Dilaton and modified gravity

Philippe Brax,1,* Carsten van de Bruck,2,† Anne-Christine Davis,3,‡ and Douglas Shaw4,x1Institut de Physique Theorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/Yvette Cedex, France

2Department of Applied Mathematics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom3Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Cambridge CB3 0WA,

United Kingdom4Queen Mary University of London, Astronomy Unit, Mile End Road, London E1 4NS, United Kingdom

(Received 1 June 2010; published 14 September 2010)

We consider the dilaton in the strong string-coupling limit and elaborate on the original idea of Damour

and Polyakov whereby the dilaton coupling to matter is minimized and vanishes at a finite value of the

dilaton field. Combining this type of coupling with an exponential potential, the effective potential of the

dilaton becomes matter density dependent. We study the background cosmology, showing that the dilaton

can play the role of dark energy. We also analyze the constraints imposed by the absence of violation of

the equivalence principle. Imposing these constraints and assuming that the dilaton plays the role of dark

energy, we consider the consequences of the dilaton on large scale structures and, in particular, the

behavior of the slip functions and the growth index at low redshift.

DOI: 10.1103/PhysRevD.82.063519 PACS numbers: 98.80.Cq, 98.70.Vc

I. INTRODUCTION

The observed late-time acceleration of the Universe hasno clear theoretical explanation yet. One of the putativecandidates is dark energy whereby a scalar field with a flatenough potential provides the potential energy leading tothe accelerating phase (for reviews and references see e.g.[1]). Although this seems like a natural scenario, it isfraught with difficulties. The most important one is cer-tainly the absence of a complete understanding of the roleof quantum corrections in such models. This is akin to thehierarchy problem of Higgs physics, albeit even moreserious due to the stringent phenomenological constraintsthat acceleration imposes on the dark energy potential. Inparticular, both the vacuum energy and the mass of thescalar field must be minute. The former is nothing but areformulation of the cosmological constant problem,whereas the latter prescribes that there should exist a newfifth force complementing the gravitational interaction atvery large scales. Reconciling such a long-range force withlocal experiments of gravity on Earth and in the SolarSystem is a difficult task. Three types of mechanisms canbe generically invoked. The first one appears in the DGP[2] modification of gravity where the Vainshtein [3] effectis present locally. In these models, gravity is modified onlarge scales and preserved close to massive bodies due tothe nonlinearities of the scalar-field kinetic terms. Theshielding of the scalar field by massive bodies is also afeature of chameleon models [4–6] [and therefore of fðRÞgravity [7]], where the mass of the scalar field becomesenvironmentally dependent. This leads to a thin shell effect

preventing any deviations from Newton’s law in the vicin-ity of massive objects.Another mechanism has been advocated in a string

theoretic context: the Damour-Polyakov effect [8].Considering the string dilaton in the strong-coupling re-gime, it turns out that no violation of general relativitywould be observed, provided the coupling of the dilaton tomatter were driven to zero by the cosmological expansion.This result stands when no potential is taken into accountfor the dilaton. It turns out that in the strong-couplingregime, one expects that the dilaton will have an exponen-tially decreasing potential akin to the ones used to describedark energy. This was noticed in [9–11], and Ref. [9]proposed a scenario whereby quintessence was the run-away dilaton, though the couplings to matter were ne-glected here. Later, this scenario was modified andgravitational tests were shown to be evaded provided thecoupling of the dilaton to matter vanished for an infinitelylarge dilaton. This is equivalent to the Damour-Polyakovmechanism with a minimum at infinity.In this paper, we will focus on the original Damour-

Polyakov setting, where the coupling vanishes for a finitevalue of the dilaton while keeping an exponentially run-away dilaton potential. In this case, the potential term tendsto displace the dilaton from the minimum with no couplingto matter. The fifth-force constraints would not be evadedanymore. In fact, this result only stands when the string andPlanck scales are of the same order of magnitude. Providedthe string scale is lower than the Planck scale by a feworders of magnitude, we find that the Damour-Polyakovmechanism is at play, albeit only locally where matterdensities are large. This allows one to evade SolarSystem constraints on gravity. This environmentally de-pendent Damour-Polyakov mechanism implies that thereexists a fifth force whose manifestation is prominent on

*philippe.brax@cea.fr†c.vandebruck@sheffield.ac.uk‡a.c.davis@damtp.cam.ac.ukxd.shaw@qmul.ac.uk

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galaxy cluster scales. This prompts the interesting possi-bility that one could probe and measure dilatonic modifi-cations of gravity with future large scale surveys.

This paper is organized as follows. In Sec. II we willrecall the main properties of dilaton models in the strong-coupling regime. Then, in Sec. III we will study the localtests and the constraints they impose on dilatonic models,and investigate the chameleon mechanism as well as theDamour-Polyakov mechanism. In Sec. IV we present theenvironmentally dependent Damour-Polyakov mechanismand discuss the cosmological evolution as well as the localconstraints. In Sec. V we will focus on the consequencesfor large scale structures in the Universe. Our conclusionscan be found in Sec. VI.

II. DILATON AND MODIFIED GRAVITY

A. Dilaton models

Our starting point is the low energy, gravi-dilaton string-frame effective action, including dilaton-dependent correc-tions, as given in [9,10] (see also [12]):

S ¼Z ffiffiffiffiffiffiffi�~g

pd4x

�e�2c ð�Þ

2l2s~Rþ Zð�Þ

2l2sð~r2�Þ � ~Vð�Þ

�

þ Smð�i; ~g��; gið�ÞÞ:Here ls is the string length scale,�i are the matter fields,

and ~R is the Ricci scalar curvature of ~g��. The gið�Þrepresent the ‘‘constants’’ of nature such as the gaugecoupling constants, which are now dilaton dependent.

We move to the arguably more physically transparentEinstein frame, by defining ~g�� ¼ A2ð�Þg��, where

Að�Þ ¼ lsec ð�Þ=�4 and �2

4 ¼ 8�GN . We are also free torescale Að�Þ by a constant factor, and so we fix its defini-tion by requiring Að�0Þ ¼ 1, where �0 is approximatelythe value of � today. We let c1 � ls=�4 ¼ expð�c ð�0ÞÞ.The Einstein frame action then becomes

S ¼Z ffiffiffiffiffiffiffi�g

pd4x

�RðgÞ2�2

4

� k2ð�Þ�24

ðr�Þ2 � Vð�Þ�

þ Smð�i; A2ð�Þg��;�Þ; (1)

where k2ð�Þ ¼ 3�2ð�Þ � A2ð�ÞZð�Þ=2c21, where�ð�Þ ¼ðlnAÞ;� and Vð�Þ ¼ A4ð�Þ ~Vð�Þ.

We will additionally assume that e��0 � 1 so that weare in the strong-coupling limit. In this limit (� ! 1), weassume (see [13] and references therein)

~Vð�Þ � ~V0e�� þOðe�2�Þ;

Zð�Þ � � 2c21�2

þ bZe�� þOðe�2�Þ;

g�2i � �g�2

i þ bie�� þOðe�2�Þ:

We will assume that e��0 is sufficiently small that theseasymptotic expansions are valid. In [9] these potentials and

couplings were derived, and it was shown that the dilatoncould, in principle, act as dark energy. Typically oneexpects bZ �Oð1Þ and bi �Oð1Þ. Natural values for �range from ��Oð1Þ to Oðc1Þ ¼ Mpl=Ms, and generally

c1 � 1. Using e��0 � 1, we have

kð�Þ � ��1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3�2�2ð�Þ

q: (2)

We define ’ by d’ ¼ kð�Þd�. The field equation for �(or equivalently ’) is then given by

h’ ¼ �24

2kð�Þ��Vð�Þ � �ð�ÞðA4ð�Þ ~Tm � 4Vð�ÞÞ

�Xi

Að�Þ�ið�Þð�ÞSi�; (3)

where

~T��m ¼ 2ffiffiffiffiffiffiffi�~g

p �Sm

�~g��

;

Að�Þ�i ¼ � lngi��

�� g2i ð�Þbie��

2;

Si ¼ �Sm

� lngi;

and ~Tm ¼ ~T��m ~g��. We have defined the conserved matter

density in the Einstein frame Tm ¼ g�T�m ¼ A3 ~Tm.

