Derivation Index
We present the minimal set of relations used throughout the project:
- State evolution: a first-order relation linking geometric state variables to their local curvature and symmetry generators; boundary terms arise from chosen domain patches.
- Constraint relation: an algebraic condition ensuring consistency between the metric-like structure and conserved measures.
- Observable map: rules converting geometric densities and distances into quantities comparable with standard cosmology (e.g., expansion rate proxies, distance moduli).
Simulations & Numerics
Numerical approach overview:
- Integrator: explicit, fixed-step with stability checks; validated on toy problems.
- Initial/boundary conditions: drawn from admissible geometric states, with edge cases tested separately.
- Outputs: trajectories, invariant checks, and observable proxies.
Parameter Estimation & Uncertainty
We estimate parameters by matching observable proxies to datasets using a robust loss and cross-validated weighting. Uncertainty is reported via bootstrapped confidence intervals and sensitivity to priors. Key steps:
- Define the likelihood-like score consistent with geometric constraints.
- Optimize with multi-start to avoid poor local minima.
- Quantify uncertainty using bootstrap and profile sweeps.
Notes, Limits, and Edge Cases
Important caveats and boundaries:
- Low-curvature limit: linearized approximations hold; deviations grow with curvature magnitude.
- Boundary terms: results depend on chosen domain; we report both inclusive and flux-corrected forms.
- Non-generic initial states: symmetry breaking can invalidate simple scaling laws; we quantify sensitivity in the parameter section.
- Numerical stability: integrator step must satisfy problem-specific constraints to preserve invariants.