Tests of chameleon gravity
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Abstract
Theories of modified gravity, where light scalars with nontrivial selfinteractions and nonminimal couplings to matter—chameleon and symmetron theories—dynamically suppress deviations from general relativity in the solar system. On other scales, the environmental nature of the screening means that such scalars may be relevant. The highlynonlinear nature of screening mechanisms means that they evade classical fifthforce searches, and there has been an intense effort towards designing new and novel tests to probe them, both in the laboratory and using astrophysical objects, and by reinterpreting existing datasets. The results of these searches are often presented using different parametrizations, which can make it difficult to compare constraints coming from different probes. The purpose of this review is to summarize the present stateoftheart searches for screened scalars coupled to matter, and to translate the current bounds into a single parametrization to survey the state of the models. Presently, commonly studied chameleon models are wellconstrained but less commonly studied models have large regions of parameter space that are still viable. Symmetron models are constrained well by astrophysical and laboratory tests, but there is a desert separating the two scales where the model is unconstrained. The coupling of chameleons to photons is tightly constrained but the symmetron coupling has yet to be explored. We also summarize the current bounds on f(R) models that exhibit the chameleon mechanism (Hu and Sawicki models). The simplest of these are well constrained by astrophysical probes, but there are currently few reported bounds for theories with higher powers of R. The review ends by discussing the future prospects for constraining screened modified gravity models further using upcoming and planned experiments.
Keywords
Scalartensor theories Modified gravity Tests of gravity1 Introduction
Since its publication in 1915, Einstein’s theory of general relativity (GR) has withstood the barrage of observational tests that have been thrown at it over the last century. From Eddington’s pioneering measurement of light bending by the Sun in 1919 to the first detection of gravitational waves by the LIGO/Virgo consortium in 2015 (Abbott et al. 2016a, b), its predictions have been perfectly consistent with our observations. To test the predictions of any theory requires alternatives with differing predictions and, for this reason, alternative theories of gravity have a history that is almost as rich and varied as that of GR itself.
The zoo of modified gravity theories is both vast and diverse (see Clifton et al. 2012; Joyce et al. 2015; Koyama 2016; Bull et al. 2016, for some compendia of popular models) but all have one thing in common: they break one of the underlying assumptions of general relativity. From a theoretical standpoint, GR is the unique lowenergy theory of a Lorentzinvariant massless spin2 particle (Weinberg 1965), and any modification must necessarily break one of these assumptions. Several interesting and viable Lorentzviolating theories exist that may have some insight for the quantum gravity problem (Blas and Lim 2015), and, similarly, healthy theories of massive spin2 particles have recently been constructed (de Rham 2014).
An alternative to these approaches is to introduce new fields that couple to gravity. One of the simplest possible options is to include a new scalar degree of freedom. These scalar–tensor theories of gravity are particularly prevalent, and are natural extensions of general relativity. Scalars coupled to gravity appear in many UV completions of GR such as string theory and other higherdimensional models, but the cosmological constant problem and the nature of dark energy, two modern mysteries that GR alone cannot account for, are driving a vigorous research effort into infrared scalar–tensor theories, with much of the effort focussing on light scalars (with cosmologically relevant masses) coupled to gravity.
Typically, the existence of such scalars are in tension with experimental bounds. If the scalar is massless, or has a Compton wavelength larger than the size of the solar system (which is certainly the case for Hubblescale scalars), the theory’s predictions typically fall within the remit of the parameterized postNewtonian (PPN) formalism for testing gravity in the solar system (see Will 2004, and references therein). Scalars whose Compton wavelengths are smaller than \(\sim \) AU predict deviations from the inversesquare law inside the solar system, which has been tested on interplanetary scales using lunar laser ranging (LLR) (Williams et al. 2004), and down to distances of \(\mathcal {O}(\upmu \mathrm {m})\) using laboratorybased experiments such as the EötWash torsion balance experiment (Adelberger et al. 2003). In many cases, scalar–tensor theories spontaneously break the equivalence principle so that objects of identical mass but differing internal compositions fall at different rates in an external gravitational field. This too can be tested with LLR and terrestrial searches.
Recently, the simultaneous observation of gravitational waves and a gamma ray burst from a binary neutron star merger (GW170817 and GRB 170817A) (Abbott et al. 2017a, b) by the LIGO/Virgo collaboration and the Fermi and INTEGRAL satellites has placed a new and stringent bound on modified gravity theories. The close arrival time of the gravitational wave and photon signal (\(\delta t<1.7\mathrm {\ s}\)) constrains the relative difference speed of photons (c) and gravitons (\(c_T\)) to be close to unity at the \(10^{15}\) level (\(3\,\times \,10^{15}<c_T^2c^2/c^2<7\,\times \,10^{16}\)) (Sakstein and Jain 2017; Ezquiaga and Zumalacárregui 2017; Creminelli and Vernizzi 2017; Baker et al. 2017; Crisostomi and Koyama 2018; Langlois et al. 2017; Dima and Vernizzi 2017; Bartolo et al. 2017), where the upper and lower bounds correspond to a \(\sim 10\) s uncertainty in the time between the emission of the photons and the emission of the gravitational waves Abbott et al. (2017b). Many scalar–tensor theories predict that the difference between the speeds of gravitons and photons is of order unity for models that act as dark energy (Bellini and Sawicki 2014; Brax et al. 2016) and so this bound represents a new hurdle for them to overcome.
These stringent bounds imply that the simplest theories with light scalars have couplings to matter that must be irrelevant on cosmological scales. Theories that try to avoid this problem using a large mass to pass solar system tests must have a Compton wavelength \(\le \mathcal {O}(\upmu \mathrm {m})\), in which case they too are cosmologically inconsequential. Ostensibly, it seems that scalar–tensor theories are trivial in a cosmological setting, but the link between solar system tests of gravity and cosmological scalar–tensor theories can be broken. Indeed, the last decade of scalar–tensor research can aptly be epitomized by two words: screening mechanisms.
Screening mechanisms utilize nonlinear dynamics to effectively decouple solar system and cosmological scales. At the heart of screening mechanisms lies the fact that there are 29 ordersofmagnitude separating the cosmological and terrestrial densities and 20 orders of magnitude separating their distance scales. As a result, the properties of the scalar can vary wildly in different environments. The quintessential example of a screening mechanism being used to ensure a dark energy scalar avoids solar system constraints is the chameleon mechanism (Khoury and Weltman 2004a, b; earlier predecessors include Gessner 1992; Pietroni 2005; Olive and Pospelov 2008). In chameleon models, the mass of the scalar is an increasing function of the ambient density. This allows it to have a submicron Compton wavelength in the solar system but be light on cosmological scales. Later, a closely related second dark energy screening mechanism was discovered: the symmetron mechanism (Hinterbichler and Khoury 2010; Hinterbichler et al. 2011a). Earlier work had studied a similar model but with a different motivation (Pietroni 2005; Olive and Pospelov 2008), and stringinspired models with similar phenomenology have also been proposed (Damour and Polyakov 1994; Brax et al. 2011a). Unlike the chameleon, the symmetron has a light mass on all scales and instead screens by driving its coupling to matter to zero when the density exceeds a certain threshold. A third mechanism, the environmentdependent dilaton was subsequently discovered that screens in a similar manner (Brax et al. 2010a).
