IUPUI – IUB
Indiana
University
Collaborative
Research
Grant
Short Range Test of the Gravitational Inverse Square Law
Evan Weisman, Simon Kelly, Trevor Leslie, Andrew Peckat, and Josh Long
Indiana University, Bloomington IN
The Project - What are we doing?
This project is a test of the Gravitational inverse square law. This law, written down by Isaac Newton in the
seventeenth century, states that the gravitational force between any two point-like or spherical masses m1 and m2
varies as the product of the masses and in inverse proportion to the square of distance r between the center of the
masses:
The Challenge – Scaling and Backgrounds
Experiment is operational at IU Cyclotron
• In order to test gravity at a given range r, the size of the experimental test masses must in general be scaled to that
range. Otherwise, the experimental signal will be overwhelmed by the Newtonian gravitational signal arising from
all the extra mass at ranges greater than r.
m1m2
F (r ) = −G 2
r
m1, ρ 1
m2, ρ 2
Source
Mass
Detector
Mass
Scale:
1 cm
~ 2r
In these expressions, the negative sign indicates attraction and G is the Newtonian gravitational constant,
6.67 ×10-11 N m2/kg2. The value of this constant implies that the gravitational force is extremely weak compared to
the other known fundamental forces in nature (the electromagnetic and nuclear forces). For example, the
gravitational force between a proton and an electron is about 1040 times weaker than the electric force.
Gm1m2 G ρ1 ρ 2 (4r )
4
~ G ρ1 ρ 2 r
F=
=
2
2
(2r )
4r
Newton claimed that these expressions are “universal,” but is this necessarily the case? This is an experimental
question. In particular, do these expressions hold for “short” ranges (r < 100 microns), where, due to its intrinsic
weakness, gravity is poorly measured (or not even measured at all)? This is the question tested by the Indiana shortrange experiment.
r = 100 µm ⇒ F ≈ 10-17 N
How do we describe deviations from 1/r2 ?
In the quantum picture, forces between particles arise from the exchange of other particles between them. The
exchange of a single massless particle results in a 1/r2 force. The exchange of multiple massless particles results in
different power-law forces, and the exchange of a massive particle results in an exponential “Yukawa” force (which
reduces to a 1/r2 force in the limit that mass of the exchange particle vanishes):
Power Law
Yukawa Interaction
50 cm
3 2
Vibration
Isolation
Stacks
ρ1 = ρ2 = 20 g/cm3, r = 10 cm ⇒ F ≈ 10-5 N
• The diagram above illustrates that, when the sizes of spherical gravitational test masses (with density ρ) are scaled
to the range r, gravitational attraction scales as r4. Thus, while an experiment with 10 cm gaps and test masses (such
as the Cavendish experiment) needs to be sensitive to dyne-sized forces, an experimental test of gravity below 100
microns must have a sensitivity on the order of the smallest forces ever measured.
• At the same time, background forces due to electrostatic surface potentials scale as 1/r2, and forces due to magnetic
contaminants and Casimir forces scale as 1/r4, quickly becoming overwhelming at short ranges.
• 60 µm stiff shield replaced with 10 µm stretched
membrane (minimum gap = 48 µm)
Vacuum bell jar base plate
• May 2012: resonant signal (probably electrostatic
background) above thermal noise, but only at 1σ level.
(1 day average ~ 5 fN sensitivity)
Readout: Capacitive transducer coupled to JFET amplifier
The IU Experiment – How do we do it?
• Torsion pendulums are very sensitive but tend to be limited by vibrations, and mechanical and thermal drifts which
limit their ability to probe forces between large masses held in very close proximity. The IU experiment takes a
different approach:
m=0
mB
Central apparatus
Shield
The potential energy associated with this interaction is:
m1m2
V (r ) = −G
r
Interaction region
m=0
m1
m2
r
m1m2
1 + α e − r / λ 
V ( r ) = −G
r
λ = h / mB c = range
a = strength relative
to gravity
m1
m2
n −1

m1m2
 r0  
V ( r ) = −G
1 + β n   
r 
 r  
~ 3 cm
r0 = experimental scale
set limits on bn for n = 2 - 5
In the literature, the most common way to parameterize deviations from the inverse square law is with a Yukawatype interaction (left), though other parameterizations are also possible (right). In the absence of a signal (no
experiment to date has detected a deviation from the 1/r2 law), experiments usually report their results in terms of
limits on the strength α of a new force as a function of the range λ.
