Screened Scalar Fields in the Laboratory and the Solar System

Abstract

The last few decades have provided abundant evidence for physics beyond the two standard models of particle physics and cosmology. As is now known, the by far largest part of our universe’s matter/energy content lies in the ‘dark’, and consists of dark energy and dark matter. Despite intensive efforts on the experimental as well as the theoretical side, the origins of both are still completely unknown. Screened scalar fields have been hypothesized as potential candidates for dark energy or dark matter. Among these, some of the most prominent models are the chameleon, symmetron, and environment-dependent dilaton. In this article, we present a summary containing the most recent experimental constraints on the parameters of these three models. For this, experimental results have been employed from the qBounce collaboration, neutron interferometry, and Lunar Laser Ranging (LLR), among others. In addition, constraints are forecast for the Casimir and Non-Newtonian force Experiment (Cannex). Combining these results with previous ones, this article collects the most up-to-date constraints on the three considered screened scalar field models.

1. Introduction

Dark energy (DE) and dark matter (DM) pose some of modern physics’ greatest open problems. Modifications of general relativity (GR) in the form of scalar-tensor theories [1], in which a scalar field is coupled to the gravitational metric tensor in the matter action, are frequently employed in attempts to solve these conundrums [2,3]. While many of these theories lead to a universal coupling between the scalar field and the trace of the energy-momentum tensor of (Standard Model) matter and, consequently, a fifth fundamental force of Nature, such emergent effects are already tightly constrained within our Solar System [4,5,6].

Screening mechanisms offer phenomenologically appealing ways for circumventing these constraints by rendering scalar fifth forces feeble in matter at least as dense as our Solar System. In turn, in environments that typically correspond to comparatively low matter densities, for example, on the edges of galaxies or galaxy clusters, screening mechanisms allow fifth forces to potentially even exceed gravity’s strength. Consequently, scalar fifth forces could themselves give rise to observable phenomena at such scales and, for example, serve as potential alternatives to particle DM [7,8,9,10]. A zoo of scalar field models with screening mechanisms have been devised in recent decades. Among those so-called screened scalar fields, some of the most prominent ones are chameleons [11,12], symmetrons [13,14,15,16,17,18,19,20], environment-dependent dilatons [15,18,21,22,23,24,25] and galileons [26,27,28]. In recent years, a large number of high-precision experiments and observations have been employed to constrain the parameter spaces of several of these screened scalar field models, see, e.g., Refs. [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61]. In addition, the first investigations of screened scalars within a quantum field theoretical framework have been carried out [62,63,64,65,66], and it has been suggested to study spacetimes with screened scalar-tensor theories in analogue gravity simulations [67].

In this article, we present an overview of existing experimental and observational constraints on chameleon, symmetron and environment-dependent dilaton models. For this, we add the most recent constraints on chameleons [53,62] and symmetrons [55,58] to the summarized data from Ref. [30], and create plots combining constraints obtained from gravity resonance spectroscopy (qBounce), Lunar Laser Ranging (LLR) [57] and neutron interferometry [58] for the environment-dependent dilaton. In addition, we present projected constraints for all three models expected to be obtained from the upcoming Casimir and Non-Newtonian force Experiment (Cannex) [61,68].

In addition to providing a brief review of existing constraints, this article updates the qBounce constraints from Refs. [33,35,46,49] for the symmetron and chameleon models. Previous analyses assumed an ideal vacuum density of ${\rho }_V=0$, exclusively used perturbation theory to compute the energy shifts that a screened scalar field would induce on a neutron bouncing in a gravitational field (which is a good approximation for the chameleon field [69] but not for the symmetron field), and neglected parameter regions for which the symmetron field is in its symmetry-broken phase, even inside the neutron mirror. This updated analysis employs the actual experimental residual gas density of ${\rho }_V=2.32\times {10}^{-7}$ kg/m³ in the vacuum chamber. For parameter regions where perturbation theory is inaccurate, a non-perturbative numerical treatment is used. Additionally, the symmetry-broken phase of the neutron is considered, as detailed in Ref. [50]. Analogous to Ref. [57], assumptions necessary for the analysis as, e.g., neglecting higher order couplings between scalar field and matter as well as neglecting the influence of the vacuum chamber on the scalar field have been properly considered. In order to physically justify these assumptions, all derived constraints have been cut off wherever this is required. As demonstrated in Appendix A, each of these improvements corrects the previous symmetron constraints by several orders of magnitude.