In this work, we will only discuss screening mechanisms of this type, which rely on nonlinear selfinteraction terms in the potential. A final class of screening, which relies on nonlinearities in the kinetic sector screen through what is known as the Vainshtein mechanism (Babichev and Deffayet 2013; Joyce et al. 2015). These theories will not be discussed here as the phenomenology of these models, and therefore the most constraining observables, are very different to that of the chameleon and symmetron models. Similarly, we will not discuss massive gravity (de Rham et al. 2011; Hinterbichler 2012; de Rham 2014; de Rham et al. 2017), which screens using the Vainshtein mechanism, for the same reason. We note, however, that many models that do screen using the Vainshtein mechanism (as well as those that predict a mass in the graviton dispersion relation such as massive gravity) are severely constrained by the new bounds from the observation of gravitational waves and photons from GW170817/GRB 170817A discussed above if they are to simultaneously act as dark energy (Sakstein and Jain 2017; Baker et al. 2017; Ezquiaga and Zumalacárregui 2017; Creminelli and Vernizzi 2017; Crisostomi and Koyama 2018; Langlois et al. 2017). [In the case of massive gravity, solar system tests are stronger than the LIGO/Fermi bound (Baker et al. 2017)]. The models we will discuss in this review (chameleon/symmetron/dilaton) predict that \(c_T=c\) identically and so this bound is irrelevant for them.
Scalar fields with screening mechanisms cannot simultaneously screen and selfaccelerate cosmologically (Wang et al. 2012), but they can act as a dark energy quintessence field (Copeland et al. 2006), i.e., they require a cosmological constant term to drive the cosmic acceleration and they are capable of producing deviations from GR on linear and nonlinear cosmological scales as well as astrophysical scales (see Jain et al. 2013b; Sakstein 2014a, and references therein). In addition to this, many candidate UV completions of GR such as string theory predict a multitude of scalars that couple to matter and screening mechanisms are a convenient method of hiding such additional degrees of freedom. For these reasons, screening mechanisms are considered interesting and novel paragon for alternative theories of gravity and, as such, there is an ongoing experimental search for screened scalars. Being designed to evade conventional tests of gravity, screening mechanisms have inspired novel and inventive approaches to search for them experimentally. These range from reinterpreting the results of experiments not designed to look for them, to designing instruments specifically adapted to testing their unique properties, to using astrophysical objects that have never before been used to test gravity, such as Cepheid stars and galaxy clusters. In many cases, new and imaginative scenarios have been concocted.
These searches typically use different parametrizations, making them difficult to compare with one another. The purpose of this review is to collect the stateoftheart constraints coming from laboratory and astrophysical tests, and to combine them into a single parametrization. This not only makes it clear which models are ruled out by different experiments, but also aides in deciding the optimum search strategy for exploring the remaining models. In many cases, we will extend the experimental results to models to which they have not previously been applied.
This review is organized as follows. In Sect. 2, we will introduce the different screening mechanisms we will consider in this review, outline their salient features, and present the parameters we will use to compare constraints. In Sect. 3, we will discuss how screening works in both astrophysical and laboratory settings. Section 4 contains a brief description of the experiments that have been used to constrain screening mechanisms, and translates the constraints into our parametrization. The crux of this review is presented in Sect. 5, where we combine all of the contemporary constraints from various experiments into a series of diagrams that show which regions of parameter space are ruled out, and how different experiments compare in the same parametrization. We do this for chameleon and symmetron modes. In Sect. 6, we conclude by discussing the implications of the constraints for screened modified gravity, and future prospects for constraining the remaining parameter space.
2 Screening mechanisms
 1.
The mass \(m_\mathrm{eff}r\gg 1\) so that the force is short ranged,
 2.
The coupling to matter \(\beta (\phi _0)\ll 1\), or
 3.
Not all of the mass sources the scalar field.
It is possible to construct models with the requisite densitydependent minimum such that one or more of the conditions above are satisfied. Models that utilize a combination of the first and third condition are typically known as chameleon models^{4} (Khoury and Weltman 2004a) and models that utilize the second are known as either symmetron (Hinterbichler and Khoury 2010) or dilaton models (Brax et al. 2010a).
2.1 Chameleon screening
2.1.1 f(R) models
2.1.2 UV properties
Screening relies on the presence of nonlinear selfinteractions of the scalar field, and on coupling the scalar to the matter energy momentum tensor. Written in the Einstein frame, this necessarily introduces nonrenormalisible operators, meaning that additional physics is required in order to UV complete the model (Joyce et al. 2015). Additionally, we might worry that integrating out physics in the UV changes the form of the lowenergy theory, either rescaling the coefficients, or introducing new terms into the Lagrangian.
For the theory to be fully predictive, it is important to understand whether the lowenergy theory we study is protected from corrections coming from UV physics. One commonly used way to estimate the size of these effects is to compute the Coleman–Weinberg (Coleman and Weinberg 1973) corrections to the scalar mass (Upadhye et al. 2012a). To do this, one computes the corrections to the scalar mass from scalar fields running in loops, these loop corrections arise precisely because the scalar field has nontrivial self interactions in its potential. The Coleman–Weinberg corrections are found to be at least logarithmically divergent with scale. Even if these corrections to the mass are assumed to be small at some scale, they may become important at another scale, or in another environment.
In Upadhye et al. (2012a), the relevance of these corrections for the EötWash experiment was computed. With some simple assumptions about the scale at which the logarithmic terms become important, it was shown that the current constraints from these experiments are computed in a regime in which the quantum corrections are indeed under control. However, as the experimental sensitivity improves these corrections will become more relevant.
Keeping track of the quantum corrections is also important in order to understand the behaviour of the chameleon in the early universe. In Erickcek et al. (2014, 2013), it was shown that, with the exception of gravitationally coupled chameleons, it is not possible to evolve the chameleon through the radiation dominated era without knowing the UV completion of the model. This is because the decoupling of standard model particles during this epoch give a large impulse to the otherwise slowly rolling chameleon field (Brax et al. 2004b). This causes the chameleon scalar to rapidly roll to the part of the potential where the field’s self interactions are large, and so high energy quantum fluctuations of the field are excited. It is possible that some nonperturbative physics could resolve this, but in the absence of a proof of this, we do not know how to evolve the chameleon model from the early universe to late times in a predictive way. One model, which can evade this problem, is the case \(n=4\) due to the absence of a lowmass scale (that is problematic in the early Universe when energies are typically high) (Miller and Erickcek 2016).
The most reliable way to compute UV corrections to the lowenergy chameleon model would be to know exactly what the UVcompletion of the theory is. A number of attempts have been made to embed the chameleon mechanism within string theory (Brax et al. 2004a; Conlon and Pedro 2011; Hinterbichler et al. 2011b; Nastase and Weltman 2015, 2013), within supersymmetry (Brax et al. 2013b, a), and using noncanonical kinetic terms (Padilla et al. 2016), but, as yet, no complete theory exists.
2.2 Symmetron screening
2.2.1 Generalized symmetrons
2.2.2 Radiativelystable symmetrons
The symmetron model, as described here, suffers the same UV stability properties as the chameleon. In particular, that Coleman–Weinberg corrections could dramatically alter the shape of the potential needed for the symmetron mechanism to work. In this case, however, the oneloop corrections can also be exploited to give rise to the screening in a radiatively stable way (Burrage et al. 2016a).
