Lecture 6 Tests of the Equivalence Principle and the Inverse-Square LawDetection of small forces 1Detection of small forces 2Torsion balanceTests of the Equivalence Principle 1Tests of the Equivalence Principle 2Search for composition-dependent forcesTheoretical motivation for 1/r2 Law testsMedium-range tests of the 1/r2 Law 1Medium-range tests of the 1/r2 Law 2Limits of the 1/r2 Law at   1 cmPaik-1Lecture 6Tests of the Equivalence Principle and the Inverse-Square LawPhysics 798G Spring 2007Ho Jung PaikUniversity of MarylandFebruary 13, 2007Paik-2Detection of small forces 1• Response of a detector:• Three regimes of operation:1)2)3)• Fluctuation-dissipation theorem:⇒ Nyquist’s theorem:HmQmkQjmFx/ ,/ e wher ,/)(/)()( 000220ωωωωωωωω≡=+−=⇒200/)()( ,spring) (stiff ωωωωωmFx ≅<<200/)()( ),(resonance ωωωωωmQjFx −==20/)()( ,spring)(soft ωωωωωmFx −≅>>jωx(ω)Q0ωω=ΔNNFFkxxHxmFH+=++→&&& ,QmTkfTHkfSBBTF0,4)(4)(ω≈=FkxxHxm=++&&&Paik-3Detection of small forces 2• Detectability of signal for an amplifier with noise PSD :1)2)3)β: energy coupling constant, : detection bandwidth• Total detection force noise PSD:1)2)3) ())(4)( ,,22000fSmQmTkfSAxBNβωωωω+=<<()[]ffSmfSfFAxTFΔ+≥<< )()()( ,,220,2S0ωββωω()[]ffSQmfSfFAxTFSΔ+≥= )(/)()( ,,220,20ωββωω()[]ffSmfSfFAxTFSΔ+≥>> )()()( ,,22,20ωββωω)( fFS)(,fSAx)(14)( ,,22000fSQmQmTkfSAxBN⎟⎟⎠⎞⎜⎜⎝⎛+==ωβωωω())(4)( ,,2200fSmQmTkfSAxBNβωωωω+=>>τ/1≈ΔfPaik-4Torsion balance• Invented by Michell (c1750) and Coulomb (c1777).• The torsional stiffness κis extremely weak, ω0/2π≈10−3Hz.⇒ A very high force sensitivity!• Used by Cavendish to measure G and weigh the Earth (1798).• Almost all the sensitive gravity experiments have been performed with torsion balances for two centuries.⎟⎠⎞⎜⎝⎛+==−=−=22202522 , ,2:torquenalGravitatiordmIIrmMGFdFggωκκθPaik-5Tests of the Equivalence Principle 1• Galileo (c1612), Gedanken exp?:Source mass: EarthDetector: two balls of different massdropped from a leaning tower.Disadvantage: τtoo short • Eötvös, Pekar and Fekete (1922):Source mass: Earth. Detector: rotating torsion balance with two masses of different composition suspended.Advantage: Null experiment detecting directly. ⇒810/−≤Δ= aaηaΔPaik-6Tests of the Equivalence Principle 2• Roll, Krotkov, and Dicke (1964):Source mass: Sun.Detector: stationary torsion balance. Earth’s spin modulates the signal. ⇒• Adelberger et al. (1998):Source mass: Earth.Detector: torsion balance on an air-bearing turntable.⇒• Lunar Laser Ranging (1972- ):Source mass: Sun.Detector: Earth-Moon system.⇒1110−≤η13105−×≤η13105−×≤ηPaik-7Search for composition-dependent forces• General quantum-exchange force: • Adelberger et al. (1998):Source masses: a sloping hill and a rotating 3-ton uranium massDetector: torsion balance on an air-bearing turntableCritical error source for short λ: metrology errors⇒m 1 for 105≥≤−λαintegral essdimensionl :)( ,amu in mass : ,4),(~~~ e~~π4)(22121/2122,1λμπαλμμμαλAAArIuGufIqqqg'aΔrqqfVrrrm±≡⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛=⇒=−−rPaik-8Theoretical motivation for 1/r2Law tests• Generalized potential:()cmrmmGrVbrh=+−=−λαλ ,e1)(/21Theory AuthorsαλspinScalar-tensor theories of gravity (scalar graviton)Wagoner (1970), Acharya & Hogan (1973)O’Hanlon (1972) +1/300Spontaneous breakdown of scale invariance (dilaton)Fujii (1971), Fujii (1974)Zee (1979)+1/3 1 km<< 1 km00Supergravity(gravitino, graviphoton)Deser & Zumino (1977)Scherk (1979), Fayet (1980)3/2, 0, 13/2, 1Unified field theories (axion)(goldstone boson)Weinberg (1978), Wilczek (1978)Chang, Mohapatra & Nussinov (1985)≤10-510~104km00Coupling to hypercharge Fishbach et al. (1986)≤10-2100 mString theory (KK graviton)(moduli) (dilaton)(radion)Arkani-Hamed, Dimopoulos, Dvali (1998)Dimopoulos & Giudice (1996)Kaplan & Wise (2000)Antoniadis, Dimopoulos & Dvali (1998)8n/3 < 1 mm≥ 100 μm20002+nnReview: Adelberger, Heckel, & Nelson, Annu. Rev. Nucl. Part. Sci. 53, 77 (2003)Paik-9Medium-range tests of the 1/r2Law 1• Spero et al. (1980): λ≈ 1 cmSource: Al cylindrical shell (null source)Detector: torsion balance• Kuroda & Hirakawa (1985): λ≈ 1 mSource: rotating dumbbellDetector: 96-Hz gravitational-wave detectorPaik-10Medium-range tests of the 1/r2Law 2• Stacey et al. (1987): λ≈ 100 mSource: EarthDetector: gravimeter moved vertically down a mine shaft⇒ Source density error• Eckardt et al. (1988):λ≈ 100 mSource: EarthDetector: gravimeter moved vertically up a TV tower⇒ Source shape error• Moore et al. (1988): λ≈ 10 mSource: pumped lake (null source)Detector: stationary beam balancePaik-11Limits of the 1/r2Law at λ≥ 1