Testing Gravity from the Dark Energy Scale to
the Moon and Beyond
C.D. Hoyle
C.D. Hoyle
for the Eöt-Wash Group at the University of Washington
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Overview
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Brief review of gravity and the Inverse-Square Law (ISL)
Motivation for precision gravitational tests
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What we don’t know about gravity
What gravity may tell us about the nature of the universe
Testing the ISL at the “Dark Energy Scale”
Using the Earth-Moon system to precisely test Einstein’s
General Relativity
Future prospects for precision gravitational tests
What We Know: Gravity in the 21st Century
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Gravity is one of the 4 known fundamental interactions
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Others: Electromagnetism, Strong and Weak Nuclear Forces
Gravity holds us to the earth (and makes things fall!)
It also holds things like the moon and satellites in orbits
Newton expressed this “unification” mathematically in the 1660’s:
+
Newton
M 1M 2
 F G
2
r
r is distance between two
bodies of mass M1 and M2
More That We Know
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Newton’s “Inverse-Square Law” worked well for about 250 years,
but troubled Einstein
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“Action at a distance” not consistent with Special Relativity
Einstein incorporated gravity and relativity with another great
unification in 1915:
General Relativity
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Gravitational attraction is just a consequence
of curved spacetime
All objects follow this curvature (fall) in the
same way, independent of composition:
The Equivalence Principle
1/r2 form of Newton’s Law has a deeper significance:
it reflects Gauss’ Law in 3-dimensional space
Very successful so far:
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Planetary precession
Deflection of light around massive objects
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What we Don’t Know
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General Relativity works well, but is fundamentally inconsistent
with the Standard Model based on quantum mechanics
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Why is gravity so weak compared to the other forces?
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Will String Theory provide us a further unification?
“Hierarchy” or “Naturalness” Problem
Why is M Planck  M EW ?
E & M force ~1040 times greater than gravitational force in an H atom!
Is gravity’s strength diluted throughout the “extra dimensions” required by
string theory?
Does an unknown property of gravity explain the mysterious
“Dark Energy” which seems to cause our universe’s expansion to
accelerate?
S. Carroll
A “Golden Age” for Gravitational Physics
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Can gravitational effects explain the Dark Energy?
What can gravity tell us about the nature of spacetime?
Are there observable effects of String Theory?
Are there new particles and forces associated with gravity’s
(unknown) quantum-mechanical nature?
Experimental prospects
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Laboratory-scale tests of the 1/r2 law and Equivalence Principle
Astronomical tests of General Relativity
Gravitational wave searches (LIGO, LISA, etc.)
Signatures of quantum gravity in high-energy collider experiments
Short-Range 1/r2 Tests
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Are there observable consequences of String Theory?
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Extra dimensions – maybe M Planck  M EW , but gravity is diluted throughout more
dimensions than the rest of the Standard Model forces. Extra dimensions could
be large (mm scale!)
e.g. N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Lett. B 436, 257 (1998)
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What is the mechanism behind the cosmic acceleration?
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“Fat” graviton - gravity may observe a cut-off length scale in the sub-mm
regime and thus does not “see” small-scale physics.
R. Sundrum, hep-th/0306106 (2003)
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Does the observed dark energy density suggest a new, fundamental
“Dark Energy Scale” in physics?
c
4
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 0.1 mm
S. Beane, hep-ph/9702419 (1997)
Vac
Are there new forces mediated by exotic particles?
e.g. S. Dimopoulos and A. Geraci, hep-phys/0306168 (2003), I. Antoniadis et al.,
hep-ph/0211409 (2003), D. Kaplan and M. Wise, hep-ph/0008116 (2000), etc.
Example: Extra Dimensions
• Test masses and ED:
From G. Landsberg
R*
Moriond ’01 Talk
• Near test mass (r  R*), we must satisfy Gauss’ Law in
3+1+n dimensions:
G3 n m1m2
V r   
r n1
• Far away (r >> R*) we must recover the usual 3-D form:
G3 n m1m2
V r   
R*n r
G3 n
G n
R*
Parameterization and Background
• General deviation from Newtonian gravity:
Gm1m2 
V r   
1   er /  


r
From Adelberger, et al., Ann. Rev.
