Chameleon dark energy models with characteristic signatures

# Chameleon dark energy models with characteristic signatures

Radouane Gannouji, Bruno Moraes, David F. Mota, David Polarski, Shinji Tsujikawa, Hans A. Winther IUCAA, Post Bag 4, Ganeshkhind, Pune 411 007, India Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan Lab. de Physique Théorique et Astroparticules, CNRS Université Montpellier II, France Institute of Theoretical Astrophysics University of Oslo, Norway
July 11, 2019
###### Abstract

In chameleon dark energy models, local gravity constraints tend to rule out parameters in which observable cosmological signatures can be found. We study viable chameleon potentials consistent with a number of recent observational and experimental bounds. A novel chameleon field potential, motivated by gravity, is constructed where observable cosmological signatures are present both at the background evolution and in the growth-rate of the perturbations. We study the evolution of matter density perturbations on low redshifts for this potential and show that the growth index today can have significant dispersion on scales relevant for large scale structures. The values of can be even smaller than with large variations of on very low redshifts for the model parameters constrained by local gravity tests. This gives a possibility to clearly distinguish these chameleon models from the -Cold-Dark-Matter (CDM) model in future high-precision observations.

###### pacs:
04.50.Kd, 95.36.+x

## I Introduction

The accelerated expansion of the Universe today is a very important challenge faced by cosmologists review (). For an isotropic comoving perfect fluid, a substantially negative pressure is required to give rise to the cosmic acceleration. One of the simplest candidates for dark energy is the cosmological constant with an equation of state , but we generally encounter a problem to explain its tiny energy density consistent with observations Weinberg ().

There are alternative models of dark energy to the cosmological constant scenario. One of such models is quintessence based on a minimally coupled scalar field with a self-interacting potential quin (). In order to realize the cosmic acceleration today, the mass of quintessence is required to be very small ( GeV). From a viewpoint of particle physics, such a light scalar field may mediate a long range force with standard model particles Carroll (). For example, the string dilaton can lead to the violation of equivalence principle through the coupling with baryons Gas (). In such cases we need to find some mechanism to suppress the fifth force for the consistency with local gravity experiments.

There are several different ways to screen the field interaction with baryons. One is the so-called run away dilaton scenario Piazza () in which the field coupling with the Ricci scalar is assumed to approach a constant value as the dilaton grows in time (e.g., as ). Another way is to consider a field potential having a large mass in the region of high density where local gravity experiments are carried out. In this case the field does not propagate freely in the local region, while on cosmological scales the field mass can be light enough to be responsible for dark energy.

The latter scenario is called the chameleon mechanism in which a density-dependent matter coupling with the field can allow the possibility to suppress an effective coupling between matter and the field outside a spherically symmetric body Khoury1 (); Khoury2 (). The chameleon mechanism can be applied to some scalar-tensor theories such as gravity fRlgc1 (); fRlgc2 () and Brans-Dicke theory TUMTY (). In gravity, for example, there have been a number of viable dark energy models fRviable () that can satisfy both cosmological and local gravity constraints. For such models the potential of an effective scalar degree of freedom (called “scalaron” star80 ()) in the Einstein frame is designed to have a large mass in the region of high density. Even with a strong coupling between the scalaron and the baryons (), the chameleon mechanism allows the models to be consistent with local gravity constraints.

The chameleon models are a kind of coupled quintessence models Amendola99 () defined in the Einstein frame Khoury1 (); Khoury2 (). While the gravitational action is described by the usual Einstein-Hilbert action, non-relativistic matter components are coupled to the Einstein frame metric multiplied by some conformal factor which depends on a scalar (chameleon) field. This is how the gravitational force felt by matter is modified. While there have been many studies for experimental and observational aspects of the chameleon models Brax:2004qh ()-Brax:2010kv (), it is not clear which chameleon potentials are viable if the same field is to be responsible for dark energy.

In this paper we identify a number of chameleon potentials that can be consistent with dark energy as well as local gravity experiments. We then constrain the viable model parameter space by using the recent experimental and observational bounds– such as the 2006 Eöt-Wash experiment Kapner:2006si (), the Lunar Laser Ranging experiment Will () and the WMAP constraint on the time-variation of particle masses wmap_constraints (). This can actually rule out some of the chameleon potentials with natural model parameters for the matter coupling of the order of unity.

