Covariant Linear Perturbation Formalism

TL;DR

These notes present a covariant, gauge-aware framework for linear cosmological perturbation theory around a Friedmann–Lemaître–Robertson–Walker background. They derive the full set of scalar, vector, and tensor perturbation equations, discuss gauge transformations and common gauges, and show how to close the system in a multicomponent universe. The text applies the formalism to inflationary perturbations (including slow-roll dynamics and quantum generation of curvature and tensor modes) and to dark matter and dark energy perturbations with generalized equations of state, as well as to the cosmic microwave background via the Boltzmann equation and line-of-sight integration to obtain angular power spectra. The framework provides practical tools for connecting fundamental microphysics to observable signatures in the CMB and large-scale structure, while emphasizing gauge-consistent interpretation and numerical stability across gauges.

Abstract

Lecture notes on covariant linear perturbation theory and its applications to inflation, dark energy or matter and the cosmic microwave background.