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

TL;DR

This paper introduces the elta Mormalism, a new perturbation approach for anisotropic inflation that simplifies tensor mode calculations in the weak shear limit, extending the separate universe concept.

Contribution

The elta Mormalism provides a novel, simplified method for evolving tensor perturbations in anisotropic inflation, analogous to the elta Normalism for curvature perturbations.

Findings

The formalism reduces tensor mode calculation complexity.

It applies to weak shear, linear shear regimes.

It extends the separate universe approach to anisotropic backgrounds.

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.