In order to distinguish the viable chameleon dark energy models from the CDM model, it is crucial to study both the modifications in the evolution of the background cosmology and the modified evolution of the cosmological density perturbations. For the former, we shall consider the evolution of the so-called statefinders introduced in Refs. Sahni1 (); Sahni2 () and show that these parameters can exhibit a peculiar behavior different from those in the CDM model.

On the other hand, the growth “index” of matter perturbations defined through , where is a scale factor and is the density parameter of non-relativistic matter, is an important quantity that allows to discriminate between different dark energy models and interest in this quantity was revived in the context of dark energy models Stein (); Linder ().

Its main importance for the study of dark energy models stems from the fact that for CDM the quantity is known to be nearly constant with respect to the redshift , i.e. to exquisite accuracy PG07 (), with Stein (). As emphasized in Ref. PG07 (), large variations of on low redshifts could signal that we are dealing with a dark energy model outside General Relativity. This was indeed found for some scalar-tensor dark energy models GP08 () and models GMP08 (); MSY10 (); Narikawa (). Such large variations can also occur in models where dark energy interacts with matter ASS09 (); HWJ09 (). This is exactly the case in chameleon models, which we investigate in this paper, because of the direct coupling between the chameleon field and all dust-like matter. This direct coupling is however not confined to the dark sector as in standard coupled quintessence.

An additional important point is the possible appearance of a scale-dependence or dispersion in . Hence the behavior of on low redshifts can be both time-dependent and scale-dependent GMP09 (); Tsujikawa:2009ku (); BT10 (). This dispersion can also be present in the models investigated here. Using the observations of large scale structure and weak lensing surveys, one can hope to detect such peculiar behaviors of (see e.g., Ref. Euclid ()). If this is the case this would signal that the gravitational law may be modified on scales relevant to large scale structures Linder (); Gannouji (); Luca (); Tsujikawa:2009ku (); Koivisto (); Nesseris (); BT10 (); SD09 (); Mota:2007zn ().

In this paper we study the evolution of as well as its dispersion, its dependence on the wavenumbers of perturbations. We shall show that some of the chameleon models investigated here can be clearly distinguished from CDM through the behavior of exhibiting both large variations and significant dispersion, with the possibility to obtain small values of today as low as .

## Ii Chameleon cosmology

### ii.1 Background equations

In this section we review the basic background evolution of chameleon cosmology. We consider the chameleon theory described by the following action Khoury1 (); Khoury2 (); EP00 ()

 S = ∫d4x√−g[R16πG−12gμν∂μϕ ∂νϕ−V(ϕ)] (1) +Sm[Ψm;A2(ϕ)gμν],

where is the determinant of the (Einstein frame) metric , is the Ricci scalar, is the bare gravitational constant, is a scalar field with a potential , and is the matter action with matter fields . At low redshifts it is sufficient to consider only non-relativistic matter (cold dark matter and baryons), but for a general dynamical analysis including high redshifts radiation must be included.

We assume that non-relativistic matter is universally coupled to the (Jordan frame) metric , the Einstein frame metric multiplied by a field-dependent (conformal) factor . This direct coupling to the field is how the gravitational interaction is modified. We can generalize this to arbitrary functions for each matter component , but in this work we will take the same function for all components. We write the function in the form

 A(ϕ)=eQϕ/Mpl, (2)

where is the reduced Planck mass and describes the strength of the coupling between the field and non-relativistic matter. In the following we shall consider the case in which is constant. In fact, the constant coupling arises for Brans-Dicke theory by a conformal transformation to the Einstein frame Khoury2 (); TUMTY (). Even when is of the order of unity, it is possible to make the effective coupling between the field and matter small through the chameleon mechanism.

Let us consider the scalar field together with non-relativistic matter (density ) and radiation (density ) in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) space-time with a time-dependent scale factor and a metric

 ds2=gμνdxμdxν=−dt2+a2(t)dx2. (3)

The corresponding background equations are given by

 3H2=(ρϕ+ρ∗m+ρr)/M2pl, (4) ¨ϕ+3H˙ϕ+V,ϕ=−Qρ∗m/Mpl, (5) ˙ρ∗m+3Hρ∗m=Qρ∗m˙ϕ/Mpl, (6) ˙ρr+4Hρr=0, (7)

where , , and a dot represents a derivative with respect to cosmic time . The quantity is the energy density of non-relativistic matter in the Einstein frame and we have kept the star to avoid any confusion. Integration of Eq. (6) gives the solution . We define the conserved matter density:

 ρm≡e−Qϕ/Mplρ∗m, (8)

which satisfies the standard continuity equation, . Then the field equation (5) can be written in the form

 ¨ϕ+3H˙ϕ+Veff,ϕ=0, (9)

where is the effective potential defined by

 Veff≡V(ϕ)+eQϕ/Mplρm. (10)

We emphasize that it is the Einstein frame which is the physical frame. Due to the coupling between the field and matter, particle masses do evolve with time in our model.

