A note on the (non-)conservation of curvature perturbation

TL;DR

This paper compares two definitions of the cosmological curvature perturbation, analyzing their conservation properties on different scales and deriving an evolution equation within a specific gauge.

Contribution

It provides a new derivation of the curvature perturbation's evolution equation using a gauge-specific approach, clarifying their non-conservation on small scales.

Findings

The curvature perturbation's conservation depends on scale.

Derived a concise evolution equation in longitudinal gauge.

Confirmed compatibility with local energy-momentum conservation.

Abstract

In this note, we compare two different definitions for the cosmological perturbation $\zeta$ which is conserved on large scales, and study their non-conservation on small scales. We derive an equation for the time evolution of the curvature perturbation on a uniform density slice through a calculation solely in longitudinal (conformal-Newtonian) gauge. The result is concise and compatible with that obtained via local conservation of energy-momentum tensor.