A High Precision Test of the Equivalence Principle
Stephan Schlamminger
Eotwash GroupCenter for Experimental Physics and Astrophysics (CENPA)
University of Washington
March 4th 2010 Physics 294
A close look at the N part of campus
Now: CENPA
Center for Experimental Nuclear Physics and Astrophysics
Machine shop
Gravity lab
Eötwash groupEric Adelberger
Jens Gundlach
Blayne Heckel
Stephan Schlamminger
Erik Swanson
Frank Fleischer
Seth Hoedl
Ted Cook
Charles Hagedorn
William Terrano
Matt Turner
Todd Wagner
Graduate students
-Do all bodies drop with the same acceleration?
-Is there a preferred frame or direction in the universe?
-Is there a large (>µm) extra dimension? (02/18/10 Talk by Charles Hagedorn)
- Search for Axions.
- Crucial measurements for LISA (Laser Interferometer Space Antenna).
- Research for LIGO
Torsion balance
Mass does not add up
Proton Electron 1H
938 272 013 eV/c2 510 999 eV/c2 938 782 999 eV/c2
938 272 013 eV /c2 510 999 eV/c2 938 783 012 eV/c2
13.6 eV/c2Mass defect = binding Energy
Binding Energy = ΔEpot-ΔEkin
The non-linearity is rather small
Mass defect for 1 kg of
Earth: Δm=0.46 μg
Mass defect for 1 kg of
Moon: Δm=0.02 μg
but occurs also in gravitationally bound systems
Newton’s Principia (1689)
Newton’s 2nd Law
Fi = mi a
Gravitational Law
FG= G mg1 mg2 /r2
Equivalence Principle (EP): mi = mg
Newton’s Principia (1689)
Newton’s 2nd Law
Fi = mi a
Gravitational Law
FG= G mg1 mg2 /r2
Equivalence Principle (EP): mi = mg
Weak Equivalence Principle: Gravitational binding energy is excluded.
Strong Equivalence Principle: Includes all 4 fundamental interactions.
The Equivalence Principle
Our Current theory of gravity, General Relativity,
is based on the Equivalence Principle:
All bodies fall in a gravitational field with the same acceleration regardless of their mass or internal structure.
Einstein realized, that:
It is impossible to distinguish between an uniform gravitational field and an acceleration.
-a
F=-ma
The Equivalence Principle
g
F=mg
Gravitational field g Acceleration a
Inertial mass = gravitational mass, mI=mG for all bodies
General Relativity and the EP
Three classical tests:
Perihelion shift of Mercury
Deflection of light by the Sun
Gravitational red shift of light
The last two can be explained and calculated with the EP alone!
The known unknowns
• GR is not a quantum theory.
• Cosmological constant problem.
• Dark matter.
EP-Testsprovide a big bang for the buck!
The unknown unknowns
• Is there another force (5th force)?
• …
Searching for New Interactions
Assumed charge
Beryllium
q/μ
Titanium q/μ
Difference
Baryon number B 0.99868 1.001077 -2.429x10-3
r / 21
2
2
1
1 e r
mm
qqG V(r)
r
1 mm (r)V 21grav G
Source Test mass
Strength relative to gravity
Interaction range
Good model for a new interaction:
r /
21 e r
1 QQ V(r) k
q/µ
Mass (u) q=B q/µ
11p 1.007 3 1 0.992 8
10n 1.008 7 1 0.991 4
94Be 9.101 2 9 0.998 7
4822Ti 47.947 9 48 1.001 1
1st Tests of the Equivalence Principle
gmF G
gm
ma
I
G
gm
ma
I
G
a2a1
gm
m
ht
I
G
2
gm
ma
I
GamF I
h
Time t to fall from h:
1600 Galileo:
1.0)( 212
1
21
aa
aa
2nd Generation Tests
G
I
m
m
g
LT 2l
Measurement of the swing periods of pendula:
Newton (1686), Bessel (1830), Potter (1923)
5102 x
The torsion balance
• A violation of the EP would yield to different plumb-line for different materials.
•A torsion balance can be used to measure the difference in plumb-lines:
Torsion fiber hangs like the average plumb line.
Difference in plumb lines produces a torque on the beam.