The Coleman–Weinberg model (Coleman and Weinberg 1973) was originally discussed as a way of using radiative corrections to generate a spontaneous symmetry breaking transition. The classical model is scale invariant, but the oneloop corrections generate a scale through dimensional transmutation of the logarithmic divergences. In the onefield model, higherorder loop corrections become important in the spontaneously broken vacuum, but in a multifield model these can be kept under control (Garbrecht and Millington 2015), and the oneloop potential can undergo a symmetry breaking transition whilst the higherorder loop corrections remain small.
2.3 Coupling to photons
A conformally coupled scalar field does not have a classical coupling to photons. This is because the scalar couples to the trace of the energy momentum tensor of the matter fields, and photons, being relativistic, have a traceless energy momentum tensor. This is not the end of the story, however, as quantum effects make it easy to generate such a coupling. One way to do this, is to assume the presence of a new heavy fermion which has an electromagnetic charge. Then, an interaction between one conformally coupled scalar, and two photons can be mediated by a triangleloop of the heavy fermion. If the fermion is sufficiently heavy that it can be integrated out, to leave the Standard Model plus the chameleon as a lowenergy effective theory, then the lowenergy theory has a contact interaction between the chameleon and two photons (Brax et al. 2010c). Such heavy, charged fermions are ubiquitous in theories of physics beyond that Standard Model, including, string theory, supersymmetry and GUTs. It can also be shown that the Weyl rescaling that allows us to change from Jordan to Einstein frame, gives rise to a coupling to photons after quantisation of these fields, this was shown for the chameleon in Brax et al. (2011b), following earlier work by Kaplunovsky and Louis (1994) in the context of supersymmetry.
3 Screening
In this section, we discuss screening mechanisms in the context of astrophysical objects and typical laboratory configurations, and discuss some salient features that are specific to screening mechanisms.
3.1 Astrophysical screening: the thinshell effect
Astrophysical objects of interest and their Newtonian potentials
Object  \(\varPhi _\mathrm{N}\) 

Earth  \(10^{9}\) 
Moon  \(10^{11}\) 
Mainsequence stars (\(M_\odot \))  \(10^{6}\) 
post Mainsequence stars (1–\(10 M_\odot \))  \(10^{7}\)–\(10^{8}\) 
Spiral and elliptical galaxies  \(10^{6}\) 
Dwarf galaxies  \(10^{8}\) 
In practice, one also needs to worry about environmental screening. So far, we have only considered the screening of a single object embedded in a larger background, but real astrophysical objects are typically not isolated; galaxies are found in clusters and stars come in pairs or groups. The nonlinear nature of the field equations means that we cannot simply superimpose solutions sourced by different objects to obtain a new solution. This implies that an object’s environment can affect whether it is screened or not. The most important example of this is the screening of dwarf galaxies. Taken in isolation, the Newtonian potential for a dwarf galaxy is \(\mathcal {O}(10^{8})\) but the typical potential associated with clusters of galaxies is \(\mathcal {O}(10^{4})\) so that only values of \(\chi _0\) larger than this can be tested. The ideal probes are, therefore, dwarf galaxies located in voids that do not suffer from environmental screening. There has been a great effort towards determining the criteria for environmental screening (Li et al. 2012; Lombriser et al. 2012a, 2013; Cai et al. 2015). Most of these rely on numerical Nbody simulations, whose description lies outside the scope of this review, but the end result is a screening map (Cabre et al. 2012) of the local universe that classifies galaxies as either screened, partially screened, or unscreened. To date, all astrophysical tests using dwarf galaxies have been taken from this screening map.
3.1.1 Screening in f(R) theories
3.1.2 Gravitational lensing: dynamical versus lensing masses
3.2 Solarsystem tests
Classical tests of GR use the PPN formalism applied to solarsystem objects and so, in this section, we will illustrate how these tests apply to screened modified gravity, and why they yield only weak constraints.
3.2.1 PPN parameters
General expressions for \(\gamma \) and \(\beta \) in screened scalar–tensor theories can be found in Hees and Fuzfa (2012) and Zhang et al. (2016). It is more instructive, however, to consider the solution for the fifthforce profile of a static object derived in (3.6). We will ignore the mass of the scalar for simplicity but including it does not change any of what follows. The calculation of the fifthforce was performed in the Einstein frame but the PPN metric is defined in the Jordan frame, since it is the metric that controls the geodesics of matter and so our task is to calculate the Jordan frame metric given \(\phi \) to \(\mathcal {O}(v^2/c^2)\) to find \(\gamma \). The calculation of \(\beta \) is analogous except one continues to \(\mathcal {O}(v^4/c^4)\); this calculation is long and tedious, and one does not gain any additional insight. For this reason, we will only calculate \(\gamma \).
3.2.2 Lensing revisited
The careful reader will now be puzzled by a conundrum. We have already argued in Sect. 3.1.2 that screened modified gravity (in fact, our derivation above applies equally to all conformal scalar–tensor theories) does not affect the lensing of light. We have also argued in this section that the PPN parameter \(\gamma \ne 1\) so that light bending by the Sun is different than in GR, which implies that the scalar does affect lensing. In fact, both of these statements are compatible, the difference is merely a choice of coordinates.
3.3 Equivalence principle violations
3.4 Laboratory screening
Laboratory searches for screened fifth forces, and the particles that mediate them, are typically performed in a vacuum chamber. Inside this chamber, the position of the minimum of the effective potential can be different to the minimum of the effective potential in the walls of the vacuum chamber and its environment. This is the key difference between screening in the laboratory, and screening in other astrophysical environments; in a vacuum chamber there is a region of low density surrounded by a region of higher density.
The behaviour of the field in the experimental apparatus depends on its mass, as the corresponding Compton wavelength sets the scale over which the field can vary its value. The field can only change its value from the exterior of the experiment to the interior of the walls of the vacuum chamber if its Compton wavelength in the walls is of order the thickness of the walls or smaller. Similarly, the field can only vary its value from the walls to the vacuum at the center of the chamber if its Compton wavelength in the chamber is comparable to, or smaller, than the diameter of the chamber.
The chameleon field can vary its mass much more easily than the symmetron, and as a result laboratory tests constrain a much broader range of models for the chameleon. If the symmetron mass is too small it will not be able to vary its VEV over the scale of the experiment. In this case, there are no field gradients in the experiment, and no resulting fifth forces, so no constraints can be placed. As the symmetron mass increases the vev starts to vary within the experiment, and a fifth force is present, however this fifth force may then be exponentially suppressed by the Yukawa term \(e^{m r}\), where m is the mass of the symmetron in the vacuum. In general, therefore, laboratory experiments will only constrain a small range of symmetron masses (Upadhye 2013; Burrage et al. 2016b; Brax and Davis 2016).
The chameleon field can vary more easily in a laboratory vacuum, and therefore is much more amenable to laboratory constraints. Over a wide range of the chameleon parameter space, the chameleon will not be able to reach the value that minimises its potential in the interior of the vacuum chamber, and instead it will evolve to the value that sets its mass to be of order the size of the chamber. Once the corresponding Compton wavelength becomes smaller than the size of the chamber, the field is able to reach the minimum of its effective potential.
Clearly determining both the background value of the scalar field and the condition for screening becomes more complicated for nonspherical geometries, and in these cases, numerics are needed to place definitive constraints. However, the principles described here will still guide the shape of the field profile and the conditions for screening.