Typically the Si are OðTmÞ or smaller. We assume that,near �0, e

�� is sufficiently small that the composition-dependent couplings �i / e�� are suppressed to be muchsmaller than the universal coupling �.The field equation for � then simplifies to

h’ � �24

2kð�Þ ½�Vð�Þ � �ð�ÞðAð�ÞTm � 4Vð�ÞÞ�: (4)

We may think of ’ as feeling an effective potentialVeffð’;TmÞ defined by

h’ ¼ �24

2Veff;’ð’;TmÞ ¼ �2

4

2kð�ÞVeff;�ð�;TmÞ:

It follows that

Veffð’;TmÞ ¼ V0A4ð�Þe�� � Að�ÞTm: (5)

With pressureless matter, Tm ¼ �m, and this effectivepotential is minimized when � ¼ �minðmÞ, which isgiven by

�ð�minÞ ¼ Vð�minÞAð�minÞm þ 4Vð�minÞ : (6)

This implies that �ð�minÞ � 1=4. Cosmologically, today ifVð�Þ is responsible for the late-time acceleration of theUniverse, we have Am=V � 0:27=0:73 � 0:37, and socosmologically � � 0:23 today.We have used the fact that, cosmologically, for pressur-

eless matter, Tm ¼ mðtÞ ¼ m0a30=a

3 does not depend

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explicitly on �. We define m’ to be the effective mass of

small perturbations in ’,

m2’ð�;TmÞ ¼ Veff;’’ð’min;TmÞ (7)

¼ k�2ð�Þ½Veff;��ð’min;TmÞ � 12ðlnk2Þ;�Veff;��: (8)

Thus, at the minimum of the effective potential

m2’ð�min;TmÞ ¼

�24�;��

2k2ð�Þ ðAm þ 4VÞ

þ �24�ð1� 3�ÞAm

2k2ð�Þ : (9)

Cosmologically, �24Am ¼ 3�mH

2 and �24V ¼ 3��H

2,and we have � � 1=4 at the minimum of Veff . Generallythen, unless �;�� � 1, we have m’ �OðHÞ, leading to a

new long-range force over cosmological scales.

B. Dilatonic modification of gravity

Over length scales less than �’ ¼ m�1’ the scalar field

mediates a fifth force that is ð�Þ times the strength ofgravity. We calculateð�Þ by considering the conservationequation for T��

m , which reads

r�T��m ¼

��ð�ÞTm þX

i

�iSi

�r��� �ð�ÞT��

m r��:

(10)

For nonrelativistic matter with energy density m ¼�T0

m0, Tm � �m, we find that a particle of matter, with

mass m, feels an additional, or fifth, force ~F�, where

~F � ¼ �m

��ð�Þ þX

i

�i�i

�~r�

¼ �m½�ð�Þ þP

i �i�i�kð�Þ

~r’; (11)

where �i ¼ Si=Tm ¼ �Si=m. When �i�i=� � 1, it fol-lows that, if � is approximately massless (over the scale ofinterest), then it mediates a force that is ð�Þ ¼ �2=k2ð�Þtimes the strength of gravity; thus,

ð�Þ ¼ �ð�Þ2��2 þ 3�2ð�Þ : (12)

Hence when 3�2�2 � 1, the force is 1=3 the strength ofgravity (equivalent to an ! ¼ 0 Brans-Dicke theory),whereas in the opposite limit �2�2 � 1, we have ��2�2ð�Þ � 1=3.

We estimated that if, today, � lies near the minimum ofthe effective potential in the cosmological background,� � 0:23, and since we generally expect � to be Oð1Þ orlarger, � 0:05–1=3. We also noted that, unless �;�� �1, m’ �OðHÞ and the force will be long range. Solar

System limits on long-range fifth forces constrain &10�5, and so we must require that the theory possessessome mechanism so that, in the Solar System,

either the new force becomes of short range [i.e. �’ ¼m�1

’ � Oð1Þ AU], or the coupling ð�Þ is suppressed.

The former scenario, where the range of the force is sup-pressed, has been dubbed the chameleon mechanism [4–6]and relies heavily on the shape of the potential. We shallshow that the shape of Vð�Þ does not allow for a viablechameleon mechanism. On the other hand, the suppressionof ð�Þ is the essence of the Damour-Polyakov mecha-nism [8]. In Ref. [8], while ð�Þ is minimized by thecosmological evolution of �, at any given epoch, ð�Þdoes not exhibit very different values in different environ-ments; hence if fifth forces are suppressed locally, thenthey are also suppressed on cosmological scales. In thisarticle we propose a new mechanism inspired by that ofDamour and Polyakov, so that although ð�Þ is suppressedin environments, such as the Solar System, where theambient density is much larger than the average cosmo-logical density, cosmologically it is still possible forð�Þ �Oð1Þ, leading to non-negligible effective modifi-cations of gravity on large scales. In the following sectionswe investigate these points further.

III. LOCAL LIMITS ON DILATON MODELS

A. Local constraints

While it seems natural that the scalar field(s) in darkenergy theories should interact with ordinary matter, localtests of gravity tightly constrain any such coupling. If, inthe Solar System, a scalar field ’ couples to ordinarymatter with a strength times that of gravity and has amass m’, then when and m’ are both (at least approxi-

mately) constant, there is an additional force ~F’ between

bodies with separation r. When r is much larger than the

length scales of these bodies, ~F’ ¼ effðm’rÞ ~FN , where~FN is the usual Newtonian force, and eff ¼ ð1þm’rÞe�m’r.

When m’r � 1 and eff � , such a theory is approxi-

mately equivalent to Brans-Dicke theory with a Brans-Dicke parameter !BD ¼ 1=2eff � 3=2. Current trackingof the Cassini satellite provides the best limit on thisparameter: !BD * 40 000 [14,15]. In this case r was ap-proximately the orbital radius of Saturn: r � 9–10 AU.Thus,

local < 1:2 10�5 if mlocal�1’ * 9–10 AU: (13)

In Brans-Dicke models, matter is minimally coupled to ametric ~g�� ¼ A2ð�Þg��. If � appears nowhere else in the

matter action, this leads to a universal coupling betweenmatter and �. More generally, and particularly with thedilaton, we expect � will have additional couplings tomatter. For instance, the gauge coupling ‘‘constants’’ giare expected to depend on the value of the dilaton field.This leads to a nonuniversal coupling to matter, which inturn violates the equivalence of free fall i.e. the weakequivalence principle (WEP). There are tight laboratory

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constraints on any violation of WEP, which are parame-trized in terms of �b�c:

�b�c ¼ 2j ~ab � ~acjj ~ab þ ~acj ; (14)

where ~ab and ~ac are, respectively, the accelerations of testmass B and test mass C towards some third body (which isgenerally the Sun). Interpreting a given limit on �b�c interms of a limit on individual composition-dependent cou-plings is not entirely straightforward, as there is somedegeneracy between the different potential couplings, andthe result depends on the composition of the test masses.

In our model we have composition-dependent couplings�i which, respectively, couple to a fraction �i of a body’smass. In addition, there is the universal coupling � whichcouples to all of the mass. When m�r � 1, the force

between a body b and the Sun is therefore a factor �b

times that of gravity, where

�b ¼ 1

k2ð�ÞXi;j

½�þ �i�i�½�þ �j�bj�; (15)

with k2 ¼ 3�2 þ ��2. There is a similar expression for theforce between a body c and the Sun, defining �c. Wethen have

�b�c ¼ j�b � �cj

¼��������

1

3�2 þ ��2

Xi;j

ð�þ �i�iÞ�j�ðb�cÞj��������;

where �ðb�cÞj ¼ �bj � �cj.

