Equivalence Principle Implications of Modified Gravity Models ()

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arXiv:0905.2966v1[astro-ph.CO

]18May2009

Equivalence Principle Implications of Modified Gravity Models

Lam Hui and Alberto NicolisInstitute for Strings, Cosmology and Astroparticle Physics (ISCAP),

Department of Physics, Columbia University, New York, NY 10027, U.S.A.

Christopher W. StubbsDepartment of Physics, Harvard University, Cambridge, MA 02138, U.S.A.

(Dated: May 18, 2009)Theories that attempt to explain the observed cosmic acceleration by modifying general relativity

all introduce a new scalar degree of freedom that is active on large scales, but is screened on smallscales to match experiments. We demonstrate that if such screening occurrs via the chameleonmechanism, such as in f(R) theory, it is possible to have order unity violation of the equivalenceprinciple, despite the absence of explicit violation in the microscopic action. Namely, extendedobjects such as galaxies or constituents thereof do not all fall at the same rate. The chameleonmechanism can screen the scalar charge for large objects but not for small ones (large/small is definedby the depth of the gravitational potential, and is controlled by the scalar coupling). This leads toorder one fluctuations in the ratio of the inertial mass to gravitational mass. We provide derivationsin both Einstein and Jordan frames. In Jordan frame, it is no longer true that all objects move ongeodesics; only unscreened ones, such as test particles, do. In contrast, if the scalar screening occurrsvia strong coupling, such as in the DGP braneworld model, equivalence principle violation occurrsat a much reduced level. We propose several observational tests of the chameleon mechanism: 1.

small galaxies should accelerate faster than large galaxies, even in environments where dynamicalfriction is negligible; 2. voids defined by small galaxies would appear larger compared to standardexpectations; 3. stars and diffuse gas in small galaxies should have different velocities, even if theyare on the same orbits; 4. lensing and dynamical mass estimates should agree for large galaxies butdisagree for small ones. We discuss possible pitfalls in some of these tests. The cleanest is the thirdone where mass estimate from HI rotational velocity could exceed that from stars by 30% or more.To avoid blanket screening of all objects, the most promising place to look is in voids.

PACS numbers: 98.80.-k; 98.80.Es; 98.65.Dx; 98.62.Dm; 95.36.+x; 95.30.Sf; 04.50.Kd; 04.80.Cc

I. INTRODUCTION

The surprising finding of cosmic acceleration about ten

years ago has motivated a number of attempts to modifygeneral relativity (GR) on large scales. They fall roughlyinto two classes. One involves adding curvature invari-ants to the Einstein-Hilbert action, the most popular ex-ample of which is f(R) theory [1, 2]. The other involvesgiving the graviton a mass or a resonance width, the mostwell known example of which is the Dvali-Gabadadze-Porrati (DGP) braneworld model [3, 4]. The first classof theories is equivalent to the classic scalar-tensor the-ories [5, 6]. The second class of theories is more subtlebut also reduces to scalar-tensor theories on sub-Hubblescales [7, 8]. In the context of DGP, the extra scalarcan be thought of as a brane-bending mode. In fact,

it appears the extra dimension is not even necessaryfour dimensional ghost-free generalizations of the DGPmodel exist where it is the nonlinear interaction of thescalar that causes self-acceleration [9].

All examples to date, therefore, introduce a light scalar

Electronic address: [email protected] address: [email protected] address: [email protected]

(or even multiple light fields) to modify gravity 1. Whilesuch a scalar is welcome phenomenologically on cosmo-logical scales, it must be suppressed or screened on smallscales to satisfy stringent constraints from solar system

and terrestrial experiments. The two classes of theorieshave quite distinctive screening mechanisms. Curvature-invariant theories such as f(R) screen the scalar viathe so-called chameleon mechanism, essentially by giv-ing the scalar a mass that depends on the local density[17, 18, 19]. Theories such as DGP screen the scalar alaVainshtein [16], namely strong coupling effects supressthe scalar on small scales [7, 8].

Our goal in this paper is to study how extended objectsmove in the presence of these two screening mechanisms.We will keep our discussion as general as possible even

1 This is not surprising given a famous theorem due to Weinberg[10] which states that a Lorentz invariant theory of a masslessspin two field must be equivalent to GR in the low energy limit.Low energy modifications of gravity therefore necessarily involveintroducing new degrees of freedom, such as a scalar. Giving thegraviton a small mass or resonance width turns out to introducea scalar a la Stueckelberg [11]. Theories that invoke a Lorentzviolating massive graviton also involve a scalar, namely a ghostcondensate [12, 13]. Recent proposals for degravitation are yetanother example of scalar-tensor theories in the appropriate lim-its [14, 15].
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as we use f(R) and DGP as illustrative examples. In-deed, neither theory is completely satisfactory: accept-able f(R) models accelerate the universe mostly by acosmological constant rather than by a genuine modi-fied gravity effect [20]; also from an effective field theorystandpoint they are not favored in any sense over moregeneral scalar-tensor theories; DGP and generalizationsthereof [9] probably lack a relativistic ultra-violet com-

pletion [21]. However, it is likely the screening mecha-nism, chameleon or strong coupling, will remain relevantin future improvements of these theories.

The problem of motion of extended objects has a vener-able history. Newton famously showed in Principiathatthe Earths motion around the Sun can be computed byconsidering the Earth as a point mass. It is worth em-pahsizing that such a result is by no means guaranteedeven in Newtonian gravity. This result holds only if thereis a separation of scales, that the dimension of the Earthis small compared with the scale on which the Suns grav-itational field varies, in other words that tides from theSun can be ignored. More concretely, the Earth effec-

tively sees a linear gradient field from the Sun; secondderivatives of the Suns gravitational potential can besafely neglected when computing the Earths motion.

We will work within the same zero-tide approximationin this paper. It has long been known that within GR,under the same approximation, extended objects move

just like infinitesimal test particles. In other words, ne-glecting tidal effects all objects move on geodesics, orequivalently, their inertial mass and gravitational massare exactly equal. This has been referred to as the efface-ment principle, that the internal structure of an object,no matter how complicated, has no bearing on its overallmotion [22].

It is also well known that the same is not true in scalar-tensor theories. In such theories, despite the universalcoupling of the scalar to all elementary matter fields, ob-

jects with different amounts of gravitational binding en-ergy will move differently. Effectively, they have differentratios of the inertial mass to gravitational mass, knownas the Nordtvedt effect [23].2 This apparent violationof the equivalence principle 3 is generally small becausemost objects have only a small fraction of their total masscoming from their gravitational binding energy. In otherwords, such violation is suppressed by where is thegravitational potential depth of the object. In the par-

2 By comparing the free fall of the Earth and the moon towards thesun, lunar laser ranging data already place limits on such strongequivalence principle violation at the level of 104 [24], with afactor of ten improvement expected in the near future [25]. Inthis paper, we are interested in theories that respect these solarsystem bounds while exhibiting O(1) deviation from GR in otherenvironments.

3 In this paper, equivalence principle violation means no more andno less than this: that different objects can fall at different rates,or equivalently, they have different ratios of inertial to gravita-tional mass.

lance of post-Newtonian expansion, such violation is anorder 1/c2 effect (c is the speed of light). Order one viola-tion of the equivalence principle is however observable inthe most strongly bound ob jects i.e. black holes, whichdo not couple to the scalar at allthey have no scalarhair. This means black holes and stars fall at appreciablydifferent rates in scalar-tensor theories [26, 27].

