter also allow the construction of Dark Energy models, whichbehave di↵erently than conformally coupled ones and introduce new e↵ects on the growth of LargeScale Structure over cosmological scales, on which the scalar force is not screened. We consider asimple Disformally Coupled Dark Matter model in detail, in which standard model particles fol-low geodesics of the gravitational metric and only Dark Matter is a↵ected by the disformal scalarfield. This particular model is not compatible with observations in the linearly perturbed regime.Nonetheless, disformally coupled theories o↵er enough freedom to construct realistic cosmologicalscenarios, which can be distinguished from the standard model through characteristic signatures.The use of more general disformal transformations provides even further relations between scalar-tensor theories of gravity.


I. INTRODUCTION

In the standard ⇤CDM model of cosmology [1], theuniverse at the present day appears to be extremely finetuned. The energy scale of the ⇤ component is extremelysmall compared to the naıve quantum corrections [2, 3],and yet large enough to be detectable through its e↵ecton the cosmological expansion [4–6]. This mystery hastriggered numerous proposals in which the cosmologicalconstant is exchanged with scalar field sourced dynamicalDark Energy [7] or alternative theories of gravity [8].

The set of viable theories is severely limited by Ostro-gradski’s Theorem [9]. It states that there exists a lin-ear instability in any non-degenerate theory whose fun-damental dynamical variable appears in the action withhigher than 2nd order in time derivatives: the Hamilto-nian for this type of theory is not bounded from belowand therefore it accepts configurations with arbitrarilylarge negative energy [10, 11]. This result can be by-passed by considering degenerate theories, i.e. those inwhich the highest derivative term can not be written as afunction of canonical variables. In this case, the dynam-ics is described by second order equations of motion, evenwhile the action contains higher derivative terms. If grav-ity only involves a rank two tensor, Lovelock’s Theorem[12] states that the Einstein-Hilbert action with a Cos-mological Constant is the only theory based on a local,1

Lorentz-invariant Lagrangian depending on the metric

1 There is some evidence that there exist viable nonlocal theoriesameliorating the fundamental problems of gravity [13].

tensor and its derivatives which gives rise to second or-der equations of motion in four space-time dimensions.The addition of a scalar degree of freedom provides a

generous extension of the possibilities. The most generalgravitational sector for a scalar-tensor theory was firstderived by Horndeski [14] and has received considerableattention recently [15–21]. It is given by the HorndeskiLagrangian

LH =5X

i=2

Li . (1)

Up to total derivative terms that do not contribute to theequations of motion, the di↵erent pieces can be writtenas [19]

L2

= G2

(X,�) , (2)

L3

= �G3

(X,�)2� , (3)

L4

= G4

(X,�)R+G4,X

⇥(2�)2 � �

;µ⌫�;µ⌫

⇤, (4)

L5

= G5

(X,�)Gµ⌫�;µ⌫ � 1

6G

5,X

h(2�)3

�3(2�)�;µ⌫�

;µ⌫ + 2� ;⌫;µ � ;�

;⌫ � ;µ;�

i. (5)

Here R,Gµ⌫ are the Ricci scalar and the Einstein tensor,X ⌘ � 1

2

gµ⌫�,⌫�,µ is the scalar field canonical kineticterm and commas and semi-colon represent partial andcovariant derivatives respectively. On top of a generalizedk-essence term (2), the remaining pieces (3-5) fix the ten-sor contractions, which rely on the anti-symmetric struc-ture of the �

;µ⌫ terms to trade higher derivatives with theRiemann tensor in the equations of motion. Note that

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2DBI Galileons in the Einstein Frame: Local Gravity and Cosmology

Miguel Zumalacarregui1,2, Tomi S. Koivisto2, and David F. Mota21 Instituto de Fısica Teorica IFT-UAM-CSIC, Universidad Autonoma de Madrid,

C/ Nicolas Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain and2 Institute for Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway

It is shown that a disformally coupled theory in which the gravitational sector has the Einstein-Hilbert form is equivalent to a quartic DBI Galileon Lagrangian, possessing non-linear higher deriva-tive interactions, and hence allowing for the Vainshtein e↵ect. This Einstein Frame descriptionconsiderably simplifies the dynamical equations and highlights the role of the di↵erent terms. Thestudy of highly dense, non-relativistic environments within this description unravels the existence ofa disformal screening mechanism, which represents an alternative way to investigate the Vainshteine↵ect. Disformal couplings to matter also allow the construction of Dark Energy models, whichbehave di↵erently than conformally coupled ones and introduce new e↵ects on the growth of LargeScale Structure over cosmological scales, on which the scalar force is not screened. We consider asimple Disformally Coupled Dark Matter model in detail, in which standard model particles fol-low geodesics of the gravitational metric and only Dark Matter is a↵ected by the disformal scalarfield. This particular model is not compatible with observations in the linearly perturbed regime.Nonetheless, disformally coupled theories o↵er enough freedom to construct realistic cosmologicalscenarios, which can be distinguished from the standard model through characteristic signatures.The use of more general disformal transformations provides even further relations between scalar-tensor theories of gravity.


I. INTRODUCTION

In the standard ⇤CDM model of cosmology [1], theuniverse at the present day appears to be extremely finetuned. The energy scale of the ⇤ component is extremelysmall compared to the naıve quantum corrections [2, 3],and yet large enough to be detectable through its e↵ecton the cosmological expansion [4–6]. This mystery hastriggered numerous proposals in which the cosmologicalconstant is exchanged with scalar field sourced dynamicalDark Energy [7] or alternative theories of gravity [8].

The set of viable theories is severely limited by Ostro-gradski’s Theorem [9]. It states that there exists a lin-ear instability in any non-degenerate theory whose fun-damental dynamical variable appears in the action withhigher than 2nd order in time derivatives: the Hamilto-nian for this type of theory is not bounded from belowand therefore it accepts configurations with arbitrarilylarge negative energy [10, 11]. This result can be by-passed by considering degenerate theories, i.e. those inwhich the highest derivative term can not be written as afunction of canonical variables. In this case, the dynam-ics is described by second order equations of motion, evenwhile the action contains higher derivative terms. If grav-ity only involves a rank two tensor, Lovelock’s Theorem[12] states that the Einstein-Hilbert action with a Cos-mological Constant is the only theory based on a local,1

Lorentz-invariant Lagrangian depending on the metric

1 There is some evidence that there exist viable nonlocal theoriesameliorating the fundamental problems of gravity [13].

tensor and its derivatives which gives rise to second or-der equations of motion in four space-time dimensions.The addition of a scalar degree of freedom provides a

generous extension of the possibilities. The most generalgravitational sector for a scalar-tensor theory was firstderived by Horndeski [14] and has received considerableattention recently [15–21]. It is given by the HorndeskiLagrangian

LH =5X

i=2

Li . (1)

Up to total derivative terms that do not contribute to theequations of motion, the di↵erent pieces can be writtenas [19]

L2

= G2

(X,�) , (2)

L3

= �G3

(X,�)2� , (3)

L4

= G4

(X,�)R+G4,X

⇥(2�)2 � �

;µ⌫�;µ⌫

⇤, (4)

L5

= G5

(X,�)Gµ⌫�;µ⌫ � 1

6G

5,X

h(2�)3

�3(2�)�;µ⌫�

;µ⌫ + 2� ;⌫;µ � ;�

;⌫ � ;µ;�

i. (5)

Here R,Gµ⌫ are the Ricci scalar and the Einstein tensor,X ⌘ � 1

2

gµ⌫�,⌫�,µ is the scalar field canonical kineticterm and commas and semi-colon represent partial andcovariant derivatives respectively. On top of a generalizedk-essence term (2), the remaining pieces (3-5) fix the ten-sor contractions, which rely on the anti-symmetric struc-ture of the �

;µ⌫ terms to trade higher derivatives with theRiemann tensor in the equations of motion. Note that

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Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) = D0e��(��0)

V (�) = V0e��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)



Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 =

p1 + 2X (10)

Clocks made of dark matter slow down:

g00 = �1 +D

˙

�

2 ! 0 = �1 +D0e��(��0)

˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)

⌦� ⇠ �

�2(14)


Interacting matter


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) = D0e��(��0)

V (�) = V0e��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)


•  The physical proper distances differ from GR: •  The equivalence principle is violated: •  The conformal prototype, Brans-‐Dicke theory, C(Φ)=exp(-‐ακ(Φ-‐Φ0)), D=0: How to reconcile with observa7ons? 1. Make the field very massive : no DE 2. Make α very small : uninteres7ng 3. Make them species-‐dependent : coupled DE 4. Make them density-‐dependent : chameleon


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) = D0e��(��0)

V (�) = V0e��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)


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gµ! = C(!)gµ! +D(!)!,µ!,! , (1)



S =

!

d4x

"

!"g

#

1

16#GR+ L"

$

+!"gLm

%

, (2)


T µ!" #

2!"g

$ (!"gL")

$gµ!, T µ!

m #2

!"g

$&!

"gLm

'

$gµ!,


$µTµ!m # "Q! ,! , (3)

where Q =C"

2CTm "$#

#

D

C!,µT

µ#m

$

+D"

2C!,µ!,!T

µ!m



(

"gµ!xµx!$(4)(x" x(%)) . (4)


˙x2 # gµ! x

µx! = Cx2 +D(x · "!)2 . (5)


xµ + $µ$% x

$x% = 0 , (6)


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 =

p1 + 2X (10)


g00 = �1 +D

˙

�

2 ! 0 = �1 +D0e��(��0)

˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)

⌦� ⇠ �

�2(14)

Geff =

✓1 +

↵

2

2

e

�rm�

◆G (15)


Effec3ve gravita3onal coupling between mager par3cles

Newton’s force + extra 5th force mediated by scalar par3cles


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 =

p1 + 2X (10)


g00 = �1 +D

˙

�

2 ! 0 = �1 +D0e��(��0)

˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)

⌦� ⇠ �

�2(14)

Geff =

✓1 +

↵

2

2

e

�rm�

◆G (15)

x

µ= �¯

�

µ↵� x

↵x

�= ��

µ↵� x

↵x

�+ 5th forces (16)


Chameleonic screening •  Spherically symmetric NR point matter source •  The scalar potential decays beyond the Compton wavelenght 1/m •  High effective mass in high density regions


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 =

p1 + 2X (10)


g00 = �1 +D

˙

�

2 ! 0 = �1 +D0e��(��0)

˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)

⌦� ⇠ �

�2(14)

Geff =

✓1 +

↵

2

2

e

�rm�

◆G (15)


Effective potential Consider non-relativistic matter

Compton wavelength Suppose the scalar has a mass

Spherically symmetric solutions The scalar potential decays exponentially above the Compton wavelength

2 2 4m G

22

2

24

d d m Gdr r dr

( ) expGMr mrr

1m

( )r r

r1m

Compton wavelength Suppose the scalar has a mass

Spherically symmetric solutions The scalar potential decays exponentially above the Compton wavelength

2 2 4m G

22

2

24

d d m Gdr r dr

( ) expGMr mrr

1m

( )r r

r1m

[Khoury and Weltman, PRL (2004) ]

Disformal screening

•  Spherically symmetric, static NR configuration: •  Each term proportional to D vanishes identically!