Laboratory searches for fifth forces are performed with both classical and quantum experiments. To determine the condition for screening in a quantum experiment requires a little more thought. If the experiment is sufficiently low energy that the internal structure of the source is not disrupted, it must still be checked how the chameleon screening condition is affected by the delocalisation of the object’s center of mass (Burrage et al. 2015). The chameleon can respond to changes in the position of the source on timescales on the order of \(1/ m_\mathrm{eff}(\phi _\mathrm{vac})\), and a delocalised source can be considered to fluctuate around with a timescale \(R_\mathrm{trap}/v\), where \(R_\mathrm{trap}\) is the spatial extent of the trapping potential, and v is the velocity of the particle. If \((v/R_\mathrm{trap})<m_\mathrm{vac}\), the chameleon field can respond to the quantum fluctuations of the object and, therefore, it is the object’s density and size that determine whether the object is screened, regardless of the uncertainty on its centerofmass position. Otherwise, the chameleon cannot respond to the fluctuations in the position of the source, and the relevant density in the screening condition is \(\bar{\psi }_\mathrm{obj}\psi _\mathrm{obj}\), where \(\psi _\mathrm{obj}\) is the wavefunction of the object (Burrage et al. 2015).
3.5 Screening in the Jordan frame
4 Experimental tests
In this section, we summarize the present experimental tests of chameleon and symmetron screening, which range from particlecollider and precisionlaboratory experiments to astrophysical tests using stars and galaxies.
4.1 Fifthforce searches
Fifthforce searches aim to directly measure the force between two objects and search for deviations from Newton’s law. The experiment is performed inside a vacuum chamber to reduce noise, and the geometry of the experiment is designed to minimize the Newtonian force. Recently, some experiments have been designed specifically for the task of searching for chameleons, either by adapting the geometry to maximize the chameleon force, or by varying the density inside the vacuum chamber. Typically, scales of order \(\upmu \)m or greater are probed.
4.1.1 Torsion balance experiments
Torsion balance experiments typically consist of one mass that acts as a pendulum suspended above a second that sources a gravitational field and acts as an attractor. The two masses are arranged in a manner that cancels the inversesquare contribution to the total force so that the experiment is sensitive to any deviations.
The stateoftheart in torsion balance tests is the EötWash experiment (Adelberger et al. 2003; Kapner et al. 2007; Lambrecht et al. 2005), which uses two circular disks as testmasses. The disks have holes bored into them which act as missing masses, giving rise to a net torque due to dipole (and higherorder multipole) moments. The upper disk is rotated at an angular velocity such that the contribution from any inversesquare forces to the torque is zero and, therefore, any residual force is nonNewtonian. The absence of any such forces places strong constraints on noninversesquare law modifications of gravity. This includes any scalar–tensor theory where the field is massive, including Yukawa interactions, and chameleons.
In order to reduce electromagnetic noise, the pendulum and attractor are coated in gold and a berylliumcopper membrane is placed between them. This poses no additional problems for linear theories such as Yuakawa forces, but does present several technical complications for chameleon theories. The membrane may or may not have a thin shell depending on the parameters under study, and the highly nonlinear nature of the field equations make the theoretical modelling of this nonsymmetric system difficult. Over time, several works have appeared with the aim of improving the accuracy of the theoretical calculation of the chameleon torque (Brax et al. 2008; Adelberger et al. 2007; Mota and Shaw 2006, 2007; Upadhye 2012b), the most recent being the work of Upadhye (2012a), which uses the socalled onedimensional planeparallel approximation to include the effects of the missing masses on the chameleon force profile. A similar effort has been undertaken for symmetron models, with the most stringent constraints presented in Upadhye (2013).
4.1.2 Casimirforce tests
The Casimir force (or Casimir–Polder force) is a prediction of quantum electrodynamics. Classically, two uncharged parallel plates placed in a vacuum would source no electromagnetic fields and, therefore, would feel no force; quantum mechanically, they interact with virtual photons of the vacuum resulting in a net force that can be interpreted as being due to the zeropoint energy of the field between the plates. This force scales as \(d^{4}\) (d is the distance between the plates) and is hence subdominant to the Newtonian force except at small separations.
The current generation of Casimir force experiments place strong constraints on \(n=4\) and \(n=6\) chameleon models when \(\varLambda _c\) is fixed to the dark energy scale. The constraints on other models are not presently competitive with other experiments discussed in this review. The next generation of experiments will use larger separations where the chameleon force is more pronounced (Lambrecht et al. 2005; Lamoreaux and Buttler 2005) so more stringent constraints on a broader class of models are expected.
Interestingly, experiments such as these can be adapted to the chameleon’s unique properties because one can vary the density of the partial vacuum inside the chamber where the experiment operates. By changing the pressure of the ambient gas, one can look for a densitydependent change in the force, which would be a smoking gun of chameleon models (Brax et al. 2010b; Almasi et al. 2015).
At the present time, Casimir force experiments have not been applied to symmetron models, mainly due to the lack of any theoretical calculations of the symmetron force between objects of different geometries.
4.1.3 Levitated microspheres
An experiment measuring forces using levitated microspheres has recently been applied to chameleon models resulting in new constraints on \(n=1\) models (Rider et al. 2016); other models have yet to be considered. Constraints on symmetron models are not currently competitive with other experiments (Burrage et al. 2016b).
4.2 Precision atomic tests
4.3 Atom interferometry
Atom interferometry is a hybridization of classical interferometric experiments and quantum mechanical double slit experiments. Atoms can be put into a superposition of two states, which travel along different paths and hence act like the arms of an interferometer. The two paths can be recombined later to produce an interference pattern that can be measured.
The atoms can be moved within the interferometer by shining laser light on them. If an atom absorbs a photon, it will be excited into a higher energy state and acquire the photon’s momentum, resulting in some linear motion. In the absence of any observation, the atom is in a superposition of the ground state (where it is stationary) and an excited state (where it is in motion). The atom can be put into a superposition of states that travel along different paths by repeating this process several times.
A massive object placed inside the vacuum chamber will source a gravitational field that contributes to a. If, in addition to this, the object sources a chameleon field then this too contributes and the probability of measuring excited atoms is sensitive to it. Since atoms placed in vacuum chambers are unscreened over a large range of the parameter space, this experiment is incredibly sensitive to chameleon and symmetron forces (Burrage et al. 2015; Burrage and Copeland 2016; Elder et al. 2016). Indeed, the first generation of atom interferometry experiments designed to test screened modified gravity was able to constrain any anomalous acceleration down to levels of \(10^{6}g\) (\(g\equiv GM_\oplus /R_{\oplus }\) is the gravitational acceleration at the surface of the Earth), placing new constraints on chameleons and symmetrons that vastly reduced the viable parameter space (Hamilton et al. 2015; Burrage et al. 2016b). The current generation of experiments has constrained this further to \(\lesssim 10^{8}g\), reducing the parameter space further (Jaffe et al. 2017).
4.4 Precision neutron tests
Neutrons are perfect objects for testing shortrange gravitational physics because they are electrically neutral and are, therefore, not sensitive to electromagnetic noise such as background fields and van der Walls forces.^{8} This has motivated a recent interest in using neutrons to test chameleon models, which we summarize below. At the present time, all of the constraints derived using neutron experiments fix \(\varLambda _c\) to the darkenergy scale.
