# Limits on $\boldsymbol R$ from Deviations in Gravitational Force Law INSPIRE search

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
$\bf{<30}$ 95 1
 2007
Torsion pendulum
• • • We do not use the following data for averages, fits, limits, etc. • • •
2
 2018
MICR Space accelerometer
3
 2018 A
MICR Space accelerometer
4
 2018
Neutron scattering
5
 2017 A
Torsion oscillator
6
 2013
Nuclei properties
7
 2011
Torsion oscillator
8
 2011
Torsion pendulum
9
 2010
Microcantilever
10
 2009
Torsion pendulum
11
 2008
Microcantilever
12
 2008
Newton's constant
13
 2007 A
Torsion oscillator
$<47$ 95 14
 2007
Torsion pendulum
15
 2005
Microcantilever
$<130$ 95 16
 2004
Torsion pendulum
17
 2003
Microcantilever
${ {}\lesssim{} }\text{ 200}$ 95 18
 2003
Microcantilever
$<190$ 95 19
 2001
Torsion pendulum
20
 1985
Torsion pendulum
1  KAPNER 2007 search for new forces, probing a range of $\alpha$ $≅$ $10^{-3} - 10^{5}$ 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.
2  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.
3  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 .
4  HADDOCK 2018 obtain constraints on non-Newtonian forces with strengths $10^{22}{ {}\lesssim{} }$ $\vert {{\mathit \alpha}}\vert { {}\lesssim{} }$ $10^{24}$ and length scales $\mathit R$ $≅$ $0.01 - 10$ nm. See their Figure 8 for more details. These constraints do not place limits on the size of extra flat dimensions.
5  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.
6  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.
7  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.
8  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.
9  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.
10  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.
11  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.
12  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.
13  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.
14  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.
15  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.
16  HOYLE 2004 search for new forces, probing $\alpha$ down to 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.
17  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.
18  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.
19  HOYLE 2001 search for new forces, probing $\alpha$ down to and distances down to 20$\mu$m. See their Fig.$~$4 for details on the bound. The quoted bound is for $\alpha$ ${}\geq{}$ 3.
20  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:
 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
PR D97 062002 Search for deviations from the inverse square law of gravity at nm range using a pulsed neutron beam
 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
Nature 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