Constraints on Yukawa-type deviations from Newtonian gravity at 20 microns

Recent theories of physics beyond the standard model have predicted deviations from Newtonian gravity at short distances. In order to test these theories, we have a built an apparatus that can measure attonewton-scale forces between gold masses separated by distances on the order of $25\mu \mathrm{m}$. A micromachined silicon cantilever was used as the force sensor, and its displacement was measured with a fiber interferometer. We have used our measurements to set bounds on the magnitude $\alpha$ and length scale $\lambda$ of Yukawa-type deviations from Newtonian gravity; our results presented here yield the best experimental limit in the range of $\lambda =6–20\mu \mathrm{m}$.

Figure 1

Experimental results (solid lines) and theoretical predictions shown in $\alpha –\lambda$ space. The area to the upper right of the experimental lines (from Refs. 9, 14, 15, 16, 31 and our latest results from this paper) shows where Yukawa-type deviations from Newtonian gravity have been excluded. The line labeled “Stanford 1” gives results from 9; the line labeled “Stanford 2” shows the results described in this paper. Dashed lines show roughly the predictions for the dilaton [8] and the first Kaluza-Klein mode of two simply-compactified extra dimensions as described in Ref. 2. Shaded regions to the left of the experimental lines show predictions for moduli and gauge bosons from Ref. 32.

Figure 2

Schematic (not to scale) showing the drive mass below the cantilever that bears the gold test mass. A gold-coated silicon nitride shield membrane separates the masses.

Figure 3

Scanning electron micrograph of the test mass and cantilever used in Cooldown 04. A piece of dust is visible on one corner of the test mass. The cantilever is $50\mu \mathrm{m}$ wide.

Figure 4

Optical micrograph of the polished side of a drive mass showing the meander pattern. In the experiment, the polished side of the drive mass was facing away from the test mass.

Figure 5

Drawing of the bottom half of the probe. The bimorph stage and wafer stage are shown with the wafer stage raised and rotated about the $x$-axis to reveal the geometry of the capacitive sensors. In practice, the two electrodes of a given sensor were separated by less than 1 mm in the $z$-direction. The three sensors were separated by $\sim 4\mathrm{cm}$ in the $x–y$ plane. The pads used for making electrical contact to the capacitive sensors are omitted from this drawing.

Figure 6

The calculated magnitude (top) and phase (bottom) of the predicted third harmonic ac gravitational force between masses. The larger Yukawa force for $\alpha =5$ and $\lambda =34\mu \mathrm{m}$ is shown by the dashed line; the smaller Newtonian force is the solid line. The phase is the same for a Newtonian or a Yukawa force. In this calculation, the drive mass was taken to be $27\mu \mathrm{m}$ underneath the test mass, without tilt. The middle gold bar of the drive mass is centered under the test mass at $y$-position of $0\mu \mathrm{m}$.

Figure 7

Schematic showing the geometry and alignment parameters between the test and drive masses. On the left, the top view shows the $x–y$ plane; the origin, marked with an asterisk, is the center of the drive mass in this plane. With the drive mass at equilibrium, the $x–y$ position of the test mass was defined with respect to this origin. This figure shows the masses to scale; the shield membrane and the ground plane over the drive mass are omitted. On the right, the side view shows (not to scale) the $z–x$ plane. The $z$-separation between masses is the face-to-face vertical distance. The test mass tilt, exaggerated in this figure for the purposes of illustration, is the tilt of the test mass with respect to the cantilever. The angle between the drive mass and the cantilever in this plane was defined as ${\theta }_{xz}$.

Figure 8

Measured thermal noise data without (a) and with (b) feedback are shown in comparison to the Lorentzian curves derived from the measured $k$, $Q$, effective $T$, and $f_0$. The rms amplitude of cantilever motion is plotted as a function of frequency. The offsets added to the Lorentzian curves are judged from the power spectrum. The background in the figure on the right is higher because of the larger bandwidth. With feedback, $Q\sim 10400$ and the spectrum is the result of averaging 30 power spectra from 2 min data records; the effective temperature was $\sim 3.5\mathrm{K}$. Without feedback, $Q\sim 75000$ and the spectrum is an average of 30 records of length 4 min each; the effective temperature was $\sim 30\mathrm{K}$.

Figure 9

The third harmonic force magnitude (a) and phase (b) as a function of $y$-position (relative to the first point) between the magnetic test mass and the drive mass with current on and off. The magnitude here is calculated from the mean of the real and imaginary components: $\sqrt{{\overline{F_R}}^2+{\overline{F_I}}^2}$.

Figure 10

Force magnitude (top) and phase (bottom) vs $y$-position from Cooldown 04. Day 1 data are on the left and Day 2 data are on the right. The black dashed line shows the approximate level of measured thermal noise, which was slightly less than the predicted thermal noise level. Plotted points are the means for the full averaging time (60 min, except for the first measurement, which was 48 min); the error bars show twice the standard (statistical) error on the mean. Lines connecting the measured points are guides to the eye.

Figure 11

Shown are the real and imaginary components of the Fourier transform at the frequency of interest for the second point of the Day 2 data—this is at the $y$-position at which the maximum force was measured for the day. Data here are reported in terms of the rms force, $F_I$ and $F_R$. The small crosses each show the result from one 30 s segment of time-series data. Vertical error bars on two of these points indicate the statistical error, which is taken to be the same on each point and in all directions in this complex plane. The circle shows the mean along the two axes; the error bars on the mean are smaller than the size of the circle.

Figure 12

Correspondence between parameters for the fits of the Monte Carlo results to Day 1 data. In the upper left, the best-fit $\alpha$ values are shown; from the top (larger $\alpha$), the groups of best-fit values correspond to values of $\lambda$ of 4, 6, 10, 18, 34, 66, 130, and $258\mu \mathrm{m}$. Different symbols are used for each value of $\lambda$, with the same symbols used in each plot. Plots show the mean square error (MSE) for the fit and the best-fit parameters $y_0$, ${\theta }_0$, $R_0$, and $I_0$.

Figure 13

The measured force as $\overline{F_R}$, $\overline{F_I}$, magnitude, and phase for Day 1 as compared to a typical best-fit Monte Carlo result, for $\lambda =18\mu \mathrm{m}$. The force magnitude $F$ is $\sqrt{{\overline{F_R}}^2+{\overline{F_I}}^2}$. Error bars show the statistical error on the mean.

Figure 14

The same information as in Fig. 13, except for Day 2 data.

Figure 15

An example histogram of the fully-varied best-fit $\alpha$ results for $\lambda =18\mu \mathrm{m}$ and Day 1 data. The scaled $\alpha$ is $\alpha /\overline{\alpha }$ and the relative frequency is scaled by the total number (800 000) of varied best-fit $\alpha$ values.

Figure 16

The parameter space of $\alpha –\lambda$ of Yukawa-type deviations from Newtonian gravity. Results from this paper (Stanford 2) are shown in comparison to previous results from this experiment [9] (Stanford 1). Also included are theoretical predictions (dashed line) and an experimental result (solid line) as shown Fig. 1.