The tightest current constraint on WEP violation wasfound by Schlamminger et al. [16], who measured thedifferential acceleration of two test masses composed,respectively, of Be and Ti towards the Sun and found

� ¼ ð0:3� 1:8Þ 10�13: (16)

In Ref. [17], the dependence on the �i was calculated. Forsimplicity, we assume that only the fine-structure constantem and �QCD vary and that the lepton and quark masses

are fixed. We take �� ¼ � ln�=�� and � ¼ � ln=��.For a substance composed of atoms with baryon number

A and lepton number Z, we have

� � 2:7 10�4 þ ð7:6 10�4ÞZðZ� 1ÞA4=3

;

�� � 0:95þ ð1:7 10�2ÞA�1=3:

We approximate the composition of the Sun as 75% hydro-gen and 25% helium-4, and have

� � 3:0 10�4; �� � 0:97:

The Schlamminger et al. constraint [16] gives

ð�0:72� þ 1:7��Þð�þ ð3 10�4Þ�Þ þ 0:97��Þ< 10�10ð3�2 þ ��2Þ:

Nowffiffiffiffi

p ¼ �=kð�Þ, and locally the Cassini limit gives

�=k < 3:5 10�3; hence 3�2 � ��2 and so k � ��1

locally. Assuming that �=k is only just below this limit,the WEP violation constraint would require

� 0:72� þ 1:7�� & 3��1 10�8;

i.e. unless there is some exact cancellation �i � � locally.In many cases, we would expect that �� ��. However,this is not necessarily the case, and, indeed, we shall seethat in our proposed model there would be no such link.The limit from WEP violation applies when m�1

� * 1 AU.

Local limits on the matter-to-scalar-field coupling aretherefore particularly stringent, limiting it to be� 1 unlessthe scalar field is sufficiently heavy. The Cassini and WEPviolation bounds are evaded if the field has a Comptonwavelength shorter than about AU�1, but a variety ofdifferent tests (for an excellent review see [15]) requireany gravitational strength fifth forces in the Solar Systemto have a Compton wavelength of no more than about0:1 mm�1.

B. Chameleon mechanism for the dilaton

Let us recall that dark energy models suffer from gravi-tational problems when coupled to ordinary matter. TheCassini bound is extremely stringent and implies that thedark energy scalar should be almost decoupled from mat-ter. Of course, no known well-motivated model of darkenergy constructed so far satisfies this constraint. In gen-eral, couplings tend to be of order one (for couplings withmuch larger values and their phenomenology, see [6]). Thiswould rule out most dark energy models. When the darkenergy models are scalar-tensor theories with coupling tomatter via the Jordan frame metric ~g�� ¼ A2ð�Þg��, the

chameleon mechanism can alleviate these gravitationalproblems. This mechanism uses two ingredients. The firstone springs from the coupling of the dark energy field tomatter and the resulting effective potential

Veffð�;mÞ ¼ �24ðVð�Þ þ mAð�ÞÞ=2; (17)

h� ¼ �24

2Veff;�ð�;mÞ; (18)

where is the pressureless matter density. Notice that for atypical runaway potential Vð�Þ, the matter contributioncan induce a matter-dependent minimum of the potentialwhen Að�Þ is an increasing function. The minimum �min

satisfies

V;�ð�minÞ ¼ �mA;�ð�minÞ: (19)

At this minimum, the field becomes massive, with a mass

m2� ¼ Veff;��ð�;Þ ¼ �2

4

2½V��ð�minÞ þ mA;��ð�minÞ�:

(20)

Assuming that Vð�Þ and Að�Þ are two convex functions,this mass is guaranteed to be bounded from below,

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m2� � �2

4

2mA;��ð�minÞ; (21)

which is a -dependent quantity. As m increases, thevalue of the minimum �min decreases. If the couplingfunction is such that mA;��ð�minÞ increases with m,

then the field � may be massive enough to evade thegravitational tests in a dense environment. Typically, onerequires that the mass of the scalar field should be largerthan 10�3 eV. This possibility can only be envisaged inrelatively dense media such as the atmosphere. It would notbe operative in a sparse environment such as the SolarSystem vacuum. In this case, the scalar field may still beresponsible for a distortion of planetary trajectories.Fortunately, another mechanism can be at play: the thinshell effect. Let us consider a spherical body and theinfluence that the scalar field generated by this body canhave on particle trajectories. Outside the body, the forcedue to the scalar field is given by

~F � ¼ ��ð�Þm ~r�; (22)

wherem is the mass of the test particle and� is the value ofthe coupling where the force is evaluated [we have ne-glected composition-dependent couplings; see Eq. (11)].When the dark energy field is massless, the induced force is

�2 ~FN , where ~FN is the Newtonian force. This leads to theCassini constraints we have already mentioned. For achameleon field with a density-dependent mass, the effectis drastically different. Indeed, the field � is mostly con-fined to vary within a thin shell close to the surface of thespherical body. The resulting force on a test particle isgiven by

~F � ¼ 3�R

R�2 ~FN; (23)

where we have assumed that �ð�mÞ is a very slowly

varying function of �m. The size of the shell is given by

�R

R¼ j�1 ��cj

3��NðRÞ ; (24)

where �1 is the value of the field at infinity and �c itsvalue in the center of the spherical body, and we havedefined the Newton potential at the surface of the body�NðRÞ. We can see that the external force due to the scalarfield is negligible, provided the thin shell exists, and there-fore

j�1 ��cj � 3��NðRÞ: (25)

This criterion can be easily applied to the case of dilatonicmodels.

Let us consider that the coupling function is an expo-nentially increasing function

Að�Þ ¼ e�0�: (26)

The normalized field ’ is nothing but a rescaling of�. It is

easy to see that the effective potential has a minimum:

�min ¼ 1

1� 3�0

ln

� ~V0

1� 4�0

�0

�; (27)

where �0 � 1=4. In a spherical situation, we have that

j�1 ��cj ¼ 1

1� 3�0

��������ln1c

��������: (28)

In the Solar System, the density inside the Sun does notexceed c � 1 g cm�3, while the vacuum is such that1 � 10�23 g cm�3. Now the Newtonian potential at thesurface of the Sun is of order�NðRÞ � 10�9, implying thata thin shell is not present. Hence this type of coupling forthe dilaton does not lead to a chameleon mechanism, andgravity would be greatly modified as ¼ �2

0=k2, unless�0

is essentially zero, to evade gravitational problems.All in all, we have seen that the thin shell mechanism

does not exist for a dilaton when the coupling � is nearlyconstant. Another way of satisfying the minimum equationis to compensate the large variations of the matter densityby an equivalently large variation of the coupling function�. In this case, gravity tests can be evaded using a differentmechanism: the Damour-Polyakov effect, whereby thecoupling � becomes density dependent and effectivelyvanishes in a dense environment.

