Evading equivalence principle violations, cosmological, and other experimental constraints in scalar field theories with a strong coupling to matter

We show that, as a result of nonlinear self-interactions, it is feasible, at least in light of the bounds coming from terrestrial tests of gravity, measurements of the Casimir force and those constraints imposed by the physics of compact objects, big-bang nucleosynthesis and measurements of the cosmic microwave background, for there to exist, in our Universe, one or more scalar fields that couple to matter much more strongly than gravity does. These scalar fields behave like chameleons: changing their properties to fit their surroundings. As a result these scalar fields can be not only very strongly coupled to matter, but also remain relatively light over solar-system scales. These fields could also be detected by a number of future experiments provided they are properly designed to do so. These results open up an altogether new window, which might lead to a completely different view of the rôle played by light scalar fields in particle physics and cosmology.

Figure 1

Sketch of the chameleon mechanism for a runaway potential: $V\sim {\varphi }^{-4}$. The sketch on the left is for a low-density background, whereas the drawing of the right shows what occurs when there is a high density of matter in the surroundings. We can clearly see that the position of the effective minimum, ${\varphi }_c$, and the steepness of the effective potential near that minimum, depends on the density. A shallow minimum corresponds to a low chameleon mass. The mass of the chameleon can be clearly seen to grow with the background density of matter.

Figure 2

Sketch of the chameleon mechanism for a potential with a minimum at $\varphi =0$: $V\sim {\varphi }^4$. The sketch on the left is for a low-density background, whereas the drawing of the right shows what occurs when there is a high density of matter in the surroundings. We can clearly see that the position of the effective minimum, ${\varphi }_c$, and the steepness of the effective potential near that minimum, depends on the density. A shallow minimum corresponds to a low chameleon mass. The mass of the chameleon near ${\varphi }_c$ can be clearly seen to grow with the background density of matter.

Figure 3

Illustration of the model for the microscopic structure of the body considered in this section. The constituent particles are assumed to be spherical of radius $R$, mass $m_p$ and of uniform density. They are separated by an average distance $2D$. The average density of the body is defined to be ${\rho }_c$.

Figure 4

Dependence of the critical chameleon mass on $D/R$. The above plots show how $m_{\mathrm{crit}}D$ depends on $\ln (D/R)$ for different values of $n$. The cases $n=-4$ and $n\ne -4$ are qualitatively different and are therefore shown on separate plots. $m_{\mathrm{crit}}$ is the maximal mass the chameleon can take inside a thin-shelled body. $2D$ is the average separation of the particles that comprise that body and $R$ is the average radius of the constituent particles. Typically we find that $\ln (D/R)\approx 11$ for bodies with density $\rho \sim 1–10\mathrm{g}{\mathrm{cm}}^{-3}$; $\ln (D/R)=11$ is indicated on the plots. Note that when $n>0$, $m_{\mathrm{crit}}D$ is independent of $n$. Also note that in ${\varphi }^4$ theory $m_{\mathrm{crit}}D\sim \mathcal{O}(1)$ whereas for other value of $n$ it is generically much smaller.

Figure 5

Constraints on chameleon theories coming from the Eöt-Wash bounds on deviations from Newton’s law. The shaded area shows the regions of parameter space that are allowed by the current data. The solid black lines indicate the cases where $M$ and $\lambda$ take “natural values“. The dotted-black line indicates when $M=M_{\varphi }≔M_{\mathrm{pl}}/\beta$ i.e. when the mass scale of the potential is the same as that of the matter coupling. Other $n<-4$ theories are similar to the $n=-8$ case, while the $n=4$ plot is typical of what is allowed for $n>0$ theories. The amount of allowed parameter space increases with $|n|$.

Figure 6

Constraints on chameleon theories coming from experimental searches for the Casimir force. The shaded area shows the regions of parameter space that are allowed by the current data. The solid black lines indicate the cases where $M$ and $\lambda$ take “natural values.” The dotted-black line indicates when $M=M_{\varphi }≔M_{\mathrm{pl}}/\beta$ i.e. when the mass scale of the potential is the same as that of the matter coupling. Other $n<-4$ theories are similar to the $n=-8$ case, while the $n=4$ plot is typical of what is allowed for $n>0$ theories. The amount of allowed parameter space increases with $|n|$ i.e. as the potential becomes steeper.

Figure 7

Constraints on chameleon theories coming from WEP violation searches. The whole of shaded area shows the regions of parameter space that are allowed by the current data. Future space-based tests could detect the more lightly shaded region. The solid black lines indicate the cases where $M$ and $\lambda$ take “natural values.” The dotted-black line indicates when $M=M_{\varphi }≔M_{\mathrm{pl}}/\beta$ i.e. when the mass scale of the potential is the same as that of the matter coupling. Other $n<-4$ theories are similar to the $n=-8$ case, while the $n=4$ plot is typical of what is allowed for $n>0$ theories. The amount of allowed parameter space increases with $|n|$.

