  We extend the standard theory of cosmological perturbations to homogeneous
but anisotropic universes. We present an exhaustive computation for the case of
a Bianchi I model, with a residual isotropy between two spatial dimensions,
which is undergoing complete isotropization at the onset of inflation; we also
show 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 gauge
and nondynamical degrees of freedom, and (ii) by finding the combinations of
the remaining modes in terms of which the quadratic action of the perturbations
is canonical. The three perturbations, which later 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 the anisotropic
phase. This generates nonvanishing correlations between different modes of the
CMB anisotropies, which can be particularly relevant at large scales (and,
potentially, be related to the large scale anomalies in the WMAP data). As an
example, 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 this simple set-up, fixing the initial conditions for
the perturbations appears more difficult than in the standard case, and
additional assumptions seem to be needed to provide predictions for the CMB
anisotropies.
