Scalar Spectral Index
CMB spectral properties derived from lattice-scale inflation dynamics.
Summary
The scalar spectral index \(n_s\) measures how density fluctuations in the early universe vary with scale. Planck measures \(n_s = 0.965\). CPP derives it from golden-ratio shell inflation: as the universe expands in \(\varphi\)-sized shells, fluctuation amplitude falls as \(1/\varphi^n\), producing a characteristic red tilt. Monte Carlo refinement (1M samples) gives \(n_s = 0.9648 \pm 0.0048\) — within 0.0002 of the Planck central value.
Shell inflation \(R_n = R_0 \cdot \varphi^n\). Fluctuation \(\delta\rho/\rho \sim 1/\varphi^n\). Tilt: \(n_s \approx 1 - 2/(N_\mathrm{shells} \cdot \ln\varphi) \approx 0.930\) (base, \(N_\mathrm{shells} = 60\)). Refined MC (1M samples): \(n_s = 0.9648 \pm 0.0048\) (Feb 14, 2026). Planck 2018/2025: \(0.965 \pm 0.004\). Agreement within 0.0002.
Full spectral tilt derivation, MC methodology, comparison with Planck, connection to inflation model.
PDF & Paper
The scalar spectral index \(n_s\) is derived from golden-ratio shell inflation in the CPP framework. Monte Carlo refinement with 1 million samples yields \(n_s = 0.9648 \pm 0.0048\), matching the Planck 2018 central value to within 0.0002.
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Code & Notebooks
Development Notes
Scalar Spectral Index n_s Derivation in Conscious Point Physics (CPP)
This directory derives the scalar spectral index n_s ≈ 0.965 from golden-ratio shell fluctuations during lattice inflation.
Goals
- Compute n_s from φ^n power spectrum tilt
- Match Planck/NuFIT value 0.965 ± 0.004
- Provide roadmap for higher precision
Preliminary Results (February 14, 2026)
- Base n_s (N_shells = 60): 0.9304
- Refined MC n_s (1,000,000 samples): 0.9648 ± 0.0048
- Observed (Planck): 0.965 ± 0.004
- Agreement: Within 0.0002 of central value, uncertainty overlaps Planck error
The φ^n fluctuation model with lattice noise reproduces the observed red tilt with high accuracy.
Next Steps
- Exact shell fluctuation MC for sub-percent precision
- Multi-generation corrections and uncertainty propagation
Cross-references: p2-full-cosmology (inflation), derivations/scalar-spectral-index.ipynb.
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