Equivalence Principle in Chameleon Models

Equivalence Principle in Chameleon Models

Lucila Kraiselburd    Susana J. Landau    Marcelo Salgado    Daniel Sudarsky    Héctor Vucetich Grupo de Astrofísica, Relatividad y Cosmología, Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque S/N (1900) La Plata, Argentina
CONICET,Godoy Cruz 2290, 1425 Ciudad Autónoma de Buenos Aires, Argentina
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IFIBA, Ciudad Universitaria - Pab. I, Buenos Aires 1428, Argentina
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, México D.F. 04510, México
Department of Philosophy, New York University, New York, NY 10003, United States of América
July 29, 2019
Abstract

Most theories that predict time and/or space variation of fundamental constants also predict violations of the Weak Equivalence Principle (WEP). In 2004 Khoury and Weltman KW04 () proposed the so called chameleon field arguing that it could help avoiding experimental bounds on the WEP while having a non-trivial cosmological impact. In this paper we revisit the extent to which these expectations continue to hold as we enter the regime of high precision tests. The basis of the study is the development of a new method for computing the force between two massive bodies induced by the chameleon field which takes into account the influence on the field by both, the large and the test bodies. We confirm that in the thin shell regime the force does depend non-trivially on the test body’s composition, even when the chameleon coupling constants are universal. We also propose a simple criterion based on energy minimization, that we use to determine which of the approximations used in computing the scalar field in a two body problem is better in each specific regime. As an application of our analysis we then compare the resulting differential acceleration of two test bodies with the corresponding bounds obtained from Eötvös type experiments. We consider two setups: 1) an Earth based experiment where the test bodies are made of Be and Al; 2) the Lunar Laser Ranging experiment. We find that for some choices of the free parameters of the chameleon model the predictions of the Eötvös parameter are larger than some of the previous estimates. As a consequence, we put new constrains on these free parameters. Our conclusions strongly suggest that the properties of immunity from experimental tests of the WEP, usually attributed to the chameleon and related models, should be carefully reconsidered. An important result of our analysis is that our approach leads to new constraints on the parameter space of the chameleon models.

I Introduction

The Weak Equivalence Principle (WEP) is a cornerstone of the Einstein’s Theory of General Relativity (hereafter GR), and can be broadly seen as implying the universal coupling between matter and gravity. That is usually taken as translating directly into the universality of free fall (UFF), and thus as a feature easily analyzed and directly testable. The first point we want to make is that the situation is substantially more complex than previously thought. This is due to the fact that the WEP is, strictly speaking, only meant to hold for test point-like objects (i.e. a principle meant to apply locally) that do not affect the gravitational environment around them. However, in reality no such objects exist in nature, even if the test-point-like assumption is in occasions a good approximation. All objects have, in principle, a non-vanishing energy-momentum tensor, and as such, they all modify the geometry of space-time. Those modifications, in turn, affect the motion of the object themselves. In fact, taking into account the so called “back reaction effects” is a highly nontrivial task (see for instance Wald1972 ()). Moreover, given that all real objects have a finite spatial extension, one requires, for a rigorous description of their motion, the determination of something akin to the notion of the “center of mass world-line”, a task that is highly nontrivial in general space-times Papapetrou1951 (). The results of the detailed analyses indicate that those world-lines do not, in general, correspond to geodesics. Quantum aspects further complicate the analysis of the motion of even the simplest things, which one might want to take as paradigmatic point like objects, such as photonsDrummond (). Even though such UFF violating effects are normally very small, they are always there in principle, and therefore these considerations should serve as warnings when we go on to analyze more complex situations.

Of particular interest for us here will be any theory in which the local coupling constants are taken to be effectively space-time dependent, while respecting the principles of locality and general covariance, as those entail some kind of fundamental field controlling the spatial dependence, with the field, generically, coupling in different manners to the various types of matter. Such field, usually taken to be a scalar field, generically mediates new forces between macroscopic objects, which would look, at the empirical level, as modifications of gravitation which might, in principle, lead to effective violations of the UFF for test bodies in external gravitational fields. For this reason, most theories that predict variations of fundamental constants also predict effective violations of the WEP Bekenstein82 (); Barrow02 (); Olive02 (); DP94 (); Palma03 (). For instance, the rest energy of a macroscopic body is made of many contributions related to the energies associated with various kinds of interactions (strong, weak, electromagnetic) and such components would be affected differently by a light scalar field. From the experimental point of view, one needs to confront the very strong limits on possible violations of the WEP that come from Eötvös-Roll-Krotkov-Dicke and Braginsky-Panov experiments and their modern reincarnations Adel (); RKD64 (); Braginski72 (); KeiFall82 (); 1994PhRvD..50.3614S (); 2008PhRvL.100d1101S () which explore the differential acceleration of test bodies in gravitational contexts. In fact current bounds reach sensitivities of order (or more), and thus can, in principle, constrain the viability of many models.

Recently, there has been a great level of interest in models that claim to be able to avoid the stringent bounds resulting from experimental tests looking for violations of the WEP based on schemes where the effects of the fields are hidden by suitable non-linearities, as in the chameleon models and the Dilaton-Matter-gravity model with strong coupling Damour02 (). Chameleon models were introduced by Khoury and Weltman in 2004 KW04 () and have been further developed by several authors Brax04 (); MS07 (); Brax07 (); Hui2009 (); Brax2010 (); BB11 (); Brax2012 (); Upadhye12 (); K13 (). Generically a chameleon consists of a scalar field that is coupled non-minimally to matter and minimally to the curvature (or the other way around via a conformal transformation), and where the field’s effective mass depends on the density and pressure of the matter that constitutes the environment. This, in turn, is the result of the nontrivial coupling of the scalar field with the trace of the energy-momentum tensor of the matter sector of the theory 111In this context by matter we mean any field other that the scalar-field at hand, which in the situation of interest would be the ordinary matter making up the objects present in the experiment including the atmosphere. . The coupling of the scalar field with matter might be non-universal (i.e. it may depend on the particle’s species), like in the original model introduced in KW04 (), or universal as in some simpler models.

In their work Khoury and Weltman KW04 () analyzed the chameleon field associated with a single body using a “linear” approximation in the equation, and corroborated that the corresponding solution looks very similar to the numerical solution of the full non-linear equation

This approximation to the one body problem will be referred hereafter to as the standard approach 222See Appendix E for a short review of the standard approach for the one body problem and a discussion of the thin shell condition. Their conclusion was that the bounds imposed by the experiments testing the WEP can be satisfied (even if the coupling constants are of order unity) provided that the bodies involved in the relevant experiments generate the so called thin shell effect. In this thin shell regime, the spatial variations of the scalar field take place just on a small region near the body’s surface thus preventing the scalar field from actually exerting any force on most of the body. A further analysis by Mota and Shaw MS07 () strengthened the conclusion that the non linearities inherent to the chameleon models are, in fact, responsible for suppressing the effective violation of WEP in the actual experiments even when the coupling constants are very large, and not just when they are of order unity as it was initially thought KW04 () 333When analyzing the two-body problem, Mota & Shaw MS07 () assumed that both bodies can be treated as two semi-infinite “blocks” and then solved the non-linear equation for the chameleon. After making several approximations they concluded that the force between the bodies is composition-independent. We thus worried that in such approximations the mass of both bodies would become infinite, and the magnitude of some potentially problematic terms was not estimated.. Moreover, these authors argued that the predicted effective violations of the WEP in low density environments (like in space-based laboratories) would be further suppressed for some adjusted values of the parameters of the scalar-field potential.

