Theory of cosmological perturbations in an anisotropic universe

Abstract

This paper describes the theory of cosmological perturbations around a homogeneous andanisotropic universe of the Bianchi I type. Starting from a general parametrization of theperturbed spacetime à la Bardeen, a complete set of gauge invariant variables isconstructed. Three physical degrees of freedom are identified and it is shownthat, in the case where matter is described by a scalar field, they generalize theMukhanov–Sasaki variables. In order to show that they are canonical variables, the actionfor the cosmological perturbations at second order is derived. Two major physicalimprints of the primordial anisotropy are identified: (1) a scalar–tensor ‘seesaw’mechanism arising from the fact that scalar, vector and tensor modes do notdecouple and (2) an explicit dependence of the statistical properties of the densityperturbations and gravity waves on the wavevector instead of its norm. Thisanalysis extends, but also sheds some light on, the quantization procedure that wasdeveloped under the assumption of a Friedmann–Lemaître background spacetime,and allows one to investigate the robustness of the predictions of the standardinflationary scenario with respect to the hypothesis on the symmetries of thebackground spacetime. These effects of a primordial anisotropy may be related to someanomalies of the cosmic microwave background anisotropies on large angularscales.