Primordial gravitational waves and perturbations during an inhomogeneous
  inflation
  We investigate the inhomogeneous inflation, in which the space exponentially
expands with inhomogeneities, and its cosmological perturbations. The
inhomogeneous inflation is realized by introducing scalar fields with spacelike
gradients that break the spatial symmetry. We find that the space can expand
uniformly in different direction with the same rate. By using the perturbative
method, we calculate the corrections to the power spectra of gravitational
waves and curvature perturbation up to the linear order in the background
inhomogeneities. Since the background is inhomogeneous, perturbations modes
with different wave numbers get correlated. We show that generally the power
spectra of perturbations depend on the ratio and the angle of wave numbers of
the two correlated modes. In particular, the two circular polarization modes of
the gravitational waves gain different powers when the background inhomogeneity
is of vector or tensor type.