In this paper, we wish to point out that there could be

order one violation of the equivalence principle, even forobjects that are not strongly bound, such as galaxies andmost of their constituents (|| 1). In the parlance ofpost-Newtonian expansion, this is an order 1 effect (i.e.this is not even post-Newtonian!). That this is possibleis thanks to the new twist introduced by recent variantsof scalar-tensor theories, namely their screening mecha-nisms. We will show that the chameleon mechanism leadsto order one violation of the equivalence principle evenfor small || objects, while the Vainshtein mechanismdoes not. These screening mechanisms were absent intraditional treatments of scalar-tensor theories becausethe latter typically did not have a non-trivial scalar po-

tential (needed for chameleon) or higher derivative in-teractions (needed for Vainshtein) i.e. they were of theJordan-Fierz-Brans-Dicke variety [28, 29, 30].

How such equivalence principle violation comes aboutis easiest to see in Einstein frame, where the scalar me-diates an actual fifth force. Under the chameleon mech-anism, screened objects effectively have a much smallerscalar charge compared to unscreened objects of the samemass. Hence, they respond differently to the fifth forcefrom other objects in the same environment; they thenmove differently. That such a gross violation of the equiv-alence principle is possible might come as a surprise inJordan frame. One is accustomed to thinking that all ob-

jects move on geodesics in Jordan frame. One of our maingoals is to show that this is in fact not true. Whethertwo different objects fall at the same rate under gravityis of course a frame-independent issue; nonetheless, somereaders might find a Jordan frame derivation illuminat-ing. Here, we follow the elegant methodology developedby Einstein, Infeld & Hoffmann [31] and Damour [22]and apply it to situations where the screening mecha-nisms operate. What we hope to accomplish in part isputting the arguments of Khoury and Weltman [17, 18]on a firmer footing, who first pointed out the existence ofthe chameleon mechanism, and used Einstein frame argu-ments to propose equivalence principle tests using artifi-cial satellites. We also wish to establish that if screening

occurrs via the Vainshtein mechanism, there is no analo-gous order one violation of equivalence principle.

It is worth emphasizing the different physical originsfor equivalence principle violations of the post-NewtonianO(1/c2) type and the Newtonian O(1) type. In anyscalar-tensor theory, the scalar is always sourced bythe trace of the matter energy-momentum tensor. Inother words, at the level of the Einstein frame action,the coupling between the scalar and matter is of theform Tm, where Tm is the trace of the matter energy-


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momentum. Fundamentally, equivalence principle viola-tions occur whenever different objects couple to dif-ferently. In a theory like Brans-Dicke, this happens be-cause different objects of the same mass have differentTm. The degree to which their Tm differs is of order 1/c2

(i.e. the gravitational potential), and so equivalenceprinciple violation of the Brans-Dicke type is typicallysmall, except for relativistic objects whose Tm is highly

suppressed i.e. most of their mass comes from gravita-tional binding energy. An extreme example is the blackhole whose mass is completely gravitational in origin. Intheories where the chameleon mechanism operates, thereis an additional and larger source of equivalence princi-ple violation, and that is through variations in the effec-tive scalar charge that a macroscopic object carries. Thescalar self-interaction in a chameleon theory makes thescalar charge sensitive to the internal structure of theobject; the scalar charge is not simply given by a vol-ume integral of Tm as in Brans-Dicke, but is given by ascreened version of the same. As we will see, chameleonscreening could introduce O(1) fluctuations in the scalarcharge to mass ratio, even for non-relativistic objects.What is perhaps surprising is that this happens despitean explicitly universal scalar coupling at the level of themicroscopic action. One can view this as a classical renor-malization of the scalar charge. We will also see that intheories where the Vainshtein mechanism operates, thereis no such O(1) charge renormalization. Therefore thesetheories only have equivalence principle violation of theBrans-Dicke, or post-Newtonian, type.

Equivalence principle violations of both types dis-cussed above, Newtonian or post-Newtonian, should beclearly distinguished from yet another type of violation,which is most commonly discussed in the cosmology lit-erature where there is explicit breaking of equivalenceprinciple at the level of the microscopic action, e.g. ele-mentary baryons and dark matter particles are coupled tothe scalar with different strengths [32, 33, 34, 35, 36, 37].In this paper, we are interested in theories where thisbreaking is not put in by hand i.e. it does not existat the level of the microscopic action; rather, the equiv-alence principle violation comes about for macroscopicobjects by way of differences in response to the screeningmechanism.

Further experimental tests come to mind after ourinvestigation of the problem of motion under the twoscreening mechanisms. We propose several observationaltests using either the bulk motion of galaxies or the in-ternal motion of objects within galaxies. The chameleonmechanism will predict gross violations of the equivalenceprinciple for suitably chosen galaxies while the Vainshteinmechanism will not. Care must be taken in carrying outsome of these tests, however, to avoid possible Yukawasuppression and/or confusion with astrophysical effects.We point out that the most robust test probably comesfrom the measurement of internal stellar and HI velocitiesof small galaxies in voids. Systematic differences betweenthe two, or lack thereof, will put interesting constraints

or rule out some of the current modified gravity theories.A guide for the readers. Readers who are interested

primarily in observational tests can skip directly to VIwhere relevant results from the more theoretical earliersections are summarized. II serves as a warm-up exercisewherein we derive the equality of inertial mass and gravi-tational mass in GR for exteneded objects. III demysti-fies this surprising effacement property of GR by viewing

GR through the prism of a field theory of massless spintwo particles. This section is somewhat of a digressionand can be skipped by readers not interested in the fieldtheoretic perspective. IV is where we deduce the motionof extended objects under the chameleon mechanism inEinstein frame. The Jordan frame derivation is given inthe Appendix. In V we do the same for the Vainshteinor strong-coupling mechanism. Observational tests arediscussed in VI, where we also summarize our results.

A number of technical details are relegated to the Ap-pendices. Appendix A discusses how the requisite surfaceintegral in the Einstein-Infeld-Hoffmann approach can bedone very close to the object. Appendix B maps ourparameterization of general scalar-tensor theories to thespecial cases of Brans-Dicke and f(R). Appendix C pro-vides the Jordan frame derivation. Appendix D derivesin detail the equation of motion under the Vainshteinmechanism. Appendix E supplies useful results for anexpanding universe.

A brief word on our notation: the speed of light c isset to unity unless otherwise stated; 8G and 1/M2Pl areused interchangeably; the scalar field (or ) is chosento be dimensionless rather than have the dimension ofmass.

II. THE PROBLEM OF MOTION IN GENERAL

RELATIVITY A WARM UP EXERCISE

We start with a review of an approach to the prob-lem of motion introduced by Einstein, Infeld & Hoffmann[31] and developed by Damour [22], using GR as a warmup exercise. The idea is to work out the motion of anextended object using energy momentum conservation,rather than to assume geodesic motion from the begin-ning. It helps clarify the notion that the motion of ex-tended objects is something that should be derived ratherthan assumeddepending on ones theory of gravity, theextended objects might or might not move like infinites-imal test particles.

The Einstein equation G

= 8G Tm

can be rewrit-ten as:

G(1) = 8G t

(1)

where G(1) is the part of the Einstein tensor G

that isfirst order in metric perturbations, and t

is the pseudoenergy-momentum tensor which is related to the energy-momentum tensor Tm

by

t = Tm

18G

G(2) (2)


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where G(2) is the higher order part of G

, includingbut not necessarily limited to second order terms. Thesuperscript m on Tm is there to remind us this is thematter energy-momentum, in anticipation of other kindsof energy-momentum we will encounter later. Note thatthe identity G

(1) = 0 implies t

is conserved in theordinary (as opposed to covariant) sense.

Let us draw a sphere of radius r enclosing the extended

object whose motion we are interested in. The linearmomentum in the i-th direction is given by

Pi =

d3x ti0 (3)

where the volume integral is over the interior of thesphere. We assume that ti

0 is dominated by the extendedobject such that Pi is an accurate measure of the linearmomentum of the object itself, even though we are inte-grating over a sphere that is larger than the object.