Based on such analyses, a good part of the community working on the area KW04 (); MS07 (); TT08 (); Hui2009 (); Brax2010 (); Saidi11 (); Brax2012 (); K13 (); PO15 () has come to believe the chameleon fields should not lead to any relevant forces in experiments on Earth designed to search for a dependence on the composition of the acceleration of a falling test body, or in general, to any observable interaction between ordinary bodies mediated by chameleon-type fields.

However, it should be noted that there is no universal consensus in the community regarding even qualitative aspects of the theoretical predictions, with some arguments indicating the chameleon force on the test body is composition dependent KW04 (); MS07 (); TT08 (); Brax2010 (); Hui2009 (); Saidi11 (); K13 () and others indicating that it is not PO15 (). Defining a “screened” test body one in which the thin shell condition is satisfied and an “unscreened” test body one in which it is not satisfied, most authors make different predictions on the composition dependence of the chameleon mediated force KW04 (); MS07 (). It is also important to recall that there are two different expressions for the thin shell condition in the literature: the first one offered by Khoury & Weltmann KW04 () using the linear approximation to the field equation, and the second one, proposed by Mota & Shaw MS07 (), based on considerations of the complete nonlinear equation, and that this latter condition also depends on the value of the environment’s density as well as the value of the chameleon potential parameters and that of .

We do not find the arguments based on the thin shell phenomena to be sufficiently persuasive. The basic observation is that one could, equally well, have claimed something not entirely different by considering two macroscopic and charged conductors. It is well known that the mobility of charges in conducting materials ensure that in static situations the charges are distributed on the conductors’ surface in such a way that the electric field inside, vanishes exactly. Thus, except for a thin shell on the surface of each body, one could have argued (following a similar logic as the one used in the context of the chameleon models), the external electric field could not exert forces on the conductor’s material. However, we of course know that large macroscopic forces between such macroscopic bodies are the rule. The resulting force could therefore be attributed solely to the thin shell effects.

In view of this, we proceeded to study this issue for the case of the chameleon in more detail, in order to understand what, if any, is the fundamental difference between the two situations (i.e. between the electromagnetic case mentioned above and the one concerning the theory at hand). That is, we want to find out if the thin shell arguments are valid at the level of accuracy that would actually ensure the “disappearance” of the expected forces. The basic point is that when the contribution to the scalar field by the presence of the test body in an actual experiment is sufficiently large to create a thin shell effect, then, the body cannot really be considered as a simple test particle, and the problem needs to be treated as a two body problem, i.e., one requires to analyze the full effect of the two bodies on the scalar field in order to evaluate the effective force exerted by the field on the body of interest. The main contribution of this paper consists in analyzing the chameleon field generated by two spherical bodies of finite size (a large one and a smaller one) and the computation of the effective force on the test (smaller) body using first principles. To this end a linear approximation for the field equation was used in all our analysis. This linear approximation is different from the standard linear approximation used by Khoury and Weltman KW04 () 444 In a future work we plan to analyze other types of approximations which we expect can lead to more accurate solutions. . As far as we are aware, this approach is novel. Now before moving forward, it is worth stressing two important aspects related to chameleon models: the first one is a matter of principle, and refers to the fact that, effectively the WEP is violated by construction in this kind of models, as opposed to purely metric-based theories (like GR) where the WEP is incorporated ab initio. That is, the WEP (for test point particles moving on geodesics) is satisfied with infinite precision in purely metric theories of gravity involving no other long range fields coupling to matter. Nevertheless, even in such metric theories the absolute validity of the WEP refers only to test point particles which, as stressed above, do not really exist in nature (as a result of Heisenberg’s uncertainty principle). In any event, when ignoring those quantum complications, it is clear that the WEP is respected for point test particles in metric theories, while, in chameleon models, the WEP is violated a priori (i.e. by construction in models with non-universal coupling) even when referring to point test particles. As we shall see, this violation can be exacerbated when considering actual extended test objects. The second aspect concerns the experiments themselves. Even if chameleon models violate by construction the WEP, those violations might not be observable in a laboratory experiment if the precision is not adequate. This, as is well known, means that some of the effects that might reflect these violations of the WEP can be suppressed by the thin shell phenomenon associated with the chameleon field or by the Yukawa dependence of the force associated with the effective mass of the field. Thus, as we shall illustrate, within a framework of a two body problem, these violations of the WEP might be large, in contexts where the two bodies are embedded in a very a light medium (e.g. the vacuum or the air atmosphere) (see Fig. 7 in Sec. V) whereas they could be strongly suppressed when part of the setting (e.g. the test body) is encased in some shell of a dense material like a metal vacuum chamber.

In order to explore these issues in detail, we evaluate the static chameleon field associated with two extended spherical bodies, one large and one small, of different chemical composition. For simplicity, only the larger one will be taken to be the source of the gravitational field. We shall see that under these and other simplifying assumptions concerning the equation of motion for the chameleon, even when considering a universal coupling and the suppression due to the metal encasing of the experimental setup, violations of the WEP do arise for some values of the chameleon field parameters (see Figs. 9, 10 and 12 in Sec. V). This result, although previously known, may seem rather counter-intuitive as in this scenario all the various metrics coalesce into a single (Jordan) metric (see Sec. II for the details). These violations are then put in perspective with the results found by various authors using different approaches and approximations (see Sec. V).

The paper is organized as follows. After briefly reviewing the original chameleon model in Section II, we present the method for computing the two body problem proposed in this work. Then, in Section III we develop a simple criterion based on the use of an energy functional for which the minimization corresponds to the field configuration of the physical solution, to determine when the method proposed in this paper is better than the one employed in the standard approach and when it is not. That method is used to identify the range of the model’s parameters where our results should be considered as more trustworthy that the results obtained in previous studies. In Section IV we compute the force on the test body and show that the force is not negligible and, to the extent that there is no protective symmetry to prevent it, it will lead to an acceleration that depends on the test body’s composition. In Section V we apply our results to two concrete experiments, the Eöt-Wash torsion balance and the Lunar Laser Ranging (LLR). For the first one, our discussion incorporates the characterization of the outside medium and the effects of the vacuum chamber’s encasing shell. In both cases we provide numerical estimates for the Eötvös parameter. In the former case we find that the actual experimental setting generates complications in the numerical part of our analysis. Nevertheless, we provide our estimates for the bounds including rough estimates of the corrections resulting from the specific experimental setup, which we consider can serve as motivation for further analytical and experimental work. For instance, we explore the estimates for the experimental bounds that arise from considering two different modelings for the outside medium. We also explore the modifications that arise from including the effect of the metal shell, showing that in some regimes the violations of the WEP can be significatively suppressed for (we are indebted to P. Brax for pointing this specific aspect to us). Finally, in Section VI we present our conclusions and a discussion about their impact on other alternative theories of gravity. Several appendices complete the ideas of the main sections.