4.4.1 Ultracold neutrons
4.4.2 Neutron interferometry
In an analogous manner to optical interferometry, a coherent beam of neutrons can be split and later recombined to produce interesting interference patterns (Pokotilovski 2013; Brax et al. 2013c). A monosilicone crystal plate can be used for this purpose.
4.5 Astrophysical tests
In this section, we describe tests of chameleon and symmetron models using astrophysical objects. In many cases, the constraints are phrased in terms of \(\chi _0\) and \(\beta (\phi _0)\) and so the specific model is not important. We will not include bounds from binary pulsars since they are uncompetitive and subject to astrophysical uncertainties to do with the screening level of the Milky Way (Brax et al. 2014; Zhang et al. 2017).
4.5.1 Distance indicator tests
In the context of modified gravity, it is possible that the relation used to determine the luminosity is sensitive to gravitational physics. If the relation has been calculated using general relativity, or has been determined empirically using local (screened) observations, then it will give incorrect distances when applied to unscreened galaxies. In contrast, relations that are insensitive to the theory of gravity will always give the correct distance. Comparing how well different distance estimates to theoretically unscreened galaxies agree can therefore yield new constraints.
One robust distance indicator that is not sensitive to screened modified gravity is the tip of the redgiant branch (TRGB). Lowmass postmainsequence stars ( Open image in new window ) in the process of ascending the redgiant branch (RGB) consist of an isothermal helium core surrounded by a thin hydrogenburning shell. The hydrogen in this shell is continually processed into helium that is deposited onto the core, causing its temperature to rise steadily as the RGB is ascended. When the temperature is sufficiently high, the triple\(\alpha \) process (core helium burning) can proceed efficiently, at which point the star moves to the asymptotic giant branch in a very short timescale. This leaves a visible discontinuity in the Iband. The discontinuity occurs at fixed luminosity [\(I=4.0\pm 0.1\), the error is due to a very weak metallicity dependence (Sakai 1999; Freedman and Madore 2010; Beaton et al. 2016)], making the TRGB a standard candle. Importantly, the physics of the helium flash is set by nuclear physics and is nongravitational in origin, elucidating our earlier assertion that this distance indicator is insensitive to modified gravity.^{10}
4.5.2 Rotationcurve tests
Measurements of the galactic rotation curves typically use either H\(\alpha \) emission or the 21cm line, both of which probe the gaseous component. An alternate but less prevalent method involves measuring the Mgb triplet lines, which are due to absorption in the atmosphere of K and Gstars ( Open image in new window ). At present, the screening map contains six unscreened dwarf galaxies, for which both Mgb and either H\(\alpha \) or 21cm data (or both) are available. Using this, Vikram et al. (2014) have reconstructed both the gaseous and stellar rotation curves, and have used them to test the prediction (4.19) using a separate \(\chi ^2\) fit for each galaxy. This has placed new constraints in the \(\chi _0\)–\(\beta (\phi _0)\) plane, which are comparable with the Cepheid bounds.
4.5.3 Galaxy clusters
4.6 f(R) specific tests
In this section, we will briefly summarize tests that have been specifically designed to test the Hu and Sawicki (2007) f(R) theories discussed in Sect. 2.1.1. Note that, since these theories correspond to chameleons with \(1<n<1/2\), many of these tests are unconstraining for more general chameleon models. Similarly, specific tests are needed to target this parameter range. Note also, that f(R) models are designed to be cosmologically relevant, and so the majority of the tests discussed here are astrophysical in nature. In what follows, we will only focus on \(b=1\) (\(n=1/2\)) models because the majority of tests have reported constraints for this model only. Larger values of b are more readily screened and so one would expect the constraints to be weaker. Note that some tests mentioned above report bounds on \(f_{R0}\). We will not repeat that discussion here. A full list of constraints on \(f_{R0}\) can be found in Table 1 of Lombriser (2014).
4.6.1 Solarsystem bounds
4.6.2 Strong gravitational lensing
Another method to probe the predicted discrepancy between the dynamical and lensing mass of an object is to use strong lensing by individual galaxies. In this case, one can use the stellar dispersion relation to calculate the dynamical mass. Smith (2009) has performed such a test for a sample of galaxies from the Sloan Lens ACS (SLACS) survey and find a constraint \(f_{R0}<2.5\times 10^{6}\).
4.6.3 Cluster density profiles
Nbody simulations of f(R) gravity have repeatedly predicted an enhancement in the dark matter halo density profiles around the virial radius compared with GR (Schmidt et al. 2009a; Schmidt 2009). This is an artefact of the latetime unscreening in f(R) models. The center of the galaxy is largely unaffected because it is both screened and formed earlier when the screening was more efficient. In contrast, there is a pileup of mass in the outer regions, which form at later times, due to the weaker screening. Lombriser et al. (2012b) has used weak lensing data for the MaxBCG galaxy cluster sample from the SDSS to probe this potential novel feature, finding a constraint \(f_{R0}<3.5\times 10^{3}\).
4.6.4 Cluster abundances
The statistics of galaxy clusters is very sensitive to the theory of gravity. For f(R) theories, the enhanced gravitational force results in a higher abundance of rare massive clusters compared with GR (Schmidt et al. 2009a) meaning the halo mass function is modified. Making quantitative theoretical predictions for this requires knowledge of physics deep within the nonlinear cosmological regime and so Nbody simulations and spherical collapse halo models calibrated on them are required in order to make quantitative predictions.
The first bound obtained by looking at cluster abundances yielded \(f_{R0}<1.2\times 10^{4}\) (Schmidt et al. 2009b). This was obtained by using Xray inferred clusters in combination with a variety of different cosmological datasets available at the time. A stronger bound \(f_{R0}<1.6\times 10^{5}\) has subsequently been obtained by Cataneo et al. (2015) using a full MCMC analysis of the cluster likelihood function for updated datasets from more recent cosmological surveys.
4.6.5 Cosmic microwave background
Modifications of GR change the structure of the equations describing linear cosmological perturbations, and can hence effect the cosmic microwave background (CMB) (Zhang 2006; Song et al. 2007; Dossett et al. 2014). Updating various CMB codes to include the effects of f(R) gravity, several groups have all obtained a similar bound \(f_{R0}<10^{3}\) (Song et al. 2007; Dossett et al. 2014; Raveri et al. 2014; Cataneo et al. 2015).
4.6.6 Scalar radiation
As was first pointed out by Silvestri (2011), pulsating stars should source scalar radiation and hence lose energy over time. If too much scalar monopole radiation (which is absent in GR) is emitted, then the pulsations may quench. This was investigated by Upadhye and Steffen (2013), who found that the energy loss to monopole radiation is too weak to place any meaningful bounds. They identified another scenario whereby the scalar radiation sourced by an expanding type II supernovae could drain the kinetic energy of the expanding matter and significantly impede the expansion. This places the weak constraint \(f_{R0}<10^{2}\).
4.6.7 Redshiftspace distortions
The clustering of matter can be greatly modified in f(R) cosmologies compared with GR, and this can be particularly pronounced in redshift space (Jennings et al. 2012; Bose and Koyama 2016, 2017). The possibility of testing this was first investigated by Yamamoto et al. (2010), who examined a sample of luminous red galaxies (LRGs) from the SDSS to find a bound \(f_{R0}<10^{4}\). A more recent study, combining redshiftspace distortion observations with other cosmological datasets, found the stronger bound \(f_{R0}<2.6\times 10^{6}\) (Xu 2015).
