Thick shell regime in the chameleon two-body problem

In a previous paper [Phys. Rev. D 97, 104044 (2018)] we pointed out some shortcomings of the standard approach to chameleon theories consisting in treating the small bodies used to test the weak equivalence principle (WEP) as test particles, whose presence do not modify the chameleon field configuration. In that paper we developed an alternative method to determine the relevant field configuration which takes into account the influence of both test and source bodies, and computed the chamaleon mediated force. Relying on that analysis we showed that the effective acceleration of test bodies is composition dependent even when the model is based on universal couplings. In this paper, we improve our method by using a more suitable approximation for the effective chameleon potential in situations where the bodies are in the so-called “thick shell regime.” We then find new and more restrictive bounds on the model’s parametres by confronting the new theoretical predictions with the empirical bounds on Eötvös parameter comming from the lunar laser ranging experiments.

Figure 1

Two body problem.

Figure 2

The chameleon field $\phi$ as a function of coordinate $z$ inside the test body (Earth) and in its outskirts ($z$ is the distance to the center of the large body (Sun)). The blue lines represet our approach for the Earth while the red ones depict our approach for a test body with the same radius of the Earth but with half of its density. The green lines stand for the standard approach, and the black one, the minimum value of $\phi$ outside the bodies. Left: The Earth is in the thick shell regime but the Sun is not. Right: Both, Earth and Sun, are in the thick shell regime.

Figure 3

The chameleon field $\phi$ as a function of coordinate $z$ inside the test body (Earth) and in its outskirts ($z$ is the distance to the center of the large body (Sun)). The blue lines represent our approach for the Earth while the red ones, our approach for a test body with the same density of the Earth but with half of its radius. The green lines stand for the standard approach, and the black one, the minimum value of $\phi$ outside the bodies. Left: The Earth is in the thick shell regime but the Sun is not. Right: Both, Earth and Sun, are in the thick shell regime.

Figure 4

Energy functional computed for a setup that mimics the LLR experiment using the two-body approach (red), the two-body approach including the thick shell approximation (green) and the standard one (blue) taking $M=10\mathrm{eV}$ and $n=1$. The Sun and the Earth correspond to the large and test bodies, respectively, while the environment represents the interstellar medium. The vertical dotted lines indicate the boundary of the thin shell regime for the Earth and the Sun (values of $\beta$ lower than those indicated by the vertical dotted lines point out that the bodies have a thick shell). The right panel zooms a portion of the left panel. Notice that the energy functional is minimized for the improved two body problem (green line).

Figure 5

Similar to Fig. 4 taking $n=2$. (a) $n=2$. (b) $n=2$ zoom.

Figure 6

The Eötvös parameter $\eta$ (in ${\log }_{10}$ scale) for the LLR experiment as a function of the parameter $\beta$ (in ${\log }_{10}$ scale) for different positive values of $n$ ((a) $n=1$. (b) $n=2$). All bodies are surrounded by the interstellar medium. $M=10\mathrm{eV}$. The vertical lines show the values of $\beta$ below which the thin shell condition is no longer satisfied for the Earth, Moon and Sun. The horizontal line represents the experimental bound [10]. For all values computed in this plot, the energy criterion developed in Sec. 3 indicates that the two body approach together with an improved approximation to the effective potential provides better results than the standard approach and than the two-body approach with the quadratic approximation for the potential. (a) $n=1$. (b) $n=2$.