Cosmological Unification via Axion Monodromy and Finsler Geometry: The W- Method

We propose a unified framework for the dark sector, dubbed the "W-Method", based on a single ultra-light scalar field ϕ (m ≈ 2.5 × 10−21 eV)originating from Type IIB string theory axion monodromy. This model transcends the traditional dichotomy between dark matter and modified gravity by introducing a hybrid mechanism: (1) Matter Sector: The scalar field undergoes a phase transition from a frozen dark energy state (w ≈ − 1 ) to an oscillating dark matter state ( w ≈ 0). We incorporate a strong repulsive self-interaction protected by approximate shift symmetry, which resolves the Lyman-α tension and stabilizes galactic cores (Sgr A*). (2) Geometry Sector: The gradient of the scalar field induces a preferred time-like direction, effectively modifying the spacetime geometry from Riemannian to Finslerian (Matsumoto metric) via disformal coupling. This geometric distortion provides a natural resolution to the Hubble tension by altering the distance-redshift relation. The model predicts a distinctive high-redshift galaxy luminosity cut-off at z > 10, offering a smoking gun signature for the James Webb Space Telescope (JWST).