The article is organized as follows. In Section 2, we provide the required theoretical background on the considered screened scalar field models, while in Section 3, our way of obtaining the constraints is described. Next, in Section 4.1, Section 4.2 and Section 4.3, up-to-date constraints on the chameleon, the symmetron and the environment-dependent dilaton models are represented, respectively. Finally, in Section 5, we draw our conclusions.

2. Theoretical Background

In this article, we only discuss screened scalar field models with canonical kinetic terms in the Einstein frame in contrast to galileon models that require additional kinetic terms for implementing their screening via the Vainshtein mechanism [70]. Consequently, all considered models for a scalar $\varphi$ can be described by the following action

$$\begin{array}{c}\hfill S=\int d^{4}x\sqrt{-g}\left(-\frac{m_{\mathrm{pl}}^2}{2}R+\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V\left(\varphi \right)\right)+\int d^{4}x\sqrt{-\tilde{g}}{\mathcal{L}}_{\mathrm{SM}}({\tilde{g}}_{\mu \nu },{\psi }_i),\end{array}$$

(1)

where $m_{\mathrm{pl}}$ is the reduced Planck mass, $g_{\mu \nu }$ the Einstein frame metric, ${\tilde{g}}_{\mu \nu }$ the Jordan frame metric, $V\left(\varphi \right)$ denotes the scalar’s self-interaction potential characterizing each individual screened scalar field model (see

Table 1. The three considered scalar field models characterized by their particular forms of the potential $V\left(\varphi \right)$ and the conformal factor $A\left(\varphi \right)$. Chameleon: the different chameleon models are distinguished by the parameter $n\in {\mathbb{Z}}^{+}\cup 2{\mathbb{Z}}^{-}\setminus \{-2\}$. $\mathsf{\Lambda }$ has the dimension of a mass and parameterizes the self-interaction of the chameleon. The mass scale $M_c$ describes the chameleon’s coupling to matter. Symmetron: $\mu$ is a tachyonic mass and ${\lambda }_S$ is a dimensionless self-coupling parameter. M is a mass scale describing the symmetron coupling to matter. Dilaton: The environment-dependent dilaton is characterized by a constant energy density $V_0$, a dimensionless self-coupling parameter $\lambda$, and a dimensionless constant $A_2$ parameterizing the coupling to matter.

Scalar Field	$V\left(\varphi \right)$	$A\left(\varphi \right)$
Chameleon	${\displaystyle \frac{{\mathsf{\Lambda }}^{n+4}}{{\varphi }^n}}$	${\displaystyle {\mathrm{e}}^{\varphi /M_c}}$
Symmetron	${\displaystyle -\frac{{\mu }^2}{2}{\varphi }^2+\frac{{\lambda }_S}{4}{\varphi }^4}$	${\displaystyle 1+\frac{{\varphi }^2}{2M^2}}$
Dilaton	${\displaystyle V_0{\mathrm{e}}^{-\lambda \varphi /m_{\mathrm{pl}}}}$	${\displaystyle 1+A_2\frac{{\varphi }^2}{2m_{\mathrm{pl}}^2}}$

