©Copyright 2020 John G. Lee A Fourier-Bessel Test of the Gravitational Inverse-Square Law John G. Lee A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2020 Reading Committee: Blayne Heckel, Chair Eric Adelberger, Chair Andreas Karch Program Authorized to Offer Degree: Department of Physics University of Washington Abstract A Fourier-Bessel Test of the Gravitational Inverse-Square Law John G. Lee Co-Chairs of the Supervisory Committee: Professor Blayne Heckel Department of Physics Professor Eric Adelberger Department of Physics This dissertation presents the latest Eöt-Wash group test of the gravitiational Inverse Square Law (ISL) at the shortest possible distances using a stationary torsion pendulum above a rotating attractor. It largely builds off of earlier Fourier-Bessel work of Ted19 Cook[ ] that utilized test-masses with both 18-fold and 120-fold azimuthal symmetries. In this case, the effective masses of the platinum test bodies of the 18- and 120-fold patterns onthe pendulum were 0.21 and 0.62 mg. We tested the ISL at separations ranging from 52µm to 3mm. We have excluded at 95% confidence gravitational strength Yukawa interactions with length scales λ > 39µm and set new limits between λ = 8µm and 90µm. We have resolved gravitational interactions at the closest separations ever and measured a new systematic arising from the magnetic susceptibilities of our test-mass materials. We present a full description of the instrument, the analysis methods, and the results. Hojišąnane, wiigihuhu anąga wiirokų gipįnįpįnį aašgekixjįirera wiiyaa’ųwi anąga Mąąru- tira Inverse Square Law (ISL) yaagųswi. Eöt-Wash group test aanąągre yaa’ųwi. Waagax te’e, jaagu haisunąkwira wahakjene. Ted Cookga Fourier-Bessel airera hikišeregįnį anąga woore že’e yaa’ųwi. Mąąssga ceexi wiirokųra haruwąkašąną nąga hogihižą nąga kerepąną nųųp wejąjąire. Wiigihuhura rutiire anąga peesįwįwigają hosgįgrera 0.21mg anąga 0.62mg yaagųswi. 52µm anąga 3mm hocąkeną peesįwį. 39µm anąga hiraicera hosereceja wesįwįgi, horoǧocra hąke Yukawa Interaction Mąąrutira hikisgeižą herenį hire wahi. Yukawa ną- gre 8µm anąga 90µm hocąkenaija hąke wa’ųnįakgųnį yaarawi. Aašgexjį peesįwįwi anąga yaagųswigejene wažącek yaagųsi yaare wa’ųajawi. Wiirokų yaa’ųwira, jaasge ha’ųwira, anąga hegųnegi jaagu yaapereswira wapahakjene. TABLE OF CONTENTS Page List of Figures ....................................... vi Chapter 1: Introduction ................................ 1 1.1 Theoretical Motivations ............................. 1 1.1.1 Large Extra Dimensions and the Gauge Hierarchy Problem ...... 2 1.1.2 String Theory Moduli .......................... 3 1.1.3 Dark Energy Scale ............................ 4 1.1.4 Cosmological Constant Problem and Fat Gravitons .......... 4 1.1.5 Particles with Screening Mechanisms .................. 5 1.1.6 Unparticles ................................ 6 1.2 Recent Experimental Results .......................... 6 1.2.1 Torsion Balance Tests .......................... 7 1.2.2 Casimir Force Tests ........................... 8 Chapter 2: Calculated Gravitational, Yukawa, and Magnetic Torques ....... 10 2.1 Fourier Bessel Expansion ............................. 10 2.1.1 Modified Helmholtz Green’s Function .................. 10 2.1.2 Torque Calculation from Wedge Sources ................ 12 2.2 Off-Center Calculations ............................. 14 2.2.1 Fourier-Bessel Monte-Carlo ....................... 14 2.2.2 Off-Center Empirical Functions ..................... 14 2.3 Magnetic Calculations .............................. 15 Chapter 3: Apparatus .................................. 18 3.1 Apparatus Improvements ............................. 18 3.1.1 In-Situ Screen Actuators ......................... 18 3.1.2 Apparatus Tilt .............................. 19 i 3.1.3 Vacuum Chamber and Spider ...................... 19 3.1.4 Calibration Turntable .......................... 20 3.1.5 Foil Stretcher ............................... 20 3.1.6 Attractor tip-tilt ............................. 20 3.1.7 Ion Pump ................................. 21 3.1.8 Thermal Enclosure Temperature Control ................ 21 3.2 Apparatus Preparation .............................. 22 Chapter 4: Test-Mass Fabrication ........................... 23 4.1 Thicknesses .................................... 23 4.2 Pattern Machining and Mass Removal ..................... 24 4.3 Pattern Geometry ................................ 27 4.3.1 Measurement ............................... 27 4.3.2 Radii and Concentricities ........................ 28 4.3.3 Subtended Angle ............................. 29 4.3.4 Overcut Parameter ............................ 30 4.4 Gluing Procedure ................................. 30 4.5 Surface Scans ................................... 