4.7 Tests of the coupling to photons
In this section, we summarize experimental tests of the coupling to photons discussed in Sect. 2.3. We will restrict our attention to chameleon models, for which the coupling to photons has been widely studied. Extending these constraints to other models with screening remains a topic for future work.
4.7.1 PVLAS
The PVLAS experiment (Zavattini et al 2006) studied the polarisation of light propagating through a magnetic field. The presence of an axion, or axionlike particle coupled as in Eq. (2.35) would mean that, in the presence of a magnetic field, one polarisation of the propagating photon can convert into the scalar particle and vice versa. The second polarisation will propagate through unimpeded (Raffelt and Stodolsky 1988). This induces rotation and ellipticity into the polarisation of the incoming laser beam. The PVLAS experiment bounded the induced rotation to be less than \(1.2\times 10^{8}\) rad at 5 T and \(1.0\times 10^{8}\) rad at 2.3 T, and the induced ellipticity to be less than \(1.4 \times 10^{8}\) at 2.3 T. This constraints the coupling strength \(M_{\gamma }\) of a light axionlike particle.
In such experiments chameleon particles behave very differently to standard axionlike particles, precisely because of their density dependent mass. If standard axionlike particles were produced in PVLAS, they would pass through the walls at the end of the vacuum chamber without interacting and so leave the experiment. For a chameleon to pass through the wall, the chameleon particle must have enough energy that it can adjust its mass to the higher value needed for it to exist inside the wall. If it does not have this energy, it is instead reflected from the wall and back into the vacuum chamber (Brax et al. 2007a, c). This leads to a large ratio of the rotation to the ellipticity of the polarisation which is a unique signal of chameleon models. For a chameleon with a potential \(V(\phi )=(2.3 \times 10^{3} \mathrm {\ eV})^5/\phi \), and assuming the coupling to photons is the same as the coupling to other matter fields, the results of the PVLAS experiment constrain \(M_\mathrm{c}=M_{\gamma }> 2\times 10^6 \mathrm {\ GeV}\).
4.7.2 GammeVCHASE
A second commonly used experimental design to look for axionlike particles, lightshiningthroughwalls, also needs to be modified in order to search for chameleon particles. Experiments searching for standard ALPs rely on the ability of ALPs to pass through walls which are impermeable to photons. Light is shone into a cavity across which a magnetic field is applied. A wall is then placed in this cavity; in the absence of ALPs, no light would be seen on the far side of the wall. But if a photon converts into an ALP before hitting the wall this ALP can pass through and then may reconvert into a photon on the far side of the wall.
As discussed in the previous subsection, chameleon ALPs cannot pass through walls in the way that standard ALPs do, and so lightshining through walls experiments cannot constrain chameleons. However, this inability to pass through walls can be developed into a new type of experiment specifically designed to look for chameleons; these are known as afterglow experiments (Gies et al. 2008; Ahlers et al. 2008). The basic design of the experiment is to shine a laser beam into a vacuum chamber across which a magnetic field is applied. If there is a nonzero probability of the photons converting into chameleons, then the number of chameleons trapped inside the chamber (because they cannot pass through the walls) will increase the longer the laser beam is on. The laser is then turned off, but the magnetic field is left on. Then the chameleons can reconvert into photons, leading to a detection of light, after the laser has been turned off.
This experiment was successfully performed by the GammeV collaboration, and was known as GammeVCHASE (GammeV CHameleon Afterglow SEarch) (Upadhye et al. 2010). Constraints were placed on values of the chameleon coupling to photons, as a function of the effective chameleon mass in the chamber (Chou et al. 2009). This mass depends on the choice of the chameleon potential and the strength of the coupling to other matter fields. For the lightest chameleons inside the vacuum chamber, GammeVCHASE constrains the coupling to photons to be \(M_{\gamma }>3 \times 10^7 \mathrm {\ GeV}\) (Steffen et al. 2010; Upadhye et al. 2012b). The constraints weaken if the effective mass of the chameleon is above \(10^{3} \mathrm {\ eV}\).
The modelling of how the chameleon behaves inside the experiment requires care. Whilst a semiclassical approximation would predict that the chameleon bounces off the walls of the vacuum chamber unchanged, considering the chameleons as fluctuations in a quantum field opens up the possibility that the nontrivial self interactions of the chameleon field could allow a chameleon particle to fragment into a number of lower energy chameleons as it hits the wall. This was shown not to be a significant effect in the GammeVCHASE experiment for the benchmark potentials \(V(\phi )= \lambda \phi ^4\) and \(V(\phi )=\varLambda ^5/\phi \) (Brax and Upadhye 2014). However, for steeper potentials this effect will start to become relevant.
4.7.3 ADMX
Axion Dark Matter eXperiment (ADMX), is another experiment aiming to detect axions and axionlike particles through the Primakov effect (Asztalos 2010, 2004). However, in this case, the axions come from outside the experiment, and are hypothesised to be responsible for the dark matter in our galaxy (Sikivie 1983). This set up has been used to constrain chameleon theories using the same afterglow effect discussed above (Rybka et al. 2010), but using microwave photons trapped in a cavity instead of laser light. The experiment excluded couplings \(5 \times 10^3 \mathrm {\ GeV}< M_{\gamma }< 1 \times 10^9 \mathrm {\ GeV}\) for effective chameleon masses in the cavity \(\sim 1.95\, \upmu \mathrm{eV}\).
4.7.4 CAST
The CERN Axion Solar Telescope (CAST) experiment searches for axions produced in the Sun, by looking for their reconversion into photons in the bore of a decommissioned LHC magnet (Zioutas et al. 2005). Results from this search can be applied to chameleons, if they are also produced in the Sun. At the particle level, the processes that produce chameleons are the same as those that produce scalar axionlike particles, but determining the total flux of chameleons from the Sun requires taking into account the added complication that the mass of the chameleon field varies with the density of the solar medium (Brax and Zioutas 2010).
CAST has not yet detected a signal from the Sun, and so bounds can be placed on the chameleon couplings. They exclude photon couplings \(M_{\gamma } \ge 2.6 \times 10^7 \text{ GeV }\), for a range of couplings to matter \(10^{12} \mathrm {\ GeV} \le M_\mathrm{c}\le 10^{18} \mathrm {\ GeV}\), assuming that the bare chameleon potential is \(V(\phi ) =(10^{3}\mathrm {\ eV})^5/ \phi \) (Anastassopoulos et al. 2015).
There are also proposals by the CAST collaboration to detect solar chameleons using a novel force sensor (Baum et al. 2014). While chameleons may be produced in the Sun due to the coupling to photons, the detection mechanism itself does not rely on the coupling in Eq. (2.35). The detection relies on having a force sensor sufficiently sensitive that it can measure the chameleon radiation pressure (Karuza et al. 2016), which comes about as the chameleons emitted from the Sun bounce off the sensor, for the same reason that chameleons are reflected from the walls of vacuum chambers, the chameleon particle does not have enough energy to adjust its mass sufficiently to pass through the membrane of the sensor.
4.7.5 Collider constraints
The collider constraints on chameleon models can also be extended to include the coupling to photons in Eq. (2.35). This leads to additional loops, which should be inserted into the diagrams, and allows for additional production and decay processes which should be included. Analysis of precision electroweak data from LEP constrains \(M_{\gamma }\gtrsim 10^3\mathrm {\ GeV}\) (Brax et al. 2009).
