Tests of the Gravitational Inverse-Square Law below the Dark-Energy Length ScaleD. J. Kapner,*T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle, and H. E. SwansonCenter for Experimental Nuclear Physics and Astrophysics, Box 354290, University of Washington,Seattle, Washington 98195-4290, USA(Received 16 October 2006; published 8 January 2007)We conducted three torsion-balance experiments to test the gravitational inverse-square law atseparations between 9.53 mm and 55 m, probing distances less than the dark-energy length scale d@c= d4p 85 m. We find with 95% confidence that the inverse-square law holds (jj 1)downtoalength scale   56 m and that an extra dimension must have a size R 44 m.DOI: 10.1103/PhysRevLett.98.021101 PACS numbers: 04.80.Cc, 95.36.+xRecent cosmological observations [1–3] have shownthat 70% of all of the mass and energy of the Universe isa mysterious ‘‘dark energy’’ with a density d3:8 keV=cm3and a repulsive gravitational effect. Thisdark-energy density corresponds to a distance d@c= d4p 85 m that may represent a fundamentallength scale of gravity [4,5]. Although quantum-mechanical vacuum energy should have a repulsive gravi-tational effect, the observed dis between 1060and10120times smaller than the vacuum energy density com-puted according to the standard laws of quantum mechan-ics. Sundrum [6] has suggested that this huge discrepancy(the ‘‘cosmological constant problem’’) could be resolvedif the graviton were a ‘‘fat’’ object with a size comparableto dthat would prevent it from ‘‘seeing’’ the short-distance physics that dominates the vacuum energy. Hisscenario implies that the gravitational force would weakenfor objects separated by distances s & d. Dvali,Gabadaze, and Senjanovic´[7] argue that a similar weak-ening of gravity could occur if there are extra time dimen-sions. In their scenario, the standard model particles arelocalized in ‘‘our’’ time, while the gravitons propagate inthe extra time dimension(s) as well. Other scenarios predictthe opposite behavior: The extra space dimensions ofM theory would cause the gravitational force to getstronger for s & R, where R is the size of the largestcompactified dimension [8]. These considerations, plusothers involving new forces from the exchange of proposedscalar or vector particles [9], motivated the tests of thegravitational inverse-square law we report in this Letter.Our tests were made with a substantially upgraded ver-sion of the ‘‘missing mass’’ torsion-balance instrumentused in our previous inverse-square-law tests [10,11].The instrument used in this work [12], shown in Fig. 1,consisted of a torsion-pendulum detector suspended by athin  80-cm-long tungsten fiber above an attractor thatwas rotated with a uniform angular velocity ! by a geared-down stepper motor. The detector’s 42 test bodies were4.767-mm-diameter cylindrical holes machined into a0.997-mm-thick molybdenum detector ring. The hole cen-ters were arrayed in two circles, each of which had 21-foldazimuthal symmetry. The attractor had a similar 21-foldazimuthal symmetry and consisted of a 0.997-mm-thickmolybdenum disk with 42 3.178-mm-diameter holesmounted atop a thicker tantalum disk containing21 6.352-mm-diameter holes. The gravitational interactionbetween the missing masses of the detector and attractorholes applied a torque on the detector that oscillated21 times for each revolution of the attractor, giving torquesat 21!, 42!, 63!, etc., that we measured by monitoringthe pendulum twist with an autocollimator system. Theholes in the lower attractor ring were displaced azimuthallyby 360=42 degrees and were designed to nearly cancel the21! torque if the inverse-square law holds. On the otherhand, an interaction that violated the inverse-square law,which we parametrize as a single Yukawa FIG. 1. Scale drawing of our detector and attractor. The 3 smallspheres near the top of the detector were used for a continuousgravitational calibration of the torque scale. Four rectangularplane mirrors below the spheres are part of the twist-monitoringsystem. The detector’s electrical shield is not shown.PRL 98, 021101 (2007)PHYSICAL REVIEW LETTERSweek ending12 JANUARY 20070031-9007=07=98(2)=021101(4) 021101-1 © 2007 The American Physical SocietyVrGm1m2r1   expr=; (1)would not be appreciably canceled if  is less than the1 mm thickness of the upper attractor disk. We minimizedelectromagnetic torques by coating the entire detector withgold and surrounding it by a gold-coated shield consistingof a tightly stretched, 10-m-thick, beryllium-coppermembrane between the detector and attractor plus a copperhousing that had small holes for the suspension fiber andthe autocollimator beam. The entire system was under avacuum of  106torr in a temperature-controlled andmagnetically shielded environment. The noise in ourtorque measurements was generally close to the thermalvalue expected from the finite quality factor, Q  3000,ofour torsion oscillator but increased noticeably for detector-membrane separations below 100 m (see Fig. 2). Acontinuous, absolute calibration of the torque scale wasprovided by the gravitational octupole interaction between3 small spheres mounted on the detector and 3 largerspheres mounted outside the vacuum vessel on a turntablethat rotated the spheres about the fiber axis at a steady rate!c, providing a calibration signal at 3!c.The detector-attractor separation~ x; y; s, where xand y are the horizontal displacements between the centersof the detector and attractor and s is the vertical separationbetween the bottom of the detector ring and the top of theupper attractor disk, was determined using capacitanceplus micrometer techniques for s and gravitational plusmicrometer techniques for x and y. Detector twist datawere taken at x and y close to zero (except for off-centerruns used to find the x; y center) and separations 55 m s 9:53 mm. The key parameters expn expnof ourinstrument (the masses removed in machining the holes,the hole radii, thicknesses, and locations, etc.) were deter-mined precisely using an electronic balance or acoordinate-measuring machine (CMM). The CMM read-ings were corrected for surface roughness by scanning therelevant surfaces with