A Phase-Dependent Mechanism for the Hubble Tension from Oscillatory Spacetime

We propose a mechanism for the Hubble tension in which spacetime is modelled as a self-oscillating scalar field in a non-zero ground state. The 8.3% discrepancy between CMB-inferred H₀ = 67.4 km/s/Mpc and locally measured H₀ = 73.0 km/s/Mpc is modelled as arising because each measurement probes the universal oscillation at a different phase angle. The background amplitude A = 1/2 is established as a geometric normalization choice from the Dirichlet boundary condition at the causal horizon, eliminating it as a free parameter. Because the exact damped cosmological field equation is intractable, we derive the oscillatory correction from the structural symmetries of the canonical energy-momentum tensor T₀₀ combined with the cosmological Virial theorem, yielding a unique phase correction proportional to sin⁴(Φ).

The nonlinear coupling constant is determined self-consistently from the two observed H₀ endpoints; it satisfies naturalness, but is not an independent prediction—it encodes the amplitude of the Hubble tension. The predictive content of the framework lies in the functional form of H(z): the sin⁴(Φ) phase profile at intermediate redshifts is distinguishable from all smooth w(z) interpolations. A full-covariance statistical analysis against DESI DR2 BAO measurements (incorporating the official DM/DH anti-correlations) yields a Δχ² = −8.47 improvement over flat ΛCDM across the primary 10-bin sample. The phase dependence additionally produces an effective phantom dark energy equation of state (w₀ = −1.186), consistent with the framework's bimetric class membership.

The entire framework derives from a single Z₂-symmetric master Lagrangian. The fully covariant action is:

S = ∫ d⁴x √(−g) [ (1/2) g^μν ∂μφ ∂νφ − (ω₀²/2c²)φ² − (λω₀²/4c²)φ⁴]

On a homogeneous and isotropic FLRW background, spatial gradients vanish by symmetry and the √(−g) factor contributes an a³(t) volume element. Working in cosmic time, this reduces to the effective Lagrangian density:

ℒ = (1/2c²)φ̇² − (ω₀²/2c²)φ² − (λω₀²/4c²)φ⁴

where ω₀ is the fundamental frequency set by the causal horizon, A = 1/2 is fixed geometrically by the Dirichlet boundary condition, and λ ≈ 5.09 (harmonic approximation) or 2.88 (exact Duffing solution) is determined entirely from the two observed H₀ endpoints. The Z₂ symmetry (φ → −φ) forbids all odd-powered terms, making the quartic the unique lowest-order self-coupling permitted. The Dirichlet boundary condition, Z₂ symmetry, and Virial theorem averaging all operate consistently at this reduced level. Every result in this repository — the phase-dependent resolution of the Hubble tension via the sin⁴(Φ) [or sn⁴(Φ)] correction, the Δχ² = −8.47 improvement over flat ΛCDM on DESI DR2, the effective phantom dark energy equation of state (w₀ ≈ −1.186), and the vacuum energy scale — follows directly from this Lagrangian with no additional free parameters. The framework is therefore sharply falsifiable: it is either correct in its present form, or it is not.

About This Repository

This record serves as the complete, reproducible archival package for the oscillatory spacetime framework. It contains the primary manuscript, theoretical companion papers, step-by-step mathematical derivations, and the Python code required to reproduce all tables, figures, and statistical claims from the text.

Input vs. Prediction

To evaluate this framework, readers should note the structural distinction between inputs and predictions:

Inputs: The early-universe (H_CMB) and late-universe (H_local) expansion rates are inputs used as boundary conditions. The framework uses them to verify the dressed geometric prediction for the coupling constant. They are not used to calibrate free parameters.

Predictions: The genuine, falsifiable predictions of this theory are (1) the intermediate-redshift H(z) curve shape, (2) the phantom dark energy equation of state (w(z) < −1), (3) the improved fit to intermediate BAO observables, and (4) the geometric value λ ≈ 3 for the bare quartic coupling, dressed to λ = 2.88 by the cosmological expansion history.

Definitive Statistical Baseline

The definitive statistical benchmark for this framework is the full-covariance DESI DR2 analysis (detailed in Section 5.4 and computed via the attached full_cov_chisq1.py script), which demonstrates that the framework's specific DM/DH signature aligns significantly better with the correlated DESI measurements than ΛCDM.

File Inventory

(1) Supplementary Note (Supplementary-Variational-Note-final.docx): The explicit, step-by-step variational derivation of the master field equations, including the structural derivation of the sin⁴(Φ) correction via the Z₂-symmetric quartic self-interaction and the cosmological Virial theorem (Turner 1983). This is the Lagrangian foundation from which all downstream results flow.

(2) Technical Note (Duffing-Technical-Note-final.docx): Solves the exact un-damped Duffing oscillator using Jacobi elliptic functions to rigorously prove that the oscillatory spacetime framework's quantitative conclusions are robust against the sinusoidal approximation.

(3) supplementary_integration_v3.py: Python script that performs the self-consistent Friedmann integration. Reproduces all data tables across the main and companion manuscripts.

(4) Main Manuscript (Hubble-Tension-final.docx): The primary paper detailing the mechanism, the self-consistent Friedmann integration, the DESI DR2 full-covariance comparison, and the unbroken derivation chain from the Bekenstein disformal metric to the sin⁴(Φ) correction.

(5) full_cov_chisq1.py: Python script that computes the DESI DR2 full-covariance χ² comparison.

(6) Companion Paper (Vacuum-Energy-final.docx): Demonstrates how the oscillatory spacetime framework's bounded-domain geometry reduces the standard QFT vacuum catastrophe, leaving a decomposable residual factor of ~17 without invoking UV mode summation.

