\bf $\delta M$ Formalism: A New Approach to Cosmological Perturbation Theory in Anisotropic Inflation

Abstract

We study the evolution of the metric perturbations in a Bianchi background in the long-wavelength limit. By applying the gradient expansion to the equations of motion we exhibit a generalized "Separate Universe" approach to the cosmological perturbation theory. Having found this consistent separate universe picture, we introduce the $\delta M $ formalism for calculating the evolution of the linear tensor perturbations in anisotropic inflation models in {\it almost} the same way that the so-called $\delta N$ formula is applied to the super-horizon dynamics of the curvature perturbations. Similar to her twin formula, $\delta N$, this new method can substantially reduce the amount of calculations related to the evolution of tensor modes. However, it is not as general as $\delta N$, it is a "perturbative" formula and solves the shear only to linear order. In other words, it is restricted to weak shear limit.