We consider runaway positive potentials in the region , which monotonically decrease and have a positive mass squared, i.e. and . We also demand the following conditions

 limϕ→0∣∣∣V,ϕV∣∣∣=∞,limϕ→∞V,ϕV=0. (11)

The former is required to have a large mass in the region of high density, whereas we need the latter condition to realize the late-time cosmic acceleration in the region of low density. At the potential approaches either or a finite positive value . In the limit we have either or , where is a nonzero positive constant. If the effective potential has a minimum at the field value satisfying the condition , i.e.

 V,ϕ(ϕm)+Q(ρm/Mpl)eQϕm/Mpl=0. (12)

If the potential satisfies the conditions and in the region , there exists a minimum at provided that . In fact this situation arises in the context of dark energy models fRlgc1 (); fRlgc2 (). Since the analysis in the latter is equivalent to that in the former, we shall focus on the case and in the following discussion.

### ii.2 Dynamical system

In order to discuss cosmological dynamics, it is convenient to introduce the following dimensionless variables

 x1≡˙ϕ√6HMpl,x2≡√V√3HMpl,x3≡√ρr√3HMpl. (13)

Equation (4) expresses the constraint existing between these variables, i.e.

 Ω∗m≡ρ∗m3H2M2pl=1−Ωϕ−Ωr, (14)

where

 Ωϕ≡x21+x22,Ωr≡x23. (15)

Taking the time-derivative of Eq. (4) and making use of Eqs. (5)-(7), it is straightforward to derive the following equation

 (16)

where a prime represents a derivative with respect to . A useful quantity is the effective equation of state

 weff=−1−23H′H=x21−x22+x233. (17)

We also introduce the field equation of state , as

 wϕ≡˙ϕ2/2−V(ϕ)˙ϕ2/2+V(ϕ)=x21−x22x21+x22. (18)

Using Eqs. (4)-(7), we obtain the following equations

 x′1=−3x1+√62λx22−x1H′H −√62Q(1−x21−x22−x23), (19) x′2=−√62λx1x2−x2H′H, (20) x′3=−2x3−x3H′H, (21) λ′=−√6λ2(Γ−1)x1, (22)

where

 λ≡−MplV,ϕV,Γ≡VV,ϕϕV2,ϕ. (23)

From the conditions (11) it follows that the quantity decreases from to 0 as grows from 0 to . Since in Eq. (22), the condition translates into

 Γ=VV,ϕϕV2,ϕ>1. (24)

Chameleon potentials shallower than the exponential potential () can satisfy this condition.

Once the field settles down at the minimum of the effective potential (10), we have

 x1≃0,x2≃√QλΩ∗m, (25)

which gives from Eq. (18). As the matter density decreases, the field evolves slowly along the instantaneous minima characterized by (25). We require that during radiation and deep matter eras for consistency with local gravity constraints in the region of high density. For the dynamical system (19)-(22) there is another fixed point called the “-matter-dominated era (MDE)” Amendola99 () where . However, since we are considering the case in which is of the order of unity, the effective equation of state is too large to be compatible with observations. Only when the MDE can be responsible for the matter era Amendola99 ().

When the chameleon is slow-rolling along the minimum, we obtain the following relation from Eqs. (15) and (25):

 λQ≃Ω∗mΩϕ. (26)

While during the radiation and matter eras, becomes the same order as around the present epoch. The field potential is the dominant contribution on the r.h.s. of Eq. (4) today, so that

 V(ϕ0)≃3H20M2pl≃ρc, (27)

where the subscript “0” represents present values and  g/cm is the critical density today.

## Iii Chameleon mechanism

In this section we review the chameleon mechanism as a way to escape local gravity constraints. In addition to the cosmological constraints discussed in the previous section, this will enable us to restrict the forms of chameleon potentials.

