Chameleon and Symmetron Screening

Chameleon and symmetron theories screen a scalar by changing its local effective potential. A chameleon becomes heavy and develops a thin shell in a dense environment; a symmetron restores a symmetry there, driving its matter coupling to zero. In both cases the scalar charge of an extended body depends on its size, compactness, density profile, and ambient field. Screening is therefore not a universal label attached to a particle species.

Required background. Ordinary Differential Equations supplies effective-potential minima and radial matching.

Helpful background. Einstein Equations supplies the metric comparison; Gravitational Actions and Boundary Terms supplies the field variation; Formulations of General Relativity supplies boundary and branch language; Gravity Beyond General Relativity supplies mode and cutoff checks; and Relativistic Matter and Compact Objects supplies the matter profiles beyond the constant-density approximation.

Density-dependent scalar equation

Use the reduced Planck mass $M_{\rm Pl}^{-2}=8\pi G$. Consider an Einstein-frame scalar with matter coupled to the Jordan metric $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$:

$$S=\int d^4x\sqrt{-g} \left[ \frac{M_{\rm Pl}^2}{2}R -\frac12(\nabla\phi)^2-V(\phi) \right] +S_m[A^2(\phi)g_{\mu\nu},\psi].$$

For nonrelativistic matter use the conserved rescaled density $\rho=A^3\tilde\rho$, where $\tilde\rho$ is the Jordan-frame rest density. The actual Einstein-frame energy density is $A\rho$; it is $\rho$, rather than $A\rho$, that is held fixed when differentiating the effective potential. The quasistatic equation is

$$\nabla^2\phi=V_{{\rm eff},\phi}, \qquad V_{\rm eff}(\phi;\rho)=V(\phi)+\rho A(\phi).$$

The fifth acceleration is

$$\boldsymbol a_\phi=-\nabla\ln A(\phi) =-\frac{\beta(\phi)}{M_{\rm Pl}}\nabla\phi, \qquad \beta(\phi)=M_{\rm Pl}\frac{d\ln A}{d\phi}.$$

Frame and density conventions matter: transforming the metric while holding the wrong density fixed changes intermediate formulas. Burrage and Sakstein derive this package in Burrage and Sakstein 2018, § 2, equations (2.1)–(2.12).

Chameleon minimum and mass

Take

$$V(\phi)=\frac{\Lambda_c^{4+n}}{\phi^n}, \qquad A(\phi)\simeq1+\frac{\beta\phi}{M_{\rm Pl}}, \qquad n\gt0,\ \beta\gt0.$$

Work on the branch $\phi\gt0$ and require $\lvert\beta\phi/M_{\rm Pl}\rvert\ll1$ for the linear expansion of $A$. The local minimum solves

$$-\frac{n\Lambda_c^{4+n}}{\phi_{\min}^{n+1}} +\frac{\beta\rho}{M_{\rm Pl}}=0,$$

so

$$\phi_{\min}(\rho)= \left( \frac{nM_{\rm Pl}\Lambda_c^{4+n}}{\beta\rho} \right)^{1/(n+1)}.$$

The effective mass is

$$m_{\rm eff}^2(\rho) =\frac{n(n+1)\Lambda_c^{4+n}} {\phi_{\min}^{n+2}} \propto \rho^{(n+2)/(n+1)}.$$

Increasing density lowers the equilibrium field and raises its mass. This is the chameleon mechanism. The result assumes the field can actually relax to the minimum on the object’s length and time scales; a formal minimum alone does not prove screening.

Thin-shell matching

Place a nearly uniform sphere of radius $R$, mass $M$, and interior minimum $\phi_c$ in an environment whose minimum is $\phi_\infty$. Let $\Phi_N=GM/R$. If the field remains close to $\phi_c$ until a screening radius $r_s$, only the outer layer sources the exterior flux. Set $\epsilon=1-M(r_s)/M$ and assume that the potential term is negligible compared with the matter source within that active layer. Linearizing the exterior about the ambient minimum gives a decaying Yukawa field. Matching its derivative to the surface flux gives

$$\phi(r)-\phi_\infty =-\frac{\beta M\epsilon}{4\pi M_{\rm Pl}(1+m_\infty R)} \frac{e^{-m_\infty(r-R)}}{r}.$$

Differentiation, rather than multiplication of a massless force by an exponential, therefore gives

$$\phi'(r)= \frac{\beta}{4\pi M_{\rm Pl}} \frac{M-M(r_s)}{r^2} \frac{1+m_\infty r}{1+m_\infty R} e^{-m_\infty(r-R)}, \qquad r\ge R.$$

The factor $1+m_\infty r$ is essential: a Yukawa potential and a Yukawa force have different radial dependence. The denominator makes the surface flux equal to $\beta M\epsilon/(4\pi M_{\rm Pl})$. This formula assumes an exterior in which linearization is valid; it does not replace nonlinear matching when the field remains far from its ambient minimum.