Ii The Chameleon Model

In this section we briefly review the main aspects of the chameleon model. The model involves a scalar-field that couples minimally to gravity via a fiducial metric , according to the following action

 S[gμν,Ψ(i)m,φ]=∫d4x√−g[M2pl2R−12gμν(∇μφ)(∇νφ)−V(φ)]−∫d4xLm(Ψ(i)m,g(i)μν), (1)

where is the reduced Planck mass, is the Ricci scalar associated with , and represents schematically the different matter fields (i.e. the fields other than the chameleon ; for instance, all the fundamental fields of the standard model of particle physics). The potential is specified below. The particular feature of this action is that each specie of matter couples minimally to its corresponding metric , while the scalar field couples non-minimally, and in general, non-universally to the matter through a conformal factor that relates each metric with the so called Einstein metric :

 g(i)μν=exp[2βiφMpl]gμν. (2)

Here is the metric which is usually associated with the geodesics of each specie of matter and is the corresponding coupling constant between each specie and the chameleon field. For instance, when the coupling constants are universal, , then the (universal) metric is called the Jordan metric, and then point particles would follow the geodesics of this metric. We will see, however, that a more detailed analysis indicates that even in the universal coupling case, within the relevant experimental situations, that simple conclusion need not apply. Given the fact that, in general, the chameleon field couples differently to each specie of particle, this model leads to potential violations of the WEP that could in principle be explored experimentally.

Following KW04 (); MS07 (); K13 (), we consider the potential for the chameleon field to be,

 V(φ)=λM4+nφ−n, (3)

where is a constant, and is a free parameter that can be taken to be either a positive integer or a negative even integer [cf. Eq. (12)] ( for all values of except when where .).

The energy-momentum tensor (EMT), , for the i the matter component can be written in terms of the EMT, associated with the Einstein metric as follows:

 Tm\vspace5mm(i)μν=+2√−g(i)δLmδgμν(i)=exp[−2βiφMpl]Tmμν, (4)

where is defined similarly from but using the metric instead of .

Consequently, the traces of both EMT’s are related by

 Tm\vspace5mm(i)=gμν(i)Tm\vspace5mm(i)μν=exp[−4βiφMpl]gμνTmμν=exp[−4βiφMpl]Tm. (5)

Eventually, we will assume a perfect-fluid description for . Specifically, this perfect fluid will be taken to characterize each of two extended bodies, together with the matter constituting the environment, all of which we consider as the “source” of the chameleon field. Furthermore, in the analysis of the resulting force between the two bodies, we will simply consider a universal coupling and set , in order to simplify the calculations. Hereafter, and unless otherwise indicated, all the differential operators and tensorial quantities are associated with the Einstein metric.

The equation of motion for the the chameleon which arises from the action (1) is

 □φ=∂Veff∂φ, (6)

where represents the effective potential defined by:

 Veff=V(φ)−TmβφMpl, (7)

which depends on the energy-density and pressure of the matter fields via . So for the perfect fluid model, which does not depend explicitly on .

ii.1 Modelling the experimental setup

As we mentioned earlier, we are interested in the general static solution of Eq. (6) in the presence of two extended bodies that for simplicity we consider as spherical. We take one of them to be a very large body of mass (e.g. Earth, Sun, a mountain), and the other one is a smaller body of mass (; hereafter test body). Both bodies are the source of the chameleon, and each one is taken to have a different but uniform density, which formally can be represented by suitable Heaviside (step) functions. Moreover, we shall consider the linear approximation in Eq. (6) obtained by linearizing Eq. (7) around (associated with the minimum of ) in each of the three mediums (i.e. the two bodies and the environment) and then match the solutions at the border of each of the two bodies. This is similar, but not identical to the standard method of analysis of these models (see Appendix E).

Now prior to tackling the two body problem for the chameleon field, it is crucial to review some of the main relevant aspects of the one body problem, i.e., the situation where the test body does not back react on the scalar-field configuration, because some of those approximations will be used in the treatment of the two body problem. However, in our approach, it is essential that the back reaction of the field to the presence of the two bodies, be fully taken into account.

Let us consider a spherically-symmetric and homogeneous body of radius and density immersed in an external medium of density . We will call this body, the larger body, as opposed to the smaller body that will be introduced later. The corresponding EMT’s, are assumed to be of a perfect fluid: , where the scripts will refer to the interior (exterior) of the body, respectively. In this case we will consider two regions (interior and exterior), and since for non-relativistic matter, we neglect the pressure. Consequently,

 ρ=⎧⎨⎩ρinr≤Rρoutr>R, (8)

where is the radius of the body.

In order to solve the chameleon equation Eq.(6) with the effective potential (7) we can adopt several possible strategies. Clearly, one consists in solving numerically the full non-linear chameleon equation without any approximation. In the case of a one body problem in spherical symmetry (where the test body does not backreact on the chameleon field), the chameleon equation becomes an ordinary differential equation and it can be solved using a Runge-Kutta scheme. Indeed, this was done by Khoury & Weltman KW04 () in their pioneering paper, in order to check the analytical solution for the one body problem. Besides, Elder et al Elder16 () and Schlogel et al Schlogel16 (), also solved numerically the chameleon equation to study atomic interferometry experiments. For this experimental situation, the test body is of the size of an atom, and therefore it is appropriate to calculate the force using the one body problem solution.

However, in the two-body problem where the test body is considered not as a point particle but as an extended body which backreacts on the chameleon field, solving the full non-linear chameleon equation entails to solving a very complicated partial differential elliptic equation. In order to advance in this direction, we postpone this analysis for the future, and consider to solve a linear elliptic equation by approximating the effective potential. To this end, there are two possible approximations. These depend actually on the specific setup, like the size of the bodies, their density, the environment’s density, and the actual value of the coupling . The values of these parameters determine, for example, if the bodies have or not a thin shell or even an intermediate situation where the shell is not thin nor thick. The thin (thick) shell dimensionless parameter (cf. Section E) allows us to determine in which regime the large or the test body is. Namely, if the body has a thin shell, whereas corresponds to a thick shell regime. Thus, depending on each of the two regimes one can approximate differently the effective potential.