Let us consider a spherically symmetric space-time in the weak gravitational background with the neglect of the backreaction of metric perturbations. As in the previous section we consider the case in which couplings are the same for each matter component (), i.e., in which the function is given by Eq. (2). Varying the action (1) with respect to in the Minkowski background, we obtain the field equation

 d2ϕdr2+2rdϕdr=dVeffdϕ, (28)

where is the distance from the center of symmetry and is defined in Eq. (10).

Assuming that a spherically symmetric object (radius and mass ) has a constant density with a homogeneous density outside the body, the effective potential has two minima at and satisfying the conditions

 V,ϕ(ϕA) + Q(ρA/Mpl)eQϕA/Mpl=0, (29) V,ϕ(ϕB) + Q(ρB/Mpl)eQϕB/Mpl=0. (30)

Since and for viable field potentials in the regions of high density, the conserved matter density is practically indistinguishable from the matter density in the Einstein frame.

The field profile inside and outside the body can be found analytically. Originally this was derived in Refs. Khoury1 (); Khoury2 () under the assumption that the field is frozen around in the region , where is the distance at which the field begins to evolve. It is possible, even without this assumption, to derive analytic solutions by considering boundary conditions at the center of the body Tamaki:2008mf ().

We consider the case in which the mass squared outside the body satisfies the condition , so that the -dependent terms can be negligible when we match solutions at . The resulting field profile outside the body () is given by Tamaki:2008mf ()

 ϕ(r)=ϕB−Qeff4πMplMcr, (31)

where the effective coupling between the field and matter is

 Qeff = Q[1−r31r3c+3r1rc1(mArc)2 (32) ×{mAr1(emAr1+e−mAr1)emAr1−e−mAr1−1}].

The mass is defined by . The distance is determined by the condition , which translates into

 ϕB−ϕA+QρA(r21−r2c)/(2Mpl) =6QMplΦc(mArc)2mAr1(emAr1+e−mAr1)emAr1−e−mAr1, (33)

where is the gravitational potential at the surface of the body.

The fifth force exerting on a test particle of a unit mass and a coupling is given by . Using Eq. (31), the amplitude of the fifth force in the region is

 F=2|QQeff|GMcr2. (34)

As long as , it is possible to make the fifth force suppressed relative to the gravitational force . From Eq. (32) the effective coupling can be made much smaller than provided that the conditions and are satisfied. Hence we require that the body has a thin-shell and that the field is heavy inside the body for the chameleon mechanism to work.

When the body has a thin-shell (), one can expand Eq. (33) in terms of the small parameters and . This leads to

 ϵth≡ϕB−ϕA6QMplΦc≃Δrcrc+1mArc, (35)

where is called the thin-shell parameter. As long as , this recovers the relation Khoury1 (); Khoury2 (). The effective coupling (32) is approximately given by

 Qeff≃3Qϵth. (36)

If is much smaller than 1 then one has , so that the models can be consistent with local gravity constraints.

As an example, let us consider the experimental bound that comes from the solar system tests of the equivalence principle, namely the Lunar Laser Ranging (LLR) experiment, using the free-fall acceleration of the Moon () and the Earth () toward the Sun (mass ) Khoury2 (); Tamaki:2008mf (); fRlgc2 (). The experimental bound on the difference of two accelerations is given by

 2|aMoon−a⊕|(aMoon+a⊕)<10−13. (37)

Under the conditions that the Earth, the Sun, and the Moon have thin-shells, the field profiles outside the bodies are given as in Eq. (31) with the replacement of corresponding quantities. The acceleration induced by a fifth force with the field profile and the effective coupling is . Using the thin-shell parameter for the Earth, the accelerations and are Khoury2 ()

 a⊕ ≃ GM⊙r2[1+18Q2ϵ2th,⊕Φ⊕Φ⊙], (38) aMoon ≃ GM⊙r2[1+18Q2ϵ2th,⊕Φ2⊕Φ⊙ΦMoon], (39)

where , , and are the gravitational potentials of Sun, Earth and Moon, respectively. Then the condition (37) reads

 ϵth,⊕<8.8×10−7Q. (40)

Using the value , the bound (40) translates into

 ϕB,⊕≲10−15Mpl, (41)

where we used the condition . For the Earth one has  g/cm g/cm (dark matter/baryon density in our galaxy), so that the condition is satisfied.

In Sec. IV we constrain viable chameleon potentials by employing the condition (41) together with the cosmological condition we discussed in Sec. II. In Sec. V we restrict the allowed model parameter space further by using a number of recent local gravity and observational constraints.