•  High density Dρ>>1 limit: •  The Klein-Gordon equation: •  The field indeed slows down (e.g. if , β>0) •  The evolution is independent of the density

•  The 5th force just isn’t there •  Pretty much regardless of the details of V and D

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gµ! = C(!)gµ! +D(!)!,µ!,! , (1)



S =

!

d4x

"

!"g

#

1

16#GR+ L"

$

+!"gLm

%

, (2)


T µ!" #

2!"g

$ (!"gL")

$gµ!, T µ!

m #2

!"g

$&!

"gLm

'

$gµ!,


$µTµ!m # "Q! ,! , (3)

where Q =C"

2CTm "$#

#

D

C!,µT

µ#m

$

+D"

2C!,µ!,!T

µ!m



(

"gµ!xµx!$(4)(x" x(%)) . (4)


˙x2 # gµ! x

µx! = Cx2 +D(x · "!)2 . (5)


xµ + $µ$% x

$x% = 0 , (6)

arX

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ov 2

012









gµ! = C(!)gµ! +D(!)!,µ!,! , (1)



S =

!

d4x

"

!"g

#

1

16#GR+ L"

$

+!"gLm

%

, (2)


T µ!" #

2!"g

$ (!"gL")

$gµ!, T µ!

m #2

!"g

$&!

"gLm

'

$gµ!,


$µTµ!m # "Q! ,! , (3)

where Q =C"

2CTm "$#

#

D

C!,µT

µ#m

$

+D"

2C!,µ!,!T

µ!m



(

"gµ!xµx!$(4)(x" x(%)) . (4)


˙x2 # gµ! x

µx! = Cx2 +D(x · "!)2 . (5)


xµ + $µ$% x

$x% = 0 , (6)

4

Addressing the e!ects of disformal couplings thereforerequires studying the field dynamics in high density, non-relativistic environments. This regime can be exploredusing the general scalar field equation (7) for a station-ary density distribution !("x) in the limit ! ! ", andneglecting the remaining spacetime curvature [31]. Thesame result follows from taking the limit ! # C/D, #2 in(8):

# $ %D!

2D#2 + C!

!

#2

C%

1

2D

"

= %$

2Mp#2 , (15)

where the first equality is general and the second appliesto our example model. The above expression departs sub-stantially from the simple conformal coupling, for whichthe ! ! " limit is ill-defined. Spatial derivatives becomeirrelevant, as they are suppressed by a p/! factor w.r.t.time derivatives. More importantly, the above equationbecomes independent of the local energy density, makingthe field evolution insensitive to the presence of massivebodies. As the field rolls homogeneously, spatial gradi-ents between separate objects, which would give rise tothe scalar force, do not form.In the purely disformal case with exponential D, equa-

tion (15) can be integrated directly

#(t) =Mp

$(t+ t"), t" &

Mp

$#(0). (16)

In this solution, the field time variation is approximatelyconstant while t ' t" ( #(0)#1 and slows down after-wards as ( 1/t. Since the coupling is proportional to #,stronger couplings decay earlier. Furthermore, as V is adecreasing function of the field, the interaction density! = %(#+ V !)# has at most cosmological values and itse!ect on dense systems is highly suppressed. Assuming#2 ) V ) !0, D!0 ! 1, where !0 is the average cosmicdensity, typical mass increase rates M/M are as small as) 10#6/Gy for the interstellar medium and ) 10#29/Gyfor the average Earth density. The induced mass vari-ation would have consequences e.g. in the Earth-Moonsystem, altering the orbits in a way degenerate with atime evolution of Newton’s constant. However, the pre-dicted rates are way below current lunar laser rangingbounds G/G < 10#3/Gy [25].New, potential signatures may be found in the presence

of matter velocity flows, non negligible pressure, fieldgradients of cosmological origin, or in the mildly non-linear regime, when the screening starts taking place [12].Other proposed tests involve high-precision, low-energyphoton experiments [26] and e!ects on the chemical po-tential of the baryon-photon fluid [27].This provides a novel disformal screening mechanism,

which is distinct from the Vainhstein [28], chameleon [29]and symmetron [30] mechanisms, which respectively relyon the nonlinearity of the field derivatives and the de-pendence of the field mass and coupling with the ambi-ent density. Our mechanism relies on the existence of a

well defined ! ! ", non-relativistic limit in the scalarfield equation, given by equation (15), for which the fieldevolution is independent of the matter energy density. Ifthe conformal part C is negligible, only a friction termremains and the field coupling density (3) is a decreas-ing function of time. As it evolves below its cosmologicalvalue (provided V ! < 0 and D!/D > 0), the e!ects ofthe coupling are suppressed by a factor ) !0/! and thetheory is consistent with precision gravity tests.In summary, the idea of disformal coupling survives

scrutiny with so high flying colours that the reader mustbe suspicious too. The theory has a rich structure andmany interesting applications. At cosmological scales, itcan mimic closely the "CDM evolution, providing meansto address the coincidence and cosmological constantproblems. At cluster and galactic scales, new e!ects areto be expected due to the novel manifestation of the fifthforce that could help to cure the discrepancies that thestandard model has with large scale structure observa-tions. Finally the disformal screening mechanism oper-ating on very small scales opens a completely new avenuefor fundamental scalar fields strongly coupled to matterinto our reality.We thank L. Amendola and J. Garcıa-Bellido for dis-

cussions. TK and DFM are supported by the Norwegianresearch council. MZ is supported by MICINN throughBES-2008-009090.

[1] E. J. Copeland, M. Sami, and S. Tsujikawa,Int.J.Mod.Phys. D15, 1753 (2006), hep-th/0603057.

[2] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis,Phys.Rept. 513, 1 (2012).

[3] C. de Rham and A. J. Tolley, JCAP 1005, 015 (2010),1003.5917.

[4] J. D. Bekenstein, Phys.Rev. D48, 3641 (1993).[5] J. Magueijo, Rept.Prog.Phys. 66, 2025 (2003).[6] N. Kaloper, Phys.Lett. B583, 1 (2004), hep-ph/0312002.[7] T. S. Koivisto (2008), 0811.1957.[8] M. Zumalacarregui, T. Koivisto, D. Mota, and P. Ruiz-

Lapuente, JCAP 1005, 038 (2010), 1004.2684.[9] J. D. Bekenstein, Phys.Rev. D70, 083509 (2004).

[10] M. Milgrom, Phys.Rev. D80, 123536 (2009), 0912.0790.[11] D. Bettoni, S. Liberati, and L. Sindoni, JCAP 1111, 007

(2011), 1108.1728.[12] M. Zumalacarregui, T. S. Koivisto, and D. F. Mota

(2012), see also suplemental material, 1210.8016.[13] L. Amendola, Phys.Rev. D62, 043511 (2000).[14] M. Doran, JCAP 0510, 011 (2005).[15] R. Amanullah, C. Lidman, D. Rubin, G. Aldering,

P. Astier, et al., Astrophys.J. 716, 712 (2010), 1004.1711.[16] C. Blake, E. Kazin, F. Beutler, T. Davis, D. Parkinson,

et al., Mon.Not.Roy.Astron.Soc. 418, 1707 (2011).[17] D. Larson, J. Dunkley, G. Hinshaw, E. Komatsu,

M. Nolta, et al., Astrophys.J.Suppl. 192, 16 (2011).[18] C. L. Reichardt, R. de Putter, O. Zahn, and Z. Hou,

Astrophys.J. 749, L9 (2012), 1110.5328.[19] T. Koivisto, Phys.Rev. D72, 043516 (2005).


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2= �1 +D0e

��(��0)˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e

��(13)


[TK, Mota & Zumalacarregui PRL (2012) ]

Potential signatures?

•  Matter velocity flows - Suppressed by v/c. Binary pulsars?

•  Pressure - Potential instability if p>C/D-X. Astrophysics?

•  Strong gravitational fields - Gravity coupling not suppressed by Dρ. Black holes?

•  Spatial field gradients - Potential remnants of LSS formation. Even Solar system?

Our assumptions are violated if we have:

Systematic study requires developing the PPN formalism [Work under progress with Kari Enqvist and Hannu Nyrhinen]

•  In flux compac3fica3ons of Type IIB string theory, warping can arise from the

backreac3on of fluxes/objects onto the compact space → warped throats •  Single Dp-‐branes can move as probes in this geometry, with a DBI ac3on

•  The disformal coupling arises generically from this set-‐up, as we will now see….