The gravitational force on the extended object is

Pi = d3x 0ti0 = d3x jti

j =

dSjtij (4)

where in the last equality we have used the Gauss lawto convert a volume integral to a surface integral i.e.dSj = dA xj , where dA is a surface area element andx is the unit outward normal. As expected, the gravi-tational force is given by the integrated momentum fluxthrough the surface. This is the essence of the methodintroduced by Einstein, Infeld & Hoffmann: the problemof motion is reduced to performing a surface integrali.e. there is no need to worry about internal forces withinthe object of interest.

Assuming that Tmij at the surface of the sphere is

negligible, we need only consider the contribution to tij

from G(2)ij (Eq. (2)). And if the surface is located ata sufficiently large distance away from the object of in-terest, we can assume the metric perturbations are smalland only second order contributions to G(2)ij need to beconsidered (without assuming metric perturbations aresmall close to the object). In the Newtonian gauge,

ds2 = (1 + 2)dt2 + (1 2)ijdxidxj , (5)we have:

G(2)ij = 2(ij2 ij) 2(ij2 ij)

+ij ijkk + 3ij 2ijkkij ij + 2G(1)ij (6)

ignoring time derivatives. On the right hand side, weare being cavalier about the placement of upper/lowerindicesit does not matter as long as we are keepingonly second order terms. The symbol 2 denotes kk.The first order Einstein tensor G(1)ij is given by:

G(1)ij = ij2( ) + ij( ) (7)

again, ignoring time derivatives.

The next step is to decompose and around thesurface of the sphere into two parts:

= 0 + 1(r) , = 0 + 1(r) (8)

where 0 and 0 represent the large scale fields due tothe environment/background while 1 and 1 are thefields due to the extended object itself. This decomposi-

tion can be formally defined as follows: 1 and 1 solvethe Einstein equation with the extended object as theone and only source; 0 and 0 are linear gradient fieldsthat can always be added to solutions of such an equation(because the equation consists of second derivatives). Wehave in mind 0 and 0 are generated by other sourcesin the environment, and they vary gently on the scale ofthe spherical surface that encloses the object i.e. theirsecond gradients can be ignored:

0(x) 0(0) + i0(0)xi0(x) 0(0) + i0(0)xi (9)

where 0 denotes the center of the sphere. In other words,

we assume i0 and i0 hardly vary on the scale ofthe sphere. On the other hand 1 and 1, the objectgenerated fields, have large variations within the sphere.At a sufficiently large r, we expect both to be

1, 1 GM/r (10)i.e. dominated by the monopole.

Note that we have carefully chosen the radius r of theenclosing sphere to fulfill two different requirements: thatr is smaller than the scale of variation of the backgroundfields, and that r is sufficiently large to have the monopoledominate. The latter condition simplifies our calculation

but is not necessary in fact, as is shown in Appendix A.Also we have in mind a situation where the density withinthe object of interest is much higher than its immediateenvironmentthat is the definition of object after allso that the total mass inside our sphere is dominated bythe objects.

Plugging the decomposition (Eq. (8)) into Eq. (6) andperforming the surface integral as in Eq. (4), we find 4

Pi =1

8G

dSjG(2)

ij

=r2

2Gi0

4

3

1r

23

1r

. (11)

This result has several remarkable features. First, onlycross-terms between the background fields (subscript 0)and the object fields (subscript 1) survive the surface

4 We have not assumed equality of 0 and 0, nor 1 and 1 inEq. (11). Keeping their relationship general will be useful forthe scalar-tensor case. Eq. (11) is derived assuming only that and are subject to an object-background split with a linearbackground.


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integration. A term such as

dSji0j0 vanishesbecause by assumption, i0 and j0 are both con-stant (or nearly so on the scale of the sphere). A termsuch as

dSji1j1 vanishes because both 1 and

1 are spherically symmetric. Second, only terms pro-portional to i0 remain. There are no i0 terms. Thisshould come as no surpriseby ignoring time deriva-tives, we are in essence assuming the object is mov-

ing with non-relativistic speed and so its motion shouldonly be sensitive to the time-time part of the back-ground/environmental metric.

Finally, plugging the monopole 1 and 1 into Eq.(11) gives us:

Pi = M i0 (12)which is the expected GR prediction in the Newtonianlimit.

To complete this discussion, it is useful to note thatthe mass of the object is given by

M =d3x t00 . (13)

Its time derivative M =

d3x it0i can once again be

turned into a surface integral M =

dSi t0i. Assuming

the energy flux through the surface is small, we can ap-proximate M as constant. The center of mass coordinateof the object is defined by

Xi

d3x xi t00/M . (14)

Making the constant M approximation, its time deriva-tive is given by

Xi =

d3x j(xit0j) t0i

/M = d3x t0i/M .(15)

This is precisely Pi/Mdefined in Eq. (3). Taking anothertime derivative, we can see that Eq. (12) is equivalent to

MXi = M i0 . (16)As expected, the center of mass of an extended objectmoves on a geodesic. Its mass cancels out of the equa-tion of motion i.e. inertial mass equals gravitational mass.That this holds independent of the internal structure ofthe extended object is sometimes referred to as the ef-facement principle [22].

It is useful to summarize the approximations or as-sumptions we have made to obtain this result, some ofwhich are crucial, some of which not.

1. We have taken the non-relativistic/Newtonian limiti.e. the object has the equation of state of dust and isslowly moving, and time derivatives of the metric areignored. These assumptions can be relaxed and the ob-

ject would still move on geodesics. For instance, photonssurely move on geodesics in GR, but we will not dwell onthe proof here.

2. We assume the metric perturbations are small atthe surface of the sphere that encloses the object of inter-est. The derivation does not assume however that metricperturbations are small close to the object. For instance,Eq. (16) remains valid for the motion of a black hole.Note also that the assumption of a spherical surface isnot crucialthe surface can take any shape, as discussedin Appendix A.

3. We assume there is a separation of scales, that ex-ternal tides can be ignored on the scale r of the enclosingsurface, and that the monopole contribution dominatesat r. The last assumption can be easily relaxed. Asshown in Appendix A, one could choose an r that is closeto the object, and the higher multipoles would still notcontribute to the relevant surface integral. This is an im-portant point because the enclosing surface is purely amathematical deviceit should not matter what shapeor size it takes, as long as it is not so big that too muchof the exterior energy-momentum is enclosed or, equiva-lently, external tides become non-negligible.

Let us end this section with an interesting observa-tion. This derivation makes use of the metric that theextended object sources (Eq. (10)). It is intriguing thatwhat the object sources also determines how the objectresponds to an external background field. In fact this isnot surprising from the Lagrangian viewpoint, and is ina sense the modern generalization of Newtons third law:the same object-field interaction Lagrangian term yieldsboth the source term in the field equations for gravity andthe gravitational force term in the objects equation ofmotion. This will be useful in understanding how equiv-alence principle violation comes about in scalar-tensortheories.

III. FIELD-THEORETICAL CONSIDERATIONS

The fact that when tidal effects are negligible, ex-tended objects move along geodesics of the backgoundmetric, looks somewhat magical if derived as above butis in fact forced upon us by Lorentz invariance.5 In par-ticular, it extends beyond the non-relativistic/weak-fieldregime discussed above. For example, a small black-holefalling in a very long-wavelength external gravitationalfield, will follow a geodesic of the latter.

This can be shown by imposing the requirement thatthe theory of massless spin-two particles (gravitons) in-teracting with matter be Lorentz invariant. In this sec-

tion, we will thus abandon momentarily the geometric de-scription of GR, and treat gravity as a theory of gravitonsin Minkowski space. After all, from the modern quantum

5 In this section, we digress a bit and wish to throw some lighton the effacement property of GR from a field-theoretical per-spective. Readers interested mainly in the problem of motion inscalar-tensor theories can skip to the next section without lossin the continuity of the logic.


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field theory (QFT) viewpoint this approach is what legit-imizes GR as a theory of gravity in the first place: GR isthe only consistent low-energy Lorentz-invariant theoryof interacting massless spin-two particles.

