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.

Figure 1

Schematic figure of the spherical shell system.

Figure 2

(a) Profile of the chameleon field $\varphi$ and (b) strength of the fifth force divided by the Newtonian gravitational force $\mathcal{R}$, for $N=10$ and ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^2$, $10^3$, $10^4$.

Figure 3

Variation of the fifth force for different numbers of shells $N=1$, 5 and 10 with ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^4$. We show an enlarged figure for the outside region in the right panel to show the dependence on the number of shells. Each line almost coincides with that for the smoothed density outside of the outermost shell, $x>1$.

Figure 4

Fifth-force strength divided by the Newtonian gravitational force for ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^3$ with $N=1$ and 5. The constant density case is also shown for comparison.

Figure 5

Numerical result (blue line) and analytical approximate result (green) of the field for ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^4$ and $N=10$.

Figure 6

$\mathcal{R}$ as a function of $x$. The blue lines show the result of numerical integration for $N=10$ and ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^4$. The red points show the analytic approximation at the position of each shell given by Eq. (47), substituting $i=x/N$.

Figure 7

Schematic figure of a thick outer shell and thin inner shell system.

Figure 8

(a) Field profile and (b) $\mathcal{R}=F_{\varphi }/F_{\text{Newton}}$ for $M_{\mathrm{o}}=0.8M$, $0.6M$ and $0.4M$ with ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^3$. The field profile is shown with $\delta =0.25$ and $\mathcal{R}$ is calculated at the same point, $x=2$, as a function of $\delta$. The value of $\mathcal{R}$ for a uniform density is shown by the dashed line. The vertical lines in (b) correspond to the respective positions of ${\delta }_c$.

Figure 9

(a) Profile of the chameleon field $\varphi$ and (b) strength of the fifth force divided by the Newtonian gravitational force $\mathcal{R}$ for $N=3$, ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^3$ and $a=1/12$. The shaded regions represent the shell regions.

Figure 10

$\mathcal{R}=F_{\varphi }/F_{\text{Newton}}$ for $a=0$, $1/30$, $1/12$ with ${\tilde{m}}_{\mathrm{c}}^2{L}^2={10}^3$ and $N=3$. The profiles are plotted only in the vacuum regions to make the difference easier to see.