32 4.6 Epoxy Density .................................. 36 Chapter 5: Alignment .................................. 39 5.1 Pendulum Perpedicularity ............................ 39 5.2 Runout ...................................... 41 5.3 Turntable Leveling and Bearing Rumble .................... 45 5.4 Tilted Plate Approximation ........................... 45 5.5 Attractor-Screen Leveling ............................ 47 5.6 Pendulum-Screen Leveling ............................ 49 5.7 Separation ..................................... 52 5.7.1 COMSOL Models ............................. 52 5.7.2 Z-scans .................................. 55 5.7.3 Fitting ................................... 56 5.7.4 Separation Measurement ......................... 59 5.7.5 Achieving Minimal Separations ..................... 59 ii 5.7.6 Stability .................................. 60 5.8 Gravitational Centering ............................. 61 5.9 Experimental Parameters ............................ 62 Chapter 6: Direct Measurements of Test-Mass Magnetic Properties ......... 64 6.1 Measurement ................................... 64 6.1.1 Filtering .................................. 64 6.1.2 Binning .................................. 65 6.1.3 Fitting ................................... 66 6.2 Results ....................................... 66 Chapter 7: Data Taking Procedures .......................... 70 7.1 Data Collection Procedure ............................ 70 7.2 Non-Linearity of the Twist Angle Scale ..................... 71 7.3 Fitting ....................................... 75 7.3.1 Filtering .................................. 75 7.3.2 Cut Analysis ............................... 75 7.3.3 Cut Rejection ............................... 77 7.3.4 Combining Data ............................. 79 7.3.5 Filter Corrections ............................. 79 7.4 Turntable Rotation Period ............................ 81 7.5 Missing Reads ................................... 83 Chapter 8: Calibration ................................. 84 8.1 Angle Calibration ................................. 84 8.2 Angle to Torque Conversion ........................... 85 8.2.1 Moment of Inertia ............................ 86 8.3 Calibration TurnTable .............................. 86 8.3.1 Centering the turntable laterally .................... 87 8.3.2 Centering the turntable vertically .................... 89 8.4 Calibration data ................................. 92 8.4.1 Metrology ................................. 92 8.4.2 Error table ................................ 94 iii Chapter 9: Statistical Noise Sources .......................... 96 9.1 Thermal Fiber noise ............................... 96 9.2 Autocollimator .................................. 97 9.3 Separation-Dependent Noise Sources ...................... 97 9.3.1 Seismic Patch-field Coupling ....................... 97 9.3.2 Patch-field Potentials ........................... 99 9.3.3 Squeeze-Film Damping .......................... 100 9.3.4 Magnetic Noise .............................. 101 Chapter 10: Systematic Effects ............................. 103 10.1 Magnetic Torques ................................. 104 10.1.1 Static Magnetic Fields .......................... 104 10.1.2 Dynamic Magnetic Fields: Pendulum Dipole Coupling ........ 111 10.2 Electrostatic Torques ............................... 111 10.2.1 Foil Displacement ............................. 111 10.2.2 Separation-dependent changes ...................... 113 10.3 Parameter error .................................. 118 10.3.1 z0 error .................................. 118 10.3.2 Radial centering and ϕ-top runout ................... 119 10.4 Thermal Coupling to Twist Angle ........................ 119 10.5 Gravitational Torques .............................. 121 Chapter 11: Analysis ................................... 123 11.1 Torque Measurements .............................. 123 11.1.1 Phase Alignment ............................. 123 11.2 Torque Model and Parameters .......................... 126 11.2.1 Fourier-Bessel Tables ........................... 126 11.2.2 Radial Misalignment Correction ..................... 128 11.2.3 Tilt Correction .............................. 129 11.2.4 Torque Model ............................... 129 11.3 Newtonian Fit ................................... 131 11.4 Yukawa Fits .................................... 133 11.5 Conclusions .................................... 136 iv Bibliography ........................................ 138 Appendix A: ....................................... 146 A.1 Data ........................................ 146 A.2 Gravitational Centering Data .......................... 150 A.3 Calibration Data ................................