(7) Holographic Proof (Holographic-Proof-final.docx): Auxiliary theoretical note demonstrating equivalence with holographic dark energy scaling laws, serving as the geometric consistency capstone for the cosmological sector.

(8) Quantum Foundations Paper (Quantum-Scalar-Foundations-final.docx): Establishes the bounded-domain QFT foundations of the oscillatory spacetime framework, deriving the scalar field mass, the quantum stability parameter δ_Q, and the vacuum equation of state w_Q.

(9) Quantum Entanglement Paper (OSF-Entanglement-final.docx): Derives the vacuum entanglement structure of the oscillatory spacetime framework, including the Unruh-DeWitt detector concurrence, the position anisotropy result, and Bell-CHSH analysis.

(10) Entanglement Derivation (OSF-Entanglement-Derivation-final.docx): Step-by-step mathematical derivation of the two-point Wightman function for the framework field on the Dirichlet domain, the closed-form concurrence formula, and the full multi-mode position anisotropy calculation.

(11) Observer to Observer Translation Paper (OSF_Observer-to-Observer_Translation-1.docx): Extends the fundamental observer overlap coefficient C_AB(d) into a full operator-level Bogoliubov transformation linking the scalar field mode operators of two shifted observers. Derives the spatial overlap matrix S_mn(d) in closed form for all mode pairs, constructs the required private horizon operators, and formally proves the unitarity of the full transformation. Establishes that the diagonal mode carries zero particle creation while off-diagonal β coefficients are generically non-zero — a distinctive OSF prediction testable in analogue gravity systems. The overlap gradient ∇S_mn derived here forms the mathematical backbone of the relational gravity calculation.

(12) Particle Deficit (OSF_Particle_Deficit.docx): Derives two key results for the OSF gravity sector. First, domain compression near compact objects produces a particle deficit — not an excess — scaling as |ΔN_m| ∝ (r_s/r)^√2, where the exponent √2 is a closed-form prediction of the massive dispersion relation. This is the opposite of Hawking radiation in sign, spectrum, and scaling — a falsifiable observational signature. Second, the domain compression profile R_local(r) = R_obs(1 − π²r_s/r) is derived directly from the physical scalar action, breaking the previous circularity of matching to the Schwarzschild metric.

(13) Local Gravity (OSF_Local_Gravity-1.docx): Derives local gravitational attraction from the gradient of the relational mode-overlap matrix S_mn on a spatially compressed causal domain. The Newtonian inverse-square law emerges from the compression profile R_local(r) without free parameters. The post-Newtonian effective metric enforces PPN parameters γ = 1 and β = 1, recovering Mercury's perihelion precession of 43.0 arcsec/century. In the strong-field limit, the event horizon is identified as the surface where relational overlap S_mn vanishes — a topological support boundary of the scalar field consistent with the holographic bound.

(14) Geometric Origin of the Duffing Coupling (Geometric_Origin_OSF_Duffing_Coupling.pdf): Establishes that the quartic self-coupling λ is a geometric theorem, not a fitted parameter. Starting from the disformal determinant √−g = F^{3/2} with F(φ) = 1 + (ω₀φ/c)², the bare coupling is derived as λ = 3 (homogeneous), λ = 9/4 (1D Dirichlet mode, exact rational), λ ≈ 2.15 (3D spherical volume average), and λ ≈ 2.07 (full disformal EOM via Lindstedt–Poincaré secular analysis). The cosmologically measured value λ = 2.88 is the Hubble-friction-dressed version of this geometric prediction. The 4% agreement between λ_geom = 3 and λ_obs = 2.88, with all intermediate corrections physically derived, transforms the framework from semi-predictive to fully predictive at the structural level.

To run full_cov_chisq1.py, first create a desi_dr2/ subfolder in the same directory and download two files from the CobayaSampler GitHub repository (desi_gaussian_bao_ALL_GCcomb_mean.txt and desi_gaussian_bao_ALL_GCcomb_cov.txt). Then run python3 full_cov_chisq1.py.

"Expected output for the primary 10-bin sample: Framework χ² = 18.22, ΛCDM χ² = 26.69, Δχ² = −8.47."

Special Note:

The Relational Model of Cosmology: The cosmological sector of this framework — including the geometric derivation of the quartic coupling, the relational Friedmann closure, and the Bogoliubov observer-to-observer translation — is developed in full in: Velasquez, L., 'The Relational Model of Cosmology,' https://doi.org/10.5281/zenodo.20260241

New Evidence (2025-2026):

(1) Model-independent w(z) reconstruction reveals two phantom crossings. Li & Fan (2026, A&A), using a model-independent approach that does not assume the CPL form, reconstructed w(z) from DESI DR2 BAO + supernovae and found two phantom crossings at z ~ 0.5 and z ~ 1.5. The RMC predicts exactly this structure: the Duffing phase modulation sn^4(Theta,m) creates an oscillatory w(z) with multiple crossings as the phase evolves. No other model predicted multiple crossings from first principles. The DESI DR2 collaboration (2025) separately reported 2.8-4.2 sigma evidence for dynamical dark energy with phantom crossing.

(2) Fragility of rigid parametrizations favors oscillatory models. Hergt et al. (2026) showed that the DESI preference for w_0 w_a CDM is fragile -- it disappears or weakens depending on which CMB likelihood (Plik, CamSpec, Hillipop) or supernova sample (DES-y5, Pantheon+) is used. This instability is expected if the true w(z) is oscillatory: forcing oscillatory data into the rigid CPL form w(z) = w_0 + w_a z/(1+z) produces unstable fits because different datasets sample different phases of the oscillation. An intrinsically oscillatory model like the RMC would be more stable across datasets.