Proposed Blocks Toward a UAT/UCP Boltzmann Solver: Modified Einstein Equation, Linear Perturbations, Plasma Thermodynamics, Recombination and Transfer Function, and Primordial Spectrum — Working Roadmap

Abstract

This record contains five complementary technical notes that form a working roadmap toward a future independent UAT/UCP Boltzmann solver. They are proposals, not final or validated results. The UAT/UCP framework is defined by the scalar-tensor action:S_bulk = ∫ d⁴x √-g [ M_Pl²/2 R - ½(∂φ)² - V(φ) - ξ/2 R φ² + Lm ], V=λ/4(φ²-η²)²+V0, with η=κcrit=4.978 (Ivancho limit), ξ=-0.2810, λ≈3.08×10⁻¹²², k_early=0.967, R_geom=0.279182, φ=1.618034. The five blocks are: LaTeX 1 — Toward a UAT/UCP Perturbation Theory: Proposal for linear perturbations in Newtonian gauge ds²=-(1+2Ψ)dt²+a²(1-2Φ)dx², φ=φ0+δφ. Corrected effective coupling Geff = G/(1+ξφ0²/M_Pl²) → ≈1.062 G for φ0≈0.456. Previously G/(1-ξφ0²) → 0.945 G was a sign error. Also corrected slip Φ-Ψ = ξ δ(φ²)/(M_Pl²+ξφ0²).LaTeX 2 — Proposed Modified Einstein Equation: Formal variation δS/δgμν gives (M_Pl²+ξφ²)Gμν = T_m + T_φ + ξ[∇μ∇ν - gμν□]φ². Effective Planck mass M_eff² = M_Pl²+ξφ², Geff = 1/(8πM_eff²). This note was already correct and is now the reference for Geff unification.LaTeX 3 — Proposed Thermodynamics of the Baryon-Photon Plasma: Working assumption: local thermodynamic relations unchanged. Sound speed cs²=1/3·1/(1+R), R=3ρb/4ργ, acoustic horizon rd = ∫ cs/H dz with H²=H0²[k_early Ωr(1+z)^4 + k_early Ωm(1+z)^3+ΩΛ]. Parameterization z_rec^UAT = z_rec^ΛCDM(1+δrec). Geff now unified to G/(1+ξφ²).LaTeX 4 — Proposed Recombination History and Transfer Function: τ_UAT(z)=τ_ΛCDM(z) but dη/dz=-1/H_UAT(z), visibility g(η)=-τ̇ e^{-τ}. Shift z_rec^UAT = z_rec^ΛCDM(1+δrec). Transfer T_UAT(k)=T_ΛCDM(k; Ωm^UAT=0.3133, Ωb=0.04182, h=0.7304, rd^UAT), PUAT(k)=P(k)T_UAT²(k).LaTeX 5 — Proposed Primordial Spectrum: Phenomenological extension P_UAT(k)=As(k/k*)^{ns-1}[1+ε cos(k/kosc+δ)], reduces to ΛCDM for ε=0. Heuristic kosc ~ 1/lP (κcrit^Planck)^{1/n}, n=4 analogy to M_obs(n)=2 R_geom E_Planck κcrit^{1/n}.Together they close: perturbations → modified Einstein → plasma thermodynamics → recombination & transfer → primordial spectrum. No data fitting performed. Secular drift α=0.046 Hz/day, boundary S_boundary and coherence S_coherence sectors, and full Boltzmann hierarchy remain open. Correction v2 (Aug 22, 2026): Unified sign convention for Geff across all notes. Previous LaTeX 1 used 1-ξφ0²; corrected to 1+ξφ0² consistent with action variation.The UAT/UCP framework currently lacks an independent Boltzmann solver (background.c, perturbations.c, primordial.c, transfer.c must be rewritten, as documented in DOI 10.5281/zenodo.22050113). Direct comparisons importing ΛCDM-adjusted parameters (H0, Ωm, σ8, ns, τ) are structurally biased. These five notes provide the formal blocks needed to build that solver in the future, without claiming validation.