From Inflation to Dark Energy: The Complete Cascade in the String Landscape

Paper IV of the cascade cosmology series. The cascade framework (Papers I–III) proposes that two late-time vacuum transitions in Type IIB string theory — Phase 1 at z ~ 15 and Phase 2 at z ~ 1. 3 — drive the activation and relaxation of the gravitational acceleration scale a₀ = cH/6, simultaneously addressing JWST early galaxy formation, the DESI expansion mountain, and galactic-scale modified gravity phenomenology. Paper III embedded this framework in the Large Volume Scenario on the Calabi–Yau manifold P⁴1, 1, 1, 6, 9. This paper completes the cosmic history by showing that the same construction automatically provides inflation. Kähler moduli inflation along the blow-up cycle τₛ produces ~60 e-foldings with spectral index nₛ = 1 − 2/N ≈ 0. 967, consistent with Planck 2018 at 0. 4σ, and negligible tensor-to-scalar ratio r ~ 10⁻¹²–10⁻¹⁴. The COBE normalisation of the perturbation amplitude constrains the Valley 0 compactification volume to V ~ 10⁵ string units — the first determination of this geometric parameter from observational data, connecting the inflationary epoch to the late-time cascade through a single number. The inflationary direction is orthogonal to the cascade direction in moduli space: the volume modulus τb sits in Valley 0 throughout inflation, frozen by Hubble friction exceeding the barrier gradient by more than four orders of magnitude. The cosmological moduli problem is automatically resolved: τₛ decays during reheating, τb is undisplaced in its minimum, and the C₄ axion θb is massless until Phase 1 activation. No modulus dumps energy into the post-inflationary universe. The entire cosmic history — from the first 10⁻³⁵ seconds to the present epoch — is described by a single Calabi–Yau compactification with no additional fields or mechanisms beyond those already present in the LVS potential. Three Kähler moduli (τₛ, τb, θb) become dynamical at three successive epochs spanning 63 orders of magnitude in energy scale, with the sequence determined by the stabilisation hierarchy. Together with Papers I–III, these results establish cascade cosmology as a complete framework spanning inflation, reheating, the standard thermal history, structure formation, and late-time acceleration within a single string compactification.