Force sensor for chameleon and Casimir force experiments with parallel-plate configuration

The search for non-Newtonian forces has been pursued following many different paths. Recently it was suggested that hypothetical chameleon interactions, which might explain the mechanisms behind dark energy, could be detected in a high-precision force measurement. In such an experiment, interactions between parallel plates kept at constant separation could be measured as a function of the pressure of an ambient gas, thereby identifying chameleon interactions by their unique inverse dependence on the local mass density. During the past years we have been developing a new kind of setup complying with the stringent requirements of the proposed experiment. In this article we present the first and most important part of this setup—the force sensor. We discuss its design, fabrication, and characterization. From the results of the latter, we derive limits on chameleon interaction parameters that could be set by the forthcoming experiment. Finally, we describe the opportunity to use the same setup to measure Casimir forces at large surface separations with unprecedented accuracy, thereby potentially giving unambiguous answers to long-standing open questions.

Figure 1

Top view of the force transducer. Tangential spring arms extend from a central disk onto which the measured forces act. Insert: enlarged view and geometric definitions.

Figure 2

Schematic view of the experimental setup. Note that the two different configurations for ac and dc measurements of the dynamic response and static deflections, respectively, are shown together.

Figure 3

Normalized dynamical response of the force transducer to mechanical excitation $\mathrm{\Delta }{d}_{\mathrm{pz}}$. A numerical simulation of the geometry using a constant base vibration amplitude (solid line) agrees very well with the experimental results (dots). The slight mismatch of 3% between the frequencies of the first resonance seen in experimental and numerical results ($15.20\pm 0.15\mathrm{Hz}$ and $14.85\pm 0.05\mathrm{Hz}$, respectively) is a consequence of fabrication tolerances. Contrary to that, the simple model of Eq. (1) (dashed line) fails to qualitatively predict the response.

Figure 4

Correction factor $C_{\mathrm{pp}}(d,\theta )/C_{\mathrm{pp}}(d)$ due to the relative tilt angle $\theta$ between the plates for several separations $d$, as defined in the insert.

Figure 5

Measured $\mathrm{\Delta }{C}_{\mathrm{pp}}$ as a function of the applied $V_{\mathrm{dc}}$ for three different nominal distances. The solid curves represent the best results obtained by least squares fits to the numeric model in Eqs. (2a)–(2c).

Figure 6

Theoretical prediction of RMS noise levels in the final Cannex setup for a bridge detection bandwidth of 0.86 Hz [15]. The thermal noise (dotted line) of laser-cut sensors and the bridge resolution (black line) are the main limiting factors for the achievable precision. Using an advanced version of the bridge and assuming a higher $Q$ factor for etched sensors leads to significantly better results (dashed line).

Figure 7

Limits on $\beta$ in dependence on the parameter $n$ defining the chameleon potential in Eq. (4). Dashed lines present upper bounds achievable with our experiment for the sensitivity levels estimated in this article for measurements at $d=10\mu \mathrm{m}$. For comparison the best current limits found in the literature from gravity resonance spectroscopy [24] (top, GRS) and torsion balance fifth force measurements [25] (bottom left, fifth force) are included.

Figure 8

Limits on Yukawa-type forces arising from the estimated levels of sensitivity for the upcoming Cannex experiment (dashed lines). The shaded area marks parameter ranges already excluded by the results of experiments the Eöt-Wash [26], Stanford [10], Yale [11]. and Tokyo [27] groups.