New Test of the Gravitational $1/{r}^{2}$ Law at Separations down to $52\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$

New Test of the Gravitational $1 / r^{2}$ Law at Separations down to $52 μ m$

J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer, and B. R. Heckel

Phys. Rev. Lett. 124, 101101 – Published 10 March 2020

Abstract

We tested the gravitational $1 / r^{2}$ law using a stationary torsion-balance detector and a rotating attractor containing test bodies with both 18-fold and 120-fold azimuthal symmetries that simultaneously tests the $1 / r^{2}$ law at two different length scales. We took data at detector-attractor separations between $52 μ m$ and 3.0 mm. Newtonian gravity gave an excellent fit to our data, limiting with 95% confidence any gravitational-strength Yukawa interactions to ranges $< 38.6 μ m$ .

Received 3 December 2019
Accepted 7 February 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.101101

Physics Subject Headings (PhySH)

Experimental studies of gravity Gravitation

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

J. G. Lee , E. G. Adelberger^*, T. S. Cook ^†, S. M. Fleischer ^‡, and B. R. Heckel

Center for Experimental Nuclear Physics and Astrophysics, Box 354290, University of Washington, Seattle, Washington 98195-4290 USA

^*eadelberger@gmail.com
^†Present Address: Micro Encoder Inc., Kirkland, Washington 98034, USA.
^‡Present Address: Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195, USA.

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Vol. 124, Iss. 10 — 13 March 2020

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Figure 1
Top left: detector and rotating attractor rendered with their separation much larger than actually used. An electrostatic shield that surrounds the detector and isolates it from the attractor is not shown. Top right: photos of glued generation 2 test body before and after gold coating. The hole pattern diameter is 52 mm. Bottom plot: predicted Newtonian torques (solid lines). The dashed and dotted lines are $α = 1$ Yukawa torques with $λ = 70$ and $30 μ m$ , respectively.
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Figure 2
Generation 2 data. Top plot: absolute calibration of the torque scale. Three 1.137 kg spheres on an external calibration turntable rotating at $ω_{c}$ applied a $3 ω_{c}$ gravitational torque on three 0.4816 g spheres on the detector. The detector and calibration spheres were equally spaced on 16.48 mm and 19.05 cm radius circles, respectively. ( $ω_{c}$ was chosen to put the $3 ω_{c}$ signal’s frequency close to those of the $120 ω$ signals in our science runs.) The result $[(2.137 \pm 0.009) fN m$ ] is based on the nominal autocollimator calibration. The expected gravitational torque is $(2.112 \pm 0.005) fN m$ where the error arises from uncertainties in the positions and masses of the spheres. The expected/nominal ratio $(γ = 0.988 \pm 0.005)$ provided an absolute calibration of the autocollimator scale. Calibration runs were taken over a period spanning 85 days. Bottom plot: power spectral density of the torque signal at $s = 72 μ m$ . Smooth lines show the thermal (and autocollimator) noise assuming $Q = 1000$ . The $54 ω$ peak is the third harmonic of the $n = 18$ signal. For this run, the 18 and $120 ω$ signals were set to $1 / 2$ and $10 / 3$ times the free-oscillation frequency $ω_{0}$ . The $36 ω$ peak occurs at $ω_{0}$ and is a residue of the free oscillation; it is not a second harmonic torque of the $n = 18$ pattern as all even harmonics vanish in our geometry.
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Figure 3
Generation 2 data. Top plot: horizontal centering of the detector on the attractor using the gravitational $120 ω$ torque. The center occurs at $x_{0} = (- 102 \pm 2) μ m$ , $y_{0} = (- 2121 \pm 2) μ m$ . Middle and bottom plots: the vertical separation $s$ was determined by combining two measurements of electrical capacitance plus the thicknesses of the foil and glue films on the test body faces. When error bars are not in shown this and later figures, the uncertainties are smaller than the symbols.
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Figure 4
Top plot: effect of an applied vertical magnetic field $B_{z}$ at $s = 72 μ m$ . Points on the solid and dashed lines were taken with the outermost magnetic shield removed and in place, respectively. The lines are parabolic fits. The field at the detector vanishes at $B_{z} = 65 μ T$ . The horizonal green band shows the measured $s = 72 μ m$ torque in our science data. The $(0.0165 \pm 0.0054) fN m$ difference between the $B_{z} = 0$ and $B_{z} = 65 μ T$ torques is the magnetic contribution to the torque. Bottom plot: Systematic effect of a $B_{z}$ field as a function of $s$ . Points show the difference between torques at a strong, applied field ( $B_{z} = - 250 μ T$ : outermost shield removed) and a nulled field ( $+ 65 μ T$ : all shields in place). The smooth curves show our Fourier-Bessel calculations of the spin-spin interaction between the induced magnetizations in the detector and attractor test bodies; a single normalization reproduces the 18, 54, and $120 ω$ effects.
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Figure 5
Top plot: 18, 54, and $120 ω$ torques corrected for the magnetic systematic. Unless shown otherwise, uncertainties are smaller than the plot symbols. Data points with essentially the same $s$ are combined for display purposes only. The Newtonian fit is shown and has $P = 0.654$ . Adding a Yukawa term did not improve the fit appreciably. Bottom plot: corresponding 95% confidence upper limits on $| α |$ from this and previous works [5, 10, 15, 21, 22, 23, 24, 25, 26].
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Physical Review Letters

New Test of the Gravitational 1/r2 Law at Separations down to 52 μm

J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer, and B. R. Heckel

Phys. Rev. Lett. 124, 101101 – Published 10 March 2020

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New Test of the Gravitational $1 / r^{2}$ Law at Separations down to $52 μ m$