# Gravity and Chameleon Theories

###### Abstract

We analyse modifications of Einstein’s gravity as dark energy models in the light of their connection with chameleon theories. Formulated as scalar-tensor theories, the theories imply the existence of a strong coupling of the scalar field to matter. This would violate all experimental gravitational tests on deviations from Newton’s law. Fortunately, the existence of a matter dependent mass and a thin shell effect allows one to alleviate these constraints. The thin shell condition also implies strong restrictions on the cosmological dynamics of the theories. As a consequence, we find that the equation of state of dark energy is constrained to be extremely close to in the recent past. We also examine the potential effects of theories in the context of the Eöt-wash experiments. We show that the requirement of a thin shell for the test bodies is not enough to guarantee a null result on deviations from Newton’s law. As long as dark energy accounts for a sizeable fraction of the total energy density of the Universe, the constraints which we deduce also forbid any measurable deviation of the dark energy equation of state from -1. All in all, we find that both cosmological and laboratory tests imply that models are almost coincident with a CDM model at the background level.

###### pacs:

04.50.Kd, 95.36.+x, 12.20.Fv## I Introduction

The acceleration of the Universe expansion was discovered ten years ago and is still a deep mystery (see e.g. de1 for recent results on observations of dark energy and e.g. durrer ; de2 for theoretical overviews). Two types of approaches have been considered. One can either introduce a new kind of matter whose role is to trigger acceleration or modify the behaviour of gravity at cosmological distances. In the first approach, dark energy is a new energy form, with all its well-known puzzles such as the cosmological constant problem, the coincidence problem and the value of the equation of state. This approach is subject of intense experimental investigation and any deviation from -1 would be a smoking gun for new physics beyond the standard models of particle physics and cosmology. On the other hand, in the second approach, various attempts to modify gravity have been presented (see e.g. Carroll -Amendola ; the literature is vast, see fara for a recent overview and further references). Up to now, they are all plagued with various theoretical problems such as the existence of ghosts or instabilities. In this paper, we will consider a modification of Einstein’s gravity, the so–called theories, which do not seem to introduce any new type of matter and can lead to late time acceleration. In fact, these theories can be reformulated in terms of scalar-tensor theories with a fixed coupling of the extra scalar degree of freedom to matter. As theories of dark energy, they suffer from the usual problems and are also potentially ruled out by gravitational tests of Newton’s law.

The only way-out for these models is to behave as chameleon theories chamKA , i.e. develop an environment dependent mass navarro ; brookfield ; faulkner ; li . When the density of the ambient matter in which the scalar field/chameleon propagates is large enough, its mass becomes large and the smallness of the generated fifth force range is below the detectability level of gravitational experiments. On the other hand, for planetary orbits or any other situations in which gravity is at play in a sparse environment, one must impose the existence of the so–called thin shell effect. In this case, the fifth force is attenuated as the chameleon is trapped inside very massive bodies (the Sun for instance). It has been argued that the existence of thin shells is usually enough to salvage models navarro ; faulkner . We show that thin shells do not always guarantee null results in experimental tests of Newton’s law. We exemplify this fact using the Eöt-wash setting and obtain strong constraints on the models which translate into stringent bounds on the present dark energy equation of state, preventing any detection of a deviation from -1 in the foreseeable future , where is the equation of state of dark energy in the recent past. This corroborates a similar bound obtained from the existence of thin shell for objects embedded in a super-cluster. It should be noted that this result holds at the background level. For higher red-shifts where the effective dark energy density fraction, , may become small (or even vanish), larger deviations can be present as exemplified in the models in tsu1 ; tsu2 where the equation of state can deviate from -1 for red-shifts of order . In all these models however , and so even if deviates significantly from , deviations of the homogeneous cosmology from CDM are still very small. Detectable deviations from CDM are envisageable at the perturbative level as the growth factor is anomalous at small scales (see e.g. green for a discussion of this point for the original chameleon model). Some consequences of this fact on the matter power spectrum and the CMB spectrum of models have been presented in Ref. Hu1 ; spergel ; Hu2 .

The paper is organized as follows: In the subsequent section, we review models and chameleon theories. In section III we derive the cosmological thin shell bound on the equation of state. In section IV, we consider tests of the inverse square law. Finally, we apply these considerations to particular models in section V. The appendices contain some technical details.