D3, wrapped D5…

D7

Warped throat CY3

h(r)

Disformal couplings from DBI: Flux compactifications in Type IIB string theory

•  Recall:

–  The disformal metric: this arises from the pull-‐back of the 10 dimensional metric on the worldvolume of a moving D-‐brane

–  The scalar field: the changing posi3on coordinate of the brane is a scalar field

from the four-‐dimensional point of view (we consider mo3on in one transverse direc3on only)

–  The func7ons C and D: both are given by the warp factor, 1/C=D=√h –  The disformally coupled maVer: whatever stuff resides on the moving brane

Disformal couplings from DBI: Flux compactifications in Type IIB string theory


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) = D0e��(��0)

V (�) = V0e��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)


•  Consider a probe D3-‐brane moving in the radial direc3on of an AdS5-‐type geometry induced by a stack of D3-‐branes

–  The disformal metric:

–  The scalar field ac7on:

Disformal couplings from DBI: Example – D3-brane in AdS5

+ poten3al + charge

D-brane probe in AdS5 geometry from stack of D3-

branes

•  Now lets couple the scalar to mager: –  Open string endpoints → U(1) vector fields on the world-‐volume –  These can acquire masses via Stückelberg couplings to bulk 2-‐forms → The massive vectors (or their decay products) are

-‐ dark to our standard model -‐ disformally coupled to our metric g

Disformal couplings from DBI: Coupling to Matter

D-brane probe in AdS5 geometry from stack of D3-

branes •  Finally lets summarise the geometric picture:

–  Transverse open string oscilla7ons → scalar field → dark energy in cosmology? –  Parallel open string oscilla7ons → vector field → dark maVer in cosmology?

[Work under progress with Ivonne Zavala]

Toy model: ΦDDM The dark ingredients: •  A canonical quintessence field Φ •  DDM living in •  2 extra parameters wrt ΛCDM, everything at Planck scale The Friedmann equations: The (non)conservation equations:

ΦDDM cosmology● Canonical field + DDM:

● (Non)conservation equations:

● Example:

Φ

11

to �;0i and � �,i). These e↵ects are suppressed by a rela-

tivistic v/c factor, but they may be important in certainsystems such as binary pulsars.

b. Pressure: Applications of the disformal couplingin the context of Dark Energy arguably require a valueB⇢

0

� 1, where ⇢0

is the average cosmic density. Then,even though the pressure is usually negligible with re-spect to the energy density, it should be easy to findsystems for which Bp is also much larger than one. Thismight have important consequences for the stability ofthe theory, as was briefly discussed in Section IVA.

c. Radiation: Unlike in the conformal case, thedisformal coupling has non-trivial e↵ects on ultra-relativistic fields for which T ⇡ 0, cf. (42). Some au-thors have initiated the study of the disformal couplingin scenarios featuring radiation. Brax et al. [89] con-sidered high-precision, low-energy photon experiments,which might be able to detect the influence of a disfor-mal coupling on top of a conformal one. The distortionsin the baryon-photon chemical potential induced by a dis-formal coupling and their signatures on the CMB smallscale spectrum have been studied by van de Bruck andSculthorpe [90]. Other e↵ects may follow if Electromag-netism is formulated in terms of the barred metric, suchas varying speed of light or modified gravitational lightdeflection [91].

d. Strong gravitational fields: The connection coe�-cient �µ

00

�,µ in the field derivative term is not suppressedby B⇢. It represents the e↵ects of gravity, and was ne-glected because it is small in most Solar System appli-cations, since �r

00

= GMr3 (r� 2GM) in the Schwarzschild

metric. However, this term might become relevant instrong gravitational fields, such as the vicinity of blackholes or compact objects.

e. Spatial Field Gradients: In the B⇢ � 1, ⇢ � plimit, the equation for the scalar field (54) becomes inde-pendent of the matter content and the field derivatives.Therefore, if the field acquires a spatial modulation be-fore reaching this limit, it will be preserved by the sub-sequent evolution. Spatial gradients of the field formedwhen the linear perturbation theory is valid would thenbe present today, with their actual value depending onthe details of the transition between the perturbative(e.g. the small scale limit (72)) and the screened regimes.Gradients of cosmological origin might be seen as pre-ferred direction e↵ects pointing towards cosmic struc-tures when analyzed in the Solar System. Spatial deriva-tives of the field may also be important if the field isrolling su�ciently slow as to overcome the p/⇢ factor in(53).

These and other settings might lead to characteristicsignatures and new bounds for disformally coupled theo-ries, which will be investigated in the future. It should bealso possible to obtain the coe�cients of the Parameter-ized Post Newtonian approximation, which would allowa more systematic comparison to local gravity tests.

VI. COSMOLOGY

Having addressed the viability of the theory in theSolar System, let us consider its cosmological implica-tions. Using the Einstein Frame description, the Fried-mann equations have the usual form

H2 +k

a2=

8⇡G

3(⇢+

�2

2+ V ) , (57)

H +H2 = �4⇡G

3(⇢+ 2�2 � 2V ) , (58)

but the conservation equations for matter and the scalarfield have to be computed from (43, 45):

⇢+ 3H⇢ = Q0

� , (59)

�+ 3H�+ V 0 = �Q0

, (60)

were ⇢ is the energy density of the coupled matter com-ponent and the background coupling factor reads

Q0

=A0 � 2B(3H�+ V 0 + A0

A �2) +B0�2

2⇣A+B(⇢� �2)

⌘ ⇢ , (61)

after solving away the higher derivatives. In the followingwe restrict to flat space, K = 0.At this stage it is possible to understand the di↵erence

between the pure conformal (B = 0) and disformal (A =1) cases by writing (61) in terms of the equation of stateand the scalar field energy density:

Q(c)

0

=A0

2A⇢ , (62)

Q(d)

0

⇡⇣ B0

2B(1 + w�)�

V 0

2V(1� w�)

⌘⇢�

+

p3

Mp((1 + w�)⇢tot⇢�)

1/2, (63)

where in the pure disformal case it has been assumedthat B⇢ � 1 & B�2. This approximation is satisfiedby the model presented in the next subsection when thecoupling is active, see Figure 2. The last term in (63)represents the contribution from the Hubble term, whichis subdominant when the slopes of B, V are large. Theabove expressions imply that the conformal and disfor-mal coupling between Dark Energy and Dark Matter arerelated to essentially di↵erent phenomenological param-eterizations, where the interaction is either proportionalto ⇢ = ⇢

dm

[92–94] or ⇢� [95, 96].The equations governing cosmological perturbations

can be obtained from Eq. (39), which can be used to readboth the disformal matter non-conservation and the fielddynamical equation. Working in the Newtonian gauge

ds2 = �(1 + 2�)dt2 + a2(1� 2 )d~x2 , (64)

avoids potential misinterpretations when swapping be-tween di↵erent frames, at least when these are related by

2

where !µ!" has a rather complicated form involving sec-

ond derivatives of the field [12].It is possible to derive the field equation without mat-

ter energy-momentum derivatives using projections ofEq. (3) [12]:

Mµ#!µ!#!+C

C " 2DXQµ#T

µ#m + V = 0 , (7)

where Mµ# = L$,Xgµ# + L$,XX!,µ!,# "D

C " 2DXT µ#m ,

Qµ# =C!

2Cgµ# +

!

C!D

C2"

D!

2C

"

!,µ!,# , X = "1

2("!)2,

and V = L$,$ + 2XL$,X$. A canonical field is assumedin the following, L$ = X " V .Equation (7) is a quasi-linear, diagonal, second or-

der, partial di"erential equation. Its hyperbolic char-acter depends on the signature of the tensor Mµ# , whichinvolves the coupled matter energy-momentum tensor.For a perfect fluid, in coordinates comoving with it,Mµ

# = #µ# " DC"2DX diag("$, p, p, p). Positive energy

density keeps the correct sign of the time derivative termifD > 0. However, a large pressure can flip the sign of thespatial derivatives coe#cient, introducing an instability.The present analysis focuses on non-relativistic environ-ments, and hence Dp

C"2DX # 1 will be further assumed.Future work will address the e"ects of pressure, includ-ing the circumstances under which the stability conditioncan break down dynamically [12].An example cosmological model. Let us consider an

application where the field acts as quintessence and thedisformal coupling is used to trigger cosmic acceleration.The Friedmann equations have the usual form

H2 +K =8%G

3($+

!2

2+ V ) ,

H +H2 = "4%G

3($+ 2!2 " 2V ) ,

but the conservation equations for matter and the scalarfield have to be computed from (3), (7):

$+ 3H$ = Q0! , !+ 3H!+ V ! = "Q0 , (8)

were the background order coupling factor reads

Q0 =C! " 2D(3H!+ V ! + C!

C !2) +D!!2

2#

C +D($" !2)$ $ , (9)

after solving away the higher derivatives. In the followingwe restrict to flat space, K = 0.To study the dynamics, we specify an exponential

parametrization for the disformal relation and the scalarfield potential

C = C0e!$/Mp , D = D0e

"($"$0)/Mp , V = V0e"%$/Mp

with Mp = (8%G)"1/2. Besides being motivated fromsome high energy scenarios, the exponential forms fa-cilitate the choice of natural scales for the constantprefactors by shifting the zero point of the field (e.g.D0 $ M"4

p , V0 $ M4p , C0 dimensionless). Furthermore,

these forms allow a convenient exploration of the phasespace of the system. In addition to the previously stud-ied fixed points [1, 13], we find only one new, a disformalscaling solution that is not an attractor [12].

Here we present numerical results for an examplemodel where the relation (1) is purely disformal (C0 = 1,& = 0) and only a"ects the dark matter component. Weinclude radiation and baryons to consider the full realis-tic universe model. In this scenario, the early evolutionis as in the usual exponential quintessence model, wherethe slope of the potential ' has a lower limit due to thepresence of early dark energy $ede = 3(1 + w)/'2 [1].The new features appear when the disformal factor D!2

grows towards order one. Then the clocks that tick fordark matter, g00 = "1 +D!2, slow down and make thee"ective equation of state for dark matter approach mi-nus unity asymptotically. The field also begins to freezeto avoid a singularity in the e"ective metric gµ# , and theuniverse enters into a de Sitter stage. This natural re-sistance to pathology was also observed in the disformalself-coupling scenario [7, 8].

