Inflationary perturbations in anisotropic backgrounds and their imprint on the cosmicmicrowave background

Abstract

We extend the standard theory of cosmological perturbations to homogeneous butanisotropic universes. We present an exhaustive computation for the case of a Bianchi Imodel, with a residual isotropy between two spatial dimensions, which is undergoingcomplete isotropization at the onset of inflation; we also show how the computation can befurther extended to more general backgrounds. In the presence of a single inflaton field,there are three physical perturbations (precisely as in the isotropic case), whichare obtained (i) by removing gauge and non-dynamical degrees of freedom, and(ii) by finding the combinations of the remaining modes in terms of which thequadratic action of the perturbations is canonical. The three perturbations, whichlater in the isotropic regime become a scalar mode and two tensor polarizations(gravitational wave), are coupled to each other already at the linearized level during theanisotropic phase. This generates non-vanishing correlations between different modes ofthe cosmic microwave background (CMB) anisotropies, , which can be particularly relevant at large scales (and, potentially, be related to the largescale anomalies in the WMAP (Wilkinson Microwave Anisotropy Probe) data). As anexample, we compute the spectrum of the perturbations in this Bianchi I geometry,assuming that the inflaton is in a slow roll regime also in the anisotropic phase. For thissimple set-up, fixing the initial conditions for the perturbations appears more difficult thanin the standard case, and additional assumptions seem to be needed to provide predictionsfor the CMB anisotropies.