Short-Range Searches for Non-Newtonian Gravity

Michael C. M. Varney, Univ. Colorado, Boulder Michael.Varney@colorado.edu,
Joshua C. Long, Los Alamos Neutron Science Center Collaboration josh.long@lanl.gov

In 1667, Isaac Newton proposed his famous universal law of gravity:

$$F = -\frac{G m_{1}m_{2}}{r^{2}},$$ (1)

where $F$ is the force between test masses $m_{1}$ and $m_{2}$, $r$ is their separation, and $G$ is the gravitational constant. But how universal is this law? Tests of Newtonian gravity and searches for new macroscopic forces have covered length scales from light-years to nanometers, and it has been found that new forces of gravitational strength can be excluded for ranges from 200 microns to nearly a light-year [1,2]. Limits on new forces become poor very rapidly below 100 microns.

During the last 5-7 years there has been a surge of interest in testing Newtonian gravity at sub-millimeter scales, based on many specific theoretical predictions of modifications to gravity in this regime. Most notable are possible signatures of ``large'' extra dimensions which could modify gravity directly below the millimeter range [3]. Additional sub-millimeter effects of gravitational strength and substantially stronger are predicted to arise as a consequence of new particles propagating in the extra dimensions [4]. In older string theory-inspired models with low-energy supersymmetry breaking, massive scalar particles including moduli and dilatons are predicted to mediate new short-range forces [5] Other predictions arise in models attempting to explain the observed smallness of the cosmological constant [6].

These developments have motivated a variety of novel table-top experiments, and there has been substantial progress in improving the limits on non-Newtonian effects over the past three years. Experimental results are usually parameterized with the Yukawa interaction. The potential due to gravity plus an additional Yukawa force is given by:

$$V = -\frac{G m_{1}m_{2}}{r}[1+\alpha\exp(-r/\lambda)],$$ (2)

where $\alpha$ is the strength of a possible new interaction relative to standard gravity and $\lambda$ is the range. The experiments cover a range of about seven orders of magnitude, from a few nanometers to a few centimeters, use a variety of techniques and confront different backgrounds. The authors thought it would be useful to attempt a short summary of the progress and prospects of tests for new short-range forces, covering a broad range of the recent small experiments. For more detailed reviews of various subsets of the experiments, see Refs. [7, 13, 20].

Low Frequency Experiments: The sensitive and linear response of the torsion balance has made it the instrument of choice for laboratory gravity measurements. Low frequency operation makes for low thermal noise, but presents challenges for vibration isolation which is also important for attaining small test mass separations.

1. Eric Adelberger at the University of Washington, along with Blayne Heckel and graduate students Dan Kapner and C. D. Hoyle (now at U. Trento, Italy), operated an exquisitely designed ``missing mass'' torsional pendulum experiment. This experiment provides the limits $\alpha = 10$ at $\lambda = 100$ microns to $\alpha = 10^{-2}$ at $\lambda = 3$ mm, which are the best currently published in that range [2]. The force-sensitive pendulum mass consisted of a 1 mm thick aluminum annulus with an array of 10 equally spaced holes. The source mass was a stack of two copper disks, each a few millimeters thick, with similar arrays of holes and which rotated approximately once every two hours. The source mass torqued the pendulum 10 times per revolution allowing easy discrimination from vibrations associated with the source drive. The source mass and detector pendulum were separated by a 20 micron thick beryllium copper stretched membrane, with a total test mass separation of 197 microns.

Current efforts are underway to allow increased sensitivity and closer test mass separation [2,7]. To facilitate this, a new design utilizing a higher density ring and attractor (copper and molybdenum, respectively) with 22 fold symmetry has been constructed. Noise has been improved by a factor of six over the previous experiment, and a passive ``bounce-mode'' damper has been employed, allowing test mass separation to be reduced by a factor of 2. This new experiment should be able to probe gravitational strength forces down to 60 microns.

2. The best limits in the range from 3 mm ( $\alpha = 10^{-2}$) to 3 cm ( $\alpha = 10^{-4}$) still derive from the null-geometry torsion balance experiment of R. Newman and colleagues at the University of California at Irvine [8]. Torque sensitivity was limited by tilt errors and other important systematics included magnetic and seismic effects. Recently, the Irvine group, in collaboration with P. Boynton of the University of Washington, has constructed a torsion pendulum designed to operate at $\sim$ 2 K in a seismically quiet underground site at Hanford [9]. The low temperature allows operation of the pendulum in the tilt-insensitive frequency mode [10], and for superconducting shielding to reduce magnetic backgrounds. The experiment, when operated as a test of the inverse square law (a precision measurement of $G$ is planned first), will be most sensitive to new forces at a range of about 15 cm and in the thermal noise limit is expected to achieve a sensitivity at least two orders of magnitude greater than previous experiments at that range.