-> twist in the fiber
Eötvös (1922)
9105 x
Historical overview
Galileo
Newton
Bessel
Potter
EötvösDicke
Braginsky
UW
)( 2121
21
aa
aa
drop
pendula
Type of experiment
torsionbalance
modulatedtorsionbalance
Principle of our experiment
Source Mass
EP-Violating signal
Composition dipole pendulum
(Be-Ti)Rotation
1 rev./ 20min
Autocollimator (=optical readout)
source mass λ (m)
local masses (hill) 1 - 104
entire earth 106 - 107
Sun 1011 - ∞
Milky Way (incl. DM) 1020 - ∞
aBe
aTi
13.3min
modulated
20 m diameter tungsten fiber
(length: 108 cm)
Κ=2.36 nNm
tuning screws for adjusting
tiny asymmetries
8 test masses (4 Be & 4 Ti )
4.84 g each (within 0.1 mg)
(can be removed)
5 cm
4 mirrors
EP torsion pendulum
torsional frequency: 1.261 mHz
quality factor: 4000
decay time: 11d 6.5 hrs
machining tolerance: 5 m
total mass : 70 g
The upper part of the apparatus
thermal expansion feet
to level turntable
air bearing turntable
electronics of angle encoder
feedthrough for
electric signals
thermal insulation
The lower part of the apparatusvacuum chamber (10-5 Pa)
ion pump
autocollimator
support structure for
gravity gradient
compensators
gravity gradient
compensators
Filter: F(t)=θ(t-T/4)+θ(t+T/4)
These data were taken
before the pendulum was
trimmed. The signal is a
result of the gravitational
coupling
Differential acceleration
dFdF 21
amdaamddmadma )( 2121
Deflection from equilibrium: φ
is caused by a torque:
mda
m
m
d
F1
F2
Some numbers
mda
4
κ 2.36 x 10-9 Nm
m 4.84 x 10-3 kg
d 1.9 x 10-2 m
6.41 x 10-15 m/s2 / nrad
Can be measured withAn uncertainty of3 nrad per day
Δa can be measured to20 fm/s2
A feel for numbers
φ=3 nrad
Seattle
San Francisco
Δa=20 fm/s2
g=9.81 m/s2 g=9.81 m/s2-Δa
Δh=7 nm
Δx=3 mm
Data taking sequence (1/3)We rotate the pendulum (and thus the dipole) in respect to the readout every day by 1800.
Data taking sequence (2/3)
50 days
The 12 days of data from the previous slide are condensed into a single point.
Dipole is orthogonal to readout.
Dipole is (anti-) parallel to readout.
Rotate every day
Rotate every day
Resolved signal in WE-direction!
Data taking sequence (3/3)After a 2 months of data taking and systematic checks we physically invert the dipole on the pendulum and put it back into the apparatus.
Ti-Be Be-Ti
Ti-Be differential acc.
These data points have been corrected for systematic effects.
Only statistical uncertainties shown.
Corrected result
ΔaN,Be-TI ΔaW,Be-TI
(10-15 m/s2) (10-15 m/s2)
as measured 3.3 ± 2.5 -2.4 ± 2.4
Due to gravity gradients 1.6 ± 0.2 0.3 ± 1.7
Tilt induced 1.2 ± 0.6 -0.2 ± 0.7
Temperature gradients 0 ± 1.7 0 ± 1.7
Magnetic coupling 0 ± 0.3 0 ± 0.3
Corrected 0.6 ± 3.1 -2.5 ± 3.5
l
lm
im
l
el
lmlm
0
qQ 12
1G π4 w
Gravitational potential energy between the pendulum and
the source masses is given by
11 q 12
1G 8 ll
lQ
Torque for 1 (m=1) signal:
gravity gradient field
)ˆ( )(ρ d q
)ˆ( )(ρdQ
*
pend
3
lm
)1(
source
3
lm
rYrrr
rYrrr
lm
l
lm
l
gravity multipole moment
.
q11
not possible for
a torsion pendulum
q21 q31
Gravity gradients (1/4)
Pb
Pb
Al
hillside &
local masses
360° rotatableQ21 compensators
Total mass: 880 kg
Q21= 1.8 g/cm3
Q31 compensators
Total mass: 2.4 kg
Q31 =6.710-4 g/cm4
Gravity gradients (2/4)
Gravity gradients (4/4)
Without
compensation:
Q21=1.78 gcm-3
0.2 fm/s2 N
1.7 fm/s2 W
Uncertainty due
to fluctuating
fields:
Tilt coupling
Tilt of the suspension point+ anisotropies of the fiber
will rotate
1. Thicker fiber atthe top providesmore torsionalstiffness
rotation axis
Tilt coupling
Tilt of the suspension point+ anisotropies of the fiber
will rotate
1. Thicker fiber atthe top providesmore torsionalstiffness
2. Feedback minimizes tilt of TT
rotation axis
More tiltActive foot
zturntable
PID
14 nrad
60 nrad
No signal!
1.70m
0.23m 53 nrad Remaining tilt uncertainty:
0.6 fm/s2 N
0.7 fm/s2 W
Thermal
50 nrad Signal
ΔT =7K
During data taking
ΔT≈0.053 K
0.38 nrad
1.7 fm/s2 N
1.7 fm/s2 W
Effect on the applied gradient on the signal
(measurement was done on one mirror):
Effect of a 7 K
gradient on
two different
mirrors.