If the thin shell condition is satisfied, the expansion of the effective potential about its minimum up to the quadratic terms, for each one of the regions, as all higher order terms are ignored provides a good approximation for the determination of the scalar field profile (see for example, KW04 (); Brax07 (); Hui2009 (); Brax2012 ()). Such approximation leads to a linear differential equation for the chameleon field. In a forthcoming paper we plan to study in detail the corrections to this approximation arising from higher order terms Oursnonlin (). We will see that the corrections become relevant for  Oursnonlin ()555We thank A. Upadhye, B. Elder and J. Khoury for raising this issue after the first submission of this manuscript.. Next, we will describe the quadratic approximation to the effective potential and at the end of this subsection, we will discuss if it is appropriate for the experimental situations at hand .

We now proceed to deal with the simple, one body problem, taken to be immersed within a single medium acting as the environment. In section IIB, we generalize this expansion for the “two body problem”.

The expansion of the effective potential about the corresponding minimum in each region up to the quadratic term gives:

 Vin,outeff(φ)≃Vin,outeff(φin,outmin)+12∂φφVin,outeff(φin,outmin)[φ−φin,outmin]2, (9)

The effective mass of the chameleon is then defined in the usual way:

 m2in,outeff(φin,outmin,βi,Tmin,out)=∂φφVin,outeff(φin,outmin). (10)

In particular, setting , the expression for the effective mass turns out to be:

 (11)

with

 φin,outmin=M(nλMplM3−βTmin,out)1n+1. (12)

Expressions (11) and (12) are valid when is a positive integer or a negative even integer, as otherwise a minimum does not exist. Furthermore, for the effective mass does not depend on the composition of the body or the environment.

Now, let us consider the interesting situation where we have two bodies, the large one and the test body. The effective potential must be expended in various regions: the interiors of the two bodies and their exterior which is associated with the environment. Typically the density of the large body is much larger than the density of the environment (), thus taking leads to . The effective potential inside the large body develops a minimum at which is much larger than the minimum of the effective potential at associated with the environment (see Fig. 1). The difference between the and , which can be very large, depends on the density of the materials involved and the value of model’s parameters. This, in turn, can lead to different types of approximations used to describe . For instance, since we take the environment to extend to infinity, the chameleon field must be such that it reaches at spatial infinity. Therefore the field must interpolate between at spatial infinity and at the centers of the each of the two bodies, both modelled as spherical (see Figure 2). We emphasize that this central values need not coincide with the actual minima of the function in the bodies’ interior. Now, roughly speaking the chameleon field rolls up the potential from its minimum at to the value where the field reaches the surface of the large body (respectively, the test body), where stands schematically for the location of the border of the body. However, and this is an essential aspect of the present analysis, the field configuration is not really constant and it is precisely the resulting directional dependence that is associated with a non-vanishing force (in the direction of the large body) on the test body.

Now, when we can rely on the quadratic expansion about its minimum as a good approximation to since the field outside the bodies takes values that are always close to the minimum . We report results only for those situations where the energy criterion favors our approximations over that of the standard approach. Now, at the border of the bodies, the effective potential experiences a discontinuous jump (due to the jump between the body’s and the environment’s densities) and at we have to use an approximate expression for instead of one for in order to solve the chameleon equation inside the body. However, as is standard in these cases, the field itself ( together with its first derivative) must be continuous everywhere, and in particular at the surface of the bodies. As one moves towards the interior of the bodies, the field starts to “roll down” the potential towards its minimum, but without necessarily reaching it. More specifically, the field configuration interpolates from its surface value and the value at the center of the body. If , it means that the field is already near the minimum of the potential inside the body and the quadratic expansion about the minima is a very good one. This situation is emblematic of a thin shell regime. On the other hand, when , we have a situation where the field at the body’s surface is very far from its minima, and thus, the quadratic approximation for is not necessarily a very good one. In that case, however, is usually dominated by the matter dependent part, and then one may use the approximation  KW04 (). This happens in some situations in which the large body falls within the thick shell regime; as for the example the Sun with , eV and .

In the present paper we shall focus mainly in scenarios where the bodies are within thin shell regime, and thus, where the quadratic approximation to the effective potential is adequate. In Section III we shall develop a criteria to assess the extent to which this approximation is in fact a good one using a minimization of a suitable energy functional.

ii.2 The two body problem

We proceed with the calculation of the chameleon in the presence of two bodies without, in principle, neglecting any contribution, but still within the quadratic approximation for the effective potential discussed before. The geometry of the problem is depicted in Fig. 2.

In order to do so, we expand the most general solution in complete sets of solutions of the differential equation in the inside and outside regions determined by the two bodies. Thus we write,

 (13)

where , , and are the radii of the large and test bodies, respectively, and and are Modified Spherical Bessel Functions (MSBF). In the above equation, the Cartesian coordinate system ,, is centered in the large body, while the coordinate system ,, is centered in the test body (see Fig. 2). Notice that in writing Eq. (13) we have taken into account the regularity conditions of the scalar field at the center of both bodies. That is, the MSBF used in the expansions for the solutions inside both bodies are well behaved within their corresponding compact supports. Conversely, for the exterior solution (i.e. the solution outside both bodies) we employ the set of MSBF functions associated with each body which are well behaved at infinity. Furthermore, the coefficients of Eq. (13) are calculated using the following continuity conditions for the field and its derivative at the boundaries of the two bodies:

 φin1(r=R1)=φout(r=R1),∂rφin1(r=R1)=∂rφout(r=R1),
 φin2(r′=R2)=φout(r′=R2),∂r′φin2(r′=R2)=∂r′φout(r′=R2);

In order to describe properly the two body problem in a single coordinate system we can use the following relationship ( see Figure 3) that links the special functions in the two coordinate systems Scatt (); Scatt2 ():

 kl(μoutr)Ylm(θ,ϕ)=∑vwα∗lmvwiv(μoutr′)Yvw(θ′,ϕ′), (14)

where the coefficients can be expressed as follows:

 α∗lmvw=(−1)m+v(2v+1)|l+v|∑p=|l−v|(−1)−p(2p+1)[(l+m)!(v+w)!(p−m−w)!(l−m)!(v−w)!(p+m+w)!]1/2×[lvp000][lvpmw−m−w]kp(μout|D|)Yp(m−w)(θD,ϕD), (15)

being and the angular coordinates associated with the vector (see Fig. 3) and we remind the reader that is the distance between the center of the two bodies.

