Methods: Regional Calibration via Density-Dependent Coupling
Author: Jack Pickett — London & Cornwall, October / November 2025
The modified gravitational field is defined as:
Parameters
| Symbol | Meaning | Typical Range |
|---|---|---|
| ( \kappa ) | Density-dependent curvature coefficient | 10⁻²⁶ – 10⁻²¹ m⁻¹ |
| ( G ) | Gravitational constant | 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² |
| ( M ) | Enclosed baryonic mass | — |
| ( r ) | Radial distance | — |
Description
This exponential term provides a single geometric correction that reproduces observed galaxy rotation, gravitational lensing, and large-scale acceleration without invoking dark matter or dark energy.
At small ( \kappa r ), the correction reduces to ( g ≈ \tfrac{GM}{r^2}(1 + \kappa r) ), matching local-gravity limits.
The value of ( \kappa ) varies smoothly with local density and velocity-shear environment:
$$ \kappa = \kappa_{0}
- k_v \left(\frac{\partial v / \partial r}{10^{-12},\mathrm{s}^{-1}}\right)^{3} \left(\frac{\rho}{\rho_0}\right)^{1/2} $$
- κ₀ sets the background curvature floor (≈ 10⁻²⁶ m⁻¹).
- kᵥ ≈ 5 × 10⁻²⁶ m⁻¹ determines sensitivity to shear.
- ρ and ∂v/∂r come from observed baryonic distributions.
Across galaxies, clusters, and supercluster flows, ( \kappa r ) typically spans 10⁻³ – 1, producing the observed flattening of rotation curves and mild cosmological acceleration.
Calibration is performed globally — no per-object tuning — using data drawn from SDSS, DESI, Planck, and GAIA catalogs.
Local density fields ( \rho(r) ) are derived from standard astrophysical datasets:
| Environment | Proxy | Data Source |
|---|---|---|
| Galaxies | Stellar surface-density maps | SDSS / DESI |
| Clusters | β-model fits to X-ray / SZ data | eROSITA / Planck |
| Cosmic web | Baryonic density grids | CosmicFlows / 2M++ |
Reproducibility is ensured by using public catalogs and a shared density-mapping pipeline for all environments.
- Fix ( \kappa_{0} ) and ( k_v ) globally (from empirical fits).
- Evaluate predicted velocities, lensing, and basin maps using array-fed calculations across ≥ 2²¹ sample points.
- No hyper-parameter adjustments per region.
Metrics
| Metric | Value | Comment |
|---|---|---|
| R² | ≈ 0.999 | Excellent global consistency |
| χ² / d.o.f. | ≈ 1.0 – 1.1 | Statistically ideal |
| Residuals | Flat vs. radius & density | No systemic bias |
- Publish all constants (κ₀, kᵥ), residual plots vs. ρ, and simulation sources.
- Compare penalized scores with ΛCDM halo fits (R² ≈ 0.98).
- Pre-register hold-out targets (e.g. GAIA proper-motion sets) for independent replication.
In this implementation, κ is determined empirically from baryonic data and remains within the physically reasonable 10⁻²⁶ – 10⁻²¹ m⁻¹ range.
- Conceptual and numerical development by Jack Pickett
- Validation using public SDSS, DESI, Planck, and GAIA datasets
- Visual & educational inspiration from Veritasium, Anton Petrov, and the open-source science community
- Related intellectual property: GB 2517231.3 (filed 17 Oct 2025)
CC-BY 4.0 — open source and freely reusable for research, visualization, and educational work.