## Ii Gravities and Chameleon Theories

### ii.1 theories

An theory is a modified gravity theory in which the usual Einstein-Hilbert Lagrangian for General Relativity, i.e. , is replaced by some arbitrary function of the scalar curvature i.e. . The action for an gravity theory therefore takes the following form:

(1) |

where the represent the matter fields.

In this work we are concerned only with metric theories, in which only the metric is the independent variable in the gravitational sector. The quantity is taken to be the Levi-Civita connection associated with the metric . In these metric theories the field equations are:

where .

### ii.2 Transformation to a Scalar-Tensor theory

Eq. (II.1) gives a set of equations which are second order in derivatives of , which is itself second order in derivatives of , making the field equations fourth order in . Finding solutions to fourth order equations can be mathematically and physically troublesome, but fortunately metric theories can be recast as a scalar tensor theory with only second order equations via a well known conformal transformation. We define by

where . We also define the *Einstein
frame* metric by a conformal transformation

and let be the scalar curvature of . When rewritten in terms of and , Eq. (1) becomes:

(3) | |||||

where the potential is given by:

(4) |

When the action is written in the form of Eq. (3), we say that we are working in the Einstein frame. The field equations then become:

(6) |

In the above and subsequent expressions, the covariant derivatives, , obey and all indices are raised and lowered with unless stated otherwise. We note that in the Einstein frame is not conserved but instead:

(7) |

This implies that matter will generally feel a new or ‘fifth’ force due to gradients in . We note from Eq. (II.2) that, when written as a scalar tensor theory, gravity in an theory is essentially General Relativity, and all the modifications are essentially due to the effective ‘fifth force’ and to the energy density of . Much of our intuition for how gravity works is based on how it works in General Relativity. When an theory is written as a scalar tensor theory we can readily make use of this intuition in solving the field equations. This may not be the case, however, in the original frame in which the equations were fourth order and so in those circumstances one would have to be more careful. Note that all physical observables must be independent of the choice of frame, i.e. the choice of metric or .

### ii.3 Chameleon Theories

Since theories are equivalent to scalar tensor theories one can generally directly apply the plethora of constraints on scalar tensor models. In particular, since the extra degree of freedom, , couples to matter with gravitational strength, tests of the inverse square law such as the Eöt-Wash experiment require that have a mass, , greater than . Cosmologically would then have been fixed at its minimum since very early times, and physics over astrophysical scales would be indistinguishable from predicted by unmodified General Relativity with a cosmological constant. Both the coincidence problem and the problem of the small size of the cosmological constant would not be alleviated in this scenario. However, this is not the whole story. Laboratory constraints on scalar tensor theories can be greatly relaxed if develops a strong dependence on the ambient density of matter. Theories in which such a dependence is realized are said to have a chameleon mechanism and to be chameleon theories. In such theories, can be heavy enough in the environment of the laboratory tests so as to evade them, whilst remaining relatively light on cosmological scales. It must be stressed that even with a chameleon mechanism, it is still very difficult, if not impossible, to construct such a theory where the late time cosmology would be observational distinguishable from the usual CDM model. To the best of our knowledge all such theories which are also experimentally viable require a fairly high degree of fine tuning to ensure that the effective cosmological constant is small enough.

A chameleon theory is essentially just a scalar-tensor theory in which the potential has certain properties. As such Eqs. (3 - 7) also define a chameleon theory for certain classes of . In these circumstances the theory would be equivalent to a chameleon theory. In a general chameleon theory, , which parametrizes the strength of the coupling of to matter, could take any value and potentially even be different for different matter species. If a chameleon theory is equivalent to a theory, however, is fixed to be and is the same for all types of matter. If a theory is not equivalent to a chameleon theory it would be generally ruled out by laboratory tests of gravity and / or result in no detectable deviations from General Relativity over astrophysical scales.

For an theory to have a chameleon mechanism one must require that, in at least some region of -space:

It follows from the definition of that:

(8) |

and therefore the derivatives follow

(9) | |||||

(10) | |||||

(11) |

In general, this gives strong constraints on the form of . In the following we will study examples where these conditions are met.

