Chameleon dark energy and atom interferometry

Atom interferometry experiments are searching for evidence of chameleon scalar fields with ever-increasing precision. As experiments become more precise, so too must theoretical predictions. Previous work has made numerous approximations to simplify the calculation, which in general requires solving a three-dimensional nonlinear partial differential equation. This paper calculates the chameleonic force using a numerical relaxation scheme on a uniform grid. This technique is more general than previous work, which assumed spherical symmetry to reduce the partial differential equation to a one-dimensional ordinary differential equation. We examine the effects of approximations made in previous efforts on this subject and calculate the chameleonic force in a setup that closely mimics the recent experiment of Hamilton et al. Specifically, we simulate the vacuum chamber as a cylinder with dimensions matching those of the experiment, taking into account the backreaction of the source mass, its offset from the center, and the effects of the chamber walls. Remarkably, the acceleration on a test atomic particle is found to differ by only 20% from the approximate analytical treatment. These results allow us to place rigorous constraints on the parameter space of chameleon field theories, although ultimately the constraint we find is the same as the one we reported in Hamilton et al. because we had slightly underestimated the size of the vacuum chamber. This computational technique will continue to be useful as experiments become even more precise and will also be a valuable tool in optimizing future searches for chameleon fields and related theories.

Figure 1

Schematic of the effective potential felt by a chameleon field (solid line), given by the sum of the bare potential of runaway form, $V(\varphi )$ (dashed line), and a density-dependent piece, from coupling to matter (dotted line).

Figure 2

Effective potential for low ambient matter density (left) and high ambient density (right). As the density increases, the minimum of the effective potential, ${\varphi }_{\min }$, shifts to smaller values, while the mass of small fluctuations, $m_{\varphi }$, increases.

Figure 3

Current constraints due to atom interferometry and torsion pendulum experiments. We are mainly concerned with $\mathrm{\Lambda }={\mathrm{\Lambda }}_0$, indicated by the black line on the first plot, so that the chameleon field can drive the observed accelerated expansion of the universe. The narrow light blue stripes on the left panel show the influence of varying the fudge parameter over $0.55\le \xi \le 0.68$. The second plot shows $M_{\mathrm{Pl}}/M$ vs $n$ and also assumes $\mathrm{\Lambda }={\mathrm{\Lambda }}_0$. The “torsion pendulum” region shown in green has been corrected from Ref. [54] to accurately reflect the constraints imposed by that experiment, following Ref. [39].

Figure 4

The chameleon field as a function of distance along the center of the spherical vacuum chamber. The black horizontal line marks the central value of $\varphi$ predicted inside an empty chamber using (18). The red vertical line denotes the location of the interferometer. We find essentially no difference between letting $\varphi$ minimize its potential in atmosphere vs in steel at the walls.

Figure 5

Diagram and dimensions of experimental setup. The cross marks the center of the vacuum chamber. The vacuum chamber walls are $\sim 2\mathrm{cm}$ thick, which is much greater than the Compton wavelength of the chameleon inside steel in all cases examined.

Figure 6

Spherical vs cylindrical vacuum chamber. Chameleon profile and acceleration as a function of distance from the center of the spherical source mass, for a spherical (blue curve) and cylindrical (green curve) vacuum chamber. The dimensions of the cylindrical vacuum chamber are chosen to match that of the experiment in Ref. [54] and are shown in Fig. 5. The radius of the sphere is chosen to match the inner radius of the cylinder. At the location of the interferometer (red vertical line), the acceleration in the spherical case is 18% larger than in the cylindrical chamber.

Figure 7

Source mass centered vs offset. Same plot as the previous figure, now comparing a source mass at the center (blue curve) and offset by 2.55 cm from the center (green curve), as in the actual experiment. As in the previous figure, the dimensions of the cylindrical chamber match those of the experiment. Although the field profile is altered by the offset, the acceleration at the interferometer (red vertical line) changes by less than 1%.

Figure 8

Source mass with vs without bore. Same as the previous two figures, but now comparing a solid source mass (blue curve) against one with a 3 mm diameter circular bore through the center (green curve), as in the experiment. All other dimensions are chosen to match those of the experiment. The only significant difference is inside the sphere, as the green line passes through the center of the bore, so it is still in vacuum. The acceleration at the interferometer (red vertical line) again changes by less than 1%.

Figure 9

Simulation of the experimental configuration, for values of $M$ ranging from ${10}^{-5}{M}_{\mathrm{Pl}}$ to ${1M}_{\mathrm{Pl}}$. We find that the profiles in vacuum are nearly identical, differing only in the walls. The field values inside the metal of the source mass also scale with $M$, but we are showing a path that passes through the center of the bore in the source mass. The bore acts as a miniature vacuum chamber, so instead the chameleon field goes to an $M$-independent value such that the Compton wavelength is of order the radius of the bore.

Figure 10

Spherical source mass vs cylindrical source. Comparison between two experimental setups: that of Ref. [54] (blue line) and that of an improved version of the experiment that is currently underway (green line). The main difference is that the latter employs a tungsten cylinder as the source mass, while the former used an aluminum sphere. The cylinder has a wedge cut out of it, allowing for vastly improved control over systematics. These show that the cutout comes at no cost to the chameleon signal; in fact, the cylinder confers a 5% stronger chameleon force over the previous setup.

Figure 11

Same plot as Fig. 9, but for the empty rectangular vacuum chamber of the CAL experiment. The field profiles are taken along the long axis of the vacuum chamber. Again, we find that the profiles in vacuum are nearly identical.