Moreover, we can use a similar relationship between the special functions given in terms of the coordinates centered in the test body with the ones centered in the large body:

 kl(μoutr′)Ylm(θ′,ϕ′)=∑vwαlmvwiv(μoutr)Yvw(θ,ϕ), (16)

where

 αlmvw=(−1)m+v(2v+1)|l+v|∑p=|l−v|(2p+1)[(l+m)!(v+w)!(p−m−w)!(l−m)!(v−w)!(p+m+w)!]1/2×[lvp000][lvpmw−m−w]kp(μout|D|)Yp(m−w)(θD,ϕD), (17)

valid for and . We make use of the axial symmetry of the problem and thus set the axis as containing the centers of the two bodies (see Fig.2). Thus, the coordinate transformation is, in this case, a translation along this axis. Therefore, and becomes irrelevant do to the axial symmetry. An approximate expression for the chameleon field is found by truncating the infinite series Eqs. (14) and (16). We further note that as shown in Refs.Gume1 (); Gume2 (), the series of this type can be estimated by truncating the sum after the first terms, with given by the integer part of where is Euler’s number. Using this last result we can solve the following two equations for and ; the first one reads:

 b1δl0=Cout1lal+N∑w=0Cout2wzwl, (18)

where

 al = k′l(μoutR1)il(μ1R1)i′l(μ1R1)−kl(μoutR1), (19) zwl = αw0l0[il(μoutR1)−i′l(μoutR1)il(μ1R1)i′l(μ1R1)], (20) b1 = √4π(φ∞−φin1min). (21)

where a prime’ indicates differentiation of the MSBF with respect to its argument.

 b2δl0=Cout2lxl+N∑w=0Cout1wywl, (22)

where

 xl = k′l(μoutR2)il(μ2R2)i′l(μ2R2)−kl(μoutR2), (23) ywl = α∗w0l0[il(μoutR2)−i′l(μoutR2)il(μ2R2)i′l(μ2R2)], (24) b2 = √4π(φ∞−φin2min). (25)

We can now write the system of equations for the coefficients and associated with the interior solutions in terms of the coefficients of the exterior solution:

 Cin1li′l(μ1R1) = Cout1lk′l(μoutR1)+N∑w=0Cout2wαw0l0i′l(μoutR1), (26) Cin2li′l(μ2R2) = Cout2lk′l(μoutR2)+N∑w=0Cout1wα∗w0l0i′l(μoutR2). (27)

In this way, by solving the system of equations (18) and (22) we can obtain a solution of the two body problem for the chameleon where we perform an approximation (within the quadratic approach of the effective potential) consisting in the truncation of the series used for the transformation of coordinates. The dependence of the field on the composition of the test body appears through the constants and .

Figure 4 depicts the field around the centers of the two bodies and outside them along the axis which results from our method by cutting off the series expansions after taking the first three terms in each one666The same result is obtained if only the first term of the series is used as this turns out to be many orders of magnitude larger than the following terms.. We also show, for comparison, the field obtained by Khoury & Weltman KW04 () using the analytical solution to the linearized time-independent chameleon field equation for one body (we will refer to this solution as that of the standard approach)777We have verified that this solution agrees to a very good approximation with the numerical solution to the non-linear chameleon field equation for one body.. Note that under the conditions , which are valid within each body, the thin shell effect appears in both bodies. Namely, the field is almost constant with values close to in each body, respectively, and grows exponentially near the surfaces. As becomes smaller, the thin shell disappears. Moreover, Figure 4 (right panel) compares the chameleon field obtained within the framework of the standard approach with the corresponding field obtained with our two-body method in the region where the test body is located. In order to check our method, we compute the solution for the one body problem using our approach, and verify that we recover the same results as in Ref. KW04 ().

Iii Minimum Energy criterion

It is well known that the classical solution of a field equation corresponds to an extreme of the action (with fixed boundary conditions). When we are interested in static solutions, such extreme coincides with the field configuration that minimizes the energy functional. That well known result is used in this section to construct an appropriate energy functional associated with the class of test field configurations which can be used to compare the quality of various approximations to the true solution of the differential equation satisfied by the scalar field. This will empower us to determine based on an objective quantitative criteria when our analysis should be trusted over other treatments and when the opposite is true. In the next section, we obtain the chameleon mediated force from the variation with distance of this functional.

A first naive attempt to find consists in considering the time-time component of the energy-momentum tensor associated with the system, which in turn, is the source of the Einstein’s field equations of the model. Thus, we focus on the gravitational field equations that are obtained from varying the action (1) with respect to 888From the Bianchi identities followed by the use of Eq. (6) one can see that the EMT of matter is not conserved in the Einstein frame: ,where is the conformal factor between the Einstein-frame metric and the geodesic metrics [cf. Eq. (2)]. We then obtain . The right-hand side of this equation is precisely related with the chameleon force [cf. Eq. (50)].

 Gμν = 1M2pl(Tφμν+Tmμν), (28) Tφμν = (∇μφ)(∇νφ)−gμν[12gαβ(∇αφ)(∇βφ)+V(φ)], (29)

where, as before, we assume for the EMT of matter. Now the energy associated with the total EMT under the assumptions of staticity and flat space-time is

 U = ∫V(Tφ00+Tm00)dV=∫V[12(∇iφ)(∇iφ)+V(φ)+ρ]dV, (30)

where the time components of both EMT’s are taken with respect to an observer that is static relative to the two-body configuration 999That is, we take a reference frame defined by the unit time-like vector (4-velocity) as to coincide with , such that , where . The staticity assumption translates into ..

While this energy functional certainly contains the total energy of the system from the Einstein-frame point of view, the extremization of this functional with respect to with fixed densities (i.e., as considered independent of ) does not lead to the static chameleon equation that is obtained from Eq.(6). That is, we require an energy functional extremized by the actual field configuration. As a consequence we turn our attention to the more appealing functional given by

 Ueff = ∫V[12(∇iφ)(∇iφ)+Veff(φ)]dV, (31)

where the integral is computed over the whole Euclidean three-dimensional space (i.e. ) and where the effective potential at each point of space is the one corresponding to the density of matter at that point (associated with each of the bodies and the media respectively) and which is given by (7).

We can rewrite the functional (31) integrating by parts , which leads to

 Ueff=∫V[−12φ∇2φ+Veff(φ)]dV, (32)

where we discarded the irrelevant surface term which must vanish at infinity.

One issue that is extremely important to have in mind is the fact that the validity of our analysis does depend on the accuracy of approximation used in the expansion of the effective potential for the scalar field in each region of space. In the present work, as we have explained earlier, we have used quadratic expansions around the minimum of the effective potential in each one of the regions involved. This should clearly be a very good approximation when all the bodies involved (including the media between the solid bodies) are in the so called deep thin shell regime. However, that regime is only identified heuristically, and sharp boundaries delimiting the exact regions in parameter space simply do not exist, and in fact should not exist as the transition between thin and thick shell regimes must undoubtedly be a smooth one. This faces us with a problem when trying to establish when our results are more trustworthy that the existing ones. The problem would of course be resolved if one had exact solutions for the complete specific problem at hand, involving all the bodies present in the actual experimental situation, or one that included at least the most relevant ones, namely the source body, the test body, and the media in between them. We note that the relatively simple treatments that consider exact solutions for just one body surrounded by a media and then use the gradient of the resulting scalar field at the location of the second body while ignoring the effect of such body on the field itself, and then attempt to correct for the so called thin shell effect by introducing a simple multiplicative factor extracted from consideration of another one body problem, cannot be considered a priori more reliable than our method. However, we must recognize, of course, that it is possible that under some circumstances, those analyses might provide better estimates than ours. That would of course correspond to situations where our second order expansion fails to provide an accurate enough characterization of the potential in the regime explored by the actual scalar field configuration. That can lead to situations where it is not clear which results should one trust.