## Iv Viable chameleon potentials

We now discuss the forms of viable field potentials that can be in principle consistent with both local gravity and cosmological constraints. Let us consider the potential

 V(ϕ)=M4f(ϕ), (42)

where is a mass scale and is a dimensionless function in terms of .

The local gravity constraint coming from the LLR experiment is given by Eq. (41), where is determined by solving

 ∣∣Mplf,ϕ(ϕB,⊕)∣∣≃QρB/M4. (43)

Here we take  g/cm for the homogeneous density outside the Earth. Once the form of is specified, the constraint on the model parameter, e.g., , can be derived.

From the cosmological constraint (26), is of the order of 1 today. Then it follows that

 ∣∣Mplf,ϕ(ϕ0)∣∣≃Qf(ϕ0)≃Qρc/M4, (44)

where we used Eq. (27).

We also require the condition (24), i.e.

 Γ=ff,ϕϕf2,ϕ>1, (45)

for all positive values of . We shall proceed to find viable potentials satisfying the conditions (41), (43), (44), and (45). From Eqs. (43) and (44) we obtain

 f,ϕ(ϕB,⊕)f,ϕ(ϕ0)≃ρBρc≃105. (46)

Let us consider the inverse power-law potential (), i.e.

 f(ϕ)=(M/ϕ)n. (47)

Since , the condition (45) is automatically satisfied. The cosmological constraint (44) gives

 ϕ0≃nMpl/Q. (48)

From Eq. (46) we find the relation between and  :

 ϕ0≃105/(n+1)ϕB,⊕. (49)

Using the LLR bound (41), it follows that

 ϕ0≲10−5(3n+2)n+1Mpl. (50)

This is incompatible with the cosmological constraint (48) for and . Hence the inverse power-law potential is not viable.

### iv.1 Inverse power-law potential + constant

The reason why the inverse power-law potential does not work is that the field value today required for cosmic acceleration is of the order of , while the local gravity constraint demands a much smaller value. This problem can be circumvented by taking into account a constant term to the inverse power-law potential. Let us then consider the potential (), i.e.

 f(ϕ)=1+μ(M/ϕ)n, (51)

where is a positive constant. The rescaling of the mass term always allows to normalize the constant to be unity in Eq. (51). For this potential the quantity reads

 (52)

which satisfies the condition . In the region we have that , which recovers the case of the inverse power-law potential. Meanwhile, in the region , one has . The latter property comes from the fact that the potential becomes shallower as the field increases. This modification of the potential allows a possibility that the model can be consistent with both cosmological and local gravity constraints.

The addition of a constant term to the inverse power-law potential does not affect the condition (46), which means that the resulting bounds (49) and (50) are not subject to change. On the other hand, the cosmological constraint (48) is modified. Let us consider the case where the condition is satisfied today, i.e. . From Eq. (44) it follows that

 M≃ρ1/4c≃10−12GeV, (53)

and

 ϕ0≃(nμMnMnpl)1/(n+1)Mpl≃(10−30nnμ)1/(n+1)Mpl. (54)

Hence the field value today can be much smaller than the Planck mass, unlike the inverse power-law potential. From Eqs. (50) and (54) we get the constraint

 μ≲1015n−10/n. (55)

If , for example, one has . For larger the bound on becomes even weaker. We note that the condition is satisfied for . This shows that even the field value such as satisfies the condition . Thus the term is smaller than 1 for the field values we are interested in ().

A large range of experimental bounds for this model has been derived in the literature, see Refs. Khoury2 (); Mota:2006fz (); Mota:2006ed (); Brax:2007vm (). For , it was found in Ref. Mota:2006fz () that the model is ruled out by the Eöt-Wash experiment unless . This applies for general : to obtain a viable model for of the order of unity one must impose a fine-tuning or .

The potentials, which have only one mass scale equivalent to the dark energy scale, are usually strongly constrained by the Eöt-Wash experiment. We shall look into this issue in more details in Sec. V.

### iv.2 Construction of viable chameleon potentials relevant to dark energy

The discussion given above shows that a function that monotonically decreases without a constant term is difficult to satisfy both cosmological and local gravity constraints. This is associated with the fact that for any power-law form of the condition leads to the overall scaling of the function itself, giving of the order of . The dominance of a constant term in changes this situation, which allows a much smaller value of relative to .

