Lab 1: The Gravitational Constant and Yukawa DeviationsStatic limit of the mediator propagator
Learning Objectives
- Derive the torsion-balance sensitivity function from first principles.
- Measure the torsion constant and extract G with a complete uncertainty budget.
- Implement a Yukawa-modified gravitational potential in simulation.
- Perform two-model comparison (Newton vs Newton+Yukawa) using likelihood-ratio methods.
- Translate exclusion limits in (α, λ) space into mediator-mass constraints.
Three-Step Protocol
| Step | For this lab |
|---|---|
| Entrance Session | Present the torsion-balance physics, the Yukawa modification, and the connection to mediator mass. Explain what you will measure, why, and how. Be prepared to derive τgrav for the mass geometry. |
| Active Lab | Measure κ and extract G. Run in-lab estimates to catch systematic problems early. Record everything in your lab notes. Begin the numerical simulation alongside data-taking. |
| Findings Session | Present your extracted G with uncertainties, the systematic budget, the exclusion curve, and the likelihood-ratio result from the synthetic data. Discuss what worked, what didn’t, and what remains unresolved. |
Background
The Torsion Balance
The torsion balance measures the gravitational torque between test masses and source masses suspended from a thin fibre. The equilibrium deflection angle θ relates to the gravitational constant through:
where κ is the torsion constant and τgrav depends on G, the mass geometry, and the separation distance. You must derive τgrav for the specific mass configuration in the laboratory, accounting for finite-size corrections.
The Yukawa Modification
A massive scalar mediator modifies the Newtonian potential:
where λ = ℏ/(mϕc) is the Compton wavelength and α characterises the coupling strength relative to gravity. This follows directly from Fourier-transforming the static-limit propagator (see Framework, §2). Deriving this connection is your first quantitative exercise.
The Yukawa form is one specific phenomenological parametrisation. Different BSM scenarios (ADD extra dimensions, chameleon fields, massive graviton theories) predict different functional forms. Treat V(r) as a working tool for setting constraints — not as a fundamental law. The model-comparison extension below requires you to confront this directly.
Connecting to High-Energy Scales
The length scale λ corresponds to a mediator mass via mϕ = ℏ/(λc). Representative values:
| Compton wavelength λ | Mediator mass mϕ | Status |
|---|---|---|
| 1 mm | ≈ 0.2 meV | Constrained by Eöt-Wash |
| 10 μm | ≈ 20 meV | Near current frontier |
| 1 μm | ≈ 0.2 eV | Weakly constrained |
Representative published limits — verify current values before your report.
Experimental Programme
Measurements
- Determine the torsion constant κ by measuring the free oscillation period of the pendulum.
- Record the equilibrium deflection for at least three distinct source-mass configurations.
- Extract G from each configuration independently.
- Identify and quantify at least five systematic effects (see below).
Synthetic Data Exercise: Discovery vs Exclusion
In addition to your real measurement, you receive a synthetic dataset with an embedded Yukawa signal at λ = 0.1 m and α = 10⁻⁴ — below current published limits but above laboratory noise. You should:
- Fit the synthetic data under two hypotheses: pure Newton and Newton+Yukawa.
- Compute the likelihood ratio Λ = L(Newton) / L(Newton+Yukawa).
- Determine whether the data prefer the extended model at ≥ 2σ significance.
- Extract best-fit α and λ with uncertainties, and discuss degeneracy between G and α.
If you only perform null-result exclusion analyses, you learn only half of precision physics. The synthetic discovery exercise forces engagement with model selection, parameter degeneracy, and the difference between setting limits and claiming a signal.
Numerical Programme
Simulation Tasks
- Implement V(r) with both Newtonian and Yukawa components.
- Compute the expected torque signal as a function of λ for the laboratory mass geometry.
- Generate an exclusion curve in (α, λ) space from the real measurement uncertainty.
- Overlay published Eöt-Wash limits and identify regions where your measurement is competitive or not.
Model Comparison Extension
Implement at least two distinct deviation models and show where their exclusion contours diverge. Suggested pair: standard Yukawa (massive scalar) vs power-law modification V(r) ∝ 1/r1+ε, inspired by extra-dimensional scenarios. This demonstrates that exclusion limits are model-dependent.
Statistical and Systematic Effects
Understanding and quantifying uncertainties is central to this lab. The skills developed here carry directly into the report and seminar.
Statistical Uncertainties
- Propagate all measurement uncertainties through to G using both analytic error propagation and Monte Carlo sampling.
- Report the full covariance matrix for fitted parameters (G, κ, geometric factors).
- For the synthetic data: report Λ, the corresponding p-value, and the confidence interval on α.
- For exclusion curves: construct 95% CL contours using Feldman–Cousins or profile likelihood.
Systematic Uncertainties
Identify, estimate, and tabulate the following:
| Systematic Effect | Estimation Method | Typical Magnitude |
|---|---|---|
| Fibre anelasticity | Time-series analysis of drift | ~0.1–1% on κ |
| Gravitational coupling to laboratory masses | FEA or analytic calculation of parasitic masses | Configuration-dependent |
| Temperature drift | Correlation analysis: θ vs Tambient | ~10 ppm/K on G |
| Electrostatic and magnetic backgrounds | Shielding test or polarity reversal | Must demonstrate < 1% of signal |
| Mass metrology (density inhomogeneity) | Cross-check mass vs geometric prediction | ~0.01–0.1% on m |
| Finite-size corrections | Numerical integration over extended geometry | Geometry-dependent; must compute |
| Seismic and acoustic coupling | PSD of pendulum at quiet vs noisy times | Must be below thermal noise |
Your lab notes should include a complete uncertainty budget table listing each systematic, its estimated magnitude, and how it was determined. State whether your measurement is statistically or systematically limited.
Lab Notes
Your lab notes for this experiment should include: raw data, κ determination, G extraction for each configuration, uncertainty budget table, exclusion curve in (α, λ) space, and likelihood-ratio analysis of synthetic data. These feed directly into your report and the unified exclusion plot in Lab 4.