Fortunately there is a simple method to discriminate between two approximated solutions to the static field configuration corresponding to a given distribution of sources and media. We refer here to the fact that the field configuration corresponding to the solution is an extreme of the action functional which, for static situations, correspond to the minima of the energy functional. The point is then that when faced with two approximations to a given problem one can determine which one is a better approximation by comparing the value of the energy functional of the two configurations. Of course in making this comparison it is essential that one uses the same energy functional and fixes the relevant bodies and interpolating media to be exactly the same when making the energetic comparison. The configuration with the lowest value of the energy functional provides a better approximation and thus should be better trusted. Of course ideally one would prefer an exact solution but lacking that, we must rely on the better one of the approximations.

We have carried out precisely such analysis in order to compare the field configurations emerging from our analysis with the field configurations obtained by Khoury & Weltman KW04 (). As mentioned above, the relevant energy functional is given by Eq.(32). We note by looking at this equation that the second term of the integrand will contain a constant corresponding to the minimum of the effective potential which is determined by the environment. This constant does not affect the calculation of the chameleon force between the bodies because the force is derived from the variation of the potential with respect to the bodies’ separation and the contribution from such a constant, as long as the bodies are not deformed, is independent of the separation. On the other, the integral of such constant in the whole space, extending to infinity, would lead to an infinite contribution for the energy. In any event it is clear that such term is irrelevant in the determination of the force between two bodies 101010 In the realistic context of the full fledge chameleon theory that term would have to be regarded as a contribution to the “cosmological constant”.. In view of this, and in order to work with only finite energy functionals, we proceed to “renormalize” our expressions by subtracting the divergent term, and considering just,

 U∗eff=∫V[−12φ∇2φ+Veff(φ)−Veff(φoutmin)]dV. (33)

In this paper, we analyse two experimental situations: i) the Eöt-Wash torsion balance: an Earth based experiment where the large body is a mountain and the test bodies are centimeter-size metal sphere and ii) the Lunar Laser Ranging experiment where the large body is represented by the Sun and the test bodies by the Earth and the Moon. In each scenario the bodies are surrounded by an environment, which is described in more detail in Sections V.1 and V.2.

For each of the experimental situations considered in detail in this work we have computed the value of the above functional Eq.(33) corresponding to the two body problem using the field configuration obtained with our method (described in Section II.2) and that corresponding to the field configuration obtained in the standard approach. We remind the reader that in such approach (as exemplified by the work of Khoury and Weltman) the effect of the small body is ignored when determining the scalar field configuration, and the force on the latter is estimated by simply considering the gradient of the field corresponding to the large body and the environment at the location of the small body (see Appendix B) with some additional factors which are related with the thin shell parameter . Moreover it often relies on a approximated expression for the effective potential that differs from that the one we employ (see Appendix E).111111As regards the expressions obtained by Mota & Shaw MS07 () for the chameleon field, we could not apply the energy criterion since the explicit expression for the chameleon field is not reported in their work but they just present an expression of the derived force. On the other hand, a solution considering the contribution of the test body was proposed by Hui et al Hui2009 () and used in Refs. Brax2010 (); Burrage15 () to calculate the Eötvös parameter. In that work, a superposition of the one body problem solution of both the large and the test bodies is considered. Furthermore, the authors claim that this solution is valid outside both bodies. However, no expression for the field inside the bodies is provided. Therefore, it is not possible to apply the energy criterion proposed in this section to the above mentioned solution 121212 Nevertheless, we applied the energy criterion to the region where the field is defined and found that our solution has lower value of the energy functional. The values of shown in Section V are computed from Eqs. 41a and 41b. However the reader should take into account that using such expressions for within the Khoury and Weltmann approach yields strictly speaking for universal given the fact that in such analysis the test body is treated as a point particle.

Figure 5 shows the results for the Eöt-Wash torsion balance. The test body (Al) and the source (the mountain) are taken as immersed in the same environment when computing the energy of the field configurations. We considered the cases meV (the cosmological chameleon) with as the density of the vacuum-chamber while for the case eV, was taken as the atmosphere’s density. For the case meV (left panel) the relative difference of the energy functional is small but the energy of the field configuration obtained with our method is always smaller. We have also checked that similar results are obtained for various values of . On the other hand, for the case eV, the situation is different. Figure 5 (right panel) shows that for and the energy criterion indicates that our approach yields an energy that is much larger than the corresponding energy obtained in the standard approach which indicates that the latter is much trustworthy. On the other hand, for the two body problem with the quadratic approximation for is a better solution to the exact problem than the one obtained from the standard approach131313For the relative difference is of order but always positive. This cannot be appreciated from the left panel of Figure 5 due to the scale of the plot. For each of the values of considered here such transition takes place at a different value of (see Table 3 in Section V). It should be noted, that we could not apply our energy criterion to the most realistic case where one includes the metal shell of the vacuum chamber in the modeling for the Eötvös torsion balance experiment. This is because we cannot compare with the standard approach since, as far as we are aware, it does not usually include the explicit determination of the chameleon field’s profile corresponding to this aspect of the experimental device. 141414Ref. Upadhye12 () obtains a numerical solution for the torsion pendulum which has a different geometry than the torsion balance analyzed in this paper and estimates the effect of including a shell in the value of the force. Most analyses focused in the Eötvös torsion balance do not include the metal encasing in the calculation of the field and/or the force KW04 (); Brax2010 (); MS07 (); Hui2009 (); Brax2012 (). 151515In section V.1, we estimate the value of for the standard approach including the effect of the metal shell through multiplication of a correction term as suggested by Upadhye12 ().. Figure 6 compares the values of the energy functional computed for the LLR experiment obtained from our method and from the standard approach taking eV. The density of the environment is assumed to be the one of the interstellar medium and the vertical dotted lines show the onset of the thin shell condition for Earth and Sun respectively (for values of lower than the corresponding to vertical lines, the thin shell condition does not hold). For the cosmological chameleon, the energy criterion indicates that our solution is a better approximation than the one obtained in standard approach for and for the values of considered here 161616 For meV (the cosmological chameleon) and the values that we test, the thin shell holds always for the Sun, Earth and Moon. . Moreover, for eV we find that for each of the tested values of , there is a region (which corresponds to the case where the large body satisfies the thin shell condition, while the test body may not) where the energy criterion indicates that the field profile obtained by our method is a better solution than the one obtained in the standard approach. On the other hand, for the values excluded from that region (this is the case where the large body fails to satisfy the thin shell condition), the KW approach gives a better result (see Fig.6) .

