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Tests of the Gravitational Inverse-Square Law below the Dark-Energy Length Scale

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D.J. Kapner, T.S. Cook, E.G. Adelberger, J.H. Gundlach, B.R. Heckel, C.D. Hoyle, H.E. Swanson

High Energy Physics - Phenomenology · Submitted 14 Nov 2006 · 17 references

Pith's verdict: A genuine hundredfold improvement in short-range gravity bounds, with one ambiguity — whether the detector-bow correction is measured or fitted — that a careful reader should press on.

T0 · AI review

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arXiv IDhep-ph/0611184 (version 1)
Pith IDPUW2NPT6, Pith's permanent record of this paper and its review
CitationB.R. Heckel, C.D. Hoyle, D.J. Kapner, E.G. Adelberger, H.E. Swanson, J.H. Gundlach, T.S. Cook. "Tests of the Gravitational Inverse-Square Law below the Dark-Energy Length Scale." arXiv:hep-ph/0611184 (2006). https://pith.science/paper/PUW2NPT6
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The paper

The paper claims

A torsion balance finds no deviation from Newtonian gravity at separations down to 55 micrometers, so any gravitational-strength Yukawa force must have a range shorter than 56 micrometers and any single compact extra dimension must be no 44

The reading

The paper asks whether gravity's $1/r^2$ law survives at distances smaller than the dark-energy length scale, $85~$µm, the scale set by the observed cosmic acceleration. A short-range failure of the inverse-square law is what several proposals for the cosmological constant problem predict, and what theories with extra spatial dimensions require in the opposite direction. Using an upgraded torsion balance whose attractor and detector are patterned with 21-fold-symmetric holes, the authors measure gravitational torques at separations from 9.53 mm down to 55 µm and fit the data with a Newtonian model plus one Yukawa term. They report no significant violation, and at 95% confidence bound the range of any gravitational-strength Yukawa interaction to at most 56 µm and the size of a single compact extra dimension to at most 44 µm. The interest is that a laboratory experiment now reaches the distance scale cosmologists associate with dark energy, closing a window that had been open by a factor of up to 100.

What carries the argument

The instrument is a ``missing mass'' torsion balance: a molybdenum detector ring with 42 machined holes in two 21-fold-symmetric circles hangs above an attractor built from a thin molybdenum disc with 42 holes atop a thicker tantalum disc with 21 holes, and the gravitational coupling between the holes produces a torque that oscillates 21 times per attractor revolution. The lower attractor holes are offset by $360/42$ degrees so that, if the inverse-square law holds, the $21ω$ torque nearly cancels, while a short-range Yukawa interaction with $λ$ below the 1 mm disc thickness is not cancelled. Running with the lower disc at two thicknesses (differing by 0.140 mm) makes a genuine short-range force flip sign between experiments while a systematic error would not, and a separate three-sphere gravitational octupole driven by a turntable provides a continuous absolute calibration of the torque scale.

Extended reading notes

Core claim

The authors claim that Newtonian gravity, augmented by nothing, fits their measured torques across three separate experiments spanning 9.53 mm to 55 µm. Parameterizing a deviation as $V(r) = -G m_1 m_2 [1 + α e^{-r/λ}]/r$, they find at 95% confidence that $|α| ≤ 1$ requires $λ ≤ 56~$µm, so no gravitational-strength Yukawa force with a range above that value survives. Translated into extra dimensions, the same data exclude a single compact dimension larger than $R ≤ 44~$µm, and for the two-large-extra-dimension scenario they require a $2σ$ lower limit on the unification mass $M_* ≥ 3.2$ TeV/$c^2$. The combined Newtonian fit gives $χ^2 = 407$ for 421 degrees of freedom, and adding a Yukawa term improves the fit by only $Δχ^2 = 3.5$, at $α = -0.0037$, $λ = 2$ mm, with no $2σ$ effect at any range.

Load-bearing premise

The 95% confidence bound depends on the modeled bow of the detector ring, which the authors invoke to explain the marginally significant excess attraction seen at the smallest separations in Experiments I and III.

