This section includes limits on the size of extra dimensions from deviations in the Newtonian (1/$\mathit r{}^{2}$) gravitational force law at short distances. Deviations are parametrized by a gravitational potential of the form $\mathit V~=~−(\mathit G$ $\mathit m$ $\mathit m'/\mathit r$) [1 $+$ $\alpha $ exp($−\mathit r/R$)]. For $\delta $ toroidal extra dimensions of equal size, $\alpha $ = 8$\delta $/3. Quoted bounds are for $\delta $ = 2 unless otherwise noted.

VALUE ($\mu {\mathrm {m}}$) CL% DOCUMENT ID TECN  COMMENT • • We do not use the following data for averages, fits, limits, etc. • • 1 BLAKEMORE 2021 Optical levitation 2 HEACOCK 2021 Neutron scattering 3 LEE 2020 Torsion pendulum $<37$ 95 4 TAN 2020 A Torsion pendulum 5 BERGE 2018 MICR Space accelerometer 6 FAYET 2018 A MICR Space accelerometer 7 KLIMCHITSKAYA 2017 A Torsion oscillator 8 XU 2013 Nuclei properties 9 BEZERRA 2011 Torsion oscillator 10 SUSHKOV 2011 Torsion pendulum 11 BEZERRA 2010 Microcantilever 12 MASUDA 2009 Torsion pendulum 13 GERACI 2008 Microcantilever 14 TRENKEL 2008 Newton's constant 15 DECCA 2007 A Torsion oscillator $<37$ 95 16 KAPNER 2007 Torsion pendulum $<47$ 95 17 TU 2007 Torsion pendulum 18 SMULLIN 2005 Microcantilever $<130$ 95 19 HOYLE 2004 Torsion pendulum 20 CHIAVERINI 2003 Microcantilever ${ {}\lesssim{} }\text{ 200}$ 95 21 LONG 2003 Microcantilever $<190$ 95 22 HOYLE 2001 Torsion pendulum 23 HOSKINS 1985 Torsion pendulum 1  BLAKEMORE 2021 obtain constraints on non-Newtonian forces with strengths $\vert \alpha \vert $ ${ {}\gtrsim{} }$ $10^{8}$ and length scales $\mathit R$ $>$ 10 ${{\mathit \mu}}$m. See their Fig. 4 for more details including comparison with previous searches. 2  HEACOCK 2021 obtain constraints on non-Newtonian forces with strengths $10^{18}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert { {}\lesssim{} }$ $10^{25}$ and length scales $\mathit R$ $≅$ $0.02 - 10$ nm. See their Figure 3 for more details. This improves the results of HADDOCK 2018 . These constraints do not place limits on the size of extra flat dimensions. 3  LEE 2020 search for new forces probing a range of $\vert \alpha \vert $ $≅$ and length scales $\mathit R$ $≅$ $7 - 90$ $\mu $m. For $\delta $ = 1 the bound on $\mathit R$ is 30 $\mu $m. See their Fig. 5 for details on the bound. 4  TAN 2020A search for new forces probing a range of $\vert \alpha \vert $ $≅$ $4 \times 10^{-3} - 100$ and length scales $\mathit R$ $≅$ $40 - 350$ $\mu $m. See their Fig. 6 for details on the bound. 5  BERGE 2018 uses results from the MICROSCOPE experiment to obtain constraints on non-Newtonian forces with strengths $10^{-11}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert { {}\lesssim{} }$ $10^{-7}$ and length scales $\mathit R$ ${ {}\gtrsim{} }10^{5}$ m. See their Figure 1 for more details. These constraints do not place limits on the size of extra flat dimensions. 6  FAYET 2018A uses results from the MICROSCOPE experiment to obtain constraints on an EP-violating force possibly arising from a new U(1) gauge boson. For $\mathit R$ ${ {}\gtrsim{} }10^{7}$ m the limits are $\vert {{\mathit \alpha}}\vert $ ${ {}\lesssim{} }$ a few $10^{-13}$ to a few $10^{-11}$ depending on the coupling, corresponding to $\vert {{\mathit \epsilon}}\vert $ ${ {}\lesssim{} }$ $10^{-24}$ for the coupling of the new spin-1 or spin-0 mediator. These constraints do not place limits on the size of extra flat dimensions. This extends the results of FAYET 2018 . 