) and ${\mathcal{L}}_{\mathrm{SM}}$ the Lagrangian describing the Standard Model (SM) fields ${\psi }_i$. The conformal transformation between Jordan and Einstein frame is given by ${\tilde{g}}_{\mu \nu }=A^2\left(\varphi \right)g_{\mu \nu }$, where the form of the conformal factor $A\left(\varphi \right)$ is determined by the particular scalar field model (see Table 1). In the Einstein frame, this leads to a universal coupling between scalar $\varphi$ and the trace of the energy-momentum tensor $T^{\mu \nu }$. In turn, this coupling leads to a fifth force of Nature, which, for a test particle with mass m in the non-relativistic limit is given by

$$\begin{array}{cc}\hfill {\mathit{f}}_{\varphi }& =-m\nabla \ln A\left(\varphi \right)\hfill \\ \hfill & =-\beta \left(\varphi \right)\frac{m}{m_{\mathrm{pl}}}\nabla \varphi ,\hfill \end{array}$$

(2)

with the dimensionless coupling parameter

$$\begin{array}{cc}\hfill \beta \left(\varphi \right)& :=m_{\mathrm{pl}}\frac{d\ln A}{d\varphi }.\hfill \end{array}$$

(3)

If the SM fields ${\psi }_i$ correspond to non-relativistic matter with density $\rho$, we can use $T_{\phantom{\mu }\mu }^{\mu }=\rho$. In this case, the scalar is subject to an effective potential

$$\begin{array}{cc}\hfill V_{\mathrm{eff}}(\varphi ;\rho )& :=V\left(\varphi \right)+\rho A\left(\varphi \right),\hfill \end{array}$$

(4)

which can be derived from Equation (1). Consequently, the equation of motion for the scalar $\varphi$ is given by

$$\begin{array}{cc}\hfill \square \varphi & =-V_{\mathrm{eff},\varphi }(\varphi ;\rho ).\hfill \end{array}$$

(5)

Depending on the different possible choices for $V\left(\varphi \right)$ and $A\left(\varphi \right)$, as given in Table 1, the effective potential in Equation (4) gives rise to the chameleon mechanism [11,12] or the Damour–Polyakov mechanism [15]. While the former is characteristic for chameleon models, symmetrons are subject to the latter. Initially, the environment-dependent dilaton was also thought to be mainly screened by the Damour–Polyakov mechanism, but in Ref. [57], it was demonstrated that for some parts of the dilaton parameter space the chameleon mechanism is actually dominant concerning the fifth force screening.

3. Constraint Calculation

In this section, further details concerning the newly computed or re-analyzed constraints presented in this article are provided. Wherever necessary, numerical algorithms have been employed to solve the scalar field equations of motion and, if applicable, the stationary Schrödinger equation. Refs. [58,59] provide further details concerning these algorithms. Herein, new results or re-analyses are provided for the qBounce, neutron interferometry and Cannex laboratory experiments as well as Lunar Laser Ranging. In the subsequent subsections, further details concerning the constraints from each of these experiments are presented. In order to justify the neglect of any higher order couplings, the analysis herein is restricted to

$$\begin{array}{c}\hfill A\left(\varphi \right)-1\ll 1,\end{array}$$

(6)

which requires excluding those regions of the constraint plots that do not fulfill this assumption. For qBounce and Cannex, only interaction ranges in vacuum of up to 1 mm are considered, in which case any influence from the vacuum chamber can be neglected. Furthermore, for Cannex, parameters for which the field does not decay to its potential minimum inside the upper mirror are excluded, while for neutron interferometry an interaction range of maximally $0.25$ mm within the cylindrical walls enclosing the chambers through which the superposed neutron beams traverse is considered. The latter is required to ensure that the minimum value ${\varphi }_M$ of the scalar field within the walls can be used as a boundary condition.

3.1. qBounce Constraints

The qBounce experiment is a realization of gravity resonance spectroscopy [33,71,72,73]. It exploits the fact that ultracold neutrons, i.e., neutrons with very small kinetic energies, are totally reflected from most materials. In Earth’s gravity, the quantum theoretical energy levels of a neutron are discrete and non-equidistant. This enables the realization of spectroscopy by inducing transitions between different energy levels. The very high precision enables effective searches for any hypothetical fifth forces. Specifically, measurements of the transition energies $\mathsf{\Delta }{E}_{13}:=E_3-E_1$ and $\mathsf{\Delta }{E}_{14}:=E_4-E_1$ have been carried out. The experimentally measured values are $\mathsf{\Delta }{E}_{13}=(1.9222\pm 0.0054)$ peV (expected value 1.9145 peV) and $\mathsf{\Delta }{E}_{14}=(2.6874\pm 0.0074)$ peV (expected value 2.676 peV), consistent with the transition energies expected from a Newtonian gravitational potential alone [46]. If it exists, a hypothetical scalar field would induce an additional potential.