Editorial extensions

If this is right

  • Any gravitational-strength Yukawa force mediating a new long-range interaction must have a range below 56 µm, which removes a wide band of parameter space for light scalar or vector mediators coupled to mass.
  • A single compact extra dimension must be smaller than 44 µm, and in the two-large-extra-dimension picture the unification mass must exceed 3.2 TeV/$c^2$ at $2σ$.
  • Scenarios in which gravity weakens for separations below the dark-energy length scale, such as a graviton with a size near 85 µm, are pushed to sizes below the experiment's reach.
  • The techniques demonstrated here — patterned nulled torques, multiple attractor thicknesses, and in-situ autocollimator linearity mapping — set a template for pushing torsion-balance tests to smaller separations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The bound is set by the smallest separation achieved, so the natural next step is engineering: a flatter, more rigid detector and a thinner membrane would move the 56 µm limit toward the 20 µm scale where thermal and electrostatic backgrounds begin to dominate.
  • Because the same data can be recast as a bound on any power-law or finite-range modification once the geometry integral is redone, the exclusion is broader than the single Yukawa curve shown; the authors say as much by deferring other functional forms to a separate publication.
  • The persistent excess torque noise at the smallest separations, roughly four times thermal, is the practical ceiling on this generation of the experiment; if it is instrumental rather than statistical, better vibration and patch-potential control would tighten the limit more than extra data at large separation.
  • A laboratory bound at the dark-energy length scale invites the reverse question: if no new force appears exactly where cosmic acceleration suggests one might, the coincidence between $\lambda_d$ and the reachable laboratory range becomes a constraint on model building rather than a hint.
Original abstract

We conducted three torsion-balance experiments to test the gravitational inverse-square law at separations between 9.53 mm and 55 micrometers, probing distances less than the dark-energy length scale $\lambda_{\rm d}=\sqrt[4]{\hbar c/\rho_{\rm d}}\approx 85 \mu$m. We find with 95% confidence that the inverse-square law holds ($|\alpha| \leq 1$) down to a length scale $\lambda = 56 \mu$m and that an extra dimension must have a size $R \leq 44 \mu$m.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,489 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.58 · human participants 0.02 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We conducted three torsion-balance experiments to test the gravitational inverse-square law at separations between 9.53 mm and 55 micrometers, probing distances less than the dark-energy length scale $\lambda_{\rm d}=\sqrt[4]{\hbar c/\rho_{\rm d}}\approx 85 \mu$m. We find with 95% confidence that the inverse-square law holds ($|\alpha| \leq 1$) down to a length scale $\lambda = 56 \mu$m and that an extra dimension must have a size $R \leq 44 \mu$m.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We conducted three torsion-balance experiments to test the gravitational inverse-square law at separations between 9.53 mm and 55 micrometers, probing distances less than the dark-energy length scale $\lambda_{\rm d}=\sqrt[4]{\hbar c/\rho_{\rm d}}\approx 85 \mu$m. method 0.59 0.97 0.06 asserted 0.92
s1 We find with 95% confidence that the inverse-square law holds ($|\alpha| \leq 1$) down to a length scale $\lambda = 56 \mu$m and that an extra dimension must have a size $R \leq 44 \mu$m. result 0.99 0.97 0.93 asserted 1.00

What kind of paper the classifier reports this is: empirical study 1.00 · evidence from observation or experiment 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Scale drawing of our detector and attractor. The 3
Figure 2. FIG. 2: Fourier transform of the raw twist signal in Experi
Figure 3. FIG. 3: [Color online] Experiment I torques as functions of
Figure 5. FIG. 5: [Color online] Experiment III torques. Notation is t
Figure 6. FIG. 6: [Color online] Constraints on Yukawa violations of

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/0611184 by the authors.

References
How later papers used it

259 later papers cite it

Evidence: substantial. 258 of the citing papers share no author with it, and 74 of those carry a confident label from Pith's citation classifier.

Pith read 38 of the 353 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its data or values 19 of 65 papers with this label
How the literature uses it

205 arXiv papers cite this in 288 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 188 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 1 paper
Uses its data or values 60 papers
Compares with it 6 papers
Cites it as support 2 papers
Cites it as background 17 papers

Discussion

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