7  KLIMCHITSKAYA 2017A uses an experiment that measures the difference of Casimir forces to obtain bounds on non-Newtonian forces with strengths $\vert \alpha \vert $ $≅$ $10^{5} - 10^{17}$ and length scales $\mathit R$ = $0.03 - 10$ $\mu $m. See their Fig. 3. These constraints do not place limits on the size of extra flat dimensions. 8  XU 2013 obtain constraints on non-Newtonian forces with strengths $\vert \alpha \vert $ $≅$ $10^{34} - 10^{36}$ and length scales $\mathit R$ $≅$ $1 - 10$ fm. See their Fig. 4 for more details. These constraints do not place limits on the size of extra flat dimensions. 9  BEZERRA 2011 obtain constraints on non-Newtonian forces with strengths $10^{11}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert { {}\lesssim{} }$ $10^{18}$ and length scales $\mathit R$ = $30 - 1260$ nm. See their Fig. 2 for more details. These constraints do not place limits on the size of extra flat dimensions. 10  SUSHKOV 2011 obtain improved limits on non-Newtonian forces with strengths $10^{7}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert $ ${ {}\lesssim{} }$ $10^{11}$ and length scales 0.4 ${{\mathit \mu}}$m $<$ ${{\mathit R}}$ $<$ 4 ${{\mathit \mu}}$m (95$\%$ CL). See their Fig. 2. These bounds do not place limits on the size of extra flat dimensions. However, a model dependent bound of ${{\mathit M}_{{*}}}$ $>$ 70 TeV is obtained assuming gauge bosons that couple to baryon number also propagate in (4 + ${{\mathit \delta}}$) dimensions. 11  BEZERRA 2010 obtain improved constraints on non-Newtonian forces with strengths $10^{19}{ {}\lesssim{} }$ $\vert \alpha \vert { {}\lesssim{} }$ $10^{29}$ and length scales $\mathit R$ = $1.6 - 14$ nm (95$\%$ CL). See their Fig.$~$1. This bound does not place limits on the size of extra flat dimensions. 12  MASUDA 2009 obtain improved constraints on non-Newtonian forces with strengths $10^{9}{ {}\lesssim{} }\vert \alpha \vert { {}\lesssim{} }10^{11}$ and length scales $\mathit R$ = $1.0 - 2.9$ $\mu $m (95$\%$ CL). See their Fig.$~$3. This bound does not place limits on the size of extra flat dimensions. 13  GERACI 2008 obtain improved constraints on non-Newtonian forces with strengths $\vert \alpha \vert $ $>$ 14,000 and length scales $\mathit R$ = $5 - 15$ $\mu {\mathrm {m}}$. See their Fig. 9. This bound does not place limits on the size of extra flat dimensions. 14  TRENKEL 2008 uses two independent measurements of Newton's constant $\mathit G$ to constrain new forces with strength $\vert \alpha \vert ≅10^{-4}$ and length scales $\mathit R$ = $0.02 - 1$ m. See their Fig. 1. This bound does not place limits on the size of extra flat dimensions. 15  DECCA 2007A search for new forces and obtain bounds in the region with strengths $\vert \alpha \vert $ $≅$ $10^{13} - 10^{18}$ and length scales $\mathit R$ = $20 - 86$ nm. See their Fig. 6. This bound does not place limits on the size of extra flat dimensions. 16  KAPNER 2007 search for new forces, probing a range of $\vert \alpha \vert $ $≅$ $10^{-3} - 10^{5}$ and length scales $\mathit R$ $≅$ $10 - 1000$ $\mu $m. For $\delta $ = 1 the bound on $\mathit R$ is 44 $\mu $m. For $\delta $ = 2, the bound is expressed in terms of ${{\mathit M}_{{*}}}$, here translated to a bound on the radius. See their Fig. 6 for details on the bound. 17  TU 2007 search for new forces probing a range of $\vert \alpha \vert $ $≅$ and length scales $\mathit R$ $≅$ $20 - 1000$ $\mu $m. For $\delta $ = 1 the bound on $\mathit R$ is 53 $\mu $m. See their Fig. 3 for details on the bound. 18  SMULLIN 2005 search for new forces, and obtain bounds in the region with strengths $\alpha $ $\simeq{}$ $10^{3} - 10^{8}$ and length scales ${{\mathit R}}$ = $6 - 20$ ${{\mathit \mu}}$m. See their Figs.$~$1 and 16 for details on the bound. This work does not place limits on the size of extra flat dimensions. 