qBounce’s experimental geometry, as realized in Ref. [46], can be approximated to an excellent degree by an infinitely extended single mirror (see, e.g., Refs. [43,50]), and the individual scalar field profiles computed by numerically solving the corresponding equations of motion (5). The transition energies are computed numerically by solving the stationary Schrödinger equation including Newtonian and scalar field effects

$$\begin{array}{c}\hfill -\frac{1}{2m_n}\frac{d^2{\mathsf{\Psi }}_n\left(z\right)}{d{z}^2}+U\left(z\right){\mathsf{\Psi }}_n\left(z\right)=E_n{\mathsf{\Psi }}_n\left(z\right),\end{array}$$

(7)

where $m_n$ is the mass of the neutron, and the potential is given by

$$\begin{array}{cc}\hfill U\left(z\right)& =m_{n}gz+U_X\left(z\right),\hfill \\ \hfill U_X\left(z\right)& ={\mathfrak{Q}}_X{m}_n\left(A_X\left(\varphi \right)-1\right),\hfill \end{array}$$

(8)

with $X\in \{C,S,D\}$ (as abbreviations for chameleon, symmetron and dilaton, respectively) and the screening charge ${\mathfrak{Q}}_X$, which quantifies the amount of screening of the neutron, see Appendix C. Furthermore, the same relations as in Ref. [58] have been employed herein.

Previous publications [46,49,57] explored two distinct approximations for the coupling of neutrons to screened scalar fields, both treating the neutron as a classical sphere. In the Fermi screening approximation, the neutron’s radius is set to $R=0.5$ fm, a value grounded in QCD scattering data. Conversely, the micron screening approximation sets the neutron’s radius at 5.9 μm derived from its wavefunction’s vertical extent. In this investigation, we exclusively adhere to the Fermi screening approximation. Notably, due to the absence of a well-defined micron screening approximation for neutron interferometry, such an approach becomes incomparable to other neutron experiments. Moreover, the wave function of neither experiment accurately conforms to a spherical model. Additionally, Fermi screening constraints are more conservative compared to micron screening constraints, resulting in notably weaker constraints. Consequently, the derived constraints may underestimate the true constraints. Further theoretical advancements are thus needed to determine the genuine coupling of neutrons to screened scalar fields.

Scalar field parameters are constrained, which would lead to a deviation of at least two standard deviations from the measured values, i.e.,

$$\begin{array}{cc}\hfill \mathsf{\Delta }{E}_{13}& \notin (1.9114,1.933)\mathrm{peV},\hfill \\ \hfill & \mathrm{or}\hfill \\ \hfill \mathsf{\Delta }{E}_{14}& \notin (2.6726,2.7022)\mathrm{peV}.\hfill \end{array}$$

(9)

The current constraint criterion fails to account for correlations among $\mathsf{\Delta }{E}_{13}$, $\mathsf{\Delta }{E}_{14}$ and the scalar field parameters under scrutiny. Due to persisting theoretical uncertainties in energy shift predictions, notably attributable to the screening charge approximation, conducting a rigorous statistical analysis is presently unfeasible. However, it has been carefully checked that this simplified constraint criterion is a very good approximation to a more rigorous ${\chi }^2$-analysis (see Appendix A.2), and is hence employed at the current level of theoretical description.

