Chameleon field in a spherical shell system

We study the screening mechanism of a chameleon field in a highly inhomogeneous density profile. For simplicity, we consider static and spherically symmetric systems composed of concentric thin shells. We calculate the fifth-force profile by different methods depending on the Compton wavelength of the chameleon field: we use a numerical method for relatively large values of the Compton wavelength and an analytic approximation for the small Compton wavelength limit. Our results show that if the thin-shell condition for the corresponding smoothed density profile is satisfied, the fifth force is safely screened outside the system irrespective of the configuration of the shells inside the system. In contrast to the outer region, we find that the fifth force can be comparable to the Newtonian gravitational force in the interior region. This is because each shell is unscreened in the thin-shell limit even though the density of the shell is infinitely large. Our results explicitly show that the screening mechanism is effective for a cluster of unscreened objects if the cluster itself satisfies the thin-shell condition on average. However, even when the screening mechanism is effective over the total system, its components can be unscreened and a large fifth force can appear inside it. We thus find that the criterion with an averaged density distribution in a highly inhomogeneous system is insufficient to conclude that the fifth-force field will be well behaved.