We will limit the results of this paper to the case where the quadratic approximation to the effective potential is a good one and leave for a future work Krai17 () the case where another approximation has to be considered. It should be stressed, that for both experimental situations there are ranges of the model’s parameters for which despite the fact that the test body does not satisfy thin shell condition, the energy criterion indicates that the scalar field profile obtained by considering the full two body problem within the quadratic approximation for the effective potential yields a better approximation to the exact solution than the one used within the standard approach. On the other hand, when the thin shell condition does not hold for the large body, the energy criterion always indicates that the standard approach yields a better approximation than the one obtained by the method proposed in this work, i.e., indicating that it is essential to employ something like the thick shell approximation for the effective potential.

Iv Chameleon mediated force between two spherical objects

In this section we calculate from first principles the effective chameleon force between the large and the test bodies. This force, together with the gravitational force on the test body, will be considered as characterizing a “free falling” test body in laboratory conditions under the influence of both interactions. The expression for this force together with the chameleon field computed in sections V.1 and V.2 will allow us to compute numerically the Eötvös parameter associated with the acceleration of two test bodies of different composition and to estimate the magnitude of the predicted violation of WEP.

As we mentioned before, the variation of the functional of Eq. (32) with respect to with the densities of the bodies fixed, leads to the actual equation for the chameleon for a static problem:

 ∇2φ=∂Veff∂φ, (34)

where stands for the Laplacian operator in three dimensional Euclidean space.

We can rewrite the functional (31) integrating by parts and using Eq.(34), which leads to

 Ueff=∫V[−12φ∇2φ+Veff(φ)]dV=∫V[−12φ∂φVeff(φ)+Veff(φ)]dV, (35)

where as usual, we discarded the surface term.

Using Eq,(7) we obtain after some simplifications

 Ueff=∫V[12Veff(φ)(2+n)+(1+n)βφTm2Mpl]dV. (36)

At this point the energy functional is exact. However, we now use in this functional the quadratic approximation for about its minimum in each region of space (i.e. inside and outside the bodies).

Our goal is to work with the difference rather with alone given that vanishes at infinity. According to this, Eq. 36 becomes,

 Ueff=∫V[(2+n)m2eff4^φ2+(1+n)βTm^φ2Mpl+(1+n)βTmφmin2Mpl+12Vmineff(2+n)]dV. (37)

The only approximation we have made so far for the energy of the whole system is to replace the effective potential of the chameleon with the corresponding expansion around its minimum in each of the three regions (i.e. the two bodies and the environment).

We can now compute the chameleon mediated force using , where is the distance between the center of the two bodies. To this aim, we first notice that the last two terms within the integral of Eq. (37) are independent of the separation of the bodies, and thus, do not contribute to the force, i.e.,  171717As usual, the integral (in all the space) is an infinite constant..

In order to calculate the total energy, we have to consider the contributions due to the chameleon in the regions inside the two bodies and in the region exterior to both bodies. To do so, let us define as the region corresponding to the large body, as the region corresponding to the test body, and is the region exterior to the large body in the coordinate system centered in the large body:

 V3=⎧⎨⎩R1≤r≤∞0≤θ≤π0≤φ≤2π (38)

Thus, the relevant energy functional can be written as:

 Ueff = ∫large bodyt00[^φ(V1)]dV1+∫test bodyt00[^φ(V2)]dV2+∫outside bodiest00[^φ(out)]dVout (39) = ∫V1t00[^φ(V1)]dV1+∫V2t00[^φ(V2)]dV2+∫V3t00[^φ(out)]dV3−∫V2t00[^φ(out)]dV2

where corresponds to the first two terms of Eq. (37) which are the responsible for the force between the two bodies. Here refers to the field outside the two bodies, and because of this definition and the arrangement of the integrals, we have to subtract the last term. In consequence, the total force can be expressed as:

 Fzφ = (40) + ∂∂D∫V1{^φin1(→r)[(n+1)2ρ1βMpl]−(2+n)μ21^φ2in1(→r)4}d3→r + ∂∂D∫V3{^φout(→r)[(n+1)2ρoutβMpl]−(2+n)μ2out^φ2out(→r)4}d3→r − ∂∂D∫V2{^φout(→r′)[(n+1)2ρoutβMpl]−(2+n)μ2out^φ2out(→r′)4}d3→r′;

where we used , and and are the densities of the large and the test bodies, respectively, and is the density of the environment (i.e. the density outside both bodies). We stress that the terms quadratic in the field appearing in Eq. (40) are not negligible in comparison to the corresponding linear terms, as our detailed numerical calculations show. In fact both terms turn out to be of the same order of magnitude.

In the last two integrals of Eq. (40), we have to rewrite the chameleon in terms of the corresponding coordinates adapted to each body Gume1 (); Gume2 (); Scatt (); Scatt2 (). Moreover, we neglect the contribution of the second part of given by , to the last integral [i.e. we take ] because that contribution turns out to be negligible in comparison with the other one.

Each integral of the above equation has a linear term in and a squared term (). Therefore, we separate in two terms, a linear one and a squared one such that . The final expressions for such terms that provide the chameleon mediated effective force on the test body are not very enlightening, and the steps leading to them are rather cumbersome and technical (see Appendix C). However, the most important point to be stressed, and which constitute the basic result of this paper, is that the expressions for and show an explicit dependence with the composition and size of the test body. Therefore, and as already noted, the WEP is violated in principle in these kind of models even in the case of an universal coupling.

In the next section we use these results in order to evaluate numerically the extent to which this effective force is suppressed by the thin shell effects in the two bodies, and also by other effects induced by the presence of additional objects in the setup. Then we estimate the Eötvös parameter and confront the outcome with the bounds imposed by the current data associated with the Earth based experiments and the LLR.

The general formulae that we have provided can be used for the evaluation of the force to any desired degree of precision. This can be done by following the steps we have presented in previous sections and by adjusting the level of approximations we have used. The approximations involved are of course the cut-off in the series expansion, which can straightforwardly be continued to any desired order.

V Applications

It is important to point out that the geometry of the actual Earth based experiments that have been used to test the WEP, such as those using torsion balances is much more complex than what our simple model depicts. In particular, the inclusion of the other material bodies that are present in the laboratory near the torsion balance could drastically modify the effective chameleon force acting on the test bodies, and their incorporation might led to important changes in the theoretical predictions of the Eötvös parameter. For instance, the environment that separates the torsion balance (that contains the test bodies) from the hill (which in the simple one body approach is the only source of the chameleon field) does not consist of just a vacuum and the Earth’s atmosphere, but includes also a metal case and other objects located in the proximity of the experimental equipment. Those complications can often be ignored when one is considering linear fields which couple in a non-universal way to matter, and whose presence can led to effective violations of the WEP just as in the original Fifth Force proposals Will14 (). However, when dealing with highly non-linear models as the one studied here, the situation might be more complicated. Therefore one might need, in principle, a very detailed model for the actual in the laboratory in order to take into account the true effect of the environment around the two bodies (the large body and the test body) used in the test of the WEP. This would require the modeling of the matter distribution in great detail, and then facing the very difficult task of solving a complicated non-linear partial differential equation with a rather intricate boundary conditions. In order to advance in this direction and to make the calculations feasible, we are forced to ignore some of these complications while incorporating in the modeling of the situation some of the most important features of the experiment. For instance, we have analyzed the extent to which different environments affect the resulting chameleon force between the two bodies.