So, lets describe gravitons in flat space via a tensorfield h . We want to exploit the smallness of the ob-

ject in question with respect to the typical variation-scale of the external gravitational field. What we can

do then, is treat the object as a point-particle with tra-jectory x(), and couple it to the graviton field in allpossible ways allowed by symmetry. (Here the xs areflat-space coordinates, and is flat-space proper-time:d2 = dxdx .) This yields an effective field the-ory valid for wavelengths much longer than the objectssize. The important symmetry for us is just Lorentz-invariance. When applied to gravitons, this also demandsinvariance under gauge transformations

h h + + , (17)

see e.g. ref. [38]. However at the order we will be working,this is not relevant. The low-energy effective action is

Seff = M

d + Sgrav[h ]

M f

d hS + . . . (18)

The first line is just the sum of the free point-particleaction and of the pure gravity action, which includesthe graviton free action as well as all graviton self-interactions. In the second line, we explicitly display oneinteraction term between h and the point-particle. S

is a Lorentz-tensor built out of the point-particle degreesof freedom: its trajectory x(), as well as possibly inter-

nal degrees of freedom such as spin, multipole moments,etc.; f is a coupling constant. For convenience, we pulledout of f an explicit factor of the particles mass M. Thedots stand for additional interactions, which involve ei-ther higher powers ofh , or derivatives thereof, or both.

Now, Weinberg [39] showed that for this effective the-ory to be Lorentz-invariant, in the low-energy gravitonlimit:

i) S must be totally independent of the internal de-grees of freedom, and given by

S = xx . (19)

ii) The coupling constant f must be universal, that is,the same for all particles.

This is nothing but the equivalence principle, whichfrom this viewpoint is rather a theorem. Indeed, sincef is a universal constant, we can absorb its value into aredefinition of h. Lets therefore set it to 1/2, so thatthe effective action becomes

Seff = M

d

1 + 12hxx

+ Sgrav [h ] + . . . (20)

We recognize in parentheses the first order expansion ofthe point-particle action in GR:

SGRpp = M

dg xx (21)

with metric g = + h . In fact, one can showthat because of the gauge-invariance (17) (which is also a

consequence of Lorentz-invariance), the non-linear inter-actions between h and the point-particle (which makeup part of the dots in eq. (20)), as well as the (low-energy) graviton self-interaction of Sgrav , must combineto yield the usual GR expressions. The remainder of thedotsinteractions involving higher derivatives of hencode non-minimal couplings of h with the particlesmultipole moments [40], which lead to tidal effects. Thesehowever are negligible for very long wavelength gravita-tional fields, their relative importance being suppressedby powers of a/, where a is the objects size, and thegravitons wavelength.

These are of course very well known notions. We re-

viewed them here, to contrast the very constrained situ-ation of GR with a generic scalar-tensor theory. For GR,any object will source and respond to long-wavelengthgravitational fields as if it were a point particle: theequivalence between inertial and gravitational masses isguaranteed by Lorentz invariance, and as a consequenceit is totally independent of the objects internal struc-ture, and of the forces that are keeping the object intact(including the gravitational one). It applies equally tomacroscopic objects with non-relativistic structure (likee.g. a very soft, very cold gas cloud) and to those witha very relativistic one (like e.g. a black hole). Noticethat the parameter M appearing in eq. (20) is the totalphysical mass of the object, which includes all classicaland quantum contributions to it. Among the former wehave, on top of the constituents masses, the classicalbinding energy of a macroscopic system. In fact this is

just the tree-level approximation of the quantum contri-butions, which are particularly relevant for microscopicsystems where the bulk of the physical mass is quantum-mechanical in origin, like e.g. for the proton. Wein-bergs argument [39] shows that upon quantum correc-tions, the inertial and gravitational masses of a quantum-mechanical system get renormalized by exactly the samemultiplicative factor 6.

No analogous statement can be made for scalar cou-plings. Consider indeed a scalar-tensor theory for gravity.

Also assume that it is a metric theory, which means thatthere is a conformal frame (Jordans) where matter-fieldsare all minimally coupled to the same metric. This is the

6 This is the gravitational analogue of a similarbut not quite aspowerfulstatement for electric charges: quantum correctionsrenormalize all charges by the same multiplicative factor, be-cause charge renormalization can be thought of as wave-functionrenormalization for the photon field. See e.g. ref. [38].


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7

closest we can get to implementing the equivalence prin-ciple. Now, upon demixing the scalar from the gravitonh (that is, in Einstein frame), this prescription corre-sponds to a direct coupling Tm between the scalar andmatter fields, where Tm = Tm

is the trace of the mat-ter energy-momentum tensor. The problem with such acoupling is that Tm is not associated with any conservedcharges. That is, in the above point-particle approxima-

tion, extending GR to a scalar-tensor theory (in Einsteinframe) amounts to adding an interaction term

Sint = Q

d + . . . , (22)

as well as of course a Lagrangian for , its self-interactions, and its interactions with h. The dots inSint denote interactions involving derivatives of , whichcan be thought of as multipole interactionsnegligiblefor a long-wavelength scalar field. Now, the coupling Qis the spatial integral of Tm over the objects volume,and is not a conserved charge. So, for instance, if weconsider a collapsing sphere of dust eventually forming

a black-hole, its coupling to h (the mass M) is con-served throughout the collapse, whereas its coupling to (the scalar charge Q) is not: as the dust particles be-come more and more relativistic, their Tm approacheszero, even as the total mass M remains constant, domi-nated more and more by gravitational binding energy. Infact we already know that for a black-hole, which is theend product of our example, Q must be zero, because ofthe no-hair theorem [41]: a black-hole cannot sustain ascalar profile, which in our language means that in thepoint-particle approximation it cannot couple to a long-wavelength . So, in comparing how a black-hole and anon-relativistic gas cloud fall in a very long-wavelength

scalar-tensor gravitational field, we expect order one vi-olation of the equivalence principle: the black-hole willnot follow a geodesic of the Jordan-frame metric.

In general, even if we start with a universal coupling Tm at the microscopic level, the ratio Q/M for ex-tended objects will be object dependent, and sensitiveto how relativistic the ob jects internal structure is: itwill be one for very non-relativistic objects (with a suit-able normalization of ), and zero for black holes. Thiskind of variation in Q/M, which is relativistic in nature(O(1/c2) in a post-Newtonian expansion), is available toall scalar-tensor theories. As we will see, theories wherethe chameleon mechanism is at work possess an addi-tional and more dramatic source of Q/M variationitcan be O(1) even for non-relativistic objects. This screen-ing mechanism arises from a particular kind of scalar self-interaction, and it results in a scalar charge Q not justdetermined by an integral of Tm, but also by the precisenon-linear dynamics of the scalar itself close to the ob-

ject. Essentially, the scalar field profile sourced by theobject at short distances contributes, via non-linearities,to the coupling of the object to a long-wavelength exter-nal scalar. (As we will see, for the Vainshtein mechanismthis scalar charge renormalization does not occur.)

The lesson is this: from the low-energy point-particleaction viewpoint, the charges Q for different objects arefree parameters, and are not forced to be related tothe corresponding total inertial masses M in any way.Therein lies the seed for equivalence principle violation.

IV. THE PROBLEM OF MOTION IN

SCALAR-TENSOR THEORIES SCREENING

BY THE CHAMELEON MECHANISM

The fact that different objects might fall at differentrates in scalar-tensor theories is easiest to see in Einsteinframethe scalar mediates a fifth force and the equiv-alence principle is violated whenever the scalar chargeto mass ratio fluctuates between objects. On the otherhand, that this should be true in Jordan frame mightcome as a bit of a surprise: afterall, are not all objectssupposed to move on geodesics in Jordan frame? Wetherefore present two different derivations, one in eachframe. The Einstein frame derivation is presented inIV A. The Jordan frame derivation is briefly summa-rized in IV B with details given in Appendix C. In addi-tion, Appendix B describes how to map our descriptionof general scalar-tensor theories to the special cases ofBrans-Dicke and f(R).