Figure 8

Constraints on chameleon theories coming from white dwarfs and neutron stars. The shaded area shows the regions of parameter space that are allowed assuming that any alterations to the white-dwarf mass-radius relationship are at the 5% level or smaller and that the chameleon only induces changes that are smaller than 10% in the maximum white-dwarf density, ${\rho }_{\mathrm{crit}}$. Neutron star bounds are not competitive with the white-dwarf constraints. The solid black lines indicate the cases where $M$ and $\lambda$ take “natural values.” The dotted-black line indicates when $M=M_{\varphi }≔M_{\mathrm{pl}}/\beta$ i.e. when the mass scale of the potential is the same as that of the matter coupling. Other $n<-4$ theories are similar to the $n=-8$ case, while the $n=4$ plot is typical of what is allowed for $n>0$ theories. The amount of allowed parameter space increases with $|n|$.

Figure 9

Constraints on chameleon theories coming from the particle masses at big-bang nucleosynthesis and the constraints the redshift of recombination. The shaded area shows the regions of parameter space that are allowed by the current data. The solid black lines indicate the cases where $M$ and $\lambda$ take “natural values.” The dotted-black line indicates when $M=M_{\varphi }≔M_{\mathrm{pl}}/\beta$ i.e. when the mass scale of the potential is the same as that of the matter coupling. Other $n<-4$ theories are similar to the $n=-8$ case, while the $n=4$ plot is typical of what is allowed for $n>0$ theories. The amount of allowed parameter space increases with $|n|$. In these plots we have assumed that the chameleon couples only to baryons. Slightly weaker constraints result if the chameleon additionally couples to dark matter.

Figure 10

Combined constraints on chameleon theories. The whole of shaded area shows the regions of parameter space that are allowed by the current data. Future space-based tests could detect the more lightly shaded region. The solid black lines indicate the cases where $M$ and $\lambda$ take “natural values“. For $n\ne -4$, a natural value for $M$ is required if the chameleon is to be dark energy. The dotted-black line indicates when $M=M_{\varphi }≔M_{\mathrm{pl}}/\beta$ i.e. when the mass scale of the potential is the same as that of the matter coupling. Other $n<-4$ theories are similar to the $n=-6$ and $n=-8$ cases, while the $n=4$ and $n=6$ plots are typical of what is allowed for $n>0$ theories. The amount of allowed parameter space increases with $|n|$.

Figure 11

Allowed couplings and force ranges for scalar fields which couple to baryon number and have approximately linear field equations. The shaded area shows the parameter space that is allowed by the current bounds on scalar-field theories with linear field equations. $\lambda =1/m$ is the range of the force, and $m$ is the mass of the scalar field. $\beta$ is the matter coupling of the scalar. $\beta >1$ is ruled out for all but the smallest ranges (currently $\lambda <{10}^{-4}\mathrm{m}$). We refer the reader to [29] for a plot of the allowed parameter space for $\lambda <{10}^{-3}\mathrm{m}$. Irvine, Seattle, and Boulder refers to Refs. 35, 36, 37, 62, 76 respectively.

Figure 12

Allowed couplings and cosmological force ranges for chameleon theories. The shaded area shows the allowed parameter space with all current bounds. ${\lambda }_{\cos }$ is the range of the chameleon force in the cosmological background with density $\rho \sim {10}^{-29}\mathrm{g}{\mathrm{cm}}^{-3}$. It is related to the cosmological mass of the chameleon, $m_c^{\cos }$, by ${\lambda }_{\cos }=1/m_c^{\cos }$. The solid black lines indicate the cases where $M$ and $\lambda$ take natural values. Plots for theories with $n<-4$ or $n>0$ are similar to the cases $n=-8$, $-6$ and $n=4$, 6, respectively.

Figure 13

Allowed couplings and cosmological force ranges for chameleon theories. The shaded area shows the allowed parameter space with all current bounds. ${\lambda }_{\mathrm{sol}}$ is the range of the chameleon force in the solar system, where the average density of matter is $\rho \sim {10}^{-24}\mathrm{g}{\mathrm{cm}}^{-3}$. It is related to the cosmological mass of the chameleon, $m_c^{\mathrm{sol}}$, by ${\lambda }_{\mathrm{sol}}=1/m_c^{\mathrm{sol}}$. The solid black lines indicate the cases where $M$ and $\lambda$ take natural values. Plots for theories with $n<-4$ or $n>0$ are similar to the cases $n=-8$, $-6$ and $n=4$, 6, respectively.