C. The Damour-Polyakov mechanism

Damour and Polyakov [8] proposed a mechanismwhereby the coupling �ð�Þ would naturally be moved tosmall values by the expansion of the Universe. The presentsmallness of �ð�Þ would then be a natural consequence ofthe age of the Universe.The Damour-Polyakov mechanism assumes that Að�Þ

has a minimum at some � ¼ �0. Hence near � ¼ �0 wehave

Að�Þ � A0

�1þ A2

2ð���0Þ2

�: (29)

We may always rescale A and the metric to fix A0 ¼ 1.Near the minimum where A2ð���0Þ2=2 � 1, wethen have �ð�Þ ¼ ðlnAÞ;� � A2ð���0Þ. Thus the field

is sufficiently near the minimum ¼ �ð�Þ2=k2ð�Þ ��2A2

2ð���0Þ2 � 1 and so fifth-force effects are sup-pressed. To suppress WEP violation effects one wouldalso have to ensure that the �i are � 1 near � ¼ �0.The original model assumes a negligible potential, but itwould continue to work well provided that Vð�Þ is alsominimized at �0. In either case, whatever value A2 takes,the cosmological evolution of � will drive � towards �0.If, however, as in our case, we have a potential Vð�Þ ¼A4ð�ÞV0e

�� which represents dark energy today, Vð�Þ isthen non-negligible and the exponential part of this poten-tial means that Vð�Þ is not minimized at � ¼ �0.Let us assume that at early times the matter coupling

dominates the evolution of �, and � is driven close to �0,

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with j���0j � 1. Then

V � Vc

�1� ð���0Þ þ 1þ 4A2

2ð���0Þ2

þOð���0Þ3�; (30)

where Vc ¼ V0 expð��0Þ. Requiring that Vð�Þ drives thepresent period of acceleration expansion, we have �2

4Vc �3��0H

20 .

Now since j���0j � 1 we have that ¼�2ð�Þ=k2ð�Þ ¼ 1=ð3þ 1=�2�2Þ, and to expect � 1 atlate times, we must have 3�2�2 � 1 so kð�Þ � ��1. Itfollows from d’ ¼ kð�Þd� that ’ ¼ �=�.

Defining �� ¼ ���0 and assuming A2��2=2 � 1,

the field equation for �� becomes

�� €�� 3H� _� � ��2�24

2Vce

���

þ �2�24

2½m þ 4Vce

����A2��: (31)

Now �ð�Þ ¼ A2��, and for consistency with local tests,we must require j�j � 1. Therefore, assume that at the endof the matter era, the cosmic evolution has ensured j�j �1. In the current epoch Vð�Þ � Vce

�� �OðmÞ, so thesecond term on the right-hand side of Eq. (31) is roughly afactor � smaller than the first, and hence initially ��would evolve according to

� � €�� 3H� _� � ��2�24

2Vce

���: (32)

If �ð�Þ remains � 1, then since �24Vc=2�OðH2Þ today,

�� would move by roughlyOð�2Þ, which is expected to beOð1Þ or greater, so finally we expect j��j * Oð1Þ. Forj�j ¼ jA2��j � 1 to remain the case, we would have torequire A2 � 1. With A2 small enough to satisfy con-straints from fifth-force tests, the resulting theory would,essentially, evolve just like an uncoupled quintessence fieldin an exponential potential; we would also have to enforcethat �2 is small enough that the equation of state of thisquintessence is approximately �1, as is observed.

The condition A2 � 1 follows directly from the assump-tion that j�ð�Þj � 1 cosmologically at the present time forconsistency with local tests. However, local tests onlyimply a small value of j�j locally i.e. in the SolarSystem. If the local and cosmological values of � are notequal, we would not necessarily need to require that j�j �1 and hence A2 � 1 cosmologically today. Additionally, if��Oð1Þ cosmologically while � � 1 locally, we wouldhave interesting modifications to gravity emerging on largescales. We now show that such a mechanism exists and iscompatible with local tests if A2 � 1. This is essentially anenvironmental-dependent version of the standard Damour-Polyakov mechanism, where the coupling is minimized bythe fact that the local matter density is much greater thanthe cosmological one.

IV. ENVIRONMENTALLY DEPENDENT DILATON

In this section we propose a dilaton theory whereby thedilaton acts as the dark energy particle and is consistentwith local tests but leads to non-negligible deviations fromgeneral relativity on astrophysical scales. In common withchameleon theories, the properties of the dilaton fieldexhibit an environmental dependence which ensures that,in high density environments, dilaton mediated fifth forcesare negligible while allowing them to be of gravitationalstrength in low density regions. In chameleon models, thestrength of the scalar-field-to-matter coupling is roughlyconstant, but the mass grows with the ambient density. Thisleads to a very short range fifth force in the laboratory but arelatively long-range force in the cosmological back-ground. In our environmentally dependent dilaton model,however, the scalar is generally light locally (i.e. there is along-range fifth force), but the matter coupling itself isminimized in high density regimes. This may be seen asan environmentally dependent analogue of the Damour-Polyakov mechanism. Instead of the coupling tending tozero at late times, the coupling decreases as the densityincreases. We therefore refer to the mechanism by whichthe coupling is suppressed locally as the environmentallydependent Damour-Polyakov (EDDP) mechanism.

A. The model

We consider a dilaton model described in the Einsteinframe by the action given in Eq. (1). In common with thestandard Damour-Polyakov mechanism, we assume that

Að�Þ � 1þ A2ð���0Þ2=2;when A2ð���0Þ2=2 � 1. We check that this quantity isindeed small later. It follows that

�ð�Þ � A2ð���0Þ; (33)

kð�Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3A2

2ð���0Þ2 þ ��2q

: (34)

We take A2 � 1. We shall show below that A2 � 1 isactually required for the coupling to be sufficiently sup-pressed locally as to evade fifth-force constraints.The largeness of the required value of A2 certainly

appears unnatural on a first inspection. It is, however,important to note that there is already a large parameter

in this theory, namely, c1 ¼ e�c ð�0Þ � Mpl=Ms. It is fea-

sible that the largeness of A2 is linked to the discrepancybetween the string and 4D Planck scales. For instance, letus define ¼ Mpl�, so that has the canonical units of

mass. Then

A � 1þ A2

2M2pl

ð � 0Þ2 ¼ 1þ A2

2c21

� � 0

Ms

�2:

If the minimum in A at � ¼ �0 ( ¼ 0) is due to somenonperturbative effect associated with the string mass

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scale, Ms, one would actually expect the coefficient ofðð � 0Þ=MsÞ2 to be Oð1Þ, and so A2 �Oðc21Þ � 1.Thus, although demanding A2 � 1 is certainly not asaesthetically pleasing a requirement as A2 �Oð1Þ, it isnot necessarily unnatural if it is associated with the factthat Ms � Mpl.

The effective potential Veff is given by Veff ¼V0A

4ð�Þe�� � Að�ÞTm. Thus in a background where thematter is nonrelativistic so Tm � �m, this effective po-tential is minimized when �ð�Þ½m þ 4A3ð�ÞV0e

��� ¼A3ð�ÞV0e

��. Assuming Að�Þ � 1, this is achieved when

�ð�minÞ ¼ A2ð�min ��0Þ � V0e��min

m þ 4V0e��min

� 1=4:

(35)

We note that as m ! 1, �ð�minÞ ! 0.Now if A2 � 1, �min �� � 1=4A2 � 1, and so we

may replace V0e��min by V0e

��0 ¼ Vc. Additionally,

A2

2ð�min ��0Þ2 � 1

32A2

� 1;

for all . The assumption that A2ð���0Þ2=2 � 1 andAð�Þ � 1 is therefore valid because A2 � 1.

Finally, with A2 � 1, we have that the dominant con-tribution to the mass, m’, of small perturbations in the

scalar field about its effective minimum �minðÞ in abackground with density m and � ¼ �min is given by

m2’ð�minÞ � �2

4A2

2k2ð�minÞ½m þ 4Vc�; (36)

where k2ð�minÞ ¼ 3�2ð�minÞ þ ��2 < ��2 þ 3=16. In abackground where � ¼ �b, and over scales r � 1=m’,

the fifth force mediated by � between two point masses is� times the strength of the particles’ mutual gravitational

attraction, where

�ð�bÞ ¼ �2ð�bÞk2ð�bÞ

¼ �2ð�bÞ3�2ð�bÞ þ ��2

:

Hence whenffiffiffi3

p�ð�bÞ�2 � 1, � � 1=3, and in the op-

posite limit � � �2�2 which is minimized as � ! 0 i.e.