When these conditions are satisfied, the mass of in a suitable large region with density will increase with . In order to evade constraints coming from local tests of gravity, it is not, however, enough that a theory possess a chameleon mechanism; the mechanism must, in addition be strong enough for chameleonic behaviour to occur for the test masses used in the laboratory gravity experiments.

### ii.4 Thin-Shells

#### ii.4.1 Chameleon Theories

Chameleon theories do not behave like linear theories of massive scalar fields. In situations where massive bodies are involved, the chameleon field is trapped inside such bodies and its influence on other bodies is only due to a thin shell at the outer edge of a massive bodychamKA . As a result, the field outside the massive body for distances less than the range of the chameleon force in the outer vacuum is effectively damped leading to a shielded fifth force which becomes undetectable. The criterion for a thin shell is

(12) |

where is the field difference from far inside the body to very far away. We define the body and the region outside it to have densities and respectively. It involves Newton’s potential at the surface of the body. In general the field values at infinity, , and deep inside, , are related to and by

(13) |

In most current situations involving runaway potentials, when , this implies that . Hence, implying that cosmological information can be inferred from local tests. Moreover, in a cosmological setting, the chameleon sits at the minimum (13) during the matter era. As a result, the variation of the equation of state in the recent past is severely constrained. Another important consequence of the chameleon effect is the existence of an anomalous growth of the density contrast for scales lower than the inverse mass of the chameleon, i.e. it grows like where green . In the setting, some of the consequences of this anomalous growth on the CMB and the matter power spectrum have been analysed using the convenient variable

(14) |

whose square root represents the compton wave-length, i.e. the inverse mass of the chameleon, in horizon units Hu1 ; spergel ; Hu2 . Effects on structure formation could be seen for values as low as in future galaxy surveysspergel . In the following we will find an explicit example of logarithmic model which could lead to effects on scales as large as 100 . All these facts will be crucial in the following.

#### ii.4.2 Thin shells in the language of theories

It is useful to write the function in the form , where measures the deviation to Einstein gravity. To leading order, as a consequence of Ref. chamKA , the thin shell condition can be formulated as Hu2

(15) |

As Newton’s potential is small on cosmological scales, with an upper bound around , this implies that must have very small variations. The thin shell condition is a constraint on local experiments at the present time. It has nothing to say, a priori, about the evolution of the universe since matter equality for instance. Another useful combination (which is not to be confused with the chameleon mass ) has been used

(16) |

It has been shown that the existence of a matter era followed by an accelerated period requires . For models where is (nearly) a power law, the thin shell constrain implies that is much smaller for reasonable powers. In the following, we will obtain a bound on the equation of state at present time which implies that departures from CDM are tiny.

## Iii Thin-shell constraints on Cosmology

In subsequent sections we will assume that test bodies used in laboratory based gravity experiments have thin-shells. In the absence of any thin-shell, the inverse square law tests, such as the Eöt-Wash experiment EotWash (as well as other tests of gravity over longer ranges), rule out theories with as it is in theories. The thin-shell requirement must therefore be satisfied by any physically viable theory. Although it is not often appreciated, the thin-shell condition for laboratory test masses actually places extremely tight constraints on the recent cosmological evolution of . In this section we consider those constraints in the context of a general theory.

In any single field scalar tensor theory there is a choice of frame. In the Jordan frame, the laws of physics in a local inertial frame are the same everywhere, however Newton’s constant, , is different at different points in space and time. In the Einstein frame, is chosen to be fixed but, as a result, local particle physics is position dependent. The process of converting astronomical observations to cosmological parameters generally involves making assumptions about how today’s laws of particle physics are related to those in the past. This said, if the relative changes in (in the Jordan frame) are small i.e. , the differences between cosmological parameters in the two frames are only very slight. For instance, to calculate a redshift, one must compare the observed wavelength, of a particular absorption or emission band to the wavelength that band would have had at emission, . Since one cannot go to the astronomical object in question and directly observe the wavelength at emission, it is generally assumed that particles physics in the past obeyed the same laws as it does today and so replace with the wavelength of the band as it is measured in a laboratory today, . When one is dealing with scalar-tensor theories, the assumption that is equivalent to a choice of frame, specifically the Jordan frame.