Thus, the disformal coupling provides a mechanismthat triggers the transition to an accelerated expansion.The relatively steeper the slope of the disformal functionis, i.e. the higher the ratio (/', the faster the transitionhappens, as seen in FIG.1. This transition also producesa short “bump” in the equation of state, which may haveinteresting observational consequences. We performed afull background MCMC analysis with a modified versionof CMBEasy [14] using the Union2 Supernovae compi-lation [15], WiggleZ baryon acoustic scale data [16], cos-mic microwave background angular scale [17] and boundson early dark energy [18]. The obtained constraints areshown in FIG.2. We see that for steep slopes ' and(, the background evolution becomes increasingly sim-ilar to %CDM. At this level there are no higher boundson these parameters, and the model is completely viablewith )2

disf = 538.79 versus )2!CDM = 538.91 (best fit

WMAP7 parameters). However, the model is essentiallydi"erent from %CDM, as is quite obvious when one looksat the e"ective dark matter equation of state in FIG.1.

Cosmological Perturbations. A more realistic descrip-tion requires considering cosmological perturbations. Inthe Newtonian gauge, the linearized field equation is

#!+ 3H#!+

!

k2

a2+ V !!

"

#! = "#Q" 2& (Q0 + V !)

+ !(&+ 3') , (10)

2




Mµ#!µ!#!+C

C " 2DXQµ#T

µ#m + V = 0 , (7)


C " 2DXT µ#m ,

Qµ# =C!

2Cgµ# +

!

C!D

C2"

D!

2C

"

!,µ!,# , X = "1

2("!)2,







H2 +K =8%G

3($+

!2

2+ V ) ,

H +H2 = "4%G

3($+ 2!2 " 2V ) ,


$+ 3H$ = Q0! , !+ 3H!+ V ! = "Q0 , (8)


Q0 =C! " 2D(3H!+ V ! + C!

C !2) +D!!2

2#

C +D($" !2)$ $ , (9)



C = C0e!$/Mp , D = D0e

"($"$0)/Mp , V = V0e"%$/Mp










#!+ 3H#!+

!

k2

a2+ V !!

"

#! = "#Q" 2& (Q0 + V !)

+ !(&+ 3') , (10)

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2









gµ! = C(!)gµ! +D(!)!,µ!,! , (1)



S =

!

d4x

"

!"g

#

1

16#GR+ L"

$

+!"gLm

%

, (2)


T µ!" #

2!"g

$ (!"gL")

$gµ!, T µ!

m #2

!"g

$&!

"gLm

'

$gµ!,


$µTµ!m # "Q! ,! , (3)

where Q =C"

2CTm "$#

#

D

C!,µT

µ#m

$

+D"

2C!,µ!,!T

µ!m



(

"gµ!xµx!$(4)(x" x(%)) . (4)


˙x2 # gµ! x

µx! = Cx2 +D(x · "!)2 . (5)


xµ + $µ$% x

$x% = 0 , (6)


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2= �1 +D0e

��(��0)˙

�

2(11)


* a saddle point if β>γ. Then as field rolls down the , the begins to matter:

ΦDDM: the background story

Practically arbitrary initial

conditions

The quintessence scaling solution Acceleration

Converging to the scaling attractor* Radiation era, Matter era Disformal “freezing” De Sitter era

The early evolution of the usual “self-tuning” scalar with exponential V The coupling then triggers acceleration 13

out of the gravitational metric also avoids problems withprecision gravity tests and the subtleties related to theexistence of di↵erent frames, hence simplifying the anal-ysis of cosmological observations.

To study the dynamics within a particular example,we focus on a simple Disformally Coupled Dark Matter(DCDM) model, constructed with the following prescrip-tions:

• Dark Matter disformally coupled to a canonicalscalar field, following Eq. (59-61).

• An exponential parametrization for the disformalrelation and the scalar field potential:

B = B0

e�(��0)/Mp , (73)

V = V0

e��/Mp , (74)

A = 1 . (75)

with Mp = (8⇡G)�1/2. The conformal factor Ahas been set to the trivial value in order to focuson the novel features. Furthermore, the coupling ischosen to be negligible in the early universe, andhence initial conditions and early evolution are nota↵ected.

• Uncoupled baryons, photons and neutrinos, whichfollow the usual barotropic scaling relations ⇢ =a3(1+w). Zero cosmological constant.

Besides being motivated from some high energy scenar-ios, the exponential forms (73-75) facilitate the choice ofnatural scales for the constant prefactors by shifting thezero point of the field (e.g. B

0

⇠ M�4

p , V0

⇠ M4

p , A0

dimensionless). Furthermore, these forms allow a conve-nient exploration of the phase space of the system. Inaddition to the previously studied fixed points [7, 93], wefind only one new, a disformal scaling solution that is notan attractor. The details of this analysis can be found inAppendix D.

1. Background Evolution

The model set up is similar to the uncoupled self-interacting field case described in Ref. [30]. In partic-ular, the potential ensures a tracking stage for the fieldand the value of �

0

is chosen to tune the transition timewhen the disformal coupling becomes relevant. Althoughonly Dark Matter is a↵ected by the coupling, radiationand baryons are included in order to provide a more real-istic description. The evolution at early times is then asin the usual exponential quintessence model, where thefield tracks the dominant fluid component and the slopeof the potential � determines the amount of Early DarkEnergy (EDE) [7]

⌦ede

=3

�2

(1 + wm) , (76)

0.1 0.2 0.5 1.0 2.0 5.0 10.010104

10105

10106

10107

10108

a

!!Mpc"4"

Scalar Field

Coupled Matter

# $10 # all $

%$40 #

%$15 #

%$5 #

10! 4 0.001 0.01 0.1 1 10 100!1.0

!0.5

0.0

0.5

a

w

Scalar Field

Coupled Matter

" #10 ! all "

$#40 "

$#15 "

$#5 "

Figure 1: Background evolution of disformally coupled mat-ter. Left: evolution of the energy density for the field (red,light) and coupled matter (blue, dark) for di↵erent choices ofthe coupling slope �. Right: equation of state for the field(red, light) and coupled matter (blue, dark). High values of�/� (solid, dashed) give a good fit to observations, while lowvalues (dotted) do not produce enough acceleration.

which depends on the dominant matter component equa-tion of state parameter wm. The new features ap-pear when the disformal factor B�2 becomes of or-der one. Then the clocks that tick for Dark Matter,g00

= �1+B�2, slow down and make the e↵ective equa-tion of state approach minus unity asymptotically. Thefield also slows down to avoid a singularity in the e↵ec-tive metric gµ⌫ , and the universe enters into a de Sitterstage. This natural resistance to pathology was also ob-served in the disformal self-coupling scenario describedin Refs. [29, 30]. The disformal coupling provides then amechanism that triggers the transition to an acceleratedexpansion. The relatively steeper the slope of the disfor-mal function is, i.e. the higher the ratio �/�, the fasterthe transition happens, as seen in Figures 1, 2. This tran-sition also produces a short “bump” in the equation ofstate, which a↵ects the growth of structure.The evolution of G

e↵

for the disformally coupled DarkMatter example model (73-75) is shown in Figure 3. It ischaracterized by a bump at the transition, whose heightincreases with �, and a further increase when the po-


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2= �1 +D0e

��(��0)˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)



Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2= �1 +D0e

��(��0)˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e

��(13)



Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2 ! 0 = �1 +D0e��(��0)

˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)


•  The field slows rolling exponentially •  Eventually dark matter freezes too •  However, the sign is never flipped!

ΦDDM: constraints

•  Since during the scaling era high values for γ are preferred

•  Since the freezing is then swifter, high values for β/γ are preferred

•  The expansion history a(t) then resembles ΛCDM though both Λ and CDM are very different!

3

10! 4 0.001 0.01 0.1 1 10 100!1.0

!0.5

0.0

0.5

a

w Scalar FieldCoupled Matter" #10 ! all "

$#40 "$#15 "$#5 "

FIG. 1: Equation of state for the field (red) and coupled mat-ter (blue) for di!erent choices of the coupling slope !. Highvalues of !/" (solid, dashed) give a good fit to observations,while low values (dotted) do not produce enough acceleration.

while the perturbed continuity and Euler equations forcoupled dark matter are

! +"

a+

Q0

#$! = 3!+

Q0

#!$+

!Q

#$ , (11)

" + "

!

H +Q0

#$

"

= k2!

"+Q0

#!$

"

. (12)

The general coupling perturbation !Q is a much morecumbersome combination of the fluid and field pertur-bations than in the purely conformal coupling !Q(c) =12 log(C)!#! + 1

2 log(C)!!#!$ [12].To extract the most relevant new features by analytic

means, we shall consider the subhorizon approximation.In the small scale limit, taking into account only the mat-ter perturbations and the gradients of the field, there isa simple expression for the perturbed interaction !Q. Inthis Newtonian limit, we further relate the field gradientto the matter perturbation through the field equation(10), which yields the simple expression !Q(N) = Q0!.Combining the dark matter equations (11) and (12) to-gether with the usual Poisson equation, we obtain theevolution of the coupled dark matter overdensity

! +

#

2H +Q0

#$

$

! = 4%Geff#! . (13)

The Hubble friction is accompanied with an additionalterm due to the evolution of the field, and the source termis modulated. The last e#ect is captured by defining ane#ective gravitational constant Geff that determines theclustering of dark matter particles on subhorizon scales

Geff

G! 1 =

Q20

4%G#2. (14)

This approximation has the same expression as the sim-ple conformal case, although with a significantly di#erentfunctional form of the coupling (9).

FIG. 2: Marginalized one and two-sigma regions obtainedfrom Supernovae (Blue), BAO (Green), CMB angular scale+ early dark energy bounds (Orange), and combined con-straints. All contours included a prior on H0 from the HST[21] and "bH

20 from Big Bang Nucleosynthesis [22].

For our example model, the late time dependence dur-ing dark energy domination produces a large enhance-ment of the matter growth, !Geff/G " (&V/#)2 # 1,as & ! 10 is required to avoid the e#ects of early darkenergy. Such behavior is in tension with large scale struc-ture observations, and also occurs in conformally coupledmodels that attempt to address the coincidence problem[19]. This discrepancy can be alleviated with a di#er-ent choice of the functions C,D, V . One possibility is tointroduce a modulation D($) $ f($)D($) to make Q0

small enough after the field enters the slow roll phase.This modification can render !Geff arbitrarily small, ex-cept for a relatively short time around the transition (seeFIG.1). A di#erent solution to control perturbations canbe achieved by allowing the field itself to live in a disfor-mal metric, as in the uncoupled models studied in Refs.[7, 8]. Viable variations might be exploited to alleviatethe claimed problems of $CDM with small scale struc-ture formation [20].

