Improvement for Testing the Gravitational Inverse-Square Law at the Submillimeter Range

We improve the test of the gravitational inverse-square law at the submillimeter range by suppressing the vibration of the electrostatic shielding membrane to reduce the disturbance coupled from the residual surface potential. The result shows that, at a 95% confidence level, the gravitational inverse-square law holds ($|\alpha |\le 1$) down to a length scale $\lambda =48\mu \mathrm{m}$. This work establishes the strongest bound on the magnitude $\alpha$ of the Yukawa violation in the range of $40–350\mu \mathrm{m}$, and improves the previous bounds by up to a factor of 3 at the length scale $\lambda \approx 70\mu \mathrm{m}$. Furthermore, the constraints on the power-law potentials are improved by about a factor of 2 for $k=4$ and 5.

Figure 1

Schematic drawing of the experimental setup (not to scale). Compared with our previous experiment [24], the rotary stage was replaced by a smaller and vacuum compatible one, and installed at the end of the rotary shaft. The attractor and the rotary stage, as well as the alignment glass, are supported on a 6-degree-of-freedom stage to adjust the position (not shown here). The electrostatic shielding membrane was placed on a translation stage to adjust the separation between the pendulum.

Figure 2

(a) The feedback voltage $U_{\mathrm{PID}}$ of the closed-loop torsion balance is a minimum when the potential $U_m$ applied on the shielding membrane equals to the surface potential difference between the pendulum and the membrane. The solid line is the parabolic fit to the experiment data. (b) The potential difference varies with time obviously after the chamber vacuumized, and the drift slows down as time increased. The results for the other membrane are similar.

Figure 3

“Non-null” experiment of the $8{\omega }_d$ component as the attractor’s position changed along the $x$ direction, with the separation between the test and the source masses fixed to $210\mu \mathrm{m}$. The solid points are the measured values, and the shaded belt represents the theoretical calculations, both with the $2\sigma$ error.

Figure 4

The $8{\omega }_d$ measured torques at 210, 230, 295, and $1095\text{−}\mu \mathrm{m}$ separations, respectively. Each solid dot represents a value obtained from a 10-rotation-period data segment, and the squares represent the mean values (shown in brackets) with the $1\sigma$ error of about $1\times {10}^{-17}\mathrm{N}\mathrm{m}$.

Figure 5

Example of the probability density of $\alpha$, considering $\lambda =100\mu \mathrm{m}$. The shaded area represents the region of $\alpha$ with 95% probability. ${\alpha }_e=-0.004$ at the maximum is the best estimate of $\alpha$. The constraint of $|\alpha |<0.027$ is set by the max one of the absolute values of the lower limit $({\alpha }_e-\delta \alpha )=-0.027$ and the upper limit $({\alpha }_e+\delta \alpha )=0.019$. The constraints of $|\alpha |$ for other values of $\lambda$ are calculated similarly.

Figure 6

Constraints on Yukawa violation of the Newtonian $1/r^2$ law. The shaded region is excluded at a 95% confidence level by this work, and the previous experiments from Refs. [10, 11, 12, 13, 14, 22, 23, 24], respectively. Light lines show various theoretical predictions summarized in Refs. [29, 39].