Theory of cosmological perturbations in an anisotropic universe

TL;DR

This work extends cosmological perturbation theory to a homogeneous but anisotropic Bianchi I background, deriving a full gauge-invariant SVT decomposition and showing how shear couples scalar, vector, and tensor modes. It identifies three physical degrees of freedom, generalizes Mukhanov-Sasaki variables to v and μ_λ, and constructs the second-order action in canonical form, revealing a scalar-tensor see-saw and a wave-vector dependent mass that vanish in the isotropic limit. The formalism demonstrates that sub-Hubble perturbations behave as decoupled harmonic oscillators, supporting standard quantization, while anisotropy leaves imprints on large scales through mixing and k_i dependence. The results illuminate how primordial anisotropy could affect inflationary dynamics and primordial spectra, with possible relevance to CMB anomalies and a more robust understanding of perturbations in non-Friedmann backgrounds.

Abstract

This article describes the theory of cosmological perturbations around a homogeneous and anisotropic universe of the Bianchi I type. Starting from a general parameterisation of the perturbed spacetime a la Bardeen, a complete set of gauge invariant variables is constructed. Three physical degrees of freedom are identified and it is shown that, in the case where matter is described by a scalar field, they generalize the Mukhanov-Sasaki variables. In order to show that they are canonical variables, the action for the cosmological perturbations at second order is derived. Two major physical imprints of the primordial anisotropy are identified: (1) a scalar-tensor ``see-saw'' mechanism arising from the fact that scalar, vector and tensor modes do not decouple and (2) an explicit dependence of the statistical properties of the density perturbations and gravity waves on the wave-vector instead of its norm. This analysis extends, but also sheds some light on, the quantization procedure that was developed under the assumption of a Friedmann-Lemaitre background spacetime, and allows to investigate the robustness of the predictions of the standard inflationary scenario with respect to the hypothesis on the symmetries of the background spacetime. These effects of a primordial anisotropy may be related to some anomalies of the cosmic microwave background anisotropies on large angular scales.