Statistical anisotropy of primordial gravitational waves from generalized $\delta N$ formalism

Abstract

In this letter, we demonstrate how to use the generalized $\delta N$ formalism, which enables us to compute the evolution of all the large scale fluctuations, including gravitational waves, solely by solving the evolution of the background homogeneous Universe. Using the Noether charge density, we derive an analytic formula which describes the mapping between the fluctuations at the horizon crossing and the sourced gravitational waves at the end of inflation. This formula can apply also to an inflation model with an anisotropic background. Using this formula, we discuss the condition for the non-vanishing linear polarization and the qualitative difference between single- and multi-gauge field models.