Time‑Accumulated Vacuum Cosmology v3.6 Unified Proper‑Time Tracking Model for Evolving Dark Energy ρ₍vac₎(x) = ρ₀ [1 + k Φ(x)/c²]
Time‑Accumulated Vacuum Cosmology v3.6 FEU: Proper‑Time Modulated Vacuum with Early‑Time Suppression and Density‑Dependent Screening FEU v3.6 is the latest corrected and stabilized version of the Time‑Accumulated Vacuum Cosmology framework. The model extends ΛCDM by introducing a small, smooth modulation of vacuum energy tied to the fraction of cosmic proper time elapsed. The accumulated term is governed byD(z) = t(z)/t₀,where the proper time is obtained from \frac{dt}{dz} = -\frac{1}{(1+z)H(z)}. To ensure full consistency with CMB, BBN, and pre‑recombination physics, FEU applies an early‑time suppression S(z) = \frac{1}{1 + \left(\frac{1+z}{1+z_*}\right)^n}, which forces the accumulated vacuum component to activate only at late times. A density‑dependent screening envelope f(\rho) = \frac{1}{1 + (\rho/\rho_{\rm crit})^m} suppresses the accumulated term inside galaxies and dense matter, keeping FEU cosmology‑active but locally silent. The vacuum energy density is modified to \rho_{\rm vac}(z) = \rho_\Lambda\left[1 + \alpha D^\gamma S(z)\right], and the normalized expansion law H^2(z) = H_0^2\left[\Omega_m(1+z)^3 + (1-\Omega_m)\frac{1+\alpha D^\gamma S(z)}{N_0}\right],\qquad N_0 = 1 + \alpha S(0), ensures H(0) = H₀ exactly. This corrects the v3.0/v3.1/v3.2 bias where the unnormalized expansion law produced an artificial RMS improvement by shifting the effective Hubble scale. Distances follow standard flat‑universe relations: \chi(z)=c\!\int_0^z\!\frac{dz^\prime}{H(z^\prime)},\quad D_L=(1+z)\chi,\quad \mu = 5\log_{10}(D_L)+25. --- Statistical Corrections and v3.6 Improvements FEU v3.6 incorporates all corrections introduced in v3.5: • Proper normalization of the vacuum term via N₀ = 1 + α S(0).• Full M‑marginalization for Pantheon+ likelihoods.• Removal of the spurious RMS improvement caused by the unnormalized solver.• Stabilized ODE integration for D(z) with backward relaxation from the matter tail.• Updated screening parameters and consistent late‑time behaviour. With these corrections, FEU v3.6 is statistically indistinguishable from ΛCDM across major background datasets: • Pantheon+ (1624 SNe): Δχ² ≲ 1• Standard cut (z > 0.01): Δχ² ≈ −0.75• 0.3–0.8 bin: Δχ² ≈ +0.5• Alternative suppression shape (z* = 1.2, n = 4): Δχ² ≈ −0.03 relative to baseline Posterior constraints allow α ≈ 0–0.2, with no strong preference for non‑zero accumulated vacuum. --- High‑Redshift Consistency Because S(z) → 0 for z ≫ z*, FEU reduces to ΛCDM at recombination: • D(z) ≈ 0.03 at z ≈ 1100• α D^γ S < 10⁻⁴ at recombination• r_s(z*), R, l_a match ΛCDM to better than 10⁻⁴ This ensures FEU v3.6 is fully compatible with: • CMB acoustic physics• early ISW• damping tail• baryon loading• neutrino sector• recombination history Late‑time effects (z < 2) produce only small changes in: • low‑ℓ ISW (≲2%)• CMB lensing smoothing (≲1%) Both are well below Planck cosmic‑variance limits. --- Lyα, BAO, and Growth Screening ensures FEU behaves identically to ΛCDM at cosmological densities. Lyα BAO remains compatible for α ≤ 0.2, and the DESI DR2 Lyα point at z = 2.33 is matched without degrading growth or S₈ constraints. FEU does not worsen the ΛCDM S₈ tension. --- Summary FEU v3.6 is a physically motivated, early‑time‑safe extension of ΛCDM. With proper normalization and full likelihood treatment, FEU remains statistically indistinguishable from ΛCDM across Pantheon+, BAO, RSD, CMB distance priors, and Lyα BAO. Both suppression shapes (baseline and alternative) remain allowed, and the model’s physical motivation—accumulated cosmic time, early‑time suppression, and density‑dependent screening—remains intact. --- Parameters (v3.6) Ωₘ = 0.37H₀ = 70 (best‑fit floated ≈ 73.07)α = 0.165γ = 1z* = 1.5 / 1.2n = 2 / 4m = 2 --- Citation Time‑Accumulated Vacuum Cosmology v3.6 (2026). Declaration of Generative AI Use in the Scientific Process The FEU concept, its motivation, and the decision to test it against cosmological datasets were entirely initiated by the author. The author defined the behaviour the model should exhibit and directed the development of the FEU framework. Generative AI tools (including Meta AI, Google’s AI systems, and Microsoft Copilot) were used to help translate the author’s conceptual description into formal mathematical expressions, to assist in structuring Python and YAML files, and to provide guidance on appropriate cosmological tests. These tools also supported the organisation and explanation of the material. All scientific responsibility for the FEU concept, its interpretation, and the conclusions of this work remains solely with the author.