The disformal screening mechanism. Finally, we con-sider the possibility of extending the disformal couplingto visible matter. Due to the stringent bounds on equiv-alence principle violations [23], some sort of screeningmechanism is necessary to hide the coupling in denseenvironments such as the Solar System. The e#ects ofdisformal couplings to the chameleon screening were re-cently investigated by Noller [24], who correctly notedthat the disformal contribution to the conservation equa-tions vanishes for static, pressureless configurations. Thisis obvious from (3), since only the T 00 component isnonzero for dust, and when contracting the field deriva-tives with the stress tensor a non-vanishing result re-quires time evolution of the scalar field.

ΦDDM cosmology● Evolution of Φ:

Practically ARBITRARY

initial conditions

●The "EXACT TRACKING"

of self-tuning scalar field

”Disformal freezing”:

DE SITTER expansion

V B

2




Mµ#!µ!#!+C

C " 2DXQµ#T

µ#m + V = 0 , (7)


C " 2DXT µ#m ,

Qµ# =C!

2Cgµ# +

!

C!D

C2"

D!

2C

"

!,µ!,# , X = "1

2("!)2,







H2 +K =8%G

3($+

!2

2+ V ) ,

H +H2 = "4%G

3($+ 2!2 " 2V ) ,


$+ 3H$ = Q0! , !+ 3H!+ V ! = "Q0 , (8)


Q0 =C! " 2D(3H!+ V ! + C!

C !2) +D!!2

2#

C +D($" !2)$ $ , (9)



C = C0e!$/Mp , D = D0e

"($"$0)/Mp , V = V0e"%$/Mp










#!+ 3H#!+

!

k2

a2+ V !!

"

#! = "#Q" 2& (Q0 + V !)

+ !(&+ 3') , (10)

2




Mµ#!µ!#!+C

C " 2DXQµ#T

µ#m + V = 0 , (7)


C " 2DXT µ#m ,

Qµ# =C!

2Cgµ# +

!

C!D

C2"

D!

2C

"

!,µ!,# , X = "1

2("!)2,







H2 +K =8%G

3($+

!2

2+ V ) ,

H +H2 = "4%G

3($+ 2!2 " 2V ) ,


$+ 3H$ = Q0! , !+ 3H!+ V ! = "Q0 , (8)


Q0 =C! " 2D(3H!+ V ! + C!

C !2) +D!!2

2#

C +D($" !2)$ $ , (9)



C = C0e!$/Mp , D = D0e

"($"$0)/Mp , V = V0e"%$/Mp










#!+ 3H#!+

!

k2

a2+ V !!

"

#! = "#Q" 2& (Q0 + V !)

+ !(&+ 3') , (10)

•  Supernovae Ia luminosity distance – redshift diagram •  Baryon acoustic oscillation scale •  Cosmic microwave background angular scale •  BBN constraints on early dark energy •  + priors on Hubble rate (HST) and baryon fraction (BBN)


Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2= �1 +D0e

��(��0)˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)



Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2= �1 +D0e

��(��0)˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e

��(13)



Tomi S. Koivisto

⇤



gµ⌫ = �gµ⌫ (1)



G2 = G3 = G5 = 0 , G4 = 1 (4)

gµ⌫ = gµ⌫ +

1

1 + 2X

�,µ�,⌫ (5)

C(�) = 1

D(�) =

4D0 e

�(��0)

V (�) =

�4V0e

��

S =

Zd

4x

p�g

R

16⇡G

+


�(6)

gµ⌫ = C(�, X)gµ⌫ +D(�, X)�,µ�,⌫ (7)

d

dX

✓X

C +XD

◆> 0 (8)

gµ⌫ = gµ⌫ + �,µ�,⌫ (9)

G2 = G3 = G5 = 0 , G4 = (1 + 2X)

�1(10)


g00 = �1 +D

˙

�

2 ! 0 = �1 +D0e��(��0)

˙

�

2(11)

V ⇠ e

��(12)

D ⇠ e�� (13)

⌦� ⇠ �

�2(14)


And obtained lower bounds on γ and β/γ:

We used the following data:


Cosmological disturbations •  We’ll use the longitudinal gauge: •  The field equations look like usual •  The coupling term Q affects only the conservation equations

2

will see, this further constrains the type of modified gravity theories which are allowed. It is worth pointing out thatthe µ–distortion is the earliest direct probe of modifications of gravity.

The paper is organised as follows: in Section 2 we present the action considered in this paper, write down theperturbation equations which govern the dynamics of the coupled photon-baryon fluid and derive the e↵ective couplingbetween the scalar field and matter and the e↵ective sound-speed for the coupled photon-baryon fluid. In Section 3we calculate the µ�distortion. In Section 4 we present our conclusions and an outlook of future work.

II. EVOLUTION OF PERTURBATIONS

A. General Equations

The action we are considering is of scalar-tensor form, namely

S =

Z p�gd4x

R16⇡G

� 1

2gµ⌫(@µ�)(@⌫�)� V (�)

�+ Smatter(�i, g

(i)µ⌫), (1)

in which R is the Ricci scalar, �i are the matter fields in the theory (relativistic and non-relativistic), � is an additionalscalar degree of freedom and the metrics g(i) are related to the metric g by

g(i)µ⌫ = C(i)(�)gµ⌫ +D(i)(�)@µ�@⌫� . (2)

Each matter species can, in general, couple to a di↵erent metric, in which the couplings are described by the functionsC(�) (called the conformal factor) and D(�) (called the disformal factor). However, multiple couplings greatly com-plicate the equations of motion and in this paper we will consider a coupling to a single species only. We will returnto multiple couplings in future work.

The field equations can easily be obtained from the action above [4]. The scalar field equation is given by

gµ⌫rµr⌫�� dV

d�+Q = 0, (3)

with

Q =C 0

2CTµ

µ �r⌫

✓D

C�,µT

µ⌫

◆+

D0

2C�,µ�,⌫T

µ⌫ (4)

and Tµ⌫ being the energy-momentum tensor of the species coupled to �, which is consequently not conserved:

rµTµ⌫ = Q�,⌫ . (5)

We will write down the equations for the perturbations only and work in the conformal (Newtonian) gauge in whichthe metric is given by

ds2 = a2(⌘)[�(1 + 2 )d⌘2 + (1� 2�)�ijdxidxj ]. (6)

Perturbing equation (5) yields the following equations

�i = � (1 + wi)(✓i � 3�)� 3H✓�Pi

�⇢i� wi

◆�i +

Q0

⇢i� �i � Q0

⇢i��

⇢i�Q, (7)

✓i = �H(1� 3wi)✓i � wi

1 + wi✓i + k2 +

�Pi/�⇢i1 + wi

k2�i � k2�i +Q0

⇢i�✓i � Q0

(1 + wi)⇢ik2��, (8)

whilst the scalar field perturbations obey the Klein-Gordon equation

��+ 2H��+ (k2 + a2V 00)�� = �( + 3�)� 2a2(V 0 �Q0) + a2�Q. (9)

The zero-order part of Q is

Q0 = �a2C 0(1� 3wi)� 2D(3H�(1 + wi) + a2V 0 + C0

C �2) +D0�2

2(a2C +D(a2⇢i � �2))⇢i , (10)

2






S =

Z p�gd4x

R16⇡G

� 1

2gµ⌫(@µ�)(@⌫�)� V (�)


(i)µ⌫), (1)


g(i)µ⌫ = C(i)(�)gµ⌫ +D(i)(�)@µ�@⌫� . (2)




d�+Q = 0, (3)

with

Q =C 0

2CTµ

µ �r⌫

✓D

C�,µT

µ⌫

◆+

D0

2C�,µ�,⌫T

µ⌫ (4)


rµTµ⌫ = Q�,⌫ . (5)


ds2 = a2(⌘)[�(1 + 2 )d⌘2 + (1� 2�)�ijdxidxj ]. (6)


�i = � (1 + wi)(✓i � 3�)� 3H✓�Pi

�⇢i� wi

◆�i +

Q0

⇢i� �i � Q0

⇢i��

⇢i�Q, (7)

✓i = �H(1� 3wi)✓i � wi

1 + wi✓i + k2 +

�Pi/�⇢i1 + wi

k2�i � k2�i +Q0

⇢i�✓i � Q0

(1 + wi)⇢ik2��, (8)


��+ 2H��+ (k2 + a2V 00)�� = �( + 3�)� 2a2(V 0 �Q0) + a2�Q. (9)


Q0 = �a2C 0(1� 3wi)� 2D(3H�(1 + wi) + a2V 0 + C0

C �2) +D0�2

2(a2C +D(a2⇢i � �2))⇢i , (10)

- Continuity eq. for the matter density contrast δ and Euler eq. for the matter velocity perturbation θ:

2






S =

Z p�gd4x

R16⇡G

� 1

2gµ⌫(@µ�)(@⌫�)� V (�)


(i)µ⌫), (1)


g(i)µ⌫ = C(i)(�)gµ⌫ +D(i)(�)@µ�@⌫� . (2)




d�+Q = 0, (3)

with

Q =C 0

2CTµ

µ �r⌫

✓D

C�,µT

µ⌫

◆+

D0

2C�,µ�,⌫T

µ⌫ (4)


rµTµ⌫ = Q�,⌫ . (5)


ds2 = a2(⌘)[�(1 + 2 )d⌘2 + (1� 2�)�ijdxidxj ]. (6)


�i = � (1 + wi)(✓i � 3�)� 3H✓�Pi

�⇢i� wi

◆�i +

Q0

⇢i� �i � Q0

⇢i��

⇢i�Q, (7)

✓i = �H(1� 3wi)✓i � wi

1 + wi✓i + k2 +

�Pi/�⇢i1 + wi

k2�i � k2�i +Q0

⇢i�✓i � Q0

(1 + wi)⇢ik2��, (8)