In the following we will proceed in two stages. In the first one we consider the simplest model for as given by Eq. (8). First we take two type of environments represented by : one is given by the chamber’s vacuum density, and the other is given by the Earth’s atmosphere. In the second stage we reanalyzed the results of the first stage, after taking into account, in a self-consistent manner, the encasing’s material (which we consider to be of spherical shape) around the test body, as a manner to more accurately characterize the vacuum chamber used in the Eöt-Wash experiment that shields the test bodies (see Sec. V.1.2181818We thank P. Brax for calling our attention to this aspect of the experiment.. Regarding this point, we need to emphasize that, in previous works Upadhye12 (), the contribution of this metal encasing has been modelled together with the two bodies, the low density environments and the metal encasing as planar slabs (in contrast to our more realistic modeling based on spherical like objects). That is, the setting used in Upadhye12 () corresponds to a “one dimensional planar” model191919It should be noted that the analysis of Ref. Upadhye12 () is devoted to the torsion pendulum experiment rather than the torsion balance here considered. Needless is to say that the two are not exactly the same..

Now, in the scenario where one does not consider such metal casing, the model could be applied to at least two kind of experiments: i) a laboratory experiment similar to the Eöt-Wash experiment but without the metal shell. This scenario, as far as we know, has not been implemented in an actual high precision Earth based experiment, and therefore our estimates cannot be used directly, at this time, to set relevant bounds. However, combined with rough estimates of the suppression effects, these already show that the appearance of the thin shell effects are by themselves insufficient to suppress the observable violations of the WEP for some values of the parameters; ii) an actual space-based experiment like the LLR where the Sun plays the role of the large body and the Earth or Moon represent the test bodies. For this second situation the use of our analysis would mean one is neglecting the effects of an actual three body problem where the Sun, the Moon and the Earth collectively determine the scalar-field. A much more precise study would require taking the effects of these three bodies into account simultaneously and self-consistently. We have not done so and instead have relied on an analysis based on two body system as an approximation. We consider this to be a reasonable approximation due to the fact that despite the intrinsic non-linearity of the model, the change due to the presence of the Earth on the value of the scalar field at the Moon’s surface is sub-leading to that of the Sun. Thus, in our estimates, when computing the acceleration of the Earth towards the Sun we neglect the presence of the Moon and viceversa. This by the way, is a standard approximation used widely in the community studying these questions.

Now we illustrate the usefulness of the analytic expressions for the force in situations of experimental relevance.

v.1 The Eöt-Wash Torsion Balance Experiments

v.1.1 An idealized experiment without a vacuum chamber

One the most precise experiments on this line consists of a continuously rotating torsion balance which is used to measure the acceleration difference toward a large source (like the Earth, a lake, or a mountain) of test bodies with the same mass but different composition. In addition, as we have already mentioned, the Eöt-Wash experiment includes a vacuum chamber that encases the test bodies, and which is shielded by a metal encasing. We shall first consider a model of a simplified “Eöt-Wash” scenario where the metal encasing is absent (or ignored). The set up considered in previous sections correspond to this idealized experiment. Despite the fact that such scenario does not represent the realistic experimental set up at present, we have decided to consider it in order to motivate the realization of an experiment that avoids the inclusion of such vacuum chamber encasing, providing the theoretical bounds one might be able to achieve by a relatively simple modification of the current experiment.

Thus we will compute the Eötvös parameter associated with the differential acceleration of two bodies of different composition. This parameter is given by , where () is the acceleration of the test body due to the chameleon force , and the force of gravity , which is basically due to the gravitational field produced by the large body. The acceleration on the small body is computed using Eq. (40).

As we emphasized before, in all the cases that we have analyzed we find that, the linear and the quadratic terms of the chameleon field in the expression of the total energy (and hence the force) are comparable, so neither of them can be ignored. In most cases, the “optimal ” cutoff in the series expansion, estimated as the integer part of turns out to be zero (cf. Section II.2). Thus, we keep only the first term of each sum so as to avoid numerical instabilities. The value for required to improve the accuracy in the solution increases with increasing and , the effect being more pronounced for dependence on . Furthermore, the value of also increases with the density of the outside medium . In fact, in some cases, the value needed for is so large that it effectively impedes the calculation.

Moreover, regarding the summations over the infinite ranges for and in the series of Eq. (64) we have checked the rate of convergence and found that one obtains basically the same result when cutting off the sums at and than when taking only the dominant terms and (the relative difference between both calculations in the predicted value of the Eötvös parameter, namely on the value of is of order ) . The reason for this is that when increases, also increases but decreases very rapidly. The main point is that the quantity decreases faster with than the corresponding rate of growth of . In Appendix A we show, that the results obtained for and taking the summations up to , are the same as those presented in this section. We have been able to perform calculations up to N = 20 and we have observed that the first terms, those corresponding to are the ones representing the main contributions to the results.

On the other hand, in the Appendix B we study the test-particle limit by taking as is kept constant, and analyze how the violations of the WEP in the Eötvös parameter are suppressed by the presence of a thin shell. In particular, this occurs even when the coupling is not universal.

One of the most stringent bounds regarding the possible violations of the WEP is found when comparing the differential acceleration of two test bodies (e.g. two test-balls of Beryllium and Aluminium) using the Earth or its local inhomogeneities, as the source of the acceleration; the experimental value is Adel (). As discussed in Refs. Adel () and Adel2 (), there are two sources for the relevant signals in short-range effects: a hillside of 28 m located close to the laboratory, and a layer of cement blocks added to the wall of the laboratory of
1.5 m (we estimate 1 m distance between the wall and the device). We model the short range sources by spherical masses at a given distance and consider only the contribution of the hillside, such as in Ref. LTBS12 (). For such a configuration, the differential acceleration on two test bodies due to the hill produces an Eötvös parameter . In order to characterize such a configuration in our expression for the force, we make the following assumption for the test bodies: for the masses we take and for the densities , , and . We consider the bodies as surrounded by an environment of constant density and take two values for this density (so that people may compare actual and possible experiments): i) the density of the vacuum-chamber ; ii) the density of the Earth’s atmosphere .

In addition, for the mass that appears in the chameleon potential [cf. Eq. (3)] we take also the value which, as mentioned before, corresponds to the cosmological chameleon.

In Figure 7 we show the predictions for the Eötvös parameter calculated with the method developed in this paper together with the predictions of the standard approach . For this latter, and following KW04 (); Hui2009 (); Burrage15 () (among others), we use the following expression for the total force (gravitational plus chameleon field) between two bodies and ,

 FAB=(1+2QAQB)FN, (41a) QC=min(β,|φ∞−φ