Time-Accumulated Vacuum Cosmology v3.6.1 — Unified Proper-Time Tracking Model for Evolving Dark Energy: ρ₍vac₎(x) = ρ₀ [1 + k Φ(x)/c²]

Abstract

Time-Accumulated Vacuum Cosmology v3.6.1 — Unified Proper-Time Tracking Model for Evolving Dark Energy ρ₍vac₎(x) = ρ₀ [1 + k Φ(x)/c²] Overview FEU v3.6.1 is a phenomenological cosmological framework in which the vacuum sector is allowed to respond to accumulated cosmic proper time and, in the broader FEU formulation, to the local gravitational environment. The original FEU concept is retained as a foundational local/environmental relation: ρ_vac(x) = ρ₀ [1 + k Φ(x)/c²] Here ρ_vac(x) is the local effective vacuum-energy density, ρ₀ is the reference vacuum density, Φ(x) is the local gravitational potential, c is the speed of light, and k is a phenomenological coupling parameter. The current MichaelBackground_FEU_v3.6.1.py implementation is intentionally more restricted: it is a homogeneous cosmological background solver. It does not calculate Φ(x) or fit k. Instead it tests the proper-time-dependent cosmological component using redshift, matter, radiation, early-time suppression, and a self-consistent proper-time history. FEU v3.6.1 retains the original local FEU relation while distinguishing it from the homogeneous background solver used for cosmological tests. MichaelBackground_FEU_v3.6.1.py File in downloads Cosmological background law For homogeneous cosmological calculations, FEU v3.6.1 uses: E²(z) = Ωₘ(1+z)³ + Ωᵣ(1+z)⁴ + ΩΛ [1 + α D(z)^γ S(z)] / N₀ where D(z) = t(z) / t₀ is the normalized cosmic proper-time fraction. Cosmic proper time is determined self-consistently from t(z) = ∫_z^∞ dz′ / [(1+z′)H(z′)] so D(z) depends on H(z), while H(z) itself depends on the FEU vacuum contribution. The background is therefore solved iteratively. Early-time suppression The implemented suppression function is S(z) = 1 / {1 + [(1+z)/(1+z*)]^n} The combined behaviour of D(z)^γ and S(z) suppresses the direct FEU background modification toward the early Universe while allowing it to become more relevant at later cosmic times. For the tested parameter range, the direct background modification is numerically negligible by approximately recombination redshift. This statement applies to the background model only and is not a claim that all perturbation, recombination, BBN, or primordial physics is automatically unchanged. Numerical implementation FEU v3.6.1 uses a tightened numerical implementation designed for reliability and auditability. · dense redshift sampling from z=0 to 2, · intermediate sampling from z=2 to 20, · a logarithmic grid extending to z=1100, · a matter-plus-radiation high-redshift proper-time tail, · monotonicity checks for D(z), · explicit E²(z)>0 checks, · convergence checks for E(z) and D(z), · cached comoving distances, and · precise conversion of H₀ into inverse gigayears. These changes improve numerical reliability without changing the underlying FEU concept. Relationship between the local and cosmological FEU equations FEU v3.6.1 therefore contains two conceptually related levels. The foundational local relation is ρ_vac(x) = ρ₀ [1 + k Φ(x)/c²] which expresses the original FEU proposal that the effective vacuum sector may depend on the local gravitational environment. The cosmological background implementation is ρ_vac(z) ∝ [1 + α D(z)^γ S(z)] / N₀ and this is the part currently implemented and tested against homogeneous cosmological observables. The current v3.6.1 solver therefore does not constitute a direct numerical test of the Φ(x) relation itself. A direct test of that local relation requires an independently defined gravitational-potential or environmental field and a separate spatial analysis or likelihood. Observational interpretation FEU v3.6.1 is intended to be tested with supernova distances, BAO measurements, CMB likelihoods, and related cosmological datasets. The framework is treated as a phenomenological extension of ΛCDM rather than as an established replacement for ΛCDM. Results must be tied to the exact FEU implementation, covariance treatment, nuisance parameters, parameter assumptions, priors, and likelihood used. No cosmological-background result should be interpreted as evidence for the local Φ(x) coupling unless that coupling is explicitly present in the tested model. Planck interpretation An FEU Planck 2018 TTTEEE + lowE + lensing implementation reported approximately α = 0.099 ± 0.232 with Δχ² ≈ +3.7 relative to the stated ΛCDM reference. The appropriate interpretation is that the tested FEU implementation remained statistically consistent with the Planck likelihoods. This does not constitute a CMB detection of FEU. Later FEU analyses using different normalizations or parameter conventions should be reported separately unless an explicit mapping between those parameters is demonstrated. Scientific status FEU v3.6.1 is a testable phenomenological framework. Its central hypothesis is that the vacuum sector may contain a contribution related to accumulated proper time, while the broader FEU concept retains the possible local gravitational dependence ρ_vac(x) = ρ₀ [1 + k Φ(x)/c²] The current cosmological solver tests the homogeneous time-dependent component. The local gravitational-potential component remains a separate, testable part of the framework and should not be regarded as observationally established by homogeneous cosmological results. FEU v3.6.1 therefore provides a numerically auditable basis for further tests with Planck, Pantheon+, DESI, large-scale structure, and future environmental or spatial analyses. FEU v3.6.1 release statement FEU v3.6.1 preserves the original FEU hypothesis while introducing a more reliable cosmological background implementation. The release includes a self-consistent proper-time calculation, explicit radiation contribution, early-time suppression, present-day normalization, corrected derivative structure, high-redshift tail treatment, numerical convergence safeguards, and more conservative statistical interpretation. The original local FEU relation ρ_vac(x) = ρ₀ [1 + k Φ(x)/c²] is retained as a foundational part of FEU v3.6.1 while being clearly distinguished from the homogeneous background equations implemented in the current cosmological source code. This separation allows the cosmological and environmental components 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.