3.2. Neutron Interferometry Constraints

A neutron interferometer uses the wave nature of neutrons for performing interferometric measurements [73,74,75]. Specifically, the path of a single propagating neutron is split into a superposition of two macroscopically separated beams before these paths are recombined and the neutron is finally detected. Each beam passes through a cylindrical chamber, one of which, denoted “air”, is filled with air of ambient density to suppress the field, while the other, denoted “vacuum”, actually contains helium with low pressure. In the latter, a scalar field typically takes on a larger field value than in the other chamber filled with air. If a scalar field indeed exists, the neutron would experience a different phase shift in each of the paths which, in the semi-classical limit, is given by (see, e.g., Ref. [58]):

$$\begin{array}{c}\hfill \delta {\phi }_{X;P}\left(r\right)=-\frac{m_n}{k_0}\underset{-L/2}{\overset{L/2}{\int }}{U}_{X;P}(r,z)dz,\end{array}$$

(10)

with $X\in \{C,S,D\}$, $k_0$ being the wave number of the neutron, r the radius from the center at which the neutron beam propagates through the chamber, L the length of the chamber, $U_{X;P}(r,z)={\mathfrak{Q}}_X{m}_n\left(A_X\left(\varphi \right)-A_X\left({\varphi }_{\mathrm{Air}}\right)\right)$ the scalar field potential in the chamber with pressure P, and the z-integration extending over the classical flight path (CFP) inside the corresponding chamber.

Experimentally, the following two measurement modes have been used to constrain scalar field parameters:

1.

Profile mode

In this mode, the following phase shift

$$\begin{array}{c}\hfill \mathsf{\Delta }{\phi }_{X;P}=\delta {\phi }_{X;P}\left(0\right)-\delta {\phi }_{X;P}\left(0.015\mathrm{m}\right),\end{array}$$

(11)

is evaluated for both chambers, with the vacuum chamber having its pressure at ${10}^{-4}$ mbar, which corresponds to the lowest pressure measured in the experiment. The experiment actually measured

$$\begin{array}{c}\hfill \alpha :=\mathsf{\Delta }{\phi }_{X;\mathrm{Vacuum}}-\mathsf{\Delta }{\phi }_{X;\mathrm{Air}}<0.\end{array}$$

(12)

This quantity is negative, since the potential is more suppressed close to the chamber walls and due to $|\mathsf{\Delta }{\phi }_{X;\mathrm{Air}}|<|\mathsf{\Delta }{\phi }_{X;\mathrm{Vacuum}}|$. The experiment constrains

$$\begin{array}{c}\hfill \alpha <-3.{55}^{°},\end{array}$$

(13)

as will be further detailed in Appendix B.1.

2.

Pressure mode

In this mode, the following quantity is measured instead

$$\begin{array}{c}\hfill \gamma :=\delta {\phi }_{X;P_0}\left(0\right)-\delta {\phi }_{X;P_1}\left(0\right)<0,\end{array}$$

(14)

with a vacuum pressure of $P_0=2\times {10}^{-4}$ mbar and a reference pressure of $P_1={10}^{-2}$ mbar. Scalar field parameters are constrained if

$$\begin{array}{c}\hfill \gamma <-5.{44}^{°},\end{array}$$

(15)

as is detailed in Appendix B.2.

Further details on the specific experimental setup used to obtain the constraints in the present article can be found in Ref. [58].

3.3. Computing Observables for Lunar Laser Ranging

Lunar Laser Ranging (LLR) allows for the measurement of the Moon’s orbit with high precision and consequently tests general relativity, as well as potential deviations from it; for example, due to the presence of a scalar fifth force [76]. For this, the Earth–Moon distance is measured by firing a laser beam at the retroreflectors that were installed on the Moon’s surface during the Apollo missions and detecting the reflection back on Earth. The propagation time of the laser pulse during its travel from Earth to Moon and back provides the corresponding distance to high precision. For previous investigations employing LLR in the context of symmetron and chameleon fields, see Refs. [47,77,78].

