Quantum Bianchi IX Anisotropies as Inflationary Relics in the Primordial Spectrum

Abstract

Starting from the WheelerDeWitt equation, we investigate whether quantum anisotropies can survive inflation as relics of the preinflationary geometry and leave an imprint on the primordial curvature spectrum. We perform a BornOppenheimer separation of the wave functional into slow gravitational and fast matter components, justified by the hierarchy between the Planck scale and the lower matter scale, associated here with the inflaton. The full wave functional is then treated through a semiclassical WKB expansion in M to the minus 1, where M is quadratic in the Planck mass. The gravitational sector consists of a de Sitter background supplemented by homogeneous quantum anisotropies in the smallanisotropy regime of the Bianchi IX potential, while matter is represented by a single scalar inflaton field. At order M in the expansion parameter, the WheelerDeWitt equation reproduces the vacuum HamiltonJacobi equation with a cosmological constant. At order M to 0, the internal freedom of the BornOppenheimer factorization separates quantum field theory on curved spacetime from the anisotropy dynamics, described by timedependent harmonic oscillators. At order M to the minus 1, we obtain the quantumgravitational correction to matter. The anisotropy wave function enters explicitly the corresponding operator, so the corrected dynamics retains information about the Bianchi IX quantum state. The curvature power spectrum consequently contains two distinguishable contributions: one independent of the anisotropy state and a genuinely anisotropic term. The latter survives through the quantum variance of the anisotropies, thereby identifying the mechanism through which the preinflationary quantum geometry can affect the primordial spectrum.