Anisotropic inflation reexamined: upper bound on broken rotational
  invariance during inflation
  The presence of a light vector field coupled to a scalar field during
inflation makes a distinct prediction: the observed correlation functions of
the cosmic microwave background (CMB) become statistically anisotropic. We
study the implications of the current bound on statistical anisotropy derived
from the Planck 2013 CMB temperature data for such a model. The previous
calculations based on the attractor solution indicate that the magnitude of
anisotropy in the power spectrum is proportional to $N^2$, where $N$ is the
number of $e$-folds of inflation counted from the end of inflation. In this
paper, we show that the attractor solution is not compatible with the current
bound, and derive new predictions using another branch of anisotropic
inflation. In addition, we improve upon the calculation of the mode function of
perturbations by including the leading-order slow-roll corrections. We find
that the anisotropy is roughly proportional to
$[2(\varepsilon_H+4\eta_H)/3-4(c-1)]^{-2}$, where $\varepsilon_H$ and $\eta_H$
are the usual slow-roll parameters and $c$ is the parameter in the model,
regardless of the form of potential of an inflaton field. The bound from Planck
implies that breaking of rotational invariance during inflation (characterized
by the background homogeneous shear divided by the Hubble rate) is limited to
be less than ${\cal O}(10^{-9})$. This bound is many orders of magnitude
smaller than the amplitude of breaking of time translation invariance, which is
observed to be ${\cal O}(10^{-2})$.