The measured value for violations of the equivalence principle is [79]

$$\begin{array}{c}\hfill {\delta }_{\mathrm{em}}\simeq \frac{a_{\varphi ♁}-a_{\varphi ☾}}{a_G}=(-3\pm 5)\times {10}^{-14},\end{array}$$

(16)

where $a_{\varphi ♁}$ and $a_{\varphi ☾}$ refer to the scalar field induced accelerations of the Earth and Moon towards the Sun, whereas $a_G$ is the regular Newtonian acceleration towards the Sun. For the constraint plots in this article, parameters are constrained, for which the scalar field contribution leads to at least a $2\sigma$-deviation of the measured value corresponding to

$$\begin{array}{c}\hfill {\delta }_{\mathrm{em}}\notin (-1.3\times {10}^{-13},7\times {10}^{-14}).\end{array}$$

(17)

The perihelion precession of the Moon has been computed, e.g., in Ref. [80], and leads to parameter constraints given by [5]

$$\begin{array}{c}\hfill \left|\frac{\delta \mathsf{\Omega }}{\mathsf{\Omega }}\right|\simeq \frac{R^2}{G{M}_{♁}}\left|\delta f\left(R\right)+\frac{R}{2}\delta f^{\prime }\left(R\right)\right|\le 1.6\times {10}^{-11},\end{array}$$

(18)

where R is the maximum Earth–Moon separation and use has been made of

$$\begin{array}{cc}\hfill {\delta }_{\mathrm{em}}& \simeq -\frac{({\mathfrak{Q}}_{♁}-{\mathfrak{Q}}_{☾})}{a_G}\frac{dA\left(\varphi \right)}{dr}{|}_{r=1\mathrm{AU}},\hfill \\ \hfill \delta f\left(r\right)& \simeq {\mathfrak{Q}}_{☾}\frac{dA\left(\varphi \right)}{dr}.\hfill \end{array}$$

(19)

3.4. Computing the Pressure in the Cannex Experiment

Among the high-precision experiments is the Casimir and Non-Newtonian force Experiment (Cannex), which has successfully completed its proof-of-principle phase and will soon commence operation [61,68]. It is the first experiment designed to measure the Casimir force employing two plane parallel plates. Due to this special geometry, interfacial as well as gravity-like forces are maximized leading to increased sensitivity. This experiment allows to probe a wide range of dark sector forces, Casimir forces in and out of thermal equilibrium as well as gravity.

Cannex measures the pressure

$$\begin{array}{c}\hfill P\left(d\right)=\frac{{\rho }_M}{{\rho }_M-{\rho }_V}\left(V_{\mathrm{eff}}({\varphi }_V,{\rho }_V)-V_{\mathrm{eff}}({\varphi }_0\left(d\right),{\rho }_V)\right),\end{array}$$

(20)

where ${\rho }_M$ and ${\rho }_V$ are the densities of the plates and of the surrounding vacuum, d is the distance between both plates and ${\varphi }_0\left(d\right)$ is the value of the scalar field in the middle between both plates, as well as the pressure gradient

$$\begin{array}{c}\hfill {\partial }_{d}P\left(d\right)\simeq \frac{P(d+\delta )-P(d-\delta )}{2\delta }.\end{array}$$

(21)

The latter approximation holds for small enough $\delta$. The sensitivity of these measurements at one standard deviation $\left(1\sigma \right)$ have already been analyzed in detail as function of d in Ref. [68]. Herein, parameter constraints are derived for

$$\begin{array}{cc}\hfill \left|P\left(d\right)\right|>2\sigma \left(d\right),\mathrm{or}|{\partial }_{d}P\left(d\right)|& >2\sigma \left(d\right).\hfill \end{array}$$

(22)

For the derivation of prospective constraints, it has been used that the plate separation and vacuum density can be varied between 3 μm < d < 30 μm and $5.3\times {10}^{-12}\mathrm{kg}/{\mathrm{m}}^3<{\rho }_V<2.6\mathrm{kg}/{\mathrm{m}}^3$, respectively. More details about this experiment can be found in Ref. [57].

4. Results

This section provides succinct discussions of the obtained constraints for the chameleon, symmetron and dilaton model.