Nuc. Part. Phys. (2003)
• Until recently (last few years), gravitation not even shown to
exist between test masses separated by less than about 1 mm!
Previous Short Range Limits
• 95% C.L., as of 1999 (when we started our work)
• All previous limits from torsion pendulum experiments
For references see CDH et al., Phys Rev. D. 70 (2001) 042004
Experimental Challenges
• Extreme weakness of gravity
– Electrostatic interactions
• Need extremely high charge balance (10-40) to attain gravitational
sensitivity!
• Casimir force, patch charges become strong at close distances
• Fortunately, effective shielding is possible, but at a cost of distance!
– Magnetic impurities
• Strong distance dependence
• Requires high purity materials and clean fabrication techniques
• Need to get large mass at small separations
– Alignment and characterization of masses
– Seismic noise
• Temperature fluctuations and thermal noise
• Etc., etc.
Torsion Pendulums
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Torsion Pendulum still the best instrument for measuring the ISL:
thin fiber
M1
M2
up
r
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Vary separation, r, between masses M1 and M2
Force on M1 causes the pendulum to twist
Measure twist angle
Compare with inverse-square prediction
Eöt-Wash Torsion Pendulum (best to date)
Fiber, 18m diameter, 80cm length, tungsten
Leveling mechanism
3 aluminum calibration spheres
4 mirrors for measuring angular deflection
21-fold axial symmetry, molybdenum disc,
1mm thick
s
Not pictured: 10m thick Au-coated BeCu
membrane - electrostatic shield
2.75”
Attractor : rotating pair of discs, shifted out of
phase with each other to reduce Newtonian
torque
Technique
• Attractor disks rotate below pendulum
• “Missing mass” of the holes causes pendulum to twist
• Measure the torque on pendulum at harmonics (21, 42, 63) of the
attractor rotation frequency, , as a function of S
• Compare observed torque to ISL prediction
s
• Twist angle measured to a nanoradian (imagine a pea in Seattle)
• Force measured equals 1/100 trillionth the weight of a single
postage stamp
Noise
Predicted thermal
noise for Q = 3500
(internal dissipation)
Data
Readout
Noise
 2 ( ) 
4k BT 
Q[(  I  ) 
2 2
2
Q2
]
Recent Results (Thesis of D. Kapner)
ISL
95% C.L. Bounds on ||
V r   
Gm1m2 
1   er /  



r
More Distant Future: Even Shorter Distances
• Why Look to Shorter Distances?
– Short range 1/r2 tests place model-independent constraints on:
• Single largest possible extra dimension
• New interactions (properties of exchange particles)
– Other, more specific scenarios (dilaton, moduli, etc.)
– Unexplored parameter space
New Promising Techniques
• Vertical plate “Step Pendulum”:
• Analytical expression for (very small) Newtonian
background torque
• Yukawa torque now falls as 2 instead of 3 for
small :
R
NY   G p a RA 2e s / 
• Drawbacks:
• Minimum separation may not be so small
• Possible Systematics at 1
Modulate attractor
plate/pendulum separation
Future High-sensitivity 1/r2 Test
Top view:
Attractor:
“Infinite” plane
2mm thick Mo
Homogenous
gravity field
Torsion pendulum
No change in torque
on pendulum if 1/r²
holds.
