Abstract: |
The presence of a light vector 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 upper bound on statistical anisotropy from Planck's CMB data for anisotropic inflation with a vector field. We first show that the previous calculations, which show that the magnitude of anisotropy is proportional to N^2 where N is the number of e-folds of inflation counted from the end of inflation, are invalid in the parameter space compatible with the current upper bound. We then show that the correct prediction is proportional to 1/\epsilon^2, where \epsilon = - \dot H/H^2 is the usual slow-roll parameter, regardless of the form of potential of an inflaton field. The current upper bound on statistical anisotropy implies that breaking of rotational invariance during inflation is limited to O (10^{-8}), which is many orders of magnitude smaller than the amplitude of breaking of time translation invariance, which is observed to be O(10^{-2}). |