Theory of cosmological perturbations in an anisotropic universe

Abstract

This paper describes the theory of cosmological perturbations around a homogeneous and anisotropic universe of the Bianchi I type. Starting from a general parametrization of the perturbed spacetime à 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 ‘seesaw’ 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 wavevector 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–Lemaître background spacetime, and allows one 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.