��+ 2H��+ (k2 + a2V 00)�� = �( + 3�)� 2a2(V 0 �Q0) + a2�Q. (9)


Q0 = �a2C 0(1� 3wi)� 2D(3H�(1 + wi) + a2V 0 + C0

C �2) +D0�2

2(a2C +D(a2⇢i � �2))⇢i , (10)

- Klein-Gordon eq. for the field δΦ: 3

and the perturbation of Q is

�Q = � ⇢i

a2C +D(a2⇢i � �2)[B1�i + B2�+ B3 + B4��+ B5��], (11)

where

B1 =a2C 0

2

✓1� 3

�Pi

�⇢i

◆� 3DH�

✓1 +

�Pi

�⇢i

◆�Da2(V 0 �Q0)�D�2

✓C 0

C� D0

2D

◆, (12)

B2 =3D�(1 + wi), (13)

B3 =6DH�(1 + wi) + 2D�2

✓C 0

C� D0

2D+

Q0

⇢i

◆, (14)

B4 = � 3DH�(1 + wi)� 2D�

✓C 0

C� D0

2D+

Q0

⇢i

◆, (15)

B5 =a2C 00(1� 3wi)

2�Dk2(1 + wi)�Da2V 00 �D0a2V 0 � 3D0H�(1 + wi)

�D�2

C 00

C�✓C 0

C

◆2

+C 0D0

CD� D00

2D

!+ (a2C 0 +D0a2⇢i �D0�2)

Q0

⇢i, (16)

and the subscript i denotes the species the field is coupled to. The expression for Q0 agrees with [4] for the casewi = 0.

For the rest of this paper we shall look at the case when the scalar field is coupled to baryons only. We are treatingphotons and baryons as fluids, coupled via Thomson scattering. For photons and baryons, Eqns. (7) and (8) become

�� = � 43✓� + 4�, (17)

✓� = 14k

2�� k2�� + k2 + ane�T (✓b � ✓�), (18)

�b = � ✓b + 3�+Q0

⇢b� �b � Q0

⇢b��

⇢b�Q, (19)

✓b = �H✓b + k2 +ane�T

R(✓� � ✓b) +

Q0

⇢b�✓b � Q0

⇢bk2�� , (20)

where we have added the interaction terms for Thomson scattering and R = 3⇢b/4⇢� .

B. Tight-Coupling Approximation

We are interested in scales much smaller than the horizon and on time-scales much smaller than the Hubbleexpansion rate. To derive a second order di↵erential equation for �� , we ignore therefore terms which involve theHubble expansion rate, the time-evolution of the background scalar field, and the time-derivatives of the scalar fieldperturbations and the gravitational potential. In this limit, the relevant equations read

�� = � 43✓� , (21)

✓� = 14k

2�� k2�� + k2 + ane�T (✓b � ✓�), (22)

�b = � ✓b, (23)

✓b = k2 +ane�T

R(✓� � ✓b)� Q0

⇢bk2��, (24)

(k2 + a2V 00)�� = a2�Q, (25)

where

Q0 =2DV 0 � C 0

2(C +D⇢b)⇢b, �Q = � ⇢b

a2(C +D⇢b)[B1�b + B5��], (26)

•  In general it is a mess:

•  For conformally coupled DM:

3

and the perturbation of Q is

�Q = � ⇢i

a2C +D(a2⇢i � �2)[B1�i + B2�+ B3 + B4��+ B5��], (11)

where

B1 =a2C 0

2

✓1� 3

�Pi

�⇢i

◆� 3DH�

✓1 +

�Pi

�⇢i

◆�Da2(V 0 �Q0)�D�2

✓C 0

C� D0

2D

◆, (12)

B2 =3D�(1 + wi), (13)

B3 =6DH�(1 + wi) + 2D�2

✓C 0

C� D0

2D+

Q0

⇢i

◆, (14)

B4 = � 3DH�(1 + wi)� 2D�

✓C 0

C� D0

2D+

Q0

⇢i

◆, (15)

B5 =a2C 00(1� 3wi)

2�Dk2(1 + wi)�Da2V 00 �D0a2V 0 � 3D0H�(1 + wi)

�D�2

C 00

C�✓C 0

C

◆2

+C 0D0

CD� D00

2D

!+ (a2C 0 +D0a2⇢i �D0�2)

Q0

⇢i, (16)

and the subscript i denotes the species the field is coupled to. The expression for Q0 agrees with [4] for the casewi = 0.

For the rest of this paper we shall look at the case when the scalar field is coupled to baryons only. We are treatingphotons and baryons as fluids, coupled via Thomson scattering. For photons and baryons, Eqns. (7) and (8) become

�� = � 43✓� + 4�, (17)

✓� = 14k

2�� k2�� + k2 + ane�T (✓b � ✓�), (18)

�b = � ✓b + 3�+Q0

⇢b� �b � Q0

⇢b��

⇢b�Q, (19)

✓b = �H✓b + k2 +ane�T

R(✓� � ✓b) +

Q0

⇢b�✓b � Q0

⇢bk2�� , (20)

where we have added the interaction terms for Thomson scattering and R = 3⇢b/4⇢� .

B. Tight-Coupling Approximation

We are interested in scales much smaller than the horizon and on time-scales much smaller than the Hubbleexpansion rate. To derive a second order di↵erential equation for �� , we ignore therefore terms which involve theHubble expansion rate, the time-evolution of the background scalar field, and the time-derivatives of the scalar fieldperturbations and the gravitational potential. In this limit, the relevant equations read

�� = � 43✓� , (21)

✓� = 14k

2�� k2�� + k2 + ane�T (✓b � ✓�), (22)

�b = � ✓b, (23)

✓b = k2 +ane�T

R(✓� � ✓b)� Q0

⇢bk2��, (24)

(k2 + a2V 00)�� = a2�Q, (25)

where

Q0 =2DV 0 � C 0

2(C +D⇢b)⇢b, �Q = � ⇢b

a2(C +D⇢b)[B1�b + B5��], (26)

3

10! 4 0.001 0.01 0.1 1 10 100!1.0

!0.5

0.0

0.5

aw Scalar Field

Coupled Matter" #10 ! all "

$#40 "$#15 "$#5 "



! +"

a+

Q0

#$! = 3!+

Q0

#!$+

!Q

#$ , (11)

" + "

!

H +Q0

#$

"

= k2!

"+Q0

#!$

"

. (12)




! +

#

2H +Q0

#$

$

! = 4%Geff#! . (13)


Geff

G! 1 =

Q20

4%G#2. (14)







3

10! 4 0.001 0.01 0.1 1 10 100!1.0

!0.5

0.0

0.5

a

w Scalar FieldCoupled Matter" #10 ! all "

$#40 "$#15 "$#5 "



! +"

a+

Q0

#$! = 3!+

Q0

#!$+

!Q

#$ , (11)

" + "

!

H +Q0

#$

"

= k2!

"+Q0

#!$

"

. (12)




! +

#

2H +Q0

#$

$

! = 4%Geff#! . (13)


Geff

G! 1 =

Q20

4%G#2. (14)







The Newtonian Limit

•  Consider subhorizon scales, k/H>>1 •  The coupling given by •  We obtain the evolution equation

•  Time-dependent effective gravitational constant •  An extra friction term •  In the exponential toy model: - scalar force >> gravity è ruled out by LSS!

12

conformal transformations [97]. Solving for the higher or-der derivatives, the perturbed continuity and Euler equa-tions for the disformally coupled matter contrast �

dc

=

�⇢/⇢ and the divergence of its velocity ✓ = ikjT0i(m)

=

ikjvj⇢a�1 and the scalar field read

�dc

+✓

a+

Q0

⇢��

dc

= 3 +Q

0

⇢��+

�Q

⇢� , (65)

✓ + ✓

✓H +

Q0

⇢�

◆= k2

✓�+

Q0

⇢��

◆, (66)

��+ 3H��+

✓k2

a2+ V 00

◆�� = ��Q� 2� (Q

0

+ V 0) + �(�+ 3 ) , (67)

and

�Q(d) = �✓k2

a2B⇢

M+ (2BV 00 �B00�2)

⇢

2M+

⇣2B0(V 0 + 3H�) +B02�2(⇢� �2)

⌘ ⇢

2M2

◆��

+(1�B�2)Q

0

M�dc

+⇣B0��B(3H � ⇢B0�)�B2(2V 0�+ 3H(⇢+ �2))

⌘ ⇢

M2

��

+⇣�B0�+B(6H � ⇢B0�) + 2B2(3H⇢+ V 0�)

⌘ ⇢�

M2

�+3B⇢�

M , (68)

for a purely disformal coupling A = 1, where M =1 + B⇢ � B�2. The coupling perturbation �Q is givenin Appendix C for the general case. This expression isa much more cumbersome combination of the fluid andfield perturbations than for the purely conformally cou-pled case

�Q(c) =1

2log(A)0⇢�

dc

+1

2log(A)00⇢�� . (69)

Note that, unlike in the conformal case, the first term in(68) is proportional to k2 and hence the coupling intro-duces explicit scale dependent terms at the level of theequations. This feature will be reflected in the growthof perturbations and the power spectrum, studied belowfor an example model.

To extract the most relevant new features by analyticmeans, we shall consider the subhorizon approximation.In the small scale limit, taking into account only the mat-ter perturbations and the gradients of the field, there isa simple expression for the perturbed interaction �Q. Inthis Newtonian limit, we further relate the field gradientto the matter perturbation through the field equation(67), which yields the simple expression

�Q(N) = Q0

�dc

. (70)

Combining equations (65) and (66) together with theusual Poisson equation, we obtain the evolution of thecoupled Dark Matter overdensity

�dc

+

2H +

Q0

⇢�

��dc

= 4⇡Ge↵

⇢�dc

. (71)

In addition to an extra friction term, the source term ismodulated. The last e↵ect is captured by defining ane↵ective gravitational constant G

e↵

that determines theclustering of Dark Matter particles on subhorizon scales

Ge↵

G� 1 =

Q2

0

4⇡G⇢2. (72)

This approximation has the same expression as the sim-ple conformal case, although with a significantly di↵erentfunctional form of the coupling Q

0

, which is now givenby Eq. (61).