19  HOYLE 2004 search for new forces, probing $\alpha $ down to $E-2$ and distances down to 10$\mu $m. Quoted bound on $\mathit R$ is for $\delta $ = 2. For $\delta $ = 1, bound goes to 160 $\mu $m. See their Fig. 34 for details on the bound. 20  CHIAVERINI 2003 search for new forces, probing $\alpha $ above $10^{4}$ and $\lambda $ down to 3$\mu $m, finding no signal. See their Fig.$~$4 for details on the bound. This bound does not place limits on the size of extra flat dimensions. 21  LONG 2003 search for new forces, probing $\alpha $ down to 3, and distances down to about 10$\mu $m. See their Fig.$~$4 for details on the bound. 22  HOYLE 2001 search for new forces, probing $\alpha $ down to $E-2$ and distances down to 20$\mu $m. See their Fig.$~$4 for details on the bound. The quoted bound is for $\alpha $ ${}\geq{}$ 3. 23  HOSKINS 1985 search for new forces, probing distances down to 4$~$mm. See their Fig.$~$13 for details on the bound. This bound does not place limits on the size of extra flat dimensions.

References:

BLAKEMORE 2021 PR D104 L061101 Search for non-Newtonian interactions at micrometer scale with a levitated test mass HEACOCK 2021 SCI 373 1239 LEE 2020 PRL 124 101101 New Test of the Gravitational $1/r^2$ Law at Separations down to 52 $\mu$m TAN 2020A PRL 124 051301 Improvement for Testing the Gravitational Inverse-Square Law at the Submillimeter Range BERGE 2018 PRL 120 141101 MICROSCOPE Mission: First Constraints on the Violation of the Weak Equivalence Principle by a Light Scalar Dilaton FAYET 2018A PR D99 055043 MICROSCOPE limits on the strength of a new force, with comparisons to gravity and electromagnetism KLIMCHITSKAYA 2017A PR D95 123013 Constraints on Axionlike Particles and non-Newtonian Gravity from Measuring the Difference of Casimir Forces XU 2013 JP G40 035107 Nuclear Constraints on non-Newtonian Gravity at Femtometer Scale BEZERRA 2011 PR D83 075004 Constraints on non-Newtonian Gravity from Measuring the Casimir Force in a Configuration with Nanoscale Rectangular Corrugations SUSHKOV 2011 PRL 107 171101 New Experimental Limits on Non-Newtonian Forces in the Micrometer Range BEZERRA 2010 PR D81 055003 Advance and Prospects in Constraining the Yukawa-type Corrections to Newtonian Gravity from the Casimir Effect MASUDA 2009 PRL 102 171101 Limits on Nonstandard Forces in the Submicrometer Range GERACI 2008 PR D78 022002 Improved Constraints on non-Newtonian Forces at 10 Microns TRENKEL 2008 PR D77 122001 Constraints on Composition Independent Yukawa Interactions Inferred from Measurements of the Newtonian Gravitational Constant DECCA 2007A EPJ C51 963 Novel Constraints on Light Elementary Particles and Extra-Dimensional Physics from the Casimir Effect KAPNER 2007 PRL 98 021101 Tests of the Gravitational Inverse-Square Law below the Dark-Energy Length Scale TU 2007 PRL 98 201101 Null Test of Newtonian Inverse-Square Law at Submillimeter Range with a Dual-Modulation Torsion Pendulum SMULLIN 2005 PR D72 122001 Constraints on Yukawa-type Deviations from Newtonian Gravity at 20 microns HOYLE 2004 PR D70 042004 Submillimeter Tests of the Gravitational Inverse-Square Law CHIAVERINI 2003 PRL 90 151101 New Experimental Constraints on nonNewtonian Forces below 100 Microns LONG 2003 NAT 421 922 Upper Limit to Submillimeter Range Forces from Extra Space Time Dimension HOYLE 2001 PRL 86 1418 Submillimeter Tests of the Gravitational Inverse Square Law: a Search for ``Large'' Extra Dimensions HOSKINS 1985 PR D32 3084 Experimental Tests of the Gravitational Inverse Square Law for Mass Separations from 2 to 105 cm