Motivation - Why are we doing it?
Source and Detector Oscillators
Shield for Background Suppression
• Planar geometry: concentrates as much mass as possible at range of interest
• Resonant operation to maximize signal - but heavy burden on vibration isolation
• Upgraded to differential readout with 2 probes on detector mass
• Run at 1 kHz for stiff, compact vibration isolation
• Sensitive to ~ 100 fm thermal oscillations
• Stiff conducting shield to suppress electrostatic, acoustic backgrounds
In addition to Newton’s claim that the
inverse square law is universal:
• Current experimental limits allow
for new forces in nature millions of
time stronger than gravity acting over
distances resolvable to the unaided
eye.
• Specific predictions of new forces
arise in many models that attempt to
describe gravity and the other
fundamental forces in the same
theoretical frame work, including:
• modulus forces and dilatons,
which are predicted by string
theory [1]
• signatures of extra spacetime
dimensions, into which gravity
alone “leaks” and appears weaker
than the other forces [2]
The plot above is the Yukawa parameter space, showing experimental limits on the strength α of a hypothetical new
interaction as a function of the range λ, in the “short” range for gravity (1 µm < λ < 1 cm). The experimentally
excluded region is above and to the right of the solid blue curves. In the range above 10 µm, the limits are defined
by “classical” gravity measurements using torsion pendulums similar to the original Cavendish experiment (which
first measured the Newtonian constant G in the 1790s). Below a few microns, the best limits derive from atomic
force microscopy experiments, and experiments used to measure the Casimir force, a force which arises between
conductors due to zero-point fluctuations of the electromagnetic field. The initial limits from the Indiana shortrange experiment (originally at the University of Colorado) are also indicated [3].
• Runs with source and lock-in tuned to detector resonance, interleaved with off-resonance and diagnostic runs to
monitor offsets and gain drifts
• Typical session: 8 hours with ~ 50% duty cycle
• Double-rectangle detector: high mechanical quality factor (Q) to suppress limiting thermal noise
FYukawa ~ FThermal ~ 10-16 N
FNewtonian ~ 10-17 N
Current and Projected Limits – Where are we going?
Central apparatus
• Test masses mounted
on tilt stages, which are
mounted to vibration
isolation stacks.
vibration
isolation
stacks
• With new, thinner shield, one day of
running will result in limits more
sensitive than the current best limit
by 1-2 orders of magnitude at the 10
micron range (upper dashed curve).
• Vibration
isolation
stacks: Brass disks
connected by fine
wires; make set of soft
springs which attenuate
at ~1010 at 1 kHz
(determines operating
frequency)
• Installed in 75 liter
vacuum bell jar (10-7
torr)
for
further
suppression of acoustic
forces
• Current preliminary limits (2σ) on
order of best published limits to date;
gap uncertainties remain.
Scale:
1 cm3
tilt stage
• A cryogenic experiment (operated at
T = 4 K) with thinner test masses,
currently under development, will
yield an experiment sensitive to
gravity at 20 microns (lower dashed
curve).
transducer
amp box
• Both projections assume the thermal
noise limit can be attained, with all
other backgrounds controlled.
detector mass
shield
PZT bimorph
References
source mass
[1] S. Dimopoulos, A. Geraci, Phys. Rev. D 68 (2003) 124021.
Figure: Bryan Christie (www.bryanchristie.com) for Scientific American (August 2000)
[2] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 (1998) 263.
[3] J. C. Long et al., Nature 421 (2003) 922.