Probing modified gravity with atom-interferometry: A numerical approach

Refined constraints on chameleon theories are calculated for atom-interferometry experiments, using a numerical approach consisting in solving for a four-region model the static and spherically symmetric Klein-Gordon equation for the chameleon field. By modeling not only the test mass and the vacuum chamber but also its walls and the exterior environment, the method allows one to probe new effects on the scalar field profile and the induced acceleration of atoms. In the case of a weakly perturbing test mass, the effect of the wall is to enhance the field profile and to lower the acceleration inside the chamber by up to 1 order of magnitude. In the thin-shell regime, results are found to be in good agreement with the analytical estimations, when measurements are realized in the immediate vicinity of the test mass. Close to the vacuum chamber wall, the acceleration becomes negative and potentially measurable. This prediction could be used to discriminate between fifth-force effects and systematic experimental uncertainties, by doing the experiment at several key positions inside the vacuum chamber. For the chameleon potential $V(\varphi )={\mathrm{\Lambda }}^{4+\alpha }/{\varphi }^{\alpha }$ and a coupling function $A(\varphi )=\exp (\varphi /M)$, one finds $M\gtrsim 7\times {10}^{16}\mathrm{GeV}$, independently of the power-law index. For $V(\varphi )={\mathrm{\Lambda }}^4(1+\mathrm{\Lambda }/\varphi )$, one finds $M\gtrsim {10}^{14}\mathrm{GeV}$. A sensitivity of $a\sim {10}^{-11}\mathrm{m}/{\mathrm{s}}^2$ would probe the model up to the Planck scale. Finally, a proposal for a second experimental setup, in a vacuum room, is presented. In this case, Planckian values of $M$ could be probed provided that $a\sim {10}^{-10}\mathrm{m}/{\mathrm{s}}^2$, a limit reachable by future experiments. Our method can easily be extended to constrain other models with a screening mechanism, such as symmetron, dilaton and f(R) theories.

Figure 1

Outline of the atom-interferometry experiment, simulated by a four-region model including the source mass, the vacuum chamber, its walls and the exterior environment. In light gray, the near and far positions where the acceleration on atoms is measured (note that we consider a fixed source mass to keep spherical symmetry whereas in the real experimental setup the source mass is moved [36]).

Figure 2

Numerical and analytical scalar field profiles (respectively solid and dashed lines) in absolute value for various $M$ listed in Table 3 [from top to bottom the curves correspond to increasing values of M, as reported in Table 3]. The numerical profiles obtained when lowering the wall density to ${\rho }_{\mathrm{w}}=5\times {10}^{-19}<{\rho }_{\mathrm{A}}$, are drawn in dotted lines, in order to illustrate that the wall density can perturb more or less importantly the field profile. Vertical lines mark out the four regions: the test mass, the chamber, the wall and outside the chamber.

Figure 3

Numerical and analytical profiles (respectively solid and dashed lines) of the acceleration $a_{\varphi }/g$ with $g$ the Earth gravitational acceleration, for $M$ values listed in Table 3 [from top to bottom the curves correspond to increasing values of M, as reported in Table 3]. Vertical lines mark out the four regions (test mass, chamber, wall and exterior).

Figure 4

Numerical results (solid lines) and analytical approximation (dashed lines) for the scalar field profile of the Chameleon-1 model, in the strongly perturbing (thin-shell) regime, for $\mathrm{\Lambda }=2.6\times {10}^{-6}\mathrm{GeV}$ and values of the coupling $M$ listed in Table 3 [from top to bottom the curves correspond to decreasing values of M, as reported in Table 3]. Differences between the two-region and four-region models are non-negligible inside the chamber, especially at the vicinity of the wall. Vertical lines mark out the four regions (test mass, chamber, wall and exterior).

Figure 5

Numerical results (solid lines) and analytical approximation (dashed lines) for the acceleration $|a_{\varphi }|/g$ profile, for the same model and parameters as in Fig. 4 [from top to bottom the curves correspond to increasing values of M, as reported in Table 3].The numerical profile for the four-region model shows that from the middle of the chamber to the wall, the acceleration becomes negative and increases in magnitude. Vertical lines mark out the four regions (test mass, chamber, wall and exterior).