The Einstein frame action is

S = M2Pl

d4x

g12 R 12 V()

+

d4x Lm(m, 2()g) . (23)

Here,denotes quantities in Einstein frame, R is the Ein-stein frame Ricci scalar, is supposed to remindus that the contraction is performed with the Einsteinframe metric g (there is otherwise no difference be-

tween and : = = ). The symbol denotes some scalar that would contribute to a fifthforce. Upon a field-redefinition, its kinetic term can al-ways be written as in eq. (23). For notational conve-nience for what follows, we decide to factor an M2Pl outof the whole scalar-tensor gravitational action. With thisnormalization is dimensionless. We assume has apotential V(), which given our normalization has mass-dimension two. The symbol m denotes some matterfield whose precise nature is not so important; the mostimportant point is that it is minimally coupled not to gbut to 2()g , which is precisely the Jordan framemetric:

g = 2()g . (24)

Performing the conformal transformation gives us the


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Jordan frame action:

S = M2Pl

d4x

g122()R 12h()

4()V

+

d4x Lm(m, g) (25)

where

h() 2

1 32

ln 2

2. (26)

The matter energy-momentum in the two frames (definedas the derivative of the action w.r.t. the correspondingmetric) are related by:

Tm = 2()Tm . (27)

In anticipation of the fact that we will be carrying outperturbative computations, let us point out that 2()must be close to unity for the metric perturbations to besmall in both Einstein and Jordan frames. We can thusapproximate it at linear order in ,

2() 1 2 , (28)where is a constant, and || 1. We are notassum-ing, however, that the fractional scalar field perturbation(i.e. /) is in any way small. In fact, nonlinear effectsin / are going to be crucial in enabling the chameleonmechanism. Under the same approximation, a useful re-lation is

ln 2

2 . (29)

The reader might find it helpful to keep in mind a partic-ularly common (if not particularly natural) form of theconformal factor, 2 = exp[2] with constant , inwhich case Eq. (29) is exact and Eq. (28) is a Taylorexpansion.

As elaborated in Appendix B, = 1/

6 in f(R) mod-els and BD = (1 62)/(42) in Brans-Dicke theory.

A. Derivation in Einstein Frame

The Einstein frame metric equation is

G = 8G

Tm

+ T

(30)

where

T =

1

8G

12

+ V

.(31)

The scalar field equation is

=V

+ 4G

ln 2

Tm

. (32)

Following the derivation in the GR case (II), we performan object-background split, derive solutions to the objectfields, and compute an integral of momentum flux overan enclosing sphere to find the net gravitational force onthe extended object.

Object-background Split

We work in the Newtonian gauge:

ds2 = (1 + 2)dt2 + (1 2)ijdxidxj . (33)As before (Eq. (8)), we carry out a decomposition of thefields at some radius r away from the object:

= 0 + 1(r) , = 0 + 1(r) , = 0 + 1(r)(34)

where the subscript 0 denotes the back-ground/environmental fields and the subscript 1denotes the object fields. The background fields areassumed to be well approximated by linear gradients:

0(x) 0(0) + i0(0)xi0(x) 0(0) + i0(0)xi

0(x) + i0(0)xi (35)where 0 denotes the origin centered at the object, andwe use to denote the background scalar field valuethere. The assumption here is that these backgroundfields sourced by the objects environment vary on a scalemuch larger than r, where we will draw our enclosingsphere.

Solutions for the Object Fields

To work out solutions for the object fields, we firstexamine the Einstein equation linearized in metric per-turbations:

2 = 4G 140 0 + 14i i + 12V0i = 120 i(ij 13ij2)( ) =

i j 13ijk k

20 +132( ) =1

6 320 0 12k k 3V (36)

where we have assumed the matter is non-relativistic,characterized only by its energy density .

We can ignore all second order scalar field terms onthe right hand side of these equations. This is becauseas we will see shortly, is already first order in G. Thusits contribution to Einsteins equations is on an equalfooting with post-Newtonian corrections, which we areneglecting. In addition, we will ignore V as well. Therationale is that G V inside the object of interest,


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whereas outside the object, V or any other sources ofenergy-momentum are just small (in the sense that thetotal mass within the enclosing sphere is dominated bythe object). Essentially, we assume the metric in thevicinity of the object is sourced mainly by matter ratherthan the scalar field:7

2 = 4G , 2( ) = 0 . (37)

Under the same assumptions as in II, we get1 = 1 = GM/r (38)

where M is the mass of the object.As in the case of GR, we in essence assume there is a

separation of scales: that r can be chosen small enoughthat second gradients of the background tidal fields canbe ignored, but large enough for the monopole to dom-inate. The latter condition is helpful but not essential(see Appendix A).

The next task is to compute the scalar field profilesourced by the object. The scalar field equation (Eq.(32)) can be written as

2 = V

+ 8G (39)

where we have ignored time derivatives and correctionsdue to metric perturbations, and we have used the ap-proximation Eq. (29). Here, it is crucial we do not as-sume / is small even though is. The important

V

Veff

FIG. 1: A scalar potential for the chameleon mechanism.The effective potential felt by in the presence of sources isthe sum of the self-interaction potential V() and the scalar-

matter coupling ( 8G) . This can give a large mass atlocations where the matter density is large.

nonlinearity for the chameleon mechanism is in the po-tential term. For instance, for the potential of fig. 1 the

7 Under this assumption, the anisotropic stress vanishes. This isemphatically an Einstein frame statement. The same will not betrue in Jordan frame.

chameleon mechanism could operate, meaning that for anobject of the right size and density, could be trapped atsome small value inside much of the extended objectavalue where the two terms on the right hand side of Eq.(39) roughly balance. This corresponds to having alarge mass, or a small Compton wavelength, inside theobject (see Fig. 1). The object will therefore have anexterior scalar profile that is screened i.e. only a frac-

tion of the objects mass (the fraction that happens tolive within a shell approximately a Compton wavelengththick at the objects boundary) sources the exterior scalarprofile. In other words, we expect the object to producea scalar field that at large r is dominated by a monopoleof the form:

1(r) = 2GMr

(40)

where quantifies the degree of screening.If the scalar field remains light inside the object, we

have V/ 8G in which case the exterior scalarprofile is sourced by the full mass of the object, i.e. = 1

in Eq. (40). On the other hand, suppose the scalar fieldis massive and therefore short-ranged inside the object,the exterior scalar profile is sourced only by the mass re-siding in a shell at the boundary of the object. Supposethe object has a radius rc and the shell has thickness rc.Within this shell, the right hand side of Eq. (39) is dom-inated by the density term (i.e. the scalar field is not sit-ting at some small equilibrium value where the potentialand density terms cancel). This means roughly speaking,/(rcrc) 8GM/(4r3c/3), where = c (c is the equilibrium value where the the potentialand density terms balance, and is approximately thescalar field value far away from the object). We therefore

have 3rc/rc (/2) (GM/rc)1

. This is preciselythe fraction by volume of the object that sources the ex-terior scalar field. This means the suppression factor inEq. (40) should be

2

GM

rc

1if

2

GM

rc

1< 1 (41)

where GM/rc is the gravitational potential of the object.The inequality is required for consistency: it had betterbe true that 3rc/rc < 1 for screening to occur. This isknown as the thin-shell condition. When this conditionis violated, there is no screening:

= 1 if 2

GM

rc

1> 1 , (42)

essentially meaning that 8G V/ throughoutthe object, in which case the scalar field cannot evenreach the equilibrium value c inside the object. It isinteresting to note that the thin-shell condition does notdepend explicitly on the potential V. The implicit re-quirement is that the potential is of such a form thatequilibrium (between density and potential) is possible.


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10

A more detailed derivation of the thin-shell condition canbe found in [18].