For a thin shell of thickness $\Delta R=R-r_s$ and an ambient range large compared with the body, $m_\infty R\ll1$,

$$\frac{M-M(r_s)}{M}\simeq3\frac{\Delta R}{R}, \qquad \frac{\Delta R}{R} \simeq \frac{\phi_\infty-\phi_c} {6\beta M_{\rm Pl}\Phi_N}.$$

Consequently,

$$\boxed{ \frac{F_\phi}{F_N} \simeq 2\beta^2 \left(3\frac{\Delta R}{R}\right) (1+m_\infty r) e^{-m_\infty(r-R)} }.$$

Here the test body is unscreened; a screened test body’s response introduces its own charge fraction. The thin-shell relation and exterior potential are given in Khoury and Weltman 2004, Eqs. (5)–(8), PDF p. 2. The force above follows by differentiating that potential, with the $m_\infty R\ll1$ approximation used consistently.

A deeply screened body has $\Delta R/R\ll1$; a fully unscreened body has $M-M(r_s)\to M$ and approaches $F_\phi/F_N=2\beta^2$ before Yukawa suppression. The mass-fraction form, rather than an extrapolated thin-shell formula, must be used through the transition. The radial integration and screening-radius condition appear in Burrage and Sakstein 2018, § 3.1, equations (3.1)–(3.10).

Symmetron symmetry restoration

For a symmetron, choose

$$V(\phi)=-\frac12\mu^2\phi^2+\frac{\lambda}{4}\phi^4, \qquad A(\phi)\simeq1+\frac{\phi^2}{2M_s^2}.$$

Then

$$V_{\rm eff} =\frac12\left(\frac{\rho}{M_s^2}-\mu^2\right)\phi^2 +\frac{\lambda}{4}\phi^4.$$

Here $M_s$ is a coupling scale, distinct from the source mass $M$, and $\lambda\gt0$. Above $\rho_\star=\mu^2M_s^2$, the symmetric minimum is $\phi_{\min}=0$. Because $\beta(\phi)\simeq M_{\rm Pl}\phi/M_s^2$ for $\lvert\phi\rvert\ll M_s$, the coupling also vanishes. Below $\rho_\star$, symmetry breaks and

$$\phi_{\min}^2 =\frac{\mu^2}{\lambda} \left(1-\frac{\rho}{\rho_\star}\right).$$

Thus a chameleon mainly suppresses range and scalar charge by increasing its mass, whereas a symmetron suppresses the local coupling through symmetry restoration. The original symmetron construction and spherical screening are given in Hinterbichler and Khoury 2010, §§ I–II, Eqs. (4)–(8).

Comparing the two mechanisms

Question	Chameleon	Symmetron
What changes with density?	The minimum and curvature of an asymmetric effective potential	The sign of the quadratic term and therefore the symmetry phase
Why is the force small?	The field is heavy and only a thin outer layer contributes scalar charge	The high-density vacuum has nearly zero field and therefore nearly zero matter coupling
What sets the exterior range?	The ambient chameleon mass $m_\infty$	The mass of fluctuations about the ambient broken- or unbroken-phase minimum
What can defeat screening?	A shallow potential, small body, low-density environment, or insufficient relaxation	A region large enough to enter the broken phase with a nonzero ambient vacuum expectation value

The mechanisms can produce similar exterior force profiles while assigning different meanings to their parameters. In particular, a quoted Yukawa range does not determine the scalar charge: the source may itself be screened. Conversely, a small local coupling in the symmetron’s restored phase does not show that the field is heavy enough to follow a sharp material boundary. A comparison must therefore carry the density profile, radius, ambient field, and selected minimum together. Replacing that boundary-value problem by a constant “effective coupling” erases the feature that performs the screening.

Environment, branch, and cutoff

The boundary value $\phi_\infty$ is set by the environment, not by the body in isolation. A low-density object inside a screened host can inherit a small ambient field; nonlinear solutions cannot be superposed to undo that effect. The scalar charge can also lag a time-dependent environment if the relaxation time is long.

Quantum and classical control require separate checks. The local mass and self-couplings derived from $V_{\rm eff}$ must not make loop corrections larger than the potential being used, while surface gradients and matter discontinuities must remain within the EFT cutoff. A screened background is not automatically a controlled background.

Exercise

For the inverse-power chameleon above, derive how $\phi_{\min}$ and $m_{\rm eff}$ change when the density is multiplied by a factor $q\gt1$. Then explain how the result supports, but does not by itself prove, a thin shell.

Solution

The minimum scales as $\phi_{\min}\propto\rho^{-1/(n+1)}$, so $\phi_{\min}(q\rho)/\phi_{\min}(\rho)=q^{-1/(n+1)}$. Since $m_{\rm eff}^2\propto\rho^{(n+2)/(n+1)}$, the mass ratio is $m_{\rm eff}(q\rho)/m_{\rm eff}(\rho) =q^{(n+2)/[2(n+1)]}$. A denser interior therefore favors a smaller field and a shorter Compton wavelength. A thin shell additionally requires $m_cR\gg1$, appropriate matching to $\phi_\infty$, and $(\phi_\infty-\phi_c)/(6\beta M_{\rm Pl}\Phi_N)\ll1$. Density scaling alone does not impose those size, compactness, boundary, or branch conditions.

References

• Burrage, Clare, and Jeremy Sakstein (2018), “Tests of Chameleon Gravity,” Living Reviews in Relativity 21, 1. DOI. Open PDF.
• Hinterbichler, Kurt, and Justin Khoury (2010), “Screening Long-Range Forces through Local Symmetry Restoration,” Physical Review Letters 104, 231301. DOI. Open PDF.
• Khoury, Justin, and Amanda Weltman (2004), “Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space,” Physical Review Letters 93, 171104. DOI. Open PDF.