Constraints on spatially oscillating sub-mm forces from the Stanford Optically Levitated Microsphere Experiment data

A recent analysis by one of the authors [L. Perivolaropoulos, Phys. Rev. D 95, 084050 (2017)] has indicated the presence of a $2\sigma$ signal of spatially oscillating new force residuals in the torsion balance data of the Washington experiment. We extend that study and analyze the data of the Stanford Optically Levitated Microsphere Experiment (SOLME) [A. D. Rider et al., Phys. Rev. Lett. 117, 101101 (2016)] (kindly provided by A. D. Rider et al.) searching for sub-mm spatially oscillating new force signals. We find a statistically significant oscillating signal for a force residual of the form $F(z)=\alpha \cos (\frac{2\pi }{\lambda }z+c)$ where $z$ is the distance between the macroscopic interacting masses (levitated microsphere and cantilever). The best fit parameter values are $\alpha =(1.1\pm 0.4)\times {10}^{-17}N$, $\lambda =(35.2\pm 0.6)\mu \mathrm{m}$. Monte Carlo simulation of the SOLME data under the assumption of zero force residuals has indicated that the statistical significance of this signal is at about $2\sigma$ level. The improvement of the ${\chi }^2$ fit compared to the null hypothesis (zero residual force) corresponds to $\mathrm{\Delta }{\chi }^2=13.1$. There are indications that this previously unnoticed signal is indeed in the data but is most probably induced by a systematic effect caused by diffraction of non-Gaussian tails of the laser beam. Thus the amplitude of this detected signal can only be useful as an upper bound to the amplitude of new spatially oscillating forces on sub-mm scales. In the context of gravitational origin of the signal emerging from a fundamental modification of the Newtonian potential of the form $V_{\mathrm{eff}}(r)=-G\frac{M}{r}(1+{\alpha }_O\cos (\frac{2\pi }{\lambda }r+\theta ))\equiv V_N(r)+V_{\mathrm{osc}}(r)$, we evaluate the source integral of the oscillating macroscopically induced force. If the origin of the SOLME oscillating signal is systematic, the parameter ${\alpha }_O$ is bounded as ${\alpha }_O<{10}^7$ for $\lambda \simeq 35\mu \mathrm{m}$. Thus, the SOLME data cannot provide useful constraints on the modified gravity parameter ${\alpha }_O$. However, the constraints on the general phenomenological parameter $\alpha$ ($\alpha <0.3\times {10}^{-17}N$ at $2\sigma$) can be useful in constraining other fifth force models related to dark energy (chameleon oscillating potentials etc.).

Private communication with the authors of [2].

Figure 1

The value of the minimized ${\chi }^2$ as a function of the wavelength $\lambda$ for the full data set (96 points). The red straight line corresponds zero residual force $dF=0$. The depth of the minimum is $\delta {\chi }^2=13.1$.

Figure 2

The $1\sigma$ and $2\sigma$ contours in the parameter space $(\alpha ,\lambda )$ for the oscillating parametrization with $c=7\pi /4$. For the combined data set (96 data points) there is a well-defined high quality fit at $(\alpha ,\lambda )=(0.011,35.2\mu \mathrm{m})$ corresponding to a wavelength $\lambda =35.2\mu \mathrm{m}$. This best fit is about $3\sigma$ away from the zero force residual value $\alpha =0$.

Figure 3

The residual force SOLME data with error bars along with the best fit oscillating parametrization (thin red line) for the full data set. The best fit harmonic parametrization has spatial wavelength $\lambda =35.2\mu \mathrm{m}$.

Figure 4

The value of the minimized difference $\delta {\chi }^2={\chi }_{\text{oscillating}}^2-{\chi }_{\alpha =0}^2$ as a function of the spatial wavelength $\lambda$ for the experimental data and a random Monte Carlo data set simulating the SOLME data under the assumption of zero residual force and gaussian errors. The depth of the $\delta {\chi }^2$ deepest minimum is significantly larger when the real data are fit to the oscillating parametrization.

Figure 5

The maximum $\delta {\chi }^2$ depth in the range $\lambda \in [10,100]$ for 100 Monte Carlo data sets assuming zero residual force (red dots). The horizontal line corresponds to the maximum $\delta {\chi }^2$ depth for the actual SOLME data set.

Figure 6

The cantilever approximated as a cylinder and a point mass $m$ at distance $z$ from its surface.

Figure 7

The minimized ${\chi }^2$ using the SOLME data as a function of the parameter $\lambda$ of the Yukawa force ansatz including the effects of the source integral. The improvement of the fit is marginal despite the additional two parameters ${\alpha }_Y$, $\lambda$.

Figure 8

The best fit form of the source integral for the Yukawa residual force is practically indistinguishable from the zero force residual.

Figure 9

Comparison between the source integral signal for oscillating force with the naive constant amplitude cosine oscillator. On scales $z$ larger than the disk radius ($R=40\mu \mathrm{m}$) they both behave like harmonic functions with very similar wavelength,while on small scales the signal is not periodic. Also, we have plot the Newtonian ansatz (blue line), the Yukawa ansatz (gray line) and a power law force with $n=1.5$ (magenta line). For the oscillating source integrals we have set ${\alpha }_{O9}=1$ which implies ${\alpha }_O={10}^9$.

Figure 10

Left panel: The ${\chi }^2$ value as a function of the wavelength $\lambda$ for plain harmonic ansatz. Right panel: The ${\chi }^2$ value for the source integral oscillating ansatz. In both cases the fit improvement to the data is significant compared to a null residual fit. The difference in ${\chi }^2$ is more than 13 units and the minimum appears in almost the same wavelength (about $35\mu \mathrm{m}$).

Figure 11

Left panel: The best fit plain harmonic ansatz with $\lambda =35\mu \mathrm{m}$. Right panel: The best fit source integral oscillating ansatz with $\lambda =33\mu \mathrm{m}$. As we see, in both cases the waveform is practically the same even though the amplitude for the best fit source integral decreases slowly with distance.

Figure 12

The best fit source integrals for cylindrical (3.23) and orthogonal (3.24) cantilever along with the best fit plain harmonic force residual ansatz (2.2).The three best fit parametrizations are very similar leading to practically the same quality of fit.