A. Disformally Coupled Dark Matter

In what follows it will be assumed that the field is onlycoupled to Dark Matter, while radiation and baryons fol-low geodesics of the gravitational metric and do not feelthe scalar interaction directly - If baryons are also cou-pled, then the ratio ⇢dm/⇢b remains fixed, because bothspecies feel the same e↵ective metric.11 Postulating thatthe baryonic and electromagnetic sectors are constructed

11 This can also be seen directly from (61): in the denominator ofQ0, the energy density has to be substituted by the total one⇢ ! ⇢dm + ⇢b, while the multiplicative coe�cient ⇢ would referto each individual species.

12


dc

=


=


�dc

+✓

a+

Q0

⇢��

dc

= 3 +Q

0

⇢��+

�Q

⇢� , (65)

✓ + ✓

✓H +

Q0

⇢�

◆= k2

✓�+

Q0

⇢��

◆, (66)

��+ 3H��+

✓k2

a2+ V 00

◆�� = ��Q� 2� (Q

0

+ V 0) + �(�+ 3 ) , (67)

and

�Q(d) = �✓k2

a2B⇢

M+ (2BV 00 �B00�2)

⇢

2M+

⇣2B0(V 0 + 3H�) +B02�2(⇢� �2)

⌘ ⇢

2M2

◆��

+(1�B�2)Q

0

M�dc

+⇣B0��B(3H � ⇢B0�)�B2(2V 0�+ 3H(⇢+ �2))

⌘ ⇢

M2

��

+⇣�B0�+B(6H � ⇢B0�) + 2B2(3H⇢+ V 0�)

⌘ ⇢�

M2

�+3B⇢�

M , (68)


�Q(c) =1

2log(A)0⇢�

dc

+1

2log(A)00⇢�� . (69)



�Q(N) = Q0

�dc

. (70)


�dc

+

2H +

Q0

⇢�

��dc

= 4⇡Ge↵

⇢�dc

. (71)


e↵


Ge↵

G� 1 =

Q2

0

4⇡G⇢2. (72)


0





12


dc

=


=


�dc

+✓

a+

Q0

⇢��

dc

= 3 +Q

0

⇢��+

�Q

⇢� , (65)

✓ + ✓

✓H +

Q0

⇢�

◆= k2

✓�+

Q0

⇢��

◆, (66)

��+ 3H��+

✓k2

a2+ V 00

◆�� = ��Q� 2� (Q

0

+ V 0) + �(�+ 3 ) , (67)

and

�Q(d) = �✓k2

a2B⇢

M+ (2BV 00 �B00�2)

⇢

2M+

⇣2B0(V 0 + 3H�) +B02�2(⇢� �2)

⌘ ⇢

2M2

◆��

+(1�B�2)Q

0

M�dc

+⇣B0��B(3H � ⇢B0�)�B2(2V 0�+ 3H(⇢+ �2))

⌘ ⇢

M2

��

+⇣�B0�+B(6H � ⇢B0�) + 2B2(3H⇢+ V 0�)

⌘ ⇢�

M2

�+3B⇢�

M , (68)


�Q(c) =1

2log(A)0⇢�

dc

+1

2log(A)00⇢�� . (69)



�Q(N) = Q0

�dc

. (70)


�dc

+

2H +

Q0

⇢�

��dc

= 4⇡Ge↵

⇢�dc

. (71)


e↵


Ge↵

G� 1 =

Q2

0

4⇡G⇢2. (72)


0





2




Mµ#!µ!#!+C

C " 2DXQµ#T

µ#m + V = 0 , (7)


C " 2DXT µ#m ,

Qµ# =C!

2Cgµ# +

!

C!D

C2"

D!

2C

"

!,µ!,# , X = "1

2("!)2,







H2 +K =8%G

3($+

!2

2+ V ) ,

H +H2 = "4%G

3($+ 2!2 " 2V ) ,


$+ 3H$ = Q0! , !+ 3H!+ V ! = "Q0 , (8)


Q0 =C! " 2D(3H!+ V ! + C!

C !2) +D!!2

2#

C +D($" !2)$ $ , (9)



C = C0e!$/Mp , D = D0e

"($"$0)/Mp , V = V0e"%$/Mp










#!+ 3H#!+

!

k2

a2+ V !!

"

#! = "#Q" 2& (Q0 + V !)

+ !(&+ 3') , (10)14

0.1 0.2 0.5 1.0 2.0 5.0 10.0

0.001

0.1

10

1000

105

107

a

!"10 , # "40 !

B!, t2

B"

BV

0.1 0.2 0.5 1.0 2.0 5.0 10.0

0.01

0.1

1

10

100

1000

104

a

!"10 , # "5 !

B!, t2

B"

BV

Figure 2: Evolution of the dimensionless disformal factorsB�

2 (solid), B⇢ (dashed) and BV (dot-dashed). Higher val-ues of �/↵ produce a sharper transition and lead to higher Bat late times.

tential becomes dominant. At the later stage, the depen-dence is approximately G

e↵

/G�1 ⇠ (�V/⇢)2 and yields alarge value since Dark Energy domination requires V & ⇢and � & 15 is necessary to avoid the e↵ects of early DarkEnergy (76) [98]. This enhancement occurs on observablescales and spoils the formation of large scale structure inthis particular case. Problematic growth enhancementalso occurs in conformally coupled models that attemptto address the coincidence problem [94]. The observ-able e↵ects will be analyzed using the full perturbation�Q within the disformally coupled Dark Matter model.Several alternatives to render the model viable will bedescribed in Section VIB.

2. Perturbations

The full system of linearized equations (67-68) wassolved numerically using a modified version of the Boltz-mann code CMBeasy adapted to the Disformally Cou-pled Dark Matter model described in Section VIA. Sincematter is essentially uncoupled until z . 10 there was noneed to modify the initial conditions, which have been as-sumed adiabatic. Figure 4 shows the evolution of the den-

0.10 1.000.500.20 0.300.15 0.700.1

1

10

100

1000

a

Geff!G!1

"#5 " dotted # , "#10 " solid #

$ #15 " " green # , $ #40 " " blue #

Figure 3: E↵ective gravitational constant on small scales (72)for di↵erent values of �, �. The value is large at the transitiondue to the disformal friction term B

0

�

2, and latter due to thecontribution of the potential term BV

0 (see text and compareto Figure 1).

sity contrast of disformally coupled matter. The baryons,which are uncoupled in this particular example, are alsoshown for comparison. Figure 5 displays the power spec-tra for disformally coupled matter and baryons at z = 0for di↵erent values of the parameters. Figure 6 shows theCMB power spectrum and the baryon-DM bias inducedby the coupling at z = 0.

Besides the e↵ect of early Dark Energy and late timescalar force captured in G

e↵

, the disformal couplingcauses a considerable integrated Sachs-Wolfe e↵ect, afundamental bias between disformally coupled matterand baryons and large scale oscillatory features beyondthe BAO scale. The numerical results and the discussionare restricted to the DCDM model, and focus on the roleof the potential slope �, which mostly determines the latetime value of G

e↵

. It remains to be studied whether ornot similar e↵ects occur in viable models such as the onesdescribed in Section VIB, and to what extent they mightbe observable by current or future surveys.

a. Early Dark Energy: Both the baryons and thecoupled Dark Matter are indistinguishable as long asthe coupling is negligible. They are equally a↵ected bythe presence of early Dark Energy (see Figure 4), whichproduces a departure from the matter domination result� / a: EDE increases the expansion rate without cluster-ing, reducing the formation of structure. This e↵ect wasalso found for the uncoupled scalar field [29, 30], and ismost noticeable for models with higher ⌦

ede

(e.g. � = 4).

b. Late Enhanced Growth: The growth of structureis enhanced after the transition takes place, consistentlywith the small scale approximation (72). The large valueof G

e↵

/G overcomes the additional friction term, andstructures form much faster than in the standard CDMscenario. Models with less EDE su↵er a higher en-


On the linear instability

•  Generic in conformally coupled models [TK: Phys.Rev. D72 (2005) 043516]

- Spoils LSS when the coupling is strong enough to address the coincidence problem!

- Misleadingly dubbed “adiabatic instability” [Bean, Flanagan, Trodden: Phys.Rev. D78 (2008) 023009]

- Appears also in the coupled neutrino models e.g. [Wetterich et al 2007-]

•  Worse with coupled vectors/forms - In addition to 5th force, an effective cs for DM

[TK and Nunes (2012)] •  Is it generic in disformal models? - Presumably not: e.g. Gauss-Bonnet coupling

[TK and Mota: PLB, PRD (2006)]

17

5 10 50 100 500 1000

1000

2000

3000

4000

5000

6000

l

Cl

10! 4 0.001 0.01 0.1 1

0.02

0.05

0.10

0.20

0.50

1.00

k ! h

bias"z"0#

Pk !b "

Pk !dcm "!"10

!"5

!"4

#CDM

Figure 6: CMB power spectrum (left) and bias between baryons and Disformally Coupled DM (DCM) induced by the coupling(right). The enhanced growth of Dark Matter structures on small scales produces a very large ISW e↵ect. Note that thedepartures are worse for models with less early Dark Energy (higher �), as derived in the small scale approximation (72). Unitsof k are Mpc�1.

when the coupling is active. They are most noticeable forthe models with a large early Dark Energy component,e.g. larger field energy density. Although it constitutes adistinctive feature of the model, the oscillations are notsignificantly imprinted on the baryonic power spectrum.This, together with the large survey volumes necessaryto explore such scales would make it di�cult to detectthem through LSS surveys. However, the large scale os-cillations would be a characteristic signature in modelswhere the disformal coupling is universal, in which thesame e↵ects occur to DM and Baryons.

B. Viable Scenarios

The study of cosmological perturbations within theDisformally Coupled Dark Matter model (73-75) showsvery drastic departures in the formation of large scalestructure, which seem very di�cult to reconcile with ob-servations. It would be interesting to obtain a moreprecise quantification of these discrepancies through anMCMC analysis and explore possible degeneracies (e.g.the growth suppression from early Dark Energy and theenhancement from the high G

e↵

). However, it is neces-sary to address the existence of alternative, viable sce-narios already at this stage.

