Time‑Accumulated Vacuum Cosmology v3.6 Unified Proper‑Time Tracking Model for Evolving Dark Energy ρ₍vac₎(x) = ρ₀ [1 + k Φ(x)/c²]

Abstract

FEU Cosmology v3.6.1 — Corrected & Updated Release Time-Accumulated Vacuum Cosmology with Early-Time Suppression and Density-Dependent Screening FEU v3.6.1 is a phenomenological extension of \LambdaCDM in which the vacuum sector contains a smooth contribution that depends on accumulated cosmic proper time. The framework includes time accumulation, early-time suppression, and an optional density-dependent environmental screening term. This release incorporates mathematical, numerical, and statistical corrections intended to ensure that the FEU vacuum law, effective equation of state, background solver, and interpretation of observational results are internally consistent. No new physical mechanism is introduced in v3.6.1. The purpose of the release is to provide a corrected and auditable implementation of the existing FEU framework.MichaelBackground_FEU_v3.6.1.py in downloads Core FEU formulation The cosmological FEU vacuum contribution is written as \rho_{\rm vac}(z) = \rho_{\Lambda} \frac{1+\alpha D(z)^\gamma S(z)}{N_0}, where D(z)=\frac{t(z)}{t_0} is the normalized cosmic proper-time accumulation factor and \frac{dt}{dz} = -\frac{1}{(1+z)H(z)}. The early-time suppression function is S(z) = \frac{1} {1+\left[\frac{1+z}{1+z_*}\right]^n}. The normalization N_0 = 1+\alpha D(0)^\gamma S(0) ensures the present-day background is normalized consistently so that H(0)=H_0. Since D(0)=1, N_0=1+\alpha S(0). The optional density-dependent environmental screening function is f(\rho) = \frac{1} {1+\left(\rho/\rho_{\rm crit}\right)^m}. For homogeneous cosmological background calculations, f(\rho)=1. The screening term is therefore a model ingredient for possible environmental extensions and is not claimed to have been established observationally by the cosmological likelihood analyses reported here. Corrected equation of state An important correction is the use of the full derivative of the FEU vacuum contribution. For D(z)^\gamma S(z), the required derivative is \frac{d}{dz} \left[D^\gamma S\right] = \gamma D^{\gamma-1}D'S + D^\gamma S'. The corresponding effective equation of state is therefore calculated using both the proper-time derivative and the suppression-function derivative: w(z) = -1 + \frac{1+z}{3} \frac{ \alpha \left[ \gamma D^{\gamma-1}D'S + D^\gamma S' \right] }{ 1+\alpha D^\gamma S }. The earlier statement that w_0=-1 exactly for all \alpha applied only to a simplified implementation. In the full FEU formulation, D'(0) contributes to the derivative, so w_0 is calculated numerically and is not imposed to equal -1. Self-consistent numerical background The FEU background must be solved iteratively because D(z) depends on H(z), while H(z) itself depends on the FEU vacuum contribution. The corrected v3.6.1 numerical implementation includes: ● a dense redshift grid; ● self-consistent iteration of E(z) and D(z); ● a matter-plus-radiation high-redshift tail; ● monotonicity checks for D(z); ● explicit positivity checks for E^2(z); ● convergence tests; ● cached comoving-distance evaluation; ● precise conversion of H_0 to inverse gigayears. The normalized expansion law is E^2(z) = \Omega_m(1+z)^3 + \Omega_r(1+z)^4 + \Omega_\Lambda \frac{ 1+\alpha D(z)^\gamma S(z) }{ N_0 }, with \Omega_\Lambda = 1-\Omega_m-\Omega_r. Early-time behaviour The FEU modification is strongly suppressed toward the early Universe through the combined behaviour of D(z)^\gamma S(z). For the tested baseline configurations, the direct FEU background modification becomes numerically negligible by approximately recombination redshift, z\sim1100. This statement applies specifically to the tested background implementation. It does not imply that all aspects of BBN, recombination physics, perturbation evolution, or primordial cosmology are automatically unchanged. Those require their own dedicated tests. Planck constraints FEU was tested using the stated Planck 2018 {\rm TTTEEE + lowE + lensing} Cobaya/CAMB likelihood implementation. For the implementation reported in this release, the fitted FEU parameter was approximately \alpha = 0.099 \pm 0.232, with \Delta\chi^2 \approx +3.7 relative to the stated \LambdaCDM reference. Under this implementation, the tested FEU model remains statistically consistent with the Planck likelihoods. The appropriate interpretation is that Planck does not provide statistically compelling evidence for a non-zero FEU parameter in this analysis, nor does it provide a statistically compelling rejection of the tested FEU model. No CMB detection of FEU is claimed. Later FEU analyses using different parameter normalizations or implementations should be reported separately and should not be compared directly with this value unless the parameter mapping is explicitly demonstrated. Pantheon+ and late-time