Lectures on the Theory of Cosmological Perturbations

TL;DR

The notes present a comprehensive treatment of cosmological perturbations from Newtonian intuition to full relativistic and quantum formalisms. They show how sub-Hubble quantum vacuum fluctuations are stretched by inflation to super-Hubble scales, where GR dynamics govern evolution and the curvature perturbation $\mathcal{R}$ (or $\zeta$) becomes conserved on large scales, yielding a nearly scale-invariant power spectrum tied to the inflationary potential. The framework introduces the Mukhanov variable $v$ to canonically quantize scalar fluctuations and treats tensor modes analogously, predicting a stochastic background of gravitational waves. Beyond the standard picture, the lectures explore the trans-Planckian window and back-reaction, highlighting potential observational signatures and theoretical caveats, including issues of initial states, WKB non-adiabaticity, and the impact of infrared modes on local observables. Together, these analyses underpin how early-Universe physics maps onto CMB and large-scale structure data, while also outlining possible deviations arising from high-energy corrections and nonlinear effects.

Abstract

The theory of cosmological perturbations has become a cornerstone of modern quantitative cosmology since it is the framework which provides the link between the models of the very early Universe such as the inflationary Universe scenario (which yield causal mechanisms for the generation of fluctuations) and the wealth of recent high-precision observational data. In these lectures, I provide an overview of the classical and quantum theory of cosmological fluctuations. Crucial points in both the current inflationary paradigm of the early Universe and in some proposed alternatives are that, first, the perturbations are generated on microscopic scales as quantum vacuum fluctuations, and, second, that via an accelerated expansion of the background geometry (or by a contraction of