3. Ho Jung Paik along with M. Vol Moody and colleagues at the University of Maryland are prototyping the cryogenic ISLES (Inverse Square Law Experiment in Space) project, to be conducted on the International Space Station [11]. In this experiment, two superconducting magnetically levitated niobium test disks are suspended 100 microns on either side of a radially mounted tantalum source disk with thin superconducting shields in between to suppress electromagnetic couplings. The source is nominally 140 mm in diameter and 2 mm thick. The planar geometry permits concentration of as much mass as possible at the short range to be explored, and represents a nearly null geometry for Newtonian background forces. The differential acceleration of the test masses is measured as the source is driven magnetically into small oscillations along its axis at about 0.2 Hz, using a SQUID readout similar to the instrumentation developed for the Maryland superconducting gravity gradiometer. The projected sensitivity, ranging from $\alpha = 10$ at $\lambda = 5$ microns to $\alpha = 10^{-5}$ at 200 microns, is up to 7 orders of magnitude stronger than current limits in this range. This sensitivity is also sufficient to further constrain (or possibly detect) the axion, a light pseudoscalar proposed as a solution to the strong CP problem of QCD.

The low-g environment of the ISS is essential for the magnetic suspension of the test masses, which in turn provides the extremely soft vibration isolation partly responsible for the impressive sensitivity. A further improvement of up to three orders of magnitude might be possible in a free-flyer version of the experiment. Mechanical suspension will be used in the ground-based prototype currently under construction, reducing the projected sensitivity by up to two orders of magnitude relative to the ISS version.

High-Frequency Experiments: Recent experiments designed to operate in the range of a few hundred to a thousand Hz show promise to operate at the thermal noise limit and to attain extremely small test mass separations.

4. Early this year, the authors and their advisor John Price at the University of Colorado published results from their high frequency experiment, giving the current best limits between 20 and 100 microns [12]. Forces greater than $10^{4}$ times gravity at 20 microns and greater than 10 times gravity at 100 microns were excluded.

The apparatus uses a nominally null planar geometry and operates at room temperature under a vacuum of $\sim 10^{-7}$ torr to reduce acoustic backgrounds. A 35 mm $\times$ 7 mm $\times$ 0.305 mm tungsten ``diving board'' cantilever is driven at the resonant frequency ($\sim$ 1 kHz) of a high-Q ($\sim$ 25500) compound torsional tungsten detector oscillator. The test masses are separated by a 0.06 mm thick gold plated sapphire shield to suppress electrostatic and acoustic backgrounds. The amplitude of the detector mass is read via a capacitive transducer. Measurements were taken at a test mass separation of 108 microns with a test mass overlap area of about 58 mm$^{2}$, and were found to be thermal-noise limited.

Recently, the system has been redesigned to make use of a 10 micron thick gold plated copper stretched membrane as an electrostatic shield. This will allow for mass separations of about 50 microns, with the goal of improving the limits at that range by at least an order of magnitude [13]. Possible plans for the future include a 4.2 K version of the experiment, higher Q tungsten detectors, and improved flatness of the test masses.

5. Aharon Kapitulnik's group at Stanford recently published results from their high frequency cryogenic experiment [14]. This experiment utilized a silicon nitride microcantilever with a 50 micron gold cube mounted on its free end as a high Q detector. A planar source mass, consisting of alternating gold and silicon strips, was driven in the direction transverse to the cantilever mode of the detector. The alternating strip design permitted the source to be driven at a frequency below the cantilever resonance, reducing the burden on vibration isolation. Detector mass amplitude was read via an optical fiber. Operating at 10 K, the system obtained a sensitivity of around $8.9 \times 10^{-17}$ N at a 25 micron source/test mass gap.

A spurious signal, most likely electrostatic in nature, limited the sensitivity of their apparatus. As the phase of this signal was not consistent with a new non-Newtonian force, the results were used to derive limits $\alpha = 10^{9}$ at $\lambda = 3$ microns to $\alpha = 10^{5}$ at $\lambda = 10$ microns, the most sensitive in that range. The experiment was designed to be a ``null'' experiment limited by thermal noise with a predicted sensitivity about an order of magnitude below that of the published results. Re-design of the source mass to remedy the background is in progress.

6. S. Schiller, L. Haiberger and colleagues at the University of Dusseldorf have constructed a room temperature prototype of a high frequency experiment of great potential sensitivity. The detector mass consists of a high purity silicon wafer similar in shape and size to that of the Colorado experiment. The use of silicon is expected to lead to very high Qs, especially in a planned cryogenic version of the experiment. The source mass consists of a variable-density rotor with a harmonic corresponding to the resonance frequency ($\sim$ 5 kHz) of the detector mass. The source and detector mass are mounted vertically and are separated by a conducting plate serving as an electrostatic shield. Detector oscillations are read out via an optical system.

Recent measurements yielded signals about 10 times the detector thermal noise level, corresponding to a sensitivity ranging from about $\alpha = 10^{9}$ at $\lambda = 60$ microns to $\alpha = 5000$ at $\lambda = 5$ mm  [15]. Improvements to the vibration isolation and thermal stability are under way. The cryogenic experiment is anticipated to reach gravitational sensitivity at 50 microns.

