Anisotropic inflation reexamined: upper bound on broken rotational invariance during inflation

TL;DR

This paper reexamines anisotropic inflation models with vector fields, deriving a new upper bound on rotational invariance breaking during inflation based on Planck data, significantly tighter than previous estimates.

Contribution

It introduces a new branch of anisotropic inflation solutions and improves mode function calculations with slow-roll corrections, refining bounds on rotational invariance breaking.

Findings

Bound on rotational invariance breaking is less than 10^{-9}.

Attractor solutions are incompatible with current bounds.

Anisotropy is proportional to slow-roll parameters and model-specific factors.

Abstract

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 ${\cal O}(10^{-2})$.