Recent improvementszturntable
Lamp on a ruler
used to find the
sensitive spot
Highest sensitivity to the lamp
Additional insulation (@ 41 in) reduced the temperature feedthrough.
Interpreting our result
North: aBe-aTi= (0.6 ± 3.1) x 10-15 m/s2
West: aBe-aTi= (-2.5 ± 3.5) x 10-15 m/s2
Interpreting our result
North: aBe-aTi= (0.6 ± 3.1) x 10-15 m/s2
West: aBe-aTi= (-2.5 ± 3.5) x 10-15 m/s2
cos2rmF II
ω
θ
ε
r
gmF GG ½|a1+a2|η=
|a1-a2|
Interpreting our result
North: aBe-aTi= (0.6 ± 3.1) x 10-15 m/s2
West: aBe-aTi= (-2.5 ± 3.5) x 10-15 m/s2
cos2rmF II
ω
θ
ε
r
gmF GG
η(Be-Ti)= (0.3 ±1.8) x 10-13
½|a1+a2|η=
|a1-a2|
Acceleration to the center of our galaxy
Hypothetical signal:
(7 times larger)
20 x 10-15 m/s2
Differential acceleration to the center of the galaxy:
(-2.1 ± 3.1) x 10-15 m/s2
In quadrature:
(2.7 ± 3.1) x 10-15 m/s2
1825 h of data taken over 220 days
Galactic dark matter
Dark matter inside the Earth’s galactic orbit causes 25-30% of the total acceleration.
Halo of dark matter
Milky Way
Earth
Galactic dark matter
Our acceleration towards the galactic center is:
agal= adark+ aordinary= 1.9 10-10 m/s2 => adark= 5 10-11 m/s2
Dark matter inside the Earth’s galactic orbit causes 25-30% of the total acceleration.
Halo of dark matter
Milky Way
Earth
Galactic dark matter
Our acceleration towards the galactic center is:
agal= adark+ aordinary= 1.9 10-10 m/s2 => adark= 5 10-11 m/s2
Dark matter inside the Earth’s galactic orbit causes 25-30% of the total acceleration.
Halo of dark matter
Milky Way
Earth
Measured differential acceleration towards the galactic center is:
agal =(-2.1 ±3.1) 10-15 m/s2 => ηdark= |agal|/adark = (-4±7) 10-5
Galactic dark matter
Our acceleration towards the galactic center is:
agal= adark+ aordinary= 1.9 10-10 m/s2 => adark= 5 10-11 m/s2
Dark matter inside the Earth’s galactic orbit causes 25-30% of the total acceleration.
Halo of dark matter
Milky Way
Earth
Measured differential acceleration towards the galactic center is:
agal =(-2.1 ±3.1) 10-15 m/s2 => ηdark= |agal|/adark = (-4±7) 10-5
The acceleration of Be and Ti towards dark matter does not differ by more than 150 ppm (with 95 % confidence).
Summary
• Test of the equivalence principle with a rotating torsion balance.
• Principle of the measurement
• Main systematic effects
• Results
– Earth fixed (North): aBe-aTi=(0.6 ± 3.1) x 10-15 m/s2.
– η=(0.3 ± 1.8) x 10-13.
– Towards Galaxy: aBe-aTi=(-2.1 ± 3.1) x 10-15 m/s2.
– ηDM=(-4 ± 7) x 10-5.
– 10x improved limits on a long range interaction.
Outlook
• Ideas for improvement:
– Decouple damper systematic from pendulum.
– Integrated q21 measurement.
– Higher Q fiber to reduce thermal noise
• Earth-like and moon-like TB to keep pace with improvements in LLR.
Searching for New Interactions
x
x
xx q
u
mNq
xQ
/ r
12
21
2
2
1
1 12e r
mm
qqG V(r)
/ r
122
2
1
1
2
2
12e r
1
4 V(r)
u
g
Yukawa potential
/ r
12
21
2
12e r
1 QQ
4 V(r)
g
G
1 u = 1.66 x 10-27 kg
/ r
12
21
2
2
1
1 12e r
mm
qqG V(r)
SourceTest mass
Strength relative to gravity
Interaction range
Assumed charge
Beryllium
q/μ
Titanium q/μ
Difference
Baryon number B 0.99868 1.001077 -2.429x10-3
Lepton number L 0.44385 0.459742 -1.565x10-2
Other possible charges: B-L, 3B+L, ..
Searching for New Interactions
Signal
Free oscillation
Filtered data
Thermal noise
In frequency space
Noise level at the signal-frequency:
≈1000 nrad/√Hz ≈ 3 nrad √day