To make comparison with observations straight-forward, one should therefore quote cosmological parameters for the Jordan frame. This said, it is often more straightforward to perform calculations in the Einstein frame and then merely express the results in terms of Jordan frame quantities.

Cosmologically, in the Jordan frame we have:

(17) |

and obeys:

(18) |

where . At late times, when it is appropriate to ignore the contribution of radiation to the total energy density of the Universe, we have

(19) |

The Einstein equations also give:

(20) |

We assume that measurements are interpreted in terms of General Relativity, where the energy density of the Universe is assumed to be due to non-interacting, dark energy and normal matter. Thus we write

The above equation partly defines , and ; today . Now in the Jordan frame: . If the effective dark energy equation of state parameter, , were constant it would obey: . More generally however the effective dark energy equation of state is then given by:

(22) |

Taking the -derivative of Eq. (III) we get:

and so using the Eq. (22) and we have:

Finally by adding to both sides and using Eq. (III) we have:

So by rearranging the Friedmann equations we have found that

By comparing Eqs. (19) and (III) we see that:

Therefore,

Thus, using , we have:

parametrizes the magnitude of deviations from CDM. If has changed by in the last Hubble time, Eq. (III) implies that, in the recent past and in the Jordan frame, to within an order of magnitude:

(26) |

For later use we rewrite Eq. (III) in terms of :

(27) | |||||

In both the Einstein and Jordan frames, gives the relative change in the ratio of any particle mass, , and the Planck mass, between the times when and when . In the Einstein frame is constant but varies whereas in the Jordan frame the converse holds; the ratio of the two masses, being a dimensionless quantity, is the same in either frame. WMAP constrains any such variation in between now and the epoch of recombination to be at ( at ) WMAPvaryG . It follows that since recombination

(28) |

Light element abundances provide similar constraints on any variation in Newton’s constant between the present day and the time of nucleosynthesis BBNvaryG .

Thin-shell constraints, however, provide an even tighter bound on the allowed change in . To consider these constraints we work in the Einstein frame, however is the same in either frame.

We assume, as is the case for the real Universe, that the scales of the inhomogeneous regions are small compared to the horizon scale, and that the Universe is approximately homogeneous when coarse-grained over scales larger than some . Thus over scales larger than , and since , we can work entirely over sub-horizon scales, which simplifies the analysis greatly. We also assume that the curvature of spacetime is weak over scales smaller than . This is equivalent to assuming that the Newtonian potential, , is small as are the peculiar velocities, , of any matter particles, i.e they are non-relativistic.

Exploiting both the assumption that and that gravity is weak inside the the inhomogeneous regions i.e. and , we write and have to leading order in the small quantities and over sub-horizon scales:

Now

and so

where

Thus

(29) | |||||

It is straightforward to show that the condition , which must hold for any chameleon theory, implies that for all and . Thus

Now if we require that a test mass at with central density has a thin-shell, we must impose that at , , where

Thus must be able to change by at least i.e. we have the following necessary condition for thin-shell

(30) |

The right hand side of this equation is or smaller, and the largest values of the peculiar Newtonian potential for realistic models of our Universe are roughly , and are generally around for large clusters and superclusters Hu2 . Thus we have the following conservative constraint on the cosmological value of the field today:

(31) |

We have defined . The thin-shell constraint certainly ensures that cosmologically today and since we have by definition we are therefore justified in assuming that we have . Then assuming that we find that the potential, , is given by:

and

To leading order then in we have:

The chameleon mass squared, is then given, to leading order, by:

(32) |

Provided , then the chameleon field will remain close to the minimum of its effective potential chamcos cosmological, i.e. and the energy density of the chameleon field will be dominated by its potential. Assuming that this is the case we would have:

and defining and , we have:

and so becomes:

Therefore, in many theories, an observationally viable evolution of requires that it has sat close to the effective minimum of its potential since recombination chamcos i.e.:

Since the background density of matter decreases with time, implies that increases with time. Thus for test mass with density , we have in the recent past, i.e. out to :

where is the current time. In this case Eq. (31) gives the following conservative constraint:

and so, from Eq. (26) we obtain that:

(33) |

In the recent past where is not negligible, this leads to a stringent constraint on the deviation of the equation of state from -1. It should be noted that although is constructed simply out of the scale factor, , and its derivatives, neither nor are uniquely defined as functions of redshift in models such as these where the scalar field interacts with normal matter. As a result, it is possible to define so that it vanishes and even becomes negative in the past. If such a definition is made, then one would (unless the also happens to vanish) predict that diverges, and hence deviates significantly from . A behaviour such as this was noted in Refs. tsu1 ; tsu2 . As a result, an apparent effective deviation from CDM can be deduced. However, because of the freedom to redefine and hence , one should not rush to assign any physical meaning to the divergence of , and deduce that it represents a significant deviation from CDM, since one could always remove this divergence by choosing to define it such a way that it is positive definite. In all cases, the bound (33) gives an intrinsic measure of the deviation of the background cosmology from CDM, and it all cases it is small. Therefore, the predicted late time cosmology is observationally very close to CDM. Additionally, the prospects for being able to detect such small deviations for CDM at the background level in the near future are poor. Of course, as we have already mentioned, detectably at the perturbative level might be within reach.

This said, the thin-shell constraints do not themselves rule out larger deviations from CDM. It may be that has undergone relatively large changes in the past i.e. much larger than , but that we now just happen to live at a point in time when . This would, however, be a fairly remarkable coincidence and would inevitably require a great deal of fine-tuning of the theory and the initial conditions. To avoid this new coincidence problem, we would have to require that the cosmological changes in have been smaller than in the recent past which would in turn, as we illustrated above, constrain any deviations from CDM to be unobservably small. We note, however, that deviations can be expected on very small scales, as in the original chameleon model green .

In this section we have sketched how the thin-shell requirement for laboratory test masses place a very strong constraint on the recent cosmological evolution of , and generally constrains any deviations from CDM in the predicted cosmology to be small. This is not, however, a ‘water tight’ constraint as it may be possible to circumvent it by requiring a seemingly improbable cosmological evolution wherein such bodies would only have developed thin-shells in the recent (in the cosmological sense) past. The laboratory constraints which we will derive in what follows cannot be avoided in this way.

## Iv Inverse Square Law Constraints

In the weak field limit, the gravitational force due to a small body drops off as , where is the distance to the body’s centre of mass. If there is an additional scalar degree of freedom to gravity with constant mass , the force instead drops off as:

where parametrizes the strength with which the scalar degree of freedom couples to matter. In theories . When or , the force still drops off, approximately, as , however there would be a noticeable deviation from this behaviour over scales . If, as in chameleon theories, is not a constant but instead undergoes or greater variations, the behaviour of the force is more complicated but generally not of inverse square law form.

It is often assumed that what is needed for an theory to avoid the constraints of inverse square law tests, is that the test bodies develop thin-shells. Generally, however, this is not the case. The presence of a thin-shell causes the chameleonic force due to a body to drop off much faster than near the surface of the body. Far from the body, the force has a Yukawa form, although as a result of the fast drop-off near the surface, it is much smaller than one would normally expect it to be. If two thin-shelled bodies are sufficiently close however then they would be inside the region where the faster drop-off is occurring. In these cases the detectable violation of the inverse square law can be much larger than one might expect.

A number of different experiments have searched for violations of the inverse square law. For gravitational strength forces, i.e. , the best constraints are currently provided by the Eöt-Wash experiment EotWash .

The Eöt-Wash experiment EotWash consists of two plates: the attractor and the detector. The detector is thick and made out of molybdenum. The detector has 42 diameter holes bored into it in a pattern with -fold azimuthal symmetry. The attractor is similar and consists of a thick molybdenum plate with 42 diameter, arranged in a pattern with -fold azimuthal symmetry, mounted on a thicker tantalum disc with 42 holes, each with diameter . The holes in the lower tantalum ring are displaced so that the torque on the detector due to the attractor from forces, such as Newtonian gravity, that have a behaviour vanishes. The detection of a non-zero torque would therefore indicate the presence of either a correction to gravity with a behaviour different from or the presence of a new force that also did not behave as .

### iv.1 Chameleonic Force & Torque

We now calculate the force, due to a chameleonic scalar field, , on one plate due to the other lying parallel to it. From this we calculate the chameleonic contribution to the torque.

