Level 5 · Graduate study · advanced assessment
Vacuum-energy accounting and stability audit
Treat regulated terms, renormalized observables, and cosmological inference as distinct entries in one auditable ledger.
Complete all three challenges. Show assumptions, units, uncertainty, and the boundary between calculation, model-dependent inference, observation, and unresolved claim.
Challenge 1 · derivation
Build the regulator-to-observable ledger
For a free relativistic field, derive the leading cutoff dependence of the zero-point energy density and show where the quartic scaling enters. Re-express the calculation as bare, counterterm, renormalized, and measured entries without claiming that a regulator-dependent term is directly observed.
Required deliverables
- • Derivation with measure, degeneracy, and units shown
- • Four-column renormalization ledger
- • Comparison of two regulator choices
Scoring criteria · 20 points
- • Leading scaling and dimensions: 8 points
- • Renormalization bookkeeping: 7 points
- • Regulator comparison without overclaiming: 5 points
Challenge 2 · analysis
Stress-test radiative stability
Create a sensitivity analysis in which field multiplicities, mass thresholds, subtraction scale, and an observational target vary. Report fine-tuning or mismatch diagnostics across the grid and identify which conclusions are prescription dependent.
Required deliverables
- • A machine-readable grid of at least 24 rows
- • A sensitivity ranking with uncertainty bands
- • A claims table labeled calculation, inference, or observation
Scoring criteria · 20 points
- • Complete and reproducible parameter grid: 7 points
- • Sensitivity logic and uncertainty: 7 points
- • Evidence labels remain accurate: 6 points
Challenge 3 · design
Design a model-comparison program
Propose a cosmological model-comparison study that tests a restricted vacuum-energy mechanism against a baseline model. State the shared data, nuisance parameters, priors, forecast metric, and result that would count against the mechanism.
Required deliverables
- • A bounded hypothesis and baseline
- • Prior and nuisance-parameter table
- • Mock-data validation, forecast, and falsification rule
Scoring criteria · 20 points
- • Hypotheses are quantitatively distinguishable: 7 points
- • Prior sensitivity and data leakage controls: 7 points
- • Falsification and reporting rules: 6 points
Instructor solution outlines
Look for reasoning that survives inspection
1. Build the regulator-to-observable ledger
- • Integrate the mode energy over momentum space to expose the cutoff scaling.
- • Keep the gravitational counterterm and renormalization condition explicit.
- • Note that different regulators rearrange intermediate terms while observables require a stated prescription.
2. Stress-test radiative stability
- • Track threshold contributions and signed multiplicities separately.
- • Use logarithmic mismatch measures only after declaring the comparison target.
- • Distinguish the technical naturalness problem from an experimental detection of vacuum energy.
3. Design a model-comparison program
- • Choose observables that the mechanism changes, not merely quantities it can fit after tuning.
- • Validate the inference pipeline on simulated null and injected datasets.
- • Report posterior predictive checks and prior dependence alongside any model score.