Æther Model – Inflationary Stability Against Bianchi I Anisotropies in the Quadratically Coupled Scalar–Tensor Sector

This technical monograph (written in Spanish) presents a fully explicit and self-contained analysis of the stability of the inflationary regime against homogeneous anisotropies in the scalar–tensor Æther framework. The study considers a cosmological model defined by the action S = ∫ d⁴x √−g [ (M² + α φ²) R − 1/2 g^μν ∂_μ φ ∂_ν φ − λ φ⁴ ], where the effective gravitational coupling is determined by f(φ) = M² + α φ². The analysis focuses on homogeneous anisotropic geometries of Bianchi I type in order to determine whether the inflationary dynamics of the model suppresses anisotropic expansion. The work derives the full gravitational field equations and isolates the traceless dynamical sector governing anisotropic expansion. The anisotropies are characterized by the shear scalar σ² = 1/2 [ (H₁ − H)² + (H₂ − H)² + (H₃ − H)² ], where H₁, H₂, H₃ are the directional expansion rates and H = (H₁ + H₂ + H₃) / 3 is the mean expansion rate. The study provides: A derivation of the modified shear evolution equation generated by the scalar–tensor coupling, σ̇ + (3H + ḟ / f) σ = 0. An explicit integration of the differential equation leading to the closed solution σ ∝ a⁻³ / f, where a(t) is the mean cosmological scale factor. A demonstration that the anisotropic expansion is damped both by the standard relativistic term 3H σ and by the additional contribution produced by the time variation of the effective gravitational coupling. A full analysis of the inflationary slow-roll regime characterized by |φ̈| ≪ |3H φ̇| |φ̇²| ≪ V(φ) with the scalar potential V(φ) = λ φ⁴. Under the inflationary approximation H ≈ constant the cosmological scale factor evolves as a(t) ∝ exp(H t), which leads to the shear evolution σ ∝ exp(−3H t) / f. Because the coupling function f(φ) = M² + α φ² varies slowly during slow-roll inflation, the dominant behavior of the shear is governed by the exponential factor exp(−3H t). As a consequence σ → 0 and therefore σ / H → 0 when the cosmological time increases during the inflationary phase. This result demonstrates that the isotropic solution acts as a dynamical attractor of the homogeneous cosmological evolution. The analysis shows that the scalar–tensor coupling introduces an additional damping contribution proportional to ḟ / f, which can enhance the suppression of anisotropies relative to the standard relativistic case. The study is restricted to the physical domain α > 0 λ > 0 f(φ) = M² + α φ² > 0. Only homogeneous Bianchi I anisotropies are considered. Spatial perturbations, inhomogeneous modes, quantum corrections, and additional matter couplings are not included. Within the declared domain, the document establishes that the inflationary regime of the scalar–tensor Æther model is dynamically stable against homogeneous anisotropic expansion and that the cosmological evolution converges toward the isotropic inflationary attractor.