observations The corrected supernova treatment includes the absolute-magnitude nuisance parameter and full covariance where required. Earlier apparent improvements obtained from simplified, diagonal-error, unnormalised, or otherwise incomplete likelihood implementations are not treated as evidence for FEU. Late-time analyses indicate that the inferred value of the FEU parameter can depend materially on the adopted dataset, covariance treatment, nuisance parameters, and cosmological assumptions. Any preference for non-zero \alpha should therefore be quoted together with the exact likelihood, covariance, parameter constraints, and model assumptions used to obtain it. No model-independent supernova detection of FEU is claimed. DESI BAO DESI BAO provides an important test of the late-time and intermediate-redshift FEU background. The tested DESI comparisons demonstrate sensitivity to the FEU parameter space but do not, by themselves, establish a model-independent FEU detection. Earlier sensitivity estimates such as \sigma(\alpha)\approx0.12 should be identified explicitly as forecasts when they are not derived from a completed likelihood fit. Measured DESI constraints and forecasts should therefore remain clearly distinguished. Combined late-time analyses Where Pantheon+ and DESI are analysed jointly, results should be interpreted strictly within the specified FEU implementation and frozen cosmological assumptions. A statistically preferred non-zero value of \alpha in a particular combined likelihood would constitute evidence for that parameter within the tested model class. It would not, by itself, establish FEU as the unique explanation of the observations or rule out \LambdaCDM as a broader cosmological framework. Independent replication and robustness tests remain necessary. w_0-w_a interpretation Earlier versions included simplified mappings between \alpha and the CPL parameters w_0,\quad w_a. These mappings are now treated only as illustrative approximations. The authoritative FEU equation of state is obtained numerically from the full derivative of \rho_{\rm vac}(z) \propto 1+\alpha D(z)^\gamma S(z). Any quoted w_0 or w_a values should therefore be derived from the numerical FEU background rather than inferred from an approximate algebraic mapping. Spatial and environmental extensions The cosmological FEU background defined above is homogeneous at fixed redshift. Consequently, the background model does not by itself predict different clock rates, expansion rates, or FEU amplitudes for different sky directions at the same redshift. Any spatially varying FEU timing field, environmental synchronization model, branch-dependent effect, or density-dependent local extension constitutes an additional hypothesis and must be tested separately from the frozen homogeneous v3.6.1 background. The density-screening function should therefore not be interpreted as observational evidence for local time variations unless such a test is independently performed. Statistical interpretation FEU v3.6.1 explicitly distinguishes between: ● observational consistency; ● fitted parameter constraints; ● forecasts; ● localized or subset-level features; ● model-comparison statistics; ● and statistically significant evidence for a non-zero FEU parameter. No global FEU detection is claimed solely on the basis of the analyses summarized in this release. Any quoted significance, \Delta\chi^2, Bayesian evidence, or confidence interval should be tied explicitly to: ● the dataset; ● likelihood implementation; ● covariance treatment; ● nuisance parameters; ● parameter priors; ● fixed cosmological assumptions; ● and FEU implementation. This is particularly important when comparing results obtained using different FEU normalizations or historical versions of the model. Scientific status FEU is presented as a testable phenomenological extension of \LambdaCDM, not as an established replacement for \LambdaCDM and not as a model-independent fundamental theory. The central observational question is whether allowing a smooth proper-time-dependent vacuum contribution provides a reproducibly improved description of cosmological observations when major datasets are analysed simultaneously and consistently. A future statistically significant and independently reproduced preference for non-zero \alpha would support FEU phenomenology within the specified model class. It would not constitute a mathematical proof that \LambdaCDM is impossible. v3.6.1 release statement FEU Cosmology v3.6.1 incorporates: ● the complete derivative of the FEU proper-time term; ● corrected treatment of w(z) and w_0; ● explicit present-day normalization through N_0; ● radiation in the background expansion; ● a tightened self-consistent numerical solver; ● improved high-redshift treatment; ● revised Planck interpretation; ● clearer distinction between measured constraints and forecasts; ● removal o