Moves back and
forth by 1mm
Be, = 1.84 g/cm ³
Pt, = 21.4 g/cm ³
Stretched metal membrane
Advantages over hole pendulum:
• True null test
• Slower fall-off with 
(³ for holes vs. ² for plates)
• Much larger signal
• Simpler machining
Current and Future Limits
V r   
Gm1m2 
1   er /  


r
Current
Step pendulum
Shooting the Moon
Testing General Relativity with Lunar
Laser Ranging
A Modern, Post-Newtonian View
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The Post-Newtonian
Parameterization (PPN) looks at
deviations from General Relativity
The main parameters are  and 
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 tells us how much spacetime
curvature is produced per unit mass
 tells us how nonlinear gravity is
(self-interaction)
 and  are identically 1.00 in GR
Current limits have:
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(–1) < 2.510-5 (Cassini)
(–1) < 1.110-4 (LLR)
Relativistic Observables in the Lunar Range
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Equivalence Principle (EP) Violation
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Weak EP
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Earth and Moon fall at different rates toward the sun
Appears as a polarization of the lunar orbit
Range signal has form of cos(D) (D is lunar phase angle)
Composition difference: e.g., iron in earth vs. silicates in moon
Probes all interactions but gravity itself
Strong EP
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Applies to gravitational “energy” itself
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Earth self-energy has equivalent mass (E = mc2)
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Amounts to 4.610-10 of earth’s total mass-energy
Does this mass have MG/MI = 1.00000?
Another way to look at it: gravity pulls on gravity
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This gets at the nonlinear aspect of gravity (PPN )
Equivalence Principle Signal
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Sluggish orbit
If, for example, Earth has
greater inertial mass than
gravitational mass (while
the moon does not):
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Nominal orbit:
Moon follows this, on average
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Sun
Earth is sluggish to move
Alternatively, pulled weakly
by gravity
Takes orbit of larger radius
(than does Moon)
Appears that Moon’s orbit is
shifted toward sun: cos(D)
signal
The Strong Equivalence Principle
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Earth’s energy of assembly amounts to 4.610-10 of its total
mass-energy
The ratio of gravitational to inertial mass for this self energy is
The resulting range signal is then
Currently  is limited by LLR to be ≤4.510-4
LLR is the best way to test the strong EP
Other Relativistic Observables
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Most sensitive test of 1/r2 force law at any length scale
Time-rate-of-change of Newton’s gravitational constant
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Geodetic precession tested to 0.35%
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Could be signature of Dark Energy (quintessence)
Currently limited to less than 1% change over age of Universe
Precession of inertial frame in curved spacetime of sun
Gravitomagnetism (frame-dragging) is also seen to be
true to 0.1% precision via LLR
LLR through the Decades
Previously
100 meters
APOLLO
APOLLO: the New Big Thing in LLR
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APOLLO offers order-of-magnitude
improvements to LLR by:
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Using a 3.5 meter telescope
Gathering multiple photons/shot
Operating at 20 pulses/sec
Using advanced detector technology
Achieving millimeter range precision
Having the best acronym
The APOLLO Collaboration
UCSD:
U Washington:
Humboldt State: Harvard:
Tom Murphy (PI)
Eric Michelsen
Evan Million
Eric Adelberger
Erik Swanson
*Russell Owen
*Larry Carey
C.D. Hoyle
Liam Furniss
Christopher Stubbs
James Battat
JPL:
Northwest Analysis:
Lincoln Labs:
Jim Williams
Slava Turyshev
Dale Boggs
Jean Dickey
Ken Nordtvedt
Brian Aull
Bob Reich
Measuring the Lunar Distance
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It takes light 1.25 seconds to get to the moon – thanks to foresight we
can reflect light off the surface!
