The Separate Universe Approach and the Evolution of Nonlinear Superhorizon Cosmological Perturbations

TL;DR

This work surveys the nonlinear extension of superhorizon curvature perturbation conservation using a long-wavelength gradient expansion, formulating the separate universe picture in which each spatial point evolves like a local FRW universe. It introduces gauge-invariant gradient variables, notably $\zeta_i$ and $\mathcal{R}_i$, and derives their nonlinear evolution equations, showing that $\dot{\zeta}_i$ vanishes for a single dynamical degree of freedom or adiabatic equation of state, but can be driven by non-adiabatic pressure in multi-field models. The authors connect these nonlinear variables to the linear theory counterparts, present a Hamilton–Jacobi–type description for the local evolution of fields, and discuss conditions under which nonlinear curvature perturbations remain constant versus evolve, with implications for non-Gaussianity generation. The approach provides a transparent, nonperturbative framework for understanding large-scale cosmological perturbations and should have broad applicability in inflationary model analysis and beyond.

Abstract

In this letter we review the separate universe approach for cosmological perturbations and point out that it is essentially the lowest order approximation to a gradient expansion. Using this approach, one can study the nonlinear evolution of inhomogeneous spacetimes and find the conditions under which the long wavlength curvature perturbation can vary with time. When there is one degree of freedom or a well-defined equation of state the nonlinear long wavelength curvature perturbation remains constant. With more degrees of freedom it can vary and this variation is determined by the non-adiabatic pressure perturbation, exactly as in linear theory. We identify combinations of spatial vectors characterizing the curvature perturbation which are invariant under a change of time hypersurfaces.

Figures (1)

• Figure 1: Two different slicings of an inhomogeneous spacetime based on two different time variables, $T$ and $\tilde{T}$, have different normal curves defining the spatial coordinates. A particular patch of spacetime during inflation starts its life inside the horizon where it can be considered as homogeneous. Classical perturbations are generated when modes cross the horizon and freeze in. While subhorizon and homogeneous, there is a preferred time slicing in the patch to which all slicings and spatial coordinate choices should match. This fixes $f^j(x^i)=\delta^j_ix^i$. This choice separates changes in the time slicing -- which we want to consider as a gauge choice -- from possible coordinate transformations of the homogeneous spacetime.