Ellipticity weakens chameleon screening

The chameleon mechanism enables a long-range fifth force to be screened in dense environments when nontrivial self-interactions of the field cause its mass to increase with the local density. To date, chameleon fifth forces have mainly been studied for spherically symmetric sources; however, the nonlinear self-interactions mean that the chameleon responds to changes in the shape of the source differently to gravity. In this work we focus on ellipsoidal departures from spherical symmetry and compute the full form of the chameleon force, comparing its shape dependence to that of gravity. Enhancement of the chameleon force by up to 40% is possible when deforming a sphere to an ellipsoid of the same mass, with an ellipticity $\sim 0.99$.

Figure 1

The chameleon field profile decomposes into core, shell and exterior regions. The relevant regime within the potential is indicated for each of the three regions.

Figure 2

The blue line represents the scaling of the relative size of the shell region within an ellipsoidal source of size $R_{\mathrm{EFF}}$. At the far left, a constant value is approached as the shell solution pervades the entire object and the core region is absent. As the effective radius is increased, the extent of the shell region begins to diminish due to the emergence of a core region. The dotted red line shows the corresponding dependence of the magnitude of the chameleon force sourced by the compact object. The maximum force magnitude occurs when the radius of the source corresponds to the formation of a core region.

Figure 3

Comparison of the shape dependence of the gravitational and chameleon force. The mass of the source, represented by the black region, is the same for all plots. From left to right the ellipsoids have ${\xi }_0=1.01$, ${\xi }_0=1.1$ and ${\xi }_0=5$ in both subfigures, corresponding to ellipticities $\sim 0.99$, 0.91 and 0.2, respectively. The colors indicate the strength of the force, with red indicating regions of strongest force and blue indicating regions of weakest force. The color spectrum across these images has been normalized to the same limits to highlight the chameleons relative contribution. Increasing the ellipticity of the source can be seen to increase the ratio between the chameleon and the gravitational interactions. The parameter values $\mathrm{\Lambda }={10}^{-12}\mathrm{GeV}$ and $M=0.5M_{\text{Pl}}$ were chosen to generate these plots, but the shape dependence is independent of these choices. (a) The shape dependence of the gravitational force. This also represents the shape dependence of the unscreened chameleon force around objects that do not have a thin shell. (b) The shape dependence of the chameleon force characteristic of objects for which a shell region has developed.

Figure 4

Example of how the size of the core region decays as the ellipticity of the source object increases (lower ${\xi }_0$). The line asymptotes to the core volume for the spherical scenario as ${\xi }_0\to \infty$, represented by the red dashed line. This figure was generated for an object of size $R_{\mathrm{EFF}}=4\mathrm{m}$ as this value captures the whole range of scaling effects. $M$ and $\mathrm{\Lambda }$ were set as $M_{\mathrm{PL}}$ and ${10}^{-12}\mathrm{GeV}$, respectively.

Figure 5

The ratio of the chameleon to gravitational forces as a function of the size of the source mass, parametrized by the radius of a sphere of equal mass $R_{\mathrm{EFF}}$. The different lines show sources of different ellipticities, with the green line being the most spherical and the blue line the most ellipsoidal. The right-hand plot is a magnification of a region of the left-hand plot to show that the shape enhancement of the chameleon force persists as the size of the source is increased. To obtain numerical values the parameters were chosen to be ${\rho }_{\text{obj}}={10}^3$, ${\rho }_{\text{bg}}={10}^{-14}{\mathrm{kgm}}^{-3}$, $M=M_{\text{Pl}}$ and $\mathrm{\Lambda }$ set at the dark energy scale ${10}^{-12}\mathrm{GeV}$.

Figure 6

The ratio of the chameleon to gravitational force as a function of the size of the source for different choices of $\mathrm{\Lambda }$ and ${\rho }_{\text{bg}}$. The lower edge of each band, drawn with a solid line, shows the force ratio for a spherical source. The upper edge of each band, drawn with a dashed line, shows the force ratio for an ellipsoid with ${\xi }_0=1.001$.

Figure 7

Left: The ratio of the chameleon to gravitational forces as a function of the parameters $\mathrm{\Lambda }$ and $M$ for a spherical source object with $R_{\mathrm{EFF}}=1\mathrm{cm}$. The ambient and source densities take our reference values, $1{\mathrm{gcm}}^{-3}$ and ${10}^{-17}{\mathrm{gcm}}^{-3}$. Right: The enhancement of the chameleon force when an ellipsoid with ${\xi }_0=1.01$ is compared to a sphere.