Along the lines ofIII, it is useful to introduce the ideaof a scalar charge: by analogy with 1 = GM/r forthe gravitational potential, let us write the scalar profileas 1 = 2GQ/r (the factor of 2 is motivated by theappearance of 8 rather than 4 in Eq. (39)). Byinspecting Eq. (40), we can see that the dimensionless

scalar charge Q isQ M . (43)

Screened objects carry a much reduced scalar charge asviewed from outside.

Just as in the case of the metric fluctuations, one canalways add to 1 a linear gradient of the form Eq. (35)to represent the field generated by sources in the envi-ronment. To summarize, , and are given by Eqs.(34), (35), (38) and (40).

Surface Integral of Momentum Flux

We are ready to compute the gravitational force fol-lowing Eq. (4):

Pi =

dSj tij (44)

where the surface integral is again over some sphere en-closing the object of interest. By analogy to Eqs. (1) and(2), the pseudo energy-momentum tensor is:

t = Tm

+ T 1

8GG(2)

. (45)

As before, Tmij outside the object is small, and we have

already computed the G(2)ij contribution (just put on

top of quantities in Eq. (11)). What remains to be done

is to compute the contribution from Tij (Eq. (31)).

The same tricks that work for the G(2)ij contributionwork here, and we obtain

dSjTij = 1

2Gr2 i0

1r

. (46)

Note that there should be a contribution from the poten-tial of the form: 1G

dSjV 1G (4r3/3)i0V/| ,

but this can be ignored by our assumption that the scalarfield 1 is sourced mainly by the object rather than itsimmediate environment.

Putting together all contributions (Eq. (46) & Einstein

frame analog of Eq. (11)), we therefore have

Pi =r2

2G

i0

4

3

1r

23

1r

i01

r

. (47)

Putting our solutions for the various fields into theabove expression, we obtain

MXi = Mi0 Mi0= M

i0 +

Q

Mi0

(48)

where have used Eq. (43), and we have equated Pi andMXi, where Xi is the object center of mass coordinate(Eq. (15)). This is the main result of this section. Thefact that screened and unscreened objects have differentscalar charge to mass ratios Q/M (or equivalently, dif-ferent ) means they fall at different rates. This is theorigin of apparent equivalence principle violation.

This result can be further simplified if we make one

more assumption, which does not necessarily hold in gen-eral. Both 0 and 0 are sourced by the environment.Suppose they are sourced in a similar way:

20 = 4G , 20 = 8G (49)

where is the environmental density. The latter equationfollows from Eq. (39) provided that potential terms aresmall compared to density terms on the scales of interest.Eq. (49) then implies

0 = 2 0 . (50)

Putting this into Eq. (48), we therefore obtain

MXi = M i0

1 + 2Q

M

= Mi0

1 + 22

. (51)

The fact that this result depends on the square of makes it clear that the sign of has no physicalmeaningchameleon mechanism only requires its sign beopposite to that of V/ (see Eq. (39)).

Suppose one has a scalar-tensor theory where 22 isorder unity, as is true in f(R) theories, then screened ob-

jects ( 0) and unscreened objects ( = 1) would moveon very different trajectories i.e. order unity apparent vi-olation of the equivalence principle. Note that infinites-imal test particles should go unscreened and thereforehave = 1.

Let us end with a summary of further approximationswe have made in this section to augment those listed atthe end ofII:

1. We have taken the non-relativistic or quasi-staticlimit i.e. time derivatives of the scalar field are ignored.

2. We assume the scalar field does not contributemuch to the r.h.s. of Einsteins equations (36).

3. We assume there is a separation of scales, that sec-ond derivatives of the background/environmental scalarfield can be ignored on the scale r of the enclosing sur-face, and that the monopole contribution to the objectsscalar profile dominates at r. As discussed in AppendixA, this latter assumption can be relaxed. The formerassumption implicitly requires the Compton wavelengthof the exterior scalar field to be large compared with r.The scalar field would be Yukawa suppressed if this werenot true. This has implications for observational tests aswe will discuss in VI.

To this list, which mirrors the list in II, we shouldadd:


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11

4. We assume the conformal factor 2() is close tounity, or equivalently || 1 (see Eq. (28)). This as-sumption is not strictly necessary in our Einstein framecalculation. It will become necessary when we try tocompare the Jordan frame calculation with it. One con-sequence of this assumption is that the Einstein frameand Jordan frame masses of the object are roughly equali.e.

M =

d3x

d3x (52)

since = 4, recalling that Tm00 = and Tm00 =

.In summary, the main results of this section are encap-

sulated in Eqs. (47), (48) and (51), which make differ-ent levels of approximations, starting from the minimum.The key observation is that, despite the explicitly uni-versal coupling of the scalar to matter at the microscopiclevel (Eq. (23)), scalar charges of macroscopic objectsget classically renormalized by nonlinear chameleon ef-fects. This produces fluctuations in the scalar charge tomass ratio Q/M for different objects, leading to differentrates of gravitational free fall. Screened objects, whichhave Q/M 0, accelerate less than unscreened objects.As we will see, no such (non-relativistic) renormalizationoccurrs for the Vainshtein mechanism.

B. Derivation in Jordan Frame

In principle, the simplest way to obtain the Jordanframe results is to transform the end results of IVA,Eqs. (47), (48) and (51), into Jordan frame by a con-formal transformation g =

2g . After all, whethertwo different objects fall at different rates is a frame-

independent issue. This might not, however, satisfy read-ers who are accustomed to the Jordan frame viewpoint:should not everything move on geodesics? In this sec-tion, we therefore briefly sketch a Jordan frame deriva-tion. Details can be found in Appendix C.

The Jordan frame metric in Newtonian gauge is

ds2 = (1 + 2)dt2 + (1 2)ijdxidxj . (53)From Eq. (24) and recalling 2 1 2 (Eq. (28)),the Jordan and Einstein frame metric perturbations arerelated by

= + , = . (54)

Note how = even if = .The Jordan frame metric equation is

G = 8G 2 [Tm + T

]

+22 2 (55)

where

T =

1

8G

h

12

h + 4V

. (56)

The scalar field energy-momentum follows from (the min-imally coupled portion of) the Jordan frame action Eq.(25). Note that does not have a canonical kinetic term,hence the presence of h() (Eq. (26)). In some theories,h might even be zero, such as in f(R).

The Einstein-Infeld-Hoffmann approach to the prob-lem of motion is to compute the gravitational force as asurface integral of the momentum flux (Eq. (4)). The

momentum flux can be obtained from the spatial com-ponents of the pseudo-energy-momentum tensor, whichis defined by Eq. (1):

t = 2(Tm + T

) 18G

G(2)

+2

8G

2 2 . (57)Comparing this with Eq. (45), we can see the calcula-tion should be similar to the Einstein frame case, exceptthat we have an overall conformal factor 2 multiplyingthe matter and scalar field energy-momentum, and that

there are some uniquely Jordan frame contributions tot (second line). These differences originate from thedifferences in the metric equation (55) from the Einsteinequation: a non-constant effective G, and extra sourcesfor G beyond the matter and scalar energy-momentum.

Phrased in this way, it is not obvious that the integralof momentum flux should imply geodesic motion in Jor-dan frame. In fact, it does not in general. Relegating thedetails to Appendix C, it can be shown that the centerof mass of an object moves according to:

MXi = M i0 + (1 )M i0 (58)

where 0 and 0 are respectively the back-ground/environmental (time-time) metric perturbationand scalar field, M is the inertial mass and Xi is thecenter of mass coordinate. This is manifestly consistentwith the Einstein frame equation of motion Eq. (48)

once we recognize that 0 = 0 0 according to Eq.(54).

