Level 5 · Graduate study teaching kit · Master’s and early doctoral level
Vacuum energy and the cosmological constant
Use the learner record during the live investigation, then use the instructor guide to facilitate comparison, address misconceptions, and assess evidence-bounded reasoning.
Learner lab record
Vacuum cutoff and curvature-mismatch ledger
How does a regulated zero-point estimate depend on cutoff, and how is that formal scale compared with observed gravitational curvature?
Setup
Use the vacuum-scale explorer. Sweep the hard cutoff by decades, keep units explicit, and compare the regulator-dependent quartic term with the declared observed dark-energy scale.
Predict first
- 1. Predict the density change for one decade of cutoff increase.
- 2. Predict whether changing regulators alone fixes the renormalized cosmological constant.
| Variable | Role | Unit |
|---|---|---|
| Ultraviolet cutoff | regulator input | energy |
| Field multiplicity/sign | model input | count |
| Regulated vacuum density | formal dependent | energy/volume |
| Observed curvature-equivalent density and mismatch | comparison diagnostic | energy/volume and decades |
Observation columns
Analyze
- 1. Does the numerical result show quartic scaling?
- 2. Which term is bare or scheme-dependent?
- 3. What measurement fixes the renormalized gravitational parameter?
- 4. Why is a large formal mismatch not a direct energy-extraction measurement?
Conclusion frame
Raising cutoff from ___ to ___ changed the regulated term by ___ orders; the comparison with observed curvature is ___, while the physical renormalized claim remains ___.
Instructor guide · 65–85 minutes
Teach the investigation, not the interface
Learning target: Learners quantify regulator sensitivity and distinguish formal vacuum contributions, renormalized cosmological parameters, observed curvature, and extractable work.
Prepare
- • Review energy-density dimensions and quartic scaling.
- • Declare the regulator and field content.
- • Provide the observational comparison scale with provenance.
Facilitation moves
- • Require a four-orders-per-decade prediction.
- • Label bare, counterterm, and renormalized quantities.
- • Ask what operational protocol would define work.
Accessibility and participation
- • Translate logarithmic ratios into orders of magnitude.
- • Use labeled layers for bare, counterterm, and observed values.
- • Offer dimensional-analysis scaffolding.
Evidence of learning
- • A verified quartic scaling result
- • A bare-versus-renormalized ledger
- • An observable-versus-extractable distinction
Misconception checks
The cutoff estimate is a measured reservoir density.
It is regulator- and scheme-dependent until embedded in a renormalized observable calculation.
Renormalization makes the cosmological-constant problem disappear.
It defines finite parameters; radiative stability and the observed small value remain deep explanatory problems.
Extension
Compare a hard cutoff with dimensional regularization and identify which physical sensitivity survives scheme changes.