Another example similar to is the potential Brax:2004qh ()

 V(ϕ)=M4exp[μ(M/ϕ)n], (56)

where and . For this model the quantity

 Γ=1+n+1n1μ(ϕM)n, (57)

is larger than 1. In the asymptotic regimes characterized by and we have and , respectively. When the function can be approximated as , which corresponds to Eq. (51). In this case the constraints on the model parameters are the same as those given in Eqs. (53)-(55).

There is another class of potentials that behaves as () in the region . In fact this asymptotic form corresponds to the potential that appears in dark energy models. While the potential is finite at the derivative diverges as for , so that the first of the condition (11) is satisfied. In order to keep the potential positive we need some modification of in the region .

In scalar-tensor theory it was shown in Ref. TUMTY () that the Jordan frame potential of the form (, ) can satisfy both cosmological and local gravity constraints. In this case the potential in the Einstein frame is given by , which possesses a de Sitter minimum due to the presence of the conformal factor. Cosmologically the solutions finally approach the de Sitter fixed point, so that the late-time cosmic acceleration can be realized.

Now we would like to consider a runaway positive potential in the Einstein frame. One example is

 V(ϕ)=M4[1−μ(1−e−ϕ/Mpl)n], (58)

where and . This potential behaves as for and approaches in the limit (see Fig. 1). For the potential (58) we obtain

 Γ−1=1−nx−μ(1−x)nnμx(1−x)n,x≡e−ϕ/Mpl. (59)

One can easily show that the r.h.s. is positive under the conditions , so that . In both limits and one has . Since has a minimum at a finite field value, the condition is not necessarily satisfied today (unlike the potential (56)).

Unless is very close to 1 the potential energy today is roughly of the order of , i.e. . From Eq. (44) it then follows that

 nμ(1−x0)n−1x0≃Q, (60)

where . If , we have that . From Eq. (43) we obtain

 nμ(1−xB)n−1xB≃105Q, (61)

where . Under the condition we have from Eq. (61). Then the LLR bound (41) corresponds to

 n⋅1010−15n

When and with , the constraint (62) gives and respectively.

### iv.3 Statefinder analysis

The statefinder diagnostics introduced in Refs. Sahni1 (); Sahni2 () can be a useful tool to distinguish dark energy models from the CDM model. The statefinder parameters are defined by

 r=...aaH3,s=r−13(q−1/2), (63)

where is the deceleration parameter. Defining , it follows that

 q=−1−h′2h,r=1+h′′2h+3h′2h, (64)

where a prime represents a derivative with respect to .

In the radiation dominated epoch we have , which gives . During the matter era approaches , whereas blows up from positive to negative because of the divergence of the denominator in (i.e. ). For the chameleon potentials (56) and (58) the solutions finally approach the de Sitter fixed point characterized by . Around the de Sitter point the solutions evolve along the instantaneous minima characterized by with . Using Eq. (22) as well, one has and . Then the statefinder diagnostics around the de Sitter point can be estimated as

 r≃1+(Γ−12)λ4−32λ2, (65) s≃−(2Γ−1)λ4−3λ23(3−λ2). (66)

Since finally approaches 0, Eq. (22) implies that asymptotically. For the potential in which holds today it can happen that , which gives and around the present epoch. In the upper panel of Fig. 2 we plot the evolution of the variables and for the potential (56) in the redshift regime . The statefinders evolve toward the de Sitter point characterized by from the regime and . This behavior is different from quintessence with the power-law potential () in which the statefinders are confined in the region and Sahni1 (); Sahni2 ().

The potential (58) allows the possibility that is not much larger than 1 even at the present epoch. In the regime we then have and . In fact we have numerically confirmed that the solutions enter this regime by today (see the lower panel of Fig. 2). Finally they approach the de Sitter point from the regime and . Hence one can distinguish between chameleon potentials from the evolution of statefinders.

## V Local gravity constraints on chameleon potentials

In this section we discuss a number of local gravity constraints on the chameleon potentials (56) and (58) in details. Together with the LLR bound (41) we use the constraint coming from 2006 Eöt-Wash experiments Kapner:2006si () as well as the WMAP bound on the variation of the field-dependent mass.

### v.1 The WMAP constraint on the variation of the particle mass

Due to the conformal coupling of the field to matter, any particle will acquire a -dependent mass:

 m(ϕ)=m0eQϕ/Mpl, (67)

where is a constant.