� ! �0. We note that the value of� at the minimum of theeffective potential tends to zero as the ambient density ofmatter, m, grows large: � / 1=m. Thus as m ! 1,� / 1=2

m ! 0.

With j���minj � 1 and Að�Þ � 1, �ð�Þ �A2ð���0Þ, and Vð�Þ ¼ V0e

�� � V0e��min ¼ Vc, the

field equation for � in a general background is then

h’ ¼ �24

2kð�Þ ½�Vc þ �ð�Þð4Vc � TmÞ�; (37)

where d’ ¼ kð�Þd�, kð�Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3�2ð�Þ þ ��2

s

p.

B. Background cosmological behavior

We now consider the large scale, homogeneous andisotropic cosmological behavior of � in this model. Wetake the background metric to be that of Friedmann-Roberston-Walker (FRW) spacetime with dust matter soTm ¼ �m:

ds2 ¼ �dt2 þ a2ðtÞ½dr2 þ fkðrÞ2d�2�; (38)

where fk ¼ sinð ffiffiffik

prÞ= ffiffiffi

kp

. We define �m and �� by�24m ¼ 3�mH

2 and �24Vc ¼ 3��H

2. With these defini-tions the value of � at the minimum of the effectivepotential is �min, where

�ð�minðtÞÞ ¼ A2ð�minðtÞ ��0Þ ¼ ��ðtÞ�mðtÞ þ 4��ðtÞ :

(39)

The parameters are such that the dilaton is consistent withbeing dark energy. We take A2 � 1 (which we shall see isrequired by the local test below). The field equation for �in this background becomes

� €’� 3H _’ ¼ � �24Vc

2kð�Þ þA2ð���0Þ

kð�Þ ½m þ 4Vc�:(40)

This is solved approximately by

� � �minðtÞ:This is the valid leading order approximation to the solu-tion, provided m2

’ � H2, where m’ is the mass of small

perturbations in ’ about the minimum of the effectivepotential. We have

m2’

H2� 3A2

2ð�m þ 4��Þ

���2 þ 3

��m

��

þ 4

��2��1

:

(41)

With ��Oð1Þ or greater, and �m þ�� �Oð1Þ, it isclear that typically m2

’=H2 �OðA2Þ or larger. Since A2 �

1, it follows that m’ � H as required. Thus �ð�Þ ��ð�minÞ cosmologically. Since �ð�minÞ � 1=4, it followsthat ���0 < 1=4A2 � 1, justifying the assumption thatj���0j � 1.Additionally, since m2

’ � H2, the background cosmol-

ogy is observationally indistinguishable from that of a�CDM model with � ¼ �2

4Vc. This said, provided A2 isnot too large, we shall see in Sec. V that the force mediatedby � is still sufficiently long range to have a detectableeffect on the formation of large scale structures. For linearperturbations in the matter density and over spatial scalessmaller than m�1

’ , this new force is cos times that of

gravity, where

cos ¼ �2ð�minÞk2ð�minÞ

¼ 1

3þ ��2ð4þ�m=��Þ2: (42)

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Taking �m ¼ 0:27 and �� ¼ 0:73, we have �m=�� �0:37 today, and taking � * 1, we have 0:045 & cos < 1=3today. The larger values of cos correspond to the largervalues of �. Thus cos is much larger than the Cassini limiton the local value of (i.e. 10�5).

We note that we have cos � 1=3 if � � 4:4. Thecosmological value of � today, �cos, is

�cos ¼ 1

4þ�m=��

� 0:23: (43)

Therefore, on scales much smaller than the horizon butpotentially cosmologically still important (depending onthe size of A2), the scalar field mediates a force with astrength slightly smaller than gravity. This will influencestructure formation on those scales.

C. Local behavior and constraints

Previously we found that �ð�minÞ ! 0 as m ! 1.

We now show that, provided A2 is large enough, thepresence of our Galaxy is enough to reduce phi from its

cosmological value cos �Oð1Þ to a local value local thatis small enough to evade the Cassini limit and other localtests (local < 10�5).

First, since d’ ¼ kð�Þd� ¼ kð�Þd�=A2, we can re-write Eq. (37) (with Tm ¼ �m) in terms of �:

h�þ 3�

3�2 þ ��2ðr�Þ2

¼ 4�GNA2

3�2 þ ��2½�Vc þ �ðm þ 4VcÞ�: (44)

Defining X ¼ 3�2=2þ ��2 ln�, so that dX ¼ð3�þ ��2=�Þd�, one can rewrite the last equation fur-ther:

hX þ ��2ðr ln�Þ2 ¼ 4�GNA2

��Vc

�þ ðm þ 4VcÞ

�:

(45)

1. Effect of a local overdensity

Consider the perturbation in X, �X, created by a quasi-static subhorizon perturbation in m, �m. The quasistaticand subhorizon conditions imply that the time scale overwhich �m evolves is much longer than the typical lengthscale of the perturbation. It also means that we can take

hX ! ~r2�X. Far from the perturbation we take� ! �1,

! 1, � ! �1 ¼ A2ð�1 ��0Þ.We set m ¼ 1 þ �m so that, far from the perturba-

tion, �m ! 0, and X ! X1 ¼ 3�21=2þ ��2 ln�1. Wedefine the shorthand notation �ðrÞ ¼ �ð�ðrÞÞ. We alsoassume that, far from the perturbation, � lies close to theminimum of its effective potential so that

�1 ¼ Vc

1 þ 4Vc

:

Writing � ¼ �1 þ ��, X ¼ X1 þ �X, and without lin-earizing, the field equation becomes

~r 2�X ¼ 4�GNA2�m þ 4�GNA2Vc

���

�1�

�

� ��2ð ~r ln�Þ2: (46)

We now simplify to the case where �m represents aspherically symmetric overdensity of matter, which is non-decreasing as r ! 0 i.e. d�mðrÞ=dr � 0 and hence�mðrÞ � 0. The effect of the perturbation is to move X,and hence�, to smaller values i.e.�ðrÞ<�1. This impliesthat d�mðrÞ=dr � 0 and d�=dr � 0. We now construct anecessary condition for�ðrÞ to be smaller than some� for�m � 0.From � � �1 we have

ð3�21 þ ��2Þð��Þ=� � �X � 0

and so

4�GNA2Vc

�1��

�� 4�GNA2½1 þ 4Vc�

3�21 þ ��2�X� m21�X � 0;

where this defines m21. It is clear that ���2ð ~r ln�Þ2 � 0,and so we have � �X < �X � 0, where

~r 2� �X ¼ 4�GNA2�m:

The perturbation in the Newtonian potential, ��N , due to�m is given by

~r 2��N ¼ 4�GN�m;

where ��N ! 0 as r ! 1. Hence we have � �X ¼ A2��N .�ðrÞ<� is equivalent to XðrÞ< X ¼ 3�2 þ ��2 ln� .It follows that a necessary condition for �ðrÞ<� is that� �X ¼ A2��N < X � X1:

3ð�21 � �2 Þ þ ��2 ln

��21�2

�< 2A2j��NðrÞj: (48)

In such a setup, we also have

�ðrÞ � �minðrÞ ¼ Vc

1 þ �ðrÞ þ 4Vc

:

Hence for �ðrÞ<� it is also necessary that

�minðrÞ ¼ Vc

1 þ �ðrÞ þ 4Vc

< � : (49)

Equations (48) and (49) are necessary but not sufficient toensure that �ðrÞ<� . Given a limit on local, it does,however, allow us to place a lower bound on A2 given �.It should be noted that although Eqs. (48) and (49) are onlynecessary conditions, when d=dr � 0, the combinationof the two conditions is often almost sufficient; i.e. if� �X � X � X1 and �minðrÞ � � , then typically �ðrÞ<� .