4.7.6 Galactic and extragalactic constraints
The effects of the chameleon on light propagating through magnetic fields, originating in the interaction of Eq. (2.35), can also be relevant to astrophysical observations. For many observations, light from distant sources has to propagate through galactic, intracluster, or extragalactic magnetic fields in order to reach us. Whilst the magnetic fields strengths are much lower than those achievable in the laboratory, they extend over much larger distances, meaning that the astrophysical constraints can in principle be more stringent that those achieved in the laboratory. They do, however, come with much larger uncertainties around the initial luminosity of the source, the polarisation of the light it emits, and over the structure of the magnetic fields. Astrophysical magnetic fields also display much more structure than the coherent magnetic fields used in laboratory, which adds to the complexity of the calculations.
In Burrage et al. (2009b), it was shown that chameleons coupled to photons can induce both linear and circular polarisation into light from stars. As long as the chameleon mass is smaller than the local plasma density, then it can be neglected in these calculations, meaning that the constraints are largely model independent as long as the chameleon is light on astrophysical scales. Within the galaxy this requires \(m_{\phi } <1.3 \times 10^{11} \mathrm {\ eV}\). From measurements of the polarisation of galactic stars, expected to be largely unpolarized initially, the bound \(M_\gamma > 1.1 \times 10^9 \mathrm {\ GeV}\) was derived. Assuming the magnetic field strength of the intergalactic medium is \(B\approx 3\, \upmu \mathrm {G}\) and the coherence length is \(20 \mathrm {\ pc}\). The polarisation of light from the Crab nebula, type Ia supernova, highredshift quasars, gammaray bursts, and the CMB was also analysed but the bounds were weaker than those from observations of stars.
Looking for the depletion in luminosity of astrophysical sources from photons converting into chameleons is difficult because there is, generally, no way of determining the intrinsic luminosity of the source. However, for some astrophysical objects, correlations have been observed between the luminosity of the source and a second observable that should not be affected by the coupling to chameleons. The best constraints of this form on chameleons come from looking at Active Galactic Nuclei (AGN), where the Xray luminosity at \(2 \mathrm {\ keV}\) is observed to be tightly correlated with the optical luminosity at \( 5 \mathrm {\ eV}\) (Steffen et al. 2006; Young et al. 2009). Similar luminosity relations exist for blazars and gamma ray bursts, but these give rise to weaker constraints. As the probability of a photon converting into a chameleon increases with the frequency of the photon, the effects of the chameleon on the Xray luminosity of the AGN can be significant, whilst the effects on the optical luminosity remain small. Therefore, the luminosity relation can be used to constrain the chameleon (Burrage et al. 2009a), with the current best constraint \(M_{\gamma }\gtrsim 10^{11}\,\mathrm {\ GeV}\) assuming, again, that the chameleons are sufficiently light, \(m_{\phi }<10^{12}\,\mathrm {\ eV}\), on astrophysical distance scales that the effects of their mass are negligible (Burrage et al. 2009a; Pettinari and Crittenden 2010).
The conversion of photons into chameleons also will increase the opacity of the universe at high frequencies. In Avgoustidis et al. (2010), tests of the distance duality relation, which relates luminosity distance and angular diameter distance to sources, were used to derive constraints on cosmic opacity. This can be viewed as a test of chameleons because depletion of photons from the source will change the luminosity distance, whilst leaving the angular diameter distance unaffected. Constraints are currently not competitive with those from starlight polarisations, but should be expected to improve significantly with new data from upcoming cosmological surveys.
Summary of present tests of chameleon and symmetron theories
Test  Chameleons  Symmetrons 

EötWash  ✓  ✓ 
Casimir force  ✓  ✗ 
Microspheres  ✓  ✗ 
Precision atomic tests  ✓  ✗ 
Atom interferometry  ✓  ✓ 
Cold neutrons  ✓  ✗ 
Neutron interferometry  ✓  ✗ 
Distance indicators  ✓  ✓ 
Rotation curves  ✓  ✓ 
Cluster lensing  ✓  ✗ 
4.8 Summary of tests
Here, we briefly summarize the tests that have been used to probe screened modified gravity to date. The summary is given in Table 2; we do not include f(R)specific tests, because they do not carry over to more general models.
5 Constraints
In this section, we convert the constraints discussed in the previous section into a single and familiar parametrization and combine them to show the presently allowed parameter ranges.
5.1 Chameleon constraints
The current bounds on chameleon models are shown below. We cover the two most commonly studied models \(n=1\) (Fig. 4) and \(n=4\) (Fig. 5). In these cases, we plot \(\varLambda \) versus \(M_\mathrm{c}\). Furthermore, many experiments focus on the case \(\varLambda =\varLambda _\mathrm{DE}=2.4\) meV (the dark energy scale) and so for this choice we plot \(M_\mathrm{c}\) versus n for both positive (Fig. 6) and negative n (Fig. 7).
5.1.1 f(R) constraints
We show the current constraints on the Hu and Sawicki \(b=1\) f(R) model (2.19) in Fig. 8. The xaxis labels each specific test and the yaxis shows the resultant upper limit on \(f_{R0}\). It is common to express constraints on \(f_{R0}\) showing the length scale on which they were obtained (e.g., Lombriser 2014). Whilst complementary tests on all scales are crucial consistency checks of the theory, it is important to note that this length is not a new parameter appearing in the theory, and that it is the same parameter \(f_{R0}\) being constrained no matter the test or the length scale that it probes. For this reason, we have included the typical length scale for each test in the figure.
5.1.2 Constraints on the coupling to photons
5.2 Symmetron constraints
6 Conclusions and outlook
Chameleon and symmetron models have been a paragon for viable, interesting, and relevant infrared modifications of general relativity for over a decade. The screening mechanism has resulted in theories of gravity that are perfectly consistent with general relativity’s predictions in the solar system, but are yet falsifiable using novel approaches such as astrophysical phenomena in distant galaxies, as well as specifically targeted laboratory searches. In many cases, these models may be relevant on linear (and nonlinear) cosmological scales.

\(n=1\) and \(n=4\) chameleon models (two of the most commonly studied) are tightly constrained but there is a large parameter space remaining for \(n>1\) and \(n<4\) when \(\varLambda \) is fixed to the dark energy scale. Away from this, the constraints are not as strong. In many cases, this is because bounds on other models are not reported.

Symmetron models are wellconstrained by astrophysical probes and atom interferometry but there is a lack of theoretical work translating the bounds from existing experimental results into symmetron constraints. This has resulted in a desert separating astrophysical and laboratory tests [this could be filled in partially by constraints from future spacebased tests of relativistic gravitation (Sakstein 2017)].

The coupling of chameleons to photons for \(n=1\) models is tightly constrained and there is only a narrow window remaining. The coupling of symmetrons to photons and chameleon models with \(n\ne 1\) has yet to be explored.

Hu and Sawicki f(R) models (Hu and Sawicki 2007) are wellconstrained for \(b=1\) but, presently, there are not enough reported bounds on larger values to make a meaningful comparison. For \(b=1\) the bounds on \(f_{R0}\) are at the \(10^{7}\) level. In theory, \(10^{8}\) would be achievable with better statistics; below this, dwarf galaxies begin to become screened and higherprecision tests are necessary.

At the present time, the environmentdependent dilaton, which screens in a distinct manner from chameleon and symmetron models, has not been studied sufficiently in the context of laboratory and astrophysical tests to produce any meaningful constraints.