The WMAP data constrain any variation in , between now and the epoch of recombination to be at ( at ) wmap_constraints (). We then require that

 ∣∣∣Δm(ϕ)m∣∣∣=∣∣ ∣∣eQ(ϕ0−ϕrec)Mpl−1∣∣ ∣∣≲0.05, (68)

where is the field value at the recombination epoch. If we assume that the chameleon follows the minimum since recombination then , and the field in the cosmological background today must satisfy

 Qϕ0/Mpl≲0.05. (69)

This provides a constraint on the coupling and the model parameters of chameleon potentials.

Note that the WMAP constraint is not a local gravity constraint. Nevertheless, it provides strong constraints on the potential (58) with natural parameters and is therefore considered in this section.

### v.2 Constraints from the 2006 Eöt-Wash experiment

The 2006 Eöt-Wash experiment Kapner:2006si () searched for deviations from the force law of gravity. The experiment used two parallel plates, the detector and attractor, which are separated by a (smallest) distance m. The plates have holes of different sizes bored into them, and the attractor is rotating with an angular velocity . The rotation of the attractor gives rise to a torque on the detector, and the setup of the experiment is such that this torque vanishes for any force that falls off as . In between the plates there is a m BeCu-sheet, which is for shielding the detector from electrostatic forces.

The chameleon force between two parallel plates, see e.g. Ref. Brax:2007vm (), usually falls off faster than , implying a strong signature on the experiment. However, if the matter-coupling is strong enough, the electrostatic shield will itself develop a thin-shell. When this happens, the effect of this shield is not only to shield electrostatic forces, but also to shield the chameleon force on the detector. This suppression is approximately given by a factor , where is the mass inside the electrostatic shield. Hence the experiment cannot detect strongly coupled chameleons.

The behavior of chameleons in the Eöt-Wash experiment have been explained in Refs. Mota:2006fz (); Brax:2008hh (); Brax:2010kv (). We calculate the Eöt-Wash constraints on our models numerically based on the prescription presented in Ref. Brax:2008hh ().

### v.3 Combined local gravity constraints

#### v.3.1 Potential V(ϕ)=M4exp[μ(M/ϕ)n]

Let us first consider the inverse-power law potential (56) with . From Eq. (12) the field value at the minimum of the effective potential satisfies

 (Mϕm)n+1=QμnMMplρmeQϕm/MplV(ϕm). (70)

In this model the field is in the regime for the density we are interested in. Since can be approximated as , it follows that

 ϕmM≃(QμnMMplρmM4)−1/(n+1). (71)

Using the LLR bound (41) with the homogeneous density  g/cm in our galaxy, we obtain the constraint

 n⋅1010−15n

The WMAP bound (69) gives

 Q

where is the matter density today, with . Since , this condition is well satisfied for of the order of unity.

For the Eöt-Wash experiment, the chameleon torque on the detector was found numerically to be larger than the experimental bound when are of the order of unity. Providing the electrostatic shield with a thin-shell, we require that , or to satisfy the experimental bound.

In Fig. 3 we plot the region constrained by the bounds (72), (73), and the Eöt-Wash experiment for . This shows that only the large coupling region with can be allowed for of the order of unity. A viable model can also be constructed by taking values of much smaller than 1. Note that the WMAP bound (73) is satisfied for the parameter regime shown in Fig. 3.

#### v.3.2 Potential V(ϕ)=M4[1−μ(1−e−ϕ/Mpl)n]

Let us proceed to another potential (58) with . In the regions of high density where local gravity experiments are carried out, we have and hence . In this regime the effective potential has a minimum at

 ϕm=(QμnρmM4)1/(n−1)Mpl. (74)

Recall that the LLR bound was already derived in Eq. (62), which is the same as the constraint (72) of the previous potential.

Assuming that the chameleon is at the minimum of its effective potential in the cosmological background today, the WMAP bound (69) translates into

 Q≳0.05(60nμ)1/n, (75)

where we have used in Eq. (74). However, a full numerical simulation of the background evolution shows that this is not always the case. For a large range of parameters the chameleon has started to lag behind the minimum, which again leads to a weaker constraint.

The Eöt-Wash experiment provides the strongest constraints when is of the order of unity for the potentials (51) and (56). This is not the case for the potential (58), because the electrostatic shield used in the experiment develops a thin-shell.

The mass inside the electrostatic shield is given by

 m2s≃n(1−n)μ(QρsμnM4)2−n1−nM4M2pl. (76)

Using  g/cm and m we have

 msds≃√n(1−n)μ(Qμn)2−n1−n106