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2. Application to our Galaxy

Laboratory, satellite, and other Solar System (collec-tively ‘‘local’’) tests of gravity certainly take place insidea sizable overdense region, namely, our Galaxy. The matterin our galaxy has the effect of moving � to a smaller valuelocally than it has cosmologically. There will also be addi-tional contributions to the local value of �� coming fromthe other members of the local group and local supercluster. For the purpose of the paper though, we take acautious approach and are only concerned with how big A2

must be so that the presence of the (roughly spherical) darkmatter halo of the Galaxy is, alone, enough to ensurecompatibility with local tests. Specifically, we focus onthe Cassini limit local < 10�5.

We consider the idealized situation where our Galaxy’sdark matter halo is a spherically symmetric overdensity� ¼ galðrÞ, where dgal=dr � 0. Assuming that our

Galaxy sits in a region of cosmological density, 1 ¼todaycos ¼ 3�m0H

20=2�

24, and hence �1 ¼ �

todaycos � 0:23.

We assume that all local tests take place at a galactocentricradius r, so local ¼ ð�ðrÞÞ.

We take �gal to be described by a Navarro-Frenk-White

(NFW) profile with core radius rc and virial mass Mv:

�24�mðrÞ ¼ 2GMvgðcÞ

r3c

1

xð1þ xÞ2 ; (50)

where x ¼ r=rc; gðcÞ ¼ ½lnð1þ cÞ � c=ð1þ cÞ��1. Here,the virial mass is defined as being the mass inside r500,which itself is defined as the radius inside which theaverage density is 500 times that of the cosmologicalbackground; the concentration parameter c is r500=rc.

From this we have

��NðrÞ ¼ �GMvgðcÞr

ln

�1þ r

rc

�: (51)

Now local ¼ �2ðrÞ=ð3�2ðrÞ þ ��2Þ and so local � 1requires �2ðrÞ � ��2=3 in which limit local ��2ðrÞ=�2. Thus, using Eq. (48), for local < 10�5 it isnecessary that

A2 > 106½0:08þ 4��2s þ 0:5��2

s ln�2���������1:0 10�6

��NðrÞ��������:(52)

Typically values ofMv, c, rs, and r from fitting measuredrotation curves to a NFW profile [18,19] are

Mv ¼ 0:91þ0:27�0:18 10�12M; c ¼ 12:0� 0:3;

rvir ¼ 26724�19 kpc; r ¼ 8:0� 0:5 kpc;

which give

��NðrÞ � 0:75–1:4 10�6;

galðrÞ � 0:15–0:37 GeV cm�3:

Other estimates give galðrÞ � 0:1–0:7 GeV cm�3.

Putting this value into Eq. (49), and requiring local <10�5, then gives

� & ð0:77–5:4Þ 102: (53)

We find more accurate limits on A2 by numericallyintegrating the field equations for � with a NFW matterprofile. The constraints depend on r=rc and GMvgðcÞ=rc.We take typical values r=rc ¼ 0:37 (corresponding tor ¼ 8:3 kpc, rvir ¼ 267 kpc, and c ¼ 12:0), andGMvgðcÞ=rc ¼ 1:2 10�6 (corresponding to Mv ¼0:91 1012M); this gives �ðrÞ ¼ 1:02 10�6 andðrÞ ¼ 0:22 GeV cm�3. This value for ðrÞ limits � <170. For these values of the parameters, the constraints onA2 for � 2 ½1; 170� for these are plotted in Fig. 1. For � �1, the analytic necessary (but not sufficient) lower boundon A2 is only a factor� 1:3 smaller than the necessary andsufficient lower bound found by numerically integratingthe equations. For Oð1Þ values of �, the discrepancy in-creases to roughly a factor of 2. Nonetheless, it is clear thatthe lower limit on A2 from the necessary condition ofEq. (48) is generally within a factor of a few of thenecessary and sufficient limit on A2. While the upper limiton � is sensitive to the local halo density, very similar

100

101

102

10−1

100

101

102

Value of λ

Val

ue o

f A2

× 10

−5

Allowed Parameter Space for Environmentally Dependent Dilaton

Unshaded region violates Cassini bound

Allowed Region

Lower bound on A2

× 10−5

from necessary condition

FIG. 1 (color online). Allowed parameter space for the envi-ronmentally dependent dilaton model. The shaded region iswhere the presence of our Galaxy is sufficient to ensure thatthe local value of the fifth-force coupling is smaller than theCassini probe upper bound of 10�5. We have modeled theGalaxy as a spherical dark matter halo with a NFW profile.We have taken typical values for the NFW model parameters forour Galaxy: rvir ¼ 267 kpc, c ¼ 12:0, Mv ¼ 0:91 1012M.We take the galactocentric radius of the Solar System, r, tobe r � 8:3 kpc. These choices correspond to �ðrÞ ¼ 1:0210�6 and ðrÞ ¼ 0:22 GeV cm�3. This value for ðrÞ limits� < 170, and we have plotted the constraints on A2 for � 2½1; 170�. Very similar bounds on A2 result for different realisticmodels of the Galactic halo.

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limits on A2 for given � are found for different realisticvalues of the NFW parameters.

We briefly comment on limits from WEP violation. Wefound in Sec. III that if local is only just below the Cassinilimit of� 10�5, then the WEP violating couplings � and�� would have to be less than 3 10�8��1. In our model�i / ~Vð�Þ / expð��Þ. In such a model ~Vð�Þ �M4 e��,whereM is typically of the same order of magnitude as theother mass scale in the high energy theory. Today ~Vð�Þ �M4

� � M4 (whereM� ¼ 2:4� 0:3 10�3 eV). This sup-pression is achieved by having � � 1 so expð��Þ � 1.Thus the same mechanism that suppresses the mass scaleof the effective cosmological constant would suppress theWEP violating couplings and naturally result in them hav-ing small values. For instance, �i & 10�10 would onlyimply expð��Þ � 10�10 and so M * 0:76 eV. If M ��QCD � 100 MeV, say, then we would expect �i �expð��Þ � 10�43. It is therefore natural for WEP viola-tion to be suppressed in this model.

In the previous subsection, we found [see Eq. (41)] thatthe cosmological mass squared m2

’ of small perturbations

in the dilaton field was proportional to A2, as well as havingsome dependence on �. We can now transform our lower

bounds on A2 to upper bounds on the range �cos ¼@c=mðcosÞ

’ of the dilaton mediated fifth force in the cosmo-logical background. The allowed values of �cos today (inunits of Mpc) are shown in Fig. 2. We see that for values of� and A2 allowed by local tests, �cos & 0:5–2:2 Mpc,which is similar to the scale of galaxy clusters today.This means that it is possible for an environmentallydependent dilaton field to simultaneously obey local con-straints and have a non-negligible effect on the formation

of large scale cosmological structures. We investigate thispossibility further in the following section.

V. LINEAR STRUCTURE FORMATION

We found above that the cosmological range �cos of thedilaton mediated fifth force was only constrained to be &0:5–2 Mpc today, depending on the value of �. Meanwhile,the strength of this force (relative to gravity), cos, onscales smaller than �cos was found to lie in the range0.04–0.33, for � * 10, cos * 0:3, and �cos * 1 Mpc.For such values of � we would, in the cosmological back-ground, have a fifth force of strength similar to that ofgravity propagating over a range that could be as large asthat of the typical length scales of clusters of galaxies [i.e.Oð1Þ Mpc]. Such a force would alter the formation of largescale structures in a manner that could be detected byongoing and future galaxy surveys. In this section we detailthe effect of our dilaton model on the structure formation inthe linear regime. The environmentally dependent natureof the model makes the study of nonlinear structures con-siderably more complicated than in the standard case. Wetherefore postpone a detailed analysis of nonlinear struc-ture formation in this model to future work.We define the Einstein frame energy momentum tensor

T�m� ¼ A3 ~T�

�, where ¼ ½ðlnAÞ;��2. The full field equa-

tions are then

h’ ¼ �24

2kð�Þ ½�Vð�Þ � A0ð�ÞT�m� þ 4�ð�ÞVð�Þ��

� Veff;’ð�; T�m�Þ; (54)

r�ðAð�ÞT�m�Þ ¼ T

�m�r�Að�Þ; (55)