Casimir Force Measurements: For test mass separations below about 10 microns, the Casimir effect, a force which arises between conductors due to zero-point fluctuations of the electromagnetic field, becomes significant. While of experimental interest in its own right, the Casimir force presents a dominant background to experimental searches for new effects at very short ranges, and must either be precisely characterized or suppressed (or both) if these experiments are to attain greater sensitivity.

7. Umar Mohideen and his group at the University of California at Riverside have continued to refine their precision measurements of the Casimir force using AFM techniques. In a recent experiment, a 200 micron diameter gold-plated sphere was attached to an AFM cantilever and suspended above a gold-plated sapphire disk [18] Deflection of the cantilever under the influence of the plate-sphere force was monitored optically, for plate-sphere separations ranging from 0 to 400 nm. While the absolute force errors were slightly larger than for previous measurements, the use of denser test masses led to the strongest constraints yet attained in the range considered ( $\alpha = 10^{19}$ at $\lambda = 20$ nm to $\alpha = 10^{14}$ at 100 nm) when the results were compared with theory.

More recently this group has used similar apparatus to measure the lateral Casimir force between a gold-plated sphere and a sinusoidally corrugated gold surface [19]. Good agreement with theory was obtained, though the constraints on new effects are less sensitive. Work continues on third-generation Casimir force experiments with improved precision.

8. V. Mostepanenko's collaboration with Riverside is part of a more general program in which he and his and colleagues have continued to derive constraints on new physics by comparing Casimir force measurements with the most sophisticated theoretical models. Much of their recent work is summarized in a comprehensive review [20]. They have also collaborated with E. Fischbach's group [see below] in deriving constraints in the very short range just above 1 nm  [21]. In this regime, the most sensitive limits ( $\alpha \approx 10^{25}$) come from a recent Casimir force measurement by Ederth using crossed cylinders [22].

9. A group from the INFN, Padova and Pavia, and the University of Padova run an experiment that is an interesting hybrid of high frequency cantilever and direct Casimir measurement. The force-sensitive detector is a 47 micron thick, 2 cm long silicon cantilever with a 50 nm thick chromium plating. It is driven electrically at its lowest order resonance mode of 138 Hz. The source mass is a 5 mm thick chromium-plated silicon block which is brought to within less than a micron of the detector surface. By optically monitoring the detector resonant frequency shifts induced by the static external force gradient, the group was able to obtain a measurement of the Casimir force to a precision of 15%, the first measurement of this effect for the parallel-plate geometry [16]. Limits on new effects derived from this experiment are not quite competitive with the most stringent constraints in the relevant range near 1 micron, but an optimized version of this experiment under construction is expected to improve these limits by at least an order of magnitude [17].

10. Ephraim Fischbach and colleagues at Purdue University including Dennis Krause (also at Wabash College) are pursuing experimental and theoretical programs to control the Casimir background in short-range experiments. Sub-micron measurements of the differences in forces between different isotopes of the same element are underway. These are expected be sensitive to new short-range effects as the Casimir force should be dominated by the electronic properties of the test masses and essentially independent of isotope (iso-electronic effect).

In a recent theoretical study [23], this group has quantified the isotopic dependence of the Casimir force, and estimated the fractional difference in Casimir forces between two isotopes of the same element to be on the order of $10^{-4}$. This is roughly two orders of magnitude below the resolution of recent Casimir force measurements, lending confidence to the prospect that differences observed in iso-electronic experiments will be due to other effects.

An initial experiment (designed primarily to investigate gross systematics and sample fabrication) used an AFM to measure the forces on a silicon nitride cantilever suspended a few nm above a surface consisting of alternating regions of gold and copper [24] (These metals have very similar electronic properties but different densities.) Results from these measurements were used to set limits of $\alpha \approx 10^{27}$ in the range $\lambda$ = 1-2 nm, slightly more sensitive than the previous limits in that range.

More recently this group has reported the first precise measurement of the Casimir Force between dissimilar metals [25]. This experiment used a more sensitive microelectromechanical apparatus, in which a 600 micron diameter gold-plated sphere was suspended above one side of a .25 mm$^{2}$ copper-coated torsional plate. Forces were measured as a function of plate-sphere separations from about 200 to 1200 nm, by observing both the static deflection of the plate and the change in its resonant frequency as it was driven into small oscillations. Comparison of these measurements with a detailed theoretical model leads to the limit of $\alpha \approx 10^{13}$ at $\lambda$ = 200 nm, about a factor of 4 improvement over the previous limit at that range. These results are based on an observed systematic difference between theory and experiment. The group suspects this is based on the imprecise characterization of the optical properties of the metals, and emphasizes the need for better measurements of these properties and better theoretical understanding of the Casimir force for non-ideal objects. Plans to improve the limits by comparing the force on the sphere to two isotopes of the same element are also underway.

In summary, the authors are aware of at least 10 active programs pursuing short range experiments with implications for non-Newtonian gravity. Over the next few years, prospects are good for improving the limits on new effects by several orders of magnitude in the range from nanometers to centimeters. The authors wish to thank all researchers who replied to requests for the latest news.

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