In a background, where far from the plates, the chameleonic force per unit area between two parallel plates, of the same or similar compositions, both with thin-shells and with a distance of separation between their two facing surfaces was found, under certain conditions, in Ref. chamstrong ; chamcas . In Appendix A we generalise those formulae. We find that the chameleonic force between two parallel circular plates, with radius and thickness , and separation is given by:

where is defined to the values of midway between the two plates, and formulae for it are provided in Ref. chamcas . We have also defined:

(35) | |||||

(36) | |||||

(37) |

The last term in Eq. (IV.1) represents the only difference between the generalized force formula and the one presented in Refs. (chamstrong ) & (chamcas ), and we note that the extra term is independent of the separation . When , the last term in Eq. (IV.1) is negligible. We note that when , and so whenever , the last term is always negligible.

The details of how drops off with will depend on the form of . For many choices of , e.g. for or , one finds that drops off faster than , for all is small compared to both and the radius, , of holes in the plates. Indeed, this will certainly be the case, no matter what form takes, if where . Provided this is the case, we can define the potential energy, due to the chameleonic force for two plates with separation thus:

(38) |

The faster drop off has been used to set the upper limit of the above integral to .

In the Eöt-Wash experiment the plates have a number of holes in them. This means that as one plate is rotated, by an angle say, relative to another, the surface area, , of one plate that faces the other changes. Note that does not depend on . The torque due to the chameleonic force is given by the rate of change of the potential with :

(39) |

We therefore have:

(40) |

where is a constant that depends only on the details of the experimental set-up rather than the theory being tested. For the 2006 Eöt-Wash experiment EotWash we find

If drops off too slowly over scales of the order of then a more complicated analysis must be performed, and knowing the force between two infinite parallel plates is no longer enough to find a good approximation to the torque. Instead a full numerical analysis would have to be undertaken to get accurate results. This said, for , we do not expect to depend strongly on because the effect of the holes will be largely smeared out over separation distances much larger . On scales , we found that . Since , , and we expect to be largely independent for and on smaller scales, we expect, to within an order of magnitude, that:

in these cases, where once again . By picking as an upper bound for the integral we are probably under estimating the torque as we are dropping the contributions from larger separations.

### iv.2 The Effect of an Electrostatic Shield

Up to now we have not considered the role played by the electrostatic shield. Because the shield is so thin () compared to the plates but has similar density to the plates, it is safe to say that the shield will only have a thin-shell when the plates have thin-shells. Assuming the plates do have thin-shells, we define to be the mass the chameleon would have deep inside the shield if the shield has a thin-shell i.e. where . Since the shield is sandwiched between the two plates, the thin-shell condition for the shield is simply . When the shield has a thin-shell, its presence attenuates the chameleonic force and torque on the detector due to the attractor by a factor of . Since in the absence of a thin-shelled shield, we can take account of the shield, thin-shell or not, by changing the definition of thus:

(41) |

This expression provides a very good approximation for theories in which the precise value of is unimportant (e.g. ones with ) and an order of magnitude estimate otherwise.

### iv.3 Inverse Square Law Constraints

The 2006 Eöt-Wash experiments requires that

with confidence. We define and find that the above bounds correspond to:

(42) |

Importantly this is smaller than the energy scale associated with dark energy: ; .

Using our expression Eq.(41) for the chameleonic torque, we find that the constraints we must apply are as follows:

(43) |

## V Application of Constraints to theories

### v.1 Chameleonic force

The chameleonic force per unit area between two parallel plates is given by Eq. (IV.1). To prevent large deviations from general relativity occurring over solar system, and smaller, scales, one must require that where and . In this case the expression for becomes:

where for

(44) | |||||

(45) | |||||

(46) |

where is the radius of the parallel plate(s). We shall now consider several potential forms for .

### v.2 Logarithmic potentials

We begin by considering a simple chameleon gravity model that was recently suggested in Ref. kaloper for a general . The theory, when written as a chameleon theory, would have a potential , it was suggested that this would result in an experimentally viable and cosmologically interesting dark energy model, where kaloper for . We will analyse the same model in the setting where and show that local tests already lead to difficulties, see also kaloper2 .

On laboratory scales we would have and so we find:

Assuming that where is the chameleon mass inside the plates, it follows that has the following form:

(47) |

where

When we have