Figure 6

Forecast for the normalized acceleration $a_{\varphi }/g$ measured in the “near” position, i.e. 8.8 mm far from the test mass, for various $M$ and $\alpha$. The dotted line represents the gravitational acceleration due to the test mass.

Figure 7

Numerical results (solid lines) and analytical approximation (dashed line) for the scalar field profile of the Chameleon-2 model, for various ${\rho }_{\mathrm{A}}$ and ${\rho }_{\mathrm{w}}$ reported in Table 4, $M={10}^{17}\mathrm{GeV}$ being fixed [from top to bottom the curves correspond to combination of parameters listed in Table 4].

Figure 8

Numerical results (solid lines) and analytical approximation (dashed line) for the acceleration $a_{\varphi }/g$ profile of the Chameleon-2 model, for various ${\rho }_{\mathrm{A}}$ and ${\rho }_{\mathrm{w}}$ reported in Table 4, $M={10}^{17}\mathrm{GeV}$ being fixed.

Figure 9

Numerical results (solid lines) and analytical approximation (dashed lines) for the scalar field profile of the Chameleon-2 model, in the strongly perturbing (thin-shell) regime, for $\mathrm{\Lambda }=2.4\times {10}^{-12}\mathrm{GeV}$ and values of the coupling $M$ listed in Table 3 [from top to bottom the curves correspond to decreasing values of M, as reported in Table 3]. The test mass, wall and exterior densities have been adapted for making the profile numerically tractable, with no effect inside the vacuum chamber (apart at the immediate vicinity of the borders), as explained in Sec. 6b2. The ratios $M/\rho$ were kept constant (with the same value as for the red curve of Fig. 7), which fixes the thin-shell radius, apart for $M={10}^{18}\mathrm{GeV}$ (purple) and $M={10}^{19}\mathrm{GeV}$ (beige) for which only the wall density was adapted. Noticeable deviations from the analytical estimation are observed inside the chamber, due to the wall effects. Vertical lines mark out the four regions (test mass, chamber, wall and exterior).

Figure 10

Numerical results (solid lines) and analytical approximation (dashed lines) for the acceleration $|a_{\varphi }|/g$ profile, for the same model and parameters as in Fig. 9 [from top to bottom the curves correspond to increasing values of M, as reported in Table 3]. Strong discrepancies are observed between the four-region (numerical) and the two-region (analytical) models. Vertical lines mark out the four regions (test mass, chamber, wall and exterior).

Figure 11

Numerical scalar field profiles for a vacuum room ($L=10\mathrm{m}$). The top green dashed curve is obtained for a test mass of $R_{\mathrm{A}}=1\mathrm{cm}$ with $M=m_{\mathrm{p}}$ (${\rho }_{\mathrm{w}}=2.5\times {10}^{-21}{\mathrm{GeV}}^4$) while the bottom blue and the center red ones are obtained for $R_{\mathrm{A}}=3.3\mathrm{cm}$ with $M=0.1\times m_{\mathrm{p}}$ (${\rho }_{\mathrm{w}}=2.5\times {10}^{-22}{\mathrm{GeV}}^4$) and $M=m_{\mathrm{p}}$ (${\rho }_{\mathrm{w}}=2.5\times {10}^{-21}{\mathrm{GeV}}^4$) respectively. We only consider a three regions model, neglecting the effect of the exterior of the vacuum room (see discussion in Sec. 6b).

Figure 12

Numerical acceleration profiles $|a_{\varphi }/g|$ for a vacuum room ($L=10\mathrm{m}$). The bottom green dashed curve is obtained for a test mass of $R_{\mathrm{A}}=1\mathrm{cm}$ with $M=m_{\mathrm{p}}$ (${\rho }_{\mathrm{w}}=2.5\times {10}^{-21}{\mathrm{GeV}}^4$) while the top blue and the center red ones are obtained for $R_{\mathrm{A}}=3.3\mathrm{cm}$ with $M=0.1\times m_{\mathrm{p}}$ (${\rho }_{\mathrm{w}}=2.5\times {10}^{-22}{\mathrm{GeV}}^4$) and $M=m_{\mathrm{p}}$ (${\rho }_{\mathrm{w}}=2.5\times {10}^{-21}{\mathrm{GeV}}^4$) respectively. We only consider a three regions model, neglecting the effect of the exterior of the vacuum room (see discussion in Sec. 6b).