G�� ¼ Að�Þ�2

4T�m� � �2

4Vð�Þ��� þ 2r�’r�’

� ���ðr’Þ2; (56)

where d’ ¼ kð�Þd� and we have defined an effectivepotential

Veffð�; T�m�Þ ¼ Vð�Þ � Að�ÞT�

m�:

The matter content is taken to be pressureless dust T��m ¼

u�u�; u�u� ¼ �1. We consider the background FRW

cosmology and denote background quantities by an over-bar; i.e. the background matter density is �. Hence,

H2 ¼�_a

a

�2 ¼ �2

4

3½Að ��Þ �þ Vð ��Þ� þ _�’2

3; (57)

� €�’� 3H _�’ ¼ Veff;’ð ��; �Þ; (58)

� / a�3: (59)

The background mass for ’ is given by m2’ ¼

Veff;’’ð ��; �Þ. Previously we found that the requirement

100

101

102

0

0.5

1

1.5

2

2.5

Value of λ

Cos

mol

ogic

al R

ange

of

Fift

h F

orce

: λco

s (in

Mpc

)

Allowed Values of the Cosmological Fifth Force Range: λ cos

Allowed Region

Ruled out by local tests

FIG. 2 (color online). Allowed values of the cosmologicalforce range �cos today, given compatibility with the Cassiniconstraint on local. Allowed values of � must be & 170 andare expected to be * 1, and so we show only the region � 2½1; 170�. We see that �cos & 0:5–2:2 Mpc today.

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A2 � 1 ensures that m2’ � H2, and so �’ (or equivalently

��) lies very close to the minimum of Veff . We define ��min

by Veff;�ð ��min; �Þ ¼ 0, and then �� � ��min. With A2 � 1,

m2’ � H2, we also have j _��j � H, and Að ��Þ � 1, and can

take �24Vð ��Þ � �. Here � is the effective value of the

cosmological constant. Thus,

H2 � �24

3�þ�

3; (60)

�� � �ð ��Þ ¼ ðlnAð ��ÞÞ0 � �

�24 �þ 4�2

4�¼ 1��m

4� 3�m

:

(61)

The mass squared m2’ is then given by Eq. (41). We define

� ¼ �2ð ��Þ=k2ð ��Þ ¼ ��2=ð3 ��2 þ ��2Þ.We now focus on the growth of linear perturbations in

the measured matter density Að�Þ around Að ��Þ �.Linearity here implies that �m ¼ �ðAð�ÞÞ=Að ��Þ �,j�mj � 1. In addition to the usual gravitational force,linear perturbations now feel an additional fifth forcewith range �cos ¼ 1=m’ and strength �. Since m2

’ �H2, the evolution of the background cosmology is, how-ever, �CDM to a very good approximation.

Galaxies represent nonlinear perturbations in the matterdensity and have densities much greater than �, and thematter coupling decreases as 1=2 as increases.Compatibility with local tests requires that the fifth-forcecoupling inside our Galaxy be greatly suppressed com-pared with its cosmological value. It is feasible that insmaller galaxies, the fifth force would be less suppressed;however, for our purposes here we assume that, on average, inside galaxies is much smaller than �, and so we treatgalaxies as being essentially uncoupled to the dilaton fifthforce.

Galaxies are often used as observational tracers of thelinear cold dark matter perturbation. The latter feels bothgravity and the fifth force, while the former only evolvesunder gravity. This is very similar to the scenario weconsidered in Ref. [20]. Using the calculations presentedin Ref. [20], and moving to Fourier space, we define �gðkÞand �mðkÞ (�A and �B, respectively, in Ref. [20]) to be,respectively, the linear density perturbations with comov-ing wave number k in the average density of galaxies and inthe average density of all pressureless matter. UsingRef. [20] and with p ¼ lna, we have

�g;ppðkÞ þ�2� 3�m

2

��g;pðkÞ ¼ 3

2�m�m; (62)

�m;ppðkÞ þ�2� 3�m

2

��m;pðkÞ

¼ 3

2½1þ effð�m; am’=kÞ��m�m; (63)

effð�m; xÞ ¼ �ð�mÞ1þ x2

; (64)

ð�mÞ ¼�3þ ��2

�4þ �m0

a3ð1��m0Þ�2��1

; (65)

where�m0 andH0 are the values of�m andH today whena ¼ 1 (p ¼ 0).The observables are the growth rate fgal ¼

dðln�gÞ=d lna, the slip functions, ��m and ��I measured

by weak-lensing and integrated Sachs-Wolfe (ISW) effectmeasurements, respectively, and finally the indicator ofmodified gravity EG. The former two are defined in termsof the two metric potentials � and �:

ds2 ¼ a2ð�Þð�ð1þ 2�Þd�2 þ ð1� 2�Þdx2Þ: (66)

Weak lensing measures �þ�, and the ISW effect is

proportional to _�þ _�. Note that in this model � ¼ �in the Einstein frame. ��m and ��I are then given by

k2ð�þ�Þ ¼ �8�Ga2 ���mDGR�i; (67)

H�1k2ð _�þ _�Þ ¼ �8�Ga2 ���IðfGR � 1ÞDGR�i; (68)

where �i is the primordial density perturbation (measuredfrom the CMB), and DGR is the growth factor in generalrelativity (GR); �m ¼ DGR�i, fGR ¼ d lnDGR=d lna is theGR growth rate. In this model �m ¼ Dm�i and so ��m ¼Dm=DGR and ��I ¼ ðfm � 1Þ��m=ðfGR � 1Þ; fm ¼d lnDm=d lna ¼ d ln�m=d lna.We also define the linear bias corrected galaxy growth

rate. This is defined by assuming that �g ¼ Dbcgal�i þ �0,

where �i is the initial Gaussian perturbation, Dbcgal is the

bias corrected growth factor for galaxies, and �0 is thesource of the bias. We then have that �bc

g ¼Dbc

gal�i ¼ b�1lin ð�gÞ�g, where b�1

lin ¼ 1� �0=�g. �0 and

hence the linear bias is estimated directly from galaxysurveys using higher order statistics. We define fbcgal ¼d ln�bc

g =d lna. In the absence of any deviations from GR,

fbcgal ¼ fGR � �0:545m , and ��m ¼ ��I ¼ 1.

Finally, we consider the modified gravity sensitivityparameter defined in Ref. [21]:

EG ¼ k2ð�þ�Þ�3H2

0a�1�

;

where � ¼ � _�gal=H ¼ �d�gal=d lna, and so � ¼�fbcgalD

bcgal�i and with a ¼ 1 today,

EG ¼ �m0Dm

fbcgalDbcgal

: (69)

In GR, EðGRÞG ¼ �m0=fGR � �m0�

�0:545m . In all cases, we

find that gravity is modified below a certain redshift z approximately given by ð1þ z Þ � �1=3. Hence gravity isonly modified at low redshift for scales which are small

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enough, inside the Compton wavelength of the dilaton oncosmological scales.

Figures 3 and 4 show fbcgal=fGR � 1, ��m and �Im and

EG=EðGRÞG � 1 for � ¼ 10 and � ¼ 1, respectively, and for

different values of redshift z and inverse spatial scale k. We

see that the largest deviations from GR occur on scales k >

A1=22 H0 (roughly k > mcos), and at late times. The devia-

tions from GR also scale with cos and hence with �; for� � 1, cosðz ¼ 0Þ � 1=3, whereas for � ¼ 1, cosðz ¼0Þ � 0:04. Additionally, �Im, ��m and EG display more

pronounced deviations from their GR values than does fbcgal.