6.1 Prospects for future bounds
We end by discussing the prospects for future tests of screened modified gravity.
6.1.1 Laboratory tests
As new experimental techniques are been developed, and existing ones are improved we can expect bounds on chameleon and symmetron models of screening to continue to improve. It is to be expected that this will be a combination of the reinterpretation of experimental results obtained when searching for other types of new physics, and a smaller number of experiments dedicated to directly searching for screening.
It is difficult to imagine that a single experiment could cover all of the remaining chameleon and symmetron parameter space, and so ideally a combination of techniques and searches are needed in order to fully rule out the possibility that screened scalars exist in our Universe.
6.1.2 Astrophysical tests
Astrophysical objects show strong deviations from GR when the Newtonian potential \(\varPhi _\mathrm{N}<\chi _0\) (\(\sim f_{R0}\) for f(R) theories). Given that current bounds place Open image in new window , the only objects in the Universe with a low enough Newtonian potential to exhibit novel effects are dwarf galaxies located in voids, and several tests using such galaxies have been proposed.
The rotationcurve test described in Sect. 4.5.2 suffers from a lack of unscreened galaxies, and a larger sample would improve the constraints. Future and upcoming data releases, in particular SDSSMaNGA, can provide a larger sample size that would significantly improve the bounds. Additional tests, such as the warping of galactic disks due to equivalence principle violations have been proposed (Jain and VanderPlas 2011), although a test using SDSS optical and ALFALFA radio observations did not yield any bounds on the model parameters (Vikram et al. 2013). Future radio surveys such as VLT may be more fruitful.
Finally, Nbody simulations are uncovering a variety of novel phenomena exhibited by chameleons on nonlinear cosmological scales (Jain et al. 2013b). Many of these are clear smokinggun signals that could be measured with upcoming peculiar velocity and galaxy redshift surveys (Hellwing et al. 2014).
6.1.3 Tests of the coupling to photons
The increase in interest in axions and axionlike particles as dark matter candidates has lead to a series of proposals and experiments aimed at further constraining these particles which, in many cases, focus on their interactions with photons. These experiments present an exciting opportunity for new constraints on theories with screening, but the details of how powerful these constraints can be remain to be worked out.
Footnotes
 1.
Note that one could consider a more general theory where each particle species i is conformally coupled to a different metric \(\tilde{g}^{(i)}_{\mu \nu }=A_i^2(\phi )g_{\mu \nu }\) although we will not consider such theories here since they are not wellstudied in the context of screened modified gravity. An even more general metric includes disformal terms \(\tilde{g}_{\mu \nu }=A(\phi )g_{\mu \nu } +B(\phi )\partial _{\mu }\phi \partial _{\nu }\phi \) (Bekenstein 1993). Constraints on disformal couplings to matter can be found in Brax and Burrage (2014), Sakstein (2014b), Sakstein (2015), Ip et al. (2015) and Sakstein and Verner (2015).
 2.
There are three commonly used densities in the literature: the Jordan frame density \(\tilde{\rho }=\tilde{T^{0}_0}\), the Einstein frame density \(\rho =T^0_0=A^6(\phi )\tilde{\rho }\), and the ‘conserved Einstein frame density’ \(\rho _\mathrm{conserved}=A(\phi )\rho \). The Jordan frame density is the result of statistical physics calculations and it is in this frame that one may specify an equation of state. The Einstein frame density is what arises naturally in Eq. (2.6) as a result of varying the action (2.1) and the conserved density is a quantity that is useful in cosmological contexts (Khoury and Weltman 2004a; Hui et al. 2009; Brax et al. 2012a; Brax and Davis 2012; Brax et al. 2012b; Sakstein 2014a). In particular, the conserved density satisfies a conservation equation that makes the cosmological equations look similar to those of GR. Since this review is concerned with experimental tests of chameleon theories, we have opted to work with the Einstein frame density. At the Newtonian level (weakfield limit), these densities are equivalent (Hui et al. 2009; Sakstein 2014a) and so the choice is somewhat arbitrary, but we note that one must work with the Jordan frame pressure and density if one is interested in compact objects such as neutron stars (Babichev and Langlois 2010; Minamitsuji and Silva 2016; Babichev et al. 2016; Sakstein et al. 2017; Brax et al. 2017). We will not consider such objects here.
 3.
Several definitions of the effective potential exist in the literature. If one uses the conserved Einstein frame density then one has \(V_\mathrm{eff}(\phi )=V'(\phi )+\rho A(\phi )\) (Khoury and Weltman 2004a; Sakstein 2014a). Furthermore, one often sees the effective potential written as \(V_\mathrm{eff}(\phi )=V'(\phi )+(A(\phi )1)\rho \) (using the conserved Einstein frame density). This is motivated by models that have \(A(\phi )=1+\beta (\phi _0)\phi /{M_\mathrm{{pl}}}+\cdots \) and the factor of \(1\) is used to subtract the matter density from the chameleon energy density in order to avoid double counting in cosmology. [The equation of motion (2.7) which governs the dynamics is unchanged by including such a factor]. Since we do not consider cosmology here we will not include this factor.
 4.
Chameleon models were the first example of screening mechanism that screens using this effect. The scalar blending in with its environment inspired the name.
 5.
Note that point particles do satisfy the equivalence principle because every matter species appearing in the action (2.1) is universally coupled to the Jordan frame metric and, thus, follow the same geodesics. The motion of extended objects is governed by energymomentum conservation and it is here that the difference arises. See Hui et al. (2009) for an extended discussion of this.
 6.
Note that our conventions differ from theirs. They use tildes to refer to Einstein frame quantities, whereas we use them to refer to Jordan frame quantities and their function \(\varOmega (\phi )\) is related to our coupling function via \(\varOmega (\phi )=A^{1}(\phi )\).
 7.
Technically one does have an \(\mathcal {O}(1)\) contribution to \(k(\phi )\approx 1+6{M_\mathrm{{pl}}}^2/M^2\) which can be \(\gg 1\) for some values of M considered here. In fact, it should be the canonically normalized field, \(\varphi =\sqrt{k(\phi )}\phi \sim \mathcal {O}(v^2/c^2)\) (at this order), which is why we can neglect this contribution.
 8.
Atoms are neutral as well but one advantage of neutrons is that their polarizability is 15 orders of magnitude smaller, making van der Waals forces less of a background. We are grateful to Tobias Jenke for pointing this out to us.
 9.
We thank Tobias Jenke for providing us with the numeric values.
 10.
Technically, this is only the case when Open image in new window , corresponding to parameters where the hydrogen burning shell becomes unscreened. When this happens, the core temperature increases at a faster rate leading to a reduction of the tip luminosity because the star has less time time to ascend the RGB. We will see shortly that \(\chi >10^{6}\) can be ruled out by other, independent means and so we will not dwell on this too much here.
 11.
This has the result that small compressions result in an increased opacity that in turn causes an increase in the energy absorbed. The energy dammed up by this compression drives the pulsations. This is known as the \(\kappa \)mechanism.
 12.
Metallicity and other corrections produce a positive \(\varDelta d/d\), which makes the constraints even stronger.
 13.
This assumes that the gas entirely supported by thermal pressure. In practice, one expects a small amount of nonthermal pressure but Nbody simulations of chameleon theories have shown this to be negligible (Wilcox et al. 2016).
Notes
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