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Retroreflector arrays always send light straight back at you (like hitting a
racquetball into a corner):
retroreflector
Lunar Retroreflector Arrays
Corner cubes
Apollo 11 retroreflector array
Apollo 14 retroreflector array
Apollo 15 retroreflector array
APOLLO’s Secret Weapon: Aperture
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The Apache Point Observatory’s
3.5 meter telescope
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Southern NM (Sunspot)
9,200 ft (2800 m) elevation
Great “seeing”: 1 arcsec
Flexibly scheduled, high-class
research telescope
6-university consortium (UW, U
Chicago, Princeton, Johns
Hopkins, Colorado, NMSU)
APOLLO Basics
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2.5 second round-trip time, 20 Hz laser pulse rate
(50 pulses in the air at any one time)
Outbound pulses have 3 x 1017 green photons (532 nm), 3.5 meter
diameter
We get about 1 (!) back per pulse (beam spreads to 15 km diameter)
Arrival time must be measured to less than a nanosecond
The Link Equation
 = one-way optical throughput (encountered twice)
f = receiver narrow-band filter throughput
Q = detector quantum efficiency
nrefl = number of corner cubes in array (100 or 300)
d = diameter of corner cubes (3.8 cm)
 = outgoing beam divergence (atmospheric “seeing”)
r = distance to moon
 = return beam divergence (diffraction from cubes)
D = telescope aperture (diameter)
• APOLLO should land safely in the multi-photon regime
• Current LLR gets < 1 photon per 100 pulses
• Even at 1% of expected rate, 1 photon/sec good enough for feedback
Differential Measurement Scheme
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Corner Cube at telescope exit returns
time-zero pulse
Same optical path, attenuated by 1010
Same detector, electronics
Diffused to present identical
illumination on detector elements
Result is differential over 2.5 seconds
Must correct for distance between
telescope axis intersection and corner
cube
Needle in a Haystack
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Signal is dim (19th magnitude), while full moon is bright (–
13th magnitude)
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1013 contrast ratio
We must filter in every available domain
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Spectral: 1 nm bandpass gets factor of 200
Spatial: 2 square arcsec gets factor of 106
Temporal: detector is on for 100 ns every 50 ms
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This itself is factor of 5105
But can discriminate laser return from background at the 1 ns
level5107 background suppression
In all, get about 1016 background suppression
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Yields signal-to-noise of 103
Systematic Error Sources
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We can cut the 50 mm random uncertainty (due mostly to moon
orientation) down to 1 mm with 2500 photons
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2 minutes at 20 Hz and 1 photon per pulse
Systematic uncertainties are more worrisome
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Atmospheric delay (2 meter effective path delay)
Deflection of earth’s crust by:
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Ocean: even in NM, tidal buildup on CA coast  few mm deflection
Atmosphere: 0.35 mm per millibar pressure differential
ground water: ????
Accurate modeling still needs to be done
Thermal expansion of telescope and retroreflector arrays
Radiation pressure (3.85 mm differential signal)
Implementation systematics
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Detector illumination
Strong signal bias
Temperature-dependent electronic timing
Observation schedule/sampling: danger of aliasing
Periodicity: Our Saving Grace
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If we don’t get all this supplemental metrology right, we’re still okay:
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Our science signals are at discrete, well-defined frequencies
Equivalence Principle signal at 29.53 days
Other science via 27.55 day signal (eccentricity)
Meteorological influences are broadband
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Atmospheric, ground-water loading are random
Even tides, ocean loading don’t have power at EP period
Thermal effects are seasonal
Laser Mounted on Telescope
First Light: 7/24/05
First Results: 10/19/05!
100 ns
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Two night total: 4000 photons
As many as the best previous station got in the last 3 years!
Calculated distance agrees well with JPL model
However, rate is slightly lower than expected and intermittent
Future Work
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Optimization of signal, stabilize laser
Software refinement/development
Gravimeter/Precision GPS installation
Precision geophysical modeling of site motion
Sufficient data for order-of-magnitude improvement in
EP test in ~1 year
Continued data collection/analysis for years to come
Summary
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Many reasons to test gravity, much we still do not understand
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Is there a “Grand Unified Theory” that describes all fundamental interactions?
Is gravity causing the mysterious acceleration of our universe’s expansion?
Are there possibly more than 3 dimensions of space?
We are entering a “Golden Age” of experimental gravity research
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Laboratory torsion pendulum tests:
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Astronomical tests of General Relativity
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APOLLO lunar laser ranging experiment
Gravity wave experiments
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Inverse-square law
Equivalence principle
more…
LISA
LIGO
Research is exciting for students of all levels
So far Einstein is still correct… but for how long?
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