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We extend the standard theory of cosmological perturbations to homogeneousbut anisotropic universes. We present an exhaustive computation for the case ofa Bianchi I model, with a residual isotropy between two spatial dimensions,which is undergoing complete isotropization at the onset of inflation; we alsoshow how the computation can be further extended to more general backgrounds.In presence of a single inflaton field, there are three physical perturbations(precisely as in the isotropic case), which are obtained (i) by removing gaugeand nondynamical degrees of freedom, and (ii) by finding the combinations ofthe remaining modes in terms of which the quadratic action of the perturbationsis canonical. The three perturbations, which later in the isotropic regimebecome a scalar mode and two tensor polarizations (gravitational wave), arecoupled to each other already at the linearized level during the anisotropicphase. This generates nonvanishing correlations between different modes of theCMB anisotropies, which can be particularly relevant at large scales (and,potentially, be related to the large scale anomalies in the WMAP data). As anexample, we compute the spectrum of the perturbations in this Bianchi Igeometry, assuming that the inflaton is in a slow roll regime also in theanisotropic phase. For this simple set-up, fixing the initial conditions forthe perturbations appears more difficult than in the standard case, andadditional assumptions seem to be needed to provide predictions for the CMBanisotropies.

Inflationary perturbations in anisotropic backgrounds and their imprint on the CMB