This is because galaxies are effectively decoupled from thedilaton mediated fifth force, whereas both�Im, EG and�Im

directly probe the growth rate of the large scale dark matterperturbation which feels the unsuppressed fifth force. For� ¼ 10, the largest deviations from GR occur in �Im atredshifts z� 1–2, where 1� �Im �Oð1Þ. In other pa-rameters the greatest deviations occur today (z ¼ 0),with fbcgal, ��m, and EG deviating from their GR values

by� 3:7%, 12%, and 6%, respectively. We note that these

10−1

100

101

102

103

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Linear Bias Corrected Galaxy Growth Rate, fgalbc, for λ = 10

Spatial Scale: (A21/2H

0)−1 k

Rel

ativ

e G

alax

y G

row

th R

ate:

f galbc

/f GR−

1

z=0z=1z=2

10−1

100

101

102

103

1

1.05

1.1

1.15

Weak Lensing Slip Parameter, Σκ m

, for λ = 10

Spatial Scale: (A21/2H

0)−1 k

Wea

k Le

nsin

g S

lip P

aram

eter

, Σκ

m

z=0z=1z=2

10−1

100

101

102

103

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Spatial Scale: (A21/2H

0)−1 k

ISW Slip Parameter, ΣI m

, for λ = 10

ISW

Slip

Par

amet

er, Σ

I m

z=0z=1z=2

10−1

100

101

102

103

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Spatial Scale: (A21/2H

0)−1 k

Modified Gravity Parameter, EG

, for λ = 10

Rel

ativ

e M

odifi

ed G

ravi

ty P

aram

eter

: EG

/EGG

R−

1 z=0z=1z=2

FIG. 3 (color online). Effect of the dilaton field on large scale structure formation. From top left to bottom right, the plots show thepredicted deviation of the (linear bias corrected) galaxy growth fbcgal from its GR value fGR, the predicted value of the slip parameter

��m extrapolated from weak-lensing measurements, the predicted slip parameter extrapolated from ISW measurements, �Im, and therelative deviation of the modified gravity parameter EG from its GR value. These plots are for � ¼ 10, and show the values of fbcgal, ��m

and �Im at the present day, z ¼ 0 (solid blue line), z ¼ 1 (dashed red line), and z ¼ 2 (dotted black line) for different values of theinverse spatial scale, k. We see that the largest deviations from GR are found in �Im, particularly at z� 1–2 and the smallest in thegrowth rate. This is because galaxies are effectively decoupled from the dilaton mediated fifth force, whereas �Im as well as ��m andEG directly probe the growth rate of the large scale dark matter perturbation which feels the unsuppressed fifth force.

BRAX et al. PHYSICAL REVIEW D 82, 063519 (2010)

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deviations are within present constraints but should bedetectable by future surveys of large scale structure.

VI. CONCLUSIONS

We have presented new results on dilaton models in thestrong-coupling regime. Our analysis combines a runawaypotential as suggested in [9] with a coupling to matter witha vanishing minimum, as investigated in [8]. In the absence

of any potential term, the coupling to matter is enough todrive the dilaton to its minimum cosmologically [8]. As aresult, the dilaton would evade gravitational tests on thenonexistence of fifth forces. This result is jeopardized bythe runaway dilatonic potential, in particular, if the dilatonplays the role of dark energy. In this paper we have focusedon the dilaton as a candidate for dark energy and shownhow this can be compatible with gravitational tests ofgravity, but could lead to observational tests at large scales.

10−1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3 Linear Bias Corrected Galaxy Growth Rate, fgalbc, for λ = 1

Spatial Scale: (A21/2H

0)−1 k

Rel

ativ

e G

alax

y G

row

th R

ate:

f galbc

/f GR−

1z=0z=1z=2

10−1

100

101

102

103

1

1.001

1.002

1.003

1.004

1.005

1.006

1.007

Weak Lensing Slip Parameter, Σκ m

, for λ = 1

Spatial Scale: (A21/2H

0)−1 k

Wea

k Le

nsin

g S

lip P

aram

eter

, Σκ

m

z=0z=1z=2

10−1

100

101

102

103

0.97

0.975

0.98

0.985

0.99

0.995

1

1.005

Spatial Scale: (A21/2H

0)−1 k

ISW Slip Parameter, ΣI m

, for λ = 1

ISW

Slip

Par

amet

er, Σ

I m

z=0z=1z=2

10−1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−3

Spatial Scale: (A21/2H

0)−1 k

Modified Gravity Parameter, EG

, for λ = 1

Rel

ativ

e M

odifi

ed G

ravi

ty P

aram

eter

: EG

/EGG

R−

1 z=0z=1z=2

FIG. 4 (color online). Effect of the dilaton field on large scale structure formation. From top to bottom the plots show the predicteddeviation of the (linear bias corrected) galaxy growth fbcgal from its GR value fGR, the predicted value of the slip parameter, ��m,

extrapolated from weak-lensing measurements, the predicted slip parameter extrapolated from ISW measurements, �Im, and therelative deviation of the modified gravity parameter EG from its GR value. These plots are for � ¼ 1, and show the values of fbcgal, ��m

and �Im at the present day, z ¼ 0 (solid blue line), z ¼ 1 (dashed red line), and z ¼ 2 (dotted black line) for different values of theinverse spatial scale, k. We see that the largest deviations from GR are found in �Im, particularly at z� 1–2 and the smallest in thegrowth rate. This is because galaxies are effectively decoupled from the dilaton mediated fifth force, whereas �Im as well as ��m andEG directly probe the growth rate of the large scale dark matter perturbation which feels the unsuppressed fifth force. We note that thedeviations from GR are much smaller for � ¼ 1 than for � ¼ 10 (see Fig. 3). This is because for � � 1, the cosmological fifth-forcecoupling cos � 1=3 today, whereas for � ¼ 1 it is only � 0:04.

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The model we have considered in this paper was moti-vated by the work of [8–11]. Models of dark energy with anexponential potential have been considered before; see[22–24] for earlier work and [1] for further referencesand details. The difference between the model in this paperand the one presented in [22] (i.e. the ‘‘cosmon’’ model),which is similar to dilatonic dark energy in the smallcoupling limit, is that in the model presented here the fieldis rather heavy and sits at the minimum of the effectivepotential, because the parameter A2 has to be large in orderto comply with local constraints. Furthermore and impor-tantly, the coupling to matter is constant in [22,23].Therefore, the Damour-Polyakov mechanism is not atwork in these types of theories, and the cosmologicalevolution is quite different from the one presented here.For example, wewould not expect a scaling behavior in ourmodel, as the coupling will be non-negligible and willinfluence the scalar-field evolution almost all the time [ifthe field is slightly displaced from the minimum of Að�Þ].Like in [22], we would expect a variation of the constantsof nature. However, the field variation throughout thecosmological history is very small. We therefore expectthe variation of constants to be small in our model, once wehave imposed the constraints from local experiments on thefree parameters of the model. Hence the results obtained

for the dilaton at strong coupling cannot be mapped ontothe ones for the cosmon field [22].We have shown that for models where the string scale is

lower than the Planck scale, the Damour-Polyakov mecha-nism, whereby the coupling to matter vanishes dynami-cally, is at play only locally in the presence of large enoughoverdensities such as the one present in galaxies. On largercosmological scales, the fifth force is not suppressed,implying the presence of a significant modification ofgravity. We found that this effect could be relevant ongalaxy cluster scales, where the growth of structures wouldbe affected. As a result, the future galaxy survey could givestringent constraints on dilaton models and could have thepotential to indicate the possible existence of a dilaton inthe strong-coupling regime. We also expect that the non-linear growth of structures would also be affected by thepresence of a dilaton. This is left for future work.

ACKNOWLEDGMENTS

We are grateful to G. Veneziano and M. Gasperini fordiscussions. We also thank G. Veneziano for comments onthe draft. D. S. is supported by STFC. C. v. d. B. andA. C. D. are partly supported by STFC.

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