Luckily, the action (33) is very general and there is con-siderable room for improvement through di↵erent choicesof the functions A,B and L�. There are at least two pos-sibilities

• Introduce a modulation in the disformal factorB(�) ! f(�)B(�), to make Q

0

small enough afterthe field enters the slow roll phase. This modifica-tion can render �G

e↵

arbitrarily small, except fora relatively short time around the transition (seeFigure 1). This type of models would allow us to

study the e↵ects imprinted by the transition bumpwithout the problems caused by the high G

e↵

atlate times.

• Constructing the field Lagrangian using a disfor-mal metric, as in the uncoupled model presentedin Refs. [29, 30]. In this model the transition toslow roll would be partially driven by the scalarfield Lagrangian itself, and the e↵ects on mattermay be significantly reduced. In the minimal pre-scription, the matter and field Lagrangian are con-structed using the same metric (6) and no extraparameters are introduced. If this model turnedout not to be viable, a di↵erent disformal metricfor the field and the coupled matter would o↵eran alternative that is able to interpolate betweendisformal quintessence and the disformally coupledDark Matter presented here (e.g. di↵erent disfor-mal factors with B(m) = ✏B(�)).

Other alternatives could be based on the interplay be-tween the conformal and the disformal part of the cou-pling. Viable scenarios might be exploited to alleviatethe claimed problems of ⇤CDM with small scale struc-ture formation, such as the tension between Dark Mat-ter simulations and observations with regard to both thedensity profiles of Dark Matter halos and for the numberof predicted substructures inside a given host halo, thebaryonic Tully-Fisher relation, the constant galactic sur-face density or the large scale bulk flows (See reference[101] for a summary and references therein for furtherdetails).As a final remark, let us note that the enhanced growth

rushes the modes into entering the non-linear regime atearlier times, breaking down the perturbative approachfollowed here. As it was explained in Section V, the dis-formal coupling comes equipped with a screening mech-anism, that hides the e↵ects of the additional force on

16

0.001 0.01 0.1 1100

1000

104

105

106

k ! h

Pk

!"10

!"5

!"4

#CDM

Coupled DM

z"0

0.001 0.01 0.1 1100

500

1000

5000

1 !104

5 !104

1 !105

k ! h

Pk

Baryons

z!0

10! 4 0.001 0.01 0.1 1

1

10

100

k ! h

Pkdisf!Pk"

CDM

Dark Matter ! z !0 "

relative to "CDM

10! 4 0.001 0.01 0.1 1

0.2

0.5

1.0

2.0

5.0

10.0

k ! h

Pkdisf!Pk"

CDM

baryons ! z !0 "

relative to "CDM

Figure 5: Power spectra for disformally coupled matter and (uncoupled) baryons at z = 0 for � = 20� and di↵erent values of� (units of k are Mpc�1). First line: Power spectrum for (uncoupled) baryons and (coupled) matter. Second Line: Ratioof the power spectra in disformally coupled models relative to ⇤CDM. See the text for a discussion of the di↵erent e↵ects.

of the coupled matter structures is enhanced, but in ascale independent way (cf. [95]).

The coupling also modifies the relation between bary-onic and Dark Matter structures, since DM couples di-rectly to the field while baryons are only indirectly af-fected. As baryons are dragged into the potential wellscreated by the coupled matter, they follow a scale depen-dent growth pattern, delayed with respect to the domi-nant matter component. The resulting bias between thetwo species is shown in Figure 6. The scale dependenceof the bias vanishes both on super-horizon scales (k/h .0.001 Mpc�1) and the small scales (k/h & 0.1 Mpc�1),which are well described by the scale-independent G

e↵

(72). The intermediate region shows the interplay be-tween the scale dependent growth for the coupled matterand the baryons following these structures.

Since galaxies form out of baryons, this fundamentalbias modifies the usual DM-galaxy power relation [100].Such a correction needs to be taken into account whencomparing the observed power spectrum with disformallycoupled models. On the other hand, other measurementsof the matter distribution such as weak lensing wouldprobe the structures formed by both components, and

may be used to break the degeneracy.

d. CMB: Integrated Sachs-Wolfe E↵ect: The en-hancement of the perturbations after the transitioncauses the very large Integrated Sachs-Wolfe e↵ect ap-preciated in Figure 6, which becomes most noticeable forthe models with higher values of �. The model with � = 4gives a better fit on the l . 10 multipoles, but departsconsiderably in the range 10 . l . 200 due to the e↵ect ofearly Dark Energy after recombination. The model withless early Dark Energy has the opposite problem: it pro-duces a better fit in the intermediate range 100 < l < 200due to the lower amount of early Dark Energy, but theISW enhancement explodes at lower multipoles due tothe higher value of G

e↵

. The di↵erent amounts of earlyDark Energy have an additional e↵ect on the CMB nor-malization due to the primary Sachs-Wolfe e↵ect: by re-ducing the potential wells that redshift the photons, ⌦

ede

acts increasing the height of the peaks.

e. Oscillatory Features beyond the BAO Scale: Os-cillatory features can be appreciated in the coupled mat-ter power spectrum on very large scales. These are likelycreated as field oscillations on scales near k ⇠ H(z),which are then transferred to the coupled component,

16

0.001 0.01 0.1 1100

1000

104

105

106

k ! h

Pk

!"10

!"5

!"4

#CDM

Coupled DM

z"0

0.001 0.01 0.1 1100

500

1000

5000

1 !104

5 !104

1 !105

k ! h

Pk

Baryons

z!0

10! 4 0.001 0.01 0.1 1

1

10

100

k ! h

Pkdisf!Pk"

CDM

Dark Matter ! z !0 "

relative to "CDM

10! 4 0.001 0.01 0.1 1

0.2

0.5

1.0

2.0

5.0

10.0

k ! h

Pkdisf!Pk"

CDM

baryons ! z !0 "

relative to "CDM

Figure 5: Power spectra for disformally coupled matter and (uncoupled) baryons at z = 0 for � = 20� and di↵erent values of� (units of k are Mpc�1). First line: Power spectrum for (uncoupled) baryons and (coupled) matter. Second Line: Ratioof the power spectra in disformally coupled models relative to ⇤CDM. See the text for a discussion of the di↵erent e↵ects.

of the coupled matter structures is enhanced, but in ascale independent way (cf. [95]).

The coupling also modifies the relation between bary-onic and Dark Matter structures, since DM couples di-rectly to the field while baryons are only indirectly af-fected. As baryons are dragged into the potential wellscreated by the coupled matter, they follow a scale depen-dent growth pattern, delayed with respect to the domi-nant matter component. The resulting bias between thetwo species is shown in Figure 6. The scale dependenceof the bias vanishes both on super-horizon scales (k/h .0.001 Mpc�1) and the small scales (k/h & 0.1 Mpc�1),which are well described by the scale-independent G

e↵

(72). The intermediate region shows the interplay be-tween the scale dependent growth for the coupled matterand the baryons following these structures.

Since galaxies form out of baryons, this fundamentalbias modifies the usual DM-galaxy power relation [100].Such a correction needs to be taken into account whencomparing the observed power spectrum with disformallycoupled models. On the other hand, other measurementsof the matter distribution such as weak lensing wouldprobe the structures formed by both components, and

may be used to break the degeneracy.

d. CMB: Integrated Sachs-Wolfe E↵ect: The en-hancement of the perturbations after the transitioncauses the very large Integrated Sachs-Wolfe e↵ect ap-preciated in Figure 6, which becomes most noticeable forthe models with higher values of �. The model with � = 4gives a better fit on the l . 10 multipoles, but departsconsiderably in the range 10 . l . 200 due to the e↵ect ofearly Dark Energy after recombination. The model withless early Dark Energy has the opposite problem: it pro-duces a better fit in the intermediate range 100 < l < 200due to the lower amount of early Dark Energy, but theISW enhancement explodes at lower multipoles due tothe higher value of G

e↵

. The di↵erent amounts of earlyDark Energy have an additional e↵ect on the CMB nor-malization due to the primary Sachs-Wolfe e↵ect: by re-ducing the potential wells that redshift the photons, ⌦

ede

acts increasing the height of the peaks.

e. Oscillatory Features beyond the BAO Scale: Os-cillatory features can be appreciated in the coupled mat-ter power spectrum on very large scales. These are likelycreated as field oscillations on scales near k ⇠ H(z),which are then transferred to the coupled component,

Cosmology: an outlook

•  Viable version of the ΦDDM? - Is coupled dark energy possible without the fatal instability? - The constraints on a viable ΛDDM interesting in any case!

•  Could we alleviate the small scale problems of ΛCDM? - Tensions between simulations and observations - The baryonic Tully-Fisher relation - The constant galactic surface density - The large scale bulk flows •  What will we learn from Euclid? - Probes LSS using gravitational lensing and BAO - We expect even an order of magnitude better probe of LSS

•  Addressing these requires going nonlinear! - Analytical derivation of exact Newtonian disformalism - Heavy simulations on supercomputers…

Conclusions

•  Disformal matter couplings are generic •  Appear in basically any generalised gravity theory •  Explicitly derivable from stringy compactifications

•  A novel screening mechanism •  The coupling disappears in static spherically symmetric configurations •  Subtle (non-cosmological) effects could still betray the modulus

•  Cosmological applications •  The DBI radion: quintessence. The stuff on the moving brane: DDM. •  New signatures in LSS, to be probed eventually by Euclid

Motivations: cosmological

● Could the coupling help with the coincidence problem?

- Strictly bound in the conformal case due to the 5th force in the LSS

● Could it alleviate the small-scale problems of ΛCDM?

- Tensions between DM simulations and observations

- The baryonic Tully-Fisher relation

- The constant galactic surface density or the large scale bulk flows

The main reference: [TK, Mota & Zumalacarregui: PRL (2012) ], [TK, DW, Zavala (to appear soon)]

Date post:	17-Jan-2015
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D-Branes and The Disformal Dark Sector - Danielle Wills and Tomi Koivisto

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