The parameter is controlled by the relative size ofthe objects gravitational potential and the environmen-tal scalar field; see Eqs. (41) & (42). An unscreenedobject, which has = 1, would move on a geodesic inJordan frame, just as expected for an infinitesimal testparticle. A screened object, where the chameleon mech-anism operates to make

0, would not move on a

geodesic even in Jordan frame. In other words, it ap-pears as if for screened objects, the gravitational massand the inertial mass are unequal. This can be mademore explicit if we make a simplifying assumption, thatboth 0 0 and 0 are sourced mainly by density(Eq. (49)), in which case 0 = 20/(1 + 2

2), and theequation of motion simplifies to

MXi = M

1 + 22

1 + 22

i0 (59)


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12

which is consistent with its Einstein frame counterpartEq. (51). Thus, in Jordan frame,

inertial mass = M

grav. mass = M1 + 22

1 + 22. (60)

An unscreened object is subject to a gravitational force

that is 1 + 22 larger than a screened object. For theo-ries such as f(R), this is not a small effect: = 1/

6

implies 1 + 22 = 4/3 (Appendix B). This might comeas a surprise in the Jordan frame, where the metric isminimally coupled to matter (Eq. (25)). Despite con-ventional wisdom, specifying the Jordan frame metricdoes not completely specify the motion of an extendedobject. The motion of an object should be ultimately de-termined by energy-momentum conservation. Since thescalar field contributes energy-momentum, it retains adirect influence on the motion of the object. This directinfluence magically cancels only for unscreened objects,which thus move on geodesics.8 The key physics is the

(non-relativistic and classical) renormalization of scalarcharge by the nonlinear chameleon mechanism, as dis-cussed in IV A. An objects coupling to the backgroundscalar field is suppressed if it is screened, and unsup-pressed if it is not.

V. THE PROBLEM OF MOTION IN

SCALAR-TENSOR THEORIES SCREENING

BY STRONG COUPLING A LA VAINSHTEIN

Another class of scalar-tensor theories where the scalaris efficiently screened at short distances is massive grav-

ity theories. These are not scalar-tensor theories to theletter, for at scales of order of the gravitons Comptonwavelength, and larger, there is no propagating scalardegree of freedom. However, at much shorter scales the(helicity-zero) longitudinal polarization of the gravitonbehaves like a scalar universally coupled to matter withgravitational strength, and the dynamics is accuratelydescribed by a scalar-tensor theory [11]. What makesthese theories potentially interesting is that this effec-tive scalar comes equipped with important derivativeself-interactions, which can make it decouple at shortdistances from compact objects, thus recovering agree-ment with solar system tests. This is known as the Vain-

shtein effect, and it shows up explicitly in the DGP model[3]which is a much healthier alternative to the mini-mal Fierz-Pauli theory of massive gravitons. However aswe will see shortly, unlike the chameleon mechanism theVainshtein effect does not lead to O(1) violations of the

8 The cancellation occurrs bewteen the effective Newton constantG2 and the uniquely Jordan frame terms 22 in thepseudo-energy-momentum (Eq. (57)).

equivalence principlein fact it leads to the same vio-lations one would have for a purely linear Brans-Dicketheory, which are O(1/c2).

For simplicity and definiteness, we will analyze the ex-ample of DGP. Everything we say can be straightfor-wardly extended to the galileon, which was argued inref. [9] to be the broadest generalization of (the scalar-tensor limit of) DGP that robustly implements the Vain-

shtein effect without introducing ghost degrees of free-dom. The scalar sector of DGP (in Einstein frame) hasthe Lagrangian [7]

L = 3M2Pl()2 2M2Plm2

()2 + Tm , (61)

where m is the DGP critical mass scalethe analogue ofthe gravitons massand is generally set to the currentinverse Hubble scale. We are using a dimensionless nor-malization for the scalar field , so that the Jordan-framemetric to which matter fields couple minimally is

h = h + 2 . (62)

That is, in the non-relativistic limit is a shift in theNewtonian potential:

= + . (63)

Comparing Eq. (61) with Eq. (23), we can see that

here is equivalent to /

6 there. We adopt the non-canonically normalized in this section (not to be con-fused with the number 3.141...) to ease comparison withthe DGP literature. In the language of the previoussection, DGP (and its galileon generalization) has =1/

6, meaning its Jordan frame action has no quadratic

scalar kinetic term. In this respect, DGP/generalizations

resemble f(R) or Brans-Dicke theory with BD = 0 (Ap-pendix B), but the resemblance is superficial becauseDGP/generalizations have higher derivative interactionswhich allow a completely different screening mechanism.

A compact source creates a profile that scales like1/r at large distances (linear regime). However in ap-proaching the source the non-linear term9 of Eq. (61)starts becoming important, thus changing the behaviorof to

r. This suppresses the force mediated by for

all objects that are inside this non-linear halo surround-ing the source. The size of this halo is set by the so-calledVainshtein radius,

RV MM2Plm21/3

, (64)

where M is the mass of the source, and m is the gravitonmass scale (about Hubble scale today).

Despite the qualitative similarities, this kind of non-linearity differs significantly from that in the chameleon

9 Non-linear in the equation of motion, cubic in the action.


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13

RS

V

0

=

S

S

0

RV

FIG. 2: Our mathematical trick.

case. The point is that the strong non-linearities expe-riences close to the source do not renormalize the sourcestotal scalar charge. To see this, notice that the source-free part of the Lagrangian (61) has a shift-symmetry,

+ a . (65)As a consequence, s equation of motion is the diver-gence of the associated Nothers current

J = Tm , (66)

where J is a non-linear function of first and secondderivatives of . We therefore have a non-linear Gauss-

like law for , whose source is the trace of the matterenergy-momentum tensor. For a time-independent solu-tion, this equation is readily integrated via Gausss the-orem to yield

S

J da = Qtot (67)

where S is any surface, and Qtot is the total scalarcharge 10 enclosed by S, defined as the volume integralof (minus) the matter Tmthe total mass for non-relativistic sources. So, for instance, if we consider asingle compact spherical source at the origin and we takeS to be a sphere centered at the origin, with radius muchlarger than RV, then on S is in the linear regime, andwe have

6M2Pl

S

da M 13

GM

rr RV ,

(68)

10 Qtot here is proportional to Q in Eq. (43). The difference innormalization is related to the different normalizations for and.

regardless of how non-linear s dynamics may be in-side the Vainshtein region. The mass M is thus also thescalar charge, and according to the analysis of III, Mthen is the coupling between the object and very long-wavelength external/background s. This implies thereis no chameleon-like non-relativistic renormalization ofthe scalar charge, and no O(1) violation of the equiv-alence principle. The only violations of the equivalenceprinciple we may have are associated with how relativisticthe sources internal structure is, i.e. on how the volumeintegral ofTm

differs from the total ADM mass, whichis a post-Newtonian O(1/c2) effect 11. For the benefit ofthe skeptical reader, in Appendix D we explicitly check

this statement along the lines of the previous sections.This would be the end of the story, if our universe were

made up of very isolated compact sources, with typicaldistances between one another much bigger that theirtypical Vainshtein radius. Unfortunately we live preciselyin the opposite limit. For macroscopic sources the Vain-shtein radius is so huge that treating those sources inisolation makes no sense. For instance, for the sun wehave

RV 1021 cm , (69)

much bigger than the distance to the closest stars. We

can repeat the same estimate at all scales, and checkthat nowhere in the universe is in the linear regime.This makes a comprehensive study of the effects of inour universe a formidable task, because by definition of a

11 Notice that since in Eq. (68) we are considering a huge volumewith sizable non-linearities for , we have to make sure that scontribution to the total ADM mass be negligible. It is straight-forward to check that this is indeed the case.


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non-linear system, we cannot simply superimpose the fields due to different sources. However here we want tofocus on possible violations of the equivalence principle,and for these we will be able to draw surprisingly generalconclusions.