4.1. Chameleon Constraints

In Figure 1, all currently available chameleon constraints on the parameters n and $M_c$ for $\mathsf{\Lambda }$ being set to the DE scale of $2.4$ meV are shown, as well as those on the parameters $\mathsf{\Lambda }$ and $M_c$ for the case $n=1$.

For this, the analytically exact solutions from Ref. [40] are used to compute the prospective Cannex constraints, while the one-mirror solution for qBounce is computed numerically.

Our analysis shows that qBounce has set ‘weak’ constraints for $n=1$, but no constraints for $\mathsf{\Lambda }=2.4$ meV and small positive n. The large discrepancy to previous chameleon constraints published in Refs. [33,35] are primarily rooted in a different definition of the screening charge, further elaborated on in Appendix C. For Cannex, our analysis predicts constraints both for $n=1$ as well as for other small values of n if $\mathsf{\Lambda }=2.4$ meV. However, constraints on parts of the parameter space that were not previously constrained by other experiments can only be obtained in the latter case. Neutron interferometry is currently not capable of producing new constraints for the cases considered in Figure 1. A static chameleon field always fulfills the inequality

$$\begin{array}{c}\hfill {\varphi }_M\le \varphi \left(x\right)\le {\varphi }_V.\end{array}$$

(23)

The quantity

$$\begin{array}{cc}\hfill {\delta }_{\max }{\phi }_C& :={\mathfrak{Q}}_C\frac{m_n^{2}L}{k_0{M}_c}\max \{\left(|{\varphi }_V-{\varphi }_{\mathrm{Air}}|\right),\left(|{\varphi }_M-{\varphi }_{\mathrm{Air}}|\right)\},\hfill \end{array}$$

(24)

where $L=9.4$ cm, can be computed analytically and used to bound the largest expected phase shift. It should be noted that

$$\begin{array}{cc}\hfill & \left|\alpha \right|\le 4{\delta }_{\max }{\phi }_C,\hfill \end{array}$$

(25)

as well as

$$\begin{array}{c}\hfill \left|\gamma \right|\le 2{\delta }_{\max }{\phi }_C.\end{array}$$

(26)

Over all unconstrained parts of the parameter space, one has

$$\begin{array}{c}\hfill 4{\delta }_{\max }{\phi }_C\ll 1^{°}.\end{array}$$

(27)

Consequently, no new constraints can be obtained from neutron interferometry. Obtained constraints would be many orders of magnitude ‘weaker’ than existing constraints, rendering a more detailed analysis for this experiment futile in the case of chameleons.

4.2. Symmetron Constraints

All existing constraints on the symmetron model derived from tabletop experiments are depicted in Figure 2. Notably, astrophysical experiments are unable to impose constraints on the $\mu$ values discussed in this article. For the re-analysis of the qBounce constraints originally presented in Refs. [46,49], the analytically exact solutions of Refs. [43,50] are employed. However, for the prospective Cannex constraints, the two mirror solution is computed by solving the differential equation numerically. The constraints are based on the most recent analysis of Cannex in Ref. [61]. Constraints from neutron interferometry are based on Ref. [58]. In Appendix A.1, we delineate how the newly computed symmetron constraints compare to those previously derived in Ref. [46], elucidating the impact of each innovation on the resulting constraints.

4.3. Dilaton Constraints

All existing constraints on the environment-dependent dilaton model were originally obtained in Refs. [57,58], and are based on experimental results from qBounce, LLR, and neutron interferometry. Prospective constraints for Cannex were also presented in Ref. [57]. A combined summary of these constraints can be found in Figure 3.

5. Conclusions

In this article, the most recent experimental constraints on the parameter spaces of some of the most popular screened scalar field models are summarized, i.e., the chameleon, symmetron and environment-dependent dilaton. Among others, experimental results from the qBounce collaboration, neutron interferometry, and LLR have been employed. In addition, future constraints for Cannex are predicted. The results herein collect the most up-to-date constraints on the considered screened scalar field models. Furthermore, several improvements to previously obtained constraints have been made. In some cases, this led to deviations by several orders of magnitude compared to previously obtained results.