Consider then the more realistic situation of an objectin the presence of other sources. To be as general as pos-sible, lets imagine that the object in question has sizable

multiple moments as well. The situation is schematicallydepicted in Fig. 2(left). We would like to compute theforce acting on the object following the approach of theprevious sections. In order to approximate the fieldcreated by the other sources (0, in the figure) as a lin-ear pure-gradient field, we have to draw our surface Svery close to the object. In fact since is deep in thenon-linear regime, we have two problems

1. How do we separate 0 from 1the generatedby the object? The total is not simply the sumof the two contributions.

2. What is the effect of the objects multipole mo-

ments? In a non-linear regime, the multipoles of are not directly related to the objects multipoles.Also, their behavior with r w.r.t. to the monopolesis different than in the linear case.

As to point 1, we are rescued once again by the sym-metries of the dynamics, which will also make point 2uninteresting for our purpose.

Besides the aforementioned shift-symmetry on , thesource-free part of the Lagrangian (61) is invariant undera constant shift in the derivative of ,

+ c , (70)dubbed galilean invariance in ref. [9, 21]. In fact

this is what makes DGPand suitable generalizationsthereofcapable of the Vainshtein effect in the presenceof generic localized sources, and of sustaining interestingnon-linear cosmological solutions [9]. Galilean invarianceallows us to add a pure-gradient field to anynon-linear so-lution, to get a new non-linear solution. Consider there-fore the field generated by all other sources in the ab-sence of our object, and denote this by 0. Now, if ourobject is much smaller than the typical variation scale of0, which is set by the distance to the other sources, thenwe can approximate 0 as a constant-gradient field in aneighborhood of the object. This means that in such aneighborhood, we can freely add 0 to the non-linear so-

lution 1 we would have for the object in isolation, i.e. inthe absence of the other sources. The full field

= 0 + 1 (71)

is a consistent solution to the non-linear problem in theneighborhood in question. It is in fact the solution, be-cause it has (a) the right sourcethe object, and (b)the right boundary conditions: in moving away from theobject, the perturbation in induced by the objectdecays away as 1/

r and we are left with 0.

In summary, thanks to Galilean invariance the split-ting of Eq. (71) is well defined whenever the object ismuch smaller than its typical distance to nearby sources,which is a much milder requirement than linearity. Wecan then draw our sphere S very close to the object, asin Fig. 2(left), and compute the force acting on it alongthe lines of the previous sections. We do this explicitlyin Appendix D for a spherical object. However for an

irregular object with sizable multipole moments, we donot even know the non-linear solution 1 for the objectin isolation. Nevertheless, here we can use a mathemat-ical trick. We have argued that the full solution on Sis the field the object would have in isolation, 1, plusa constant-gradient field, 0. Nothing prevents us fromconsidering a physically different situation, but with thesame total field on S. This would of course yield thesame result for the total force acting on the object. Avery convenient situation to consider is, the object in iso-lation with the same background constant-gradient 0 asin our case, but extended linearly throughout space, asdepicted in Fig. 2(right). That is, we can forget aboutthe external sources, and declare that in fact

0has a

constant gradient throughout space. In doing so, we arenot changing the total and its gradient on Sthus weare not changing the total force we would compute byintegrating over S. We can now change the radius ofS and bring it outside the Vainshtein region, where sdynamics is linear, its multipoles decay faster than themonopole, and the computation is precisely as for a linearscalar field. By changing the radius ofS, we are modify-ing the total mass enclosed by S by s contribution to it.As long as this is negligible, as can be explicitly checked,the total force computed this way does not depend onthe radius of S.

We therefore conclude that, as far as the coupling of

an extended object to a background field is concerned,theories with the Vainshtein effect behave exactlyas thelinear Brans-Dicke theory. The Vainshtein mechanismentails equivalence principle violations only of the post-Newtonian O(1/c2) type.

VI. DISCUSSION OBSERVATIONAL TESTS

Let us briefly summarize the main results so far, es-pecially those relevant for a discussion of observationaltests.

1. In theories where screening of the scalar field oc-currs via strong coupling effects (Vainshtein), such asDGP, the coupling of objects to the external scalar fieldsis unsuppressed to good approximation for all objectswhose mass is not dominated by the gravitational bind-ing energy. They therefore fall at the same rate undergravity. There remains, however, equivalence principleviolations at the level of , or O(1/c2) in the language ofpost-Newtonian expansion, just as in Brans-Dicke theory,where is the gravitational potential depth of the ob-

ject in question. Order one violation of the equivalence


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principle is visible only when comparing the motion ofrelativistic objects such as black holes (|| 1) with themotion of less compact objects such as stars (|| 1).We will focus on observational tests of the chameleonmechanism instead, where the equivalence principle vio-lations are more dramatic and readily detectable even innon-relativistic/non-compact objects.

2. The chameleon mechanism implies that the cou-

pling of screened objects to the external/backgroundscalar field is highly suppressed relative to unscreened ob-jects. They therefore fall at significantly different rates.More precisely, the equation of motion for the center ofmass of an object is, in Jordan frame (Eq. (58)):

MXi = M i0 + (1 )Mi0 (72)where Xi is the center of mass coordinate, Mis the inertial mass, 0 is the time-time part ofthe external/background metric and 0 is the exter-nal/background scalar field (through which the objectmoves). Here we have adopted a convenient parametriza-tion of the relevant conformal factor 2() in the scalar-

tensor action (Eqs. (25) & (28)):

2() = 1 2 , || 1 (73)where is a constant. Eq. (73) can be thought of asa small field expansion of e.g. exp[2]. The quantity represents the degree of screening. Unscreened objectshave = 1, while screened ob jects have < 1. Thescreening parameter is controlled by (Eqs. (41) & (42)):

2(GM/rc)

if

2(GM/rc)< 1 (screened)

1 if 2(GM/rc)

> 1 (unscreened)(74)

where rc is the size of the object in question, and is thevalue of the external scalar field. Screening is thereforecontrolled by the ratio of the exterior scalar field and thegravitational potential of the object. Unless the values ofthe two are finely tuned, one generally expects 1 ifscreening occurrs at all. Eq. (72) tells us only unscreenedobjects, those with a sufficiently shallow gravitational po-tential, have equal inertial mass and gravitational massin Jordan frame.

3. The precise ratio of inertial mass to gravitationalmass predicted by Eq. (72) depends on how the externalscalar field 0 is sourced. There are two useful limits to

consider. One is when the external scalar field is sourcedprimarily by density (Eq. (C17)) in which case the equa-tion of motion is well-approximated by

MXi = M

1 + 22

1 + 22

i0 (75)

which implies the unscreened and screened gravitationalmass are related by

unscreened g. mass = (1 + 22) screened g. mass(76)

The other limiting case is when the external scalar fieldprofile is determined primarily by the potential, in whichcase 0 is Yukawa suppressed on sufficiently large scales(scales larger than the Compton wavelength of the scalarfield 1/m). If 0 is Yukawa suppressed, the sec-ond term on the right of Eq. (72) is small regardlessof whether the object is screened or not. In this regime,one cannot observe equivalence principle violations by

comparing motions of different objects. As we will dis-cuss below, it is therefore important to avoid Yukawasuppression when performing observational tests.

4. Observationally, the single most important parame-ter besides 2 in a chameleon theory is the quantity /(see Eq. (73)). For a given ob ject, the relevant is thescalar field value in its immediate surrounding. If thisis sufficiently small, the object will be screened accord-ing to Eq. (74). In other words, if the object is sittingin some high density environment which is itself screened(i.e. small scalar field value), it is likely that the object isalso screened. We will refer to this as blanket screening.In an environment where blanket screening of all objectsapplies, violations of equivalence principle will be hardto detect.

Conversely, if the object is sitting in an environmentwhere the density is at the cosmic mean or even lower(voids), the chances are better that the object is un-screened because the relevant is larger. In otherwords, to find unscreened objects, it is useful to avoidoverdense regions and to focus on objects with a shallowgravitional potential such that (Eq. (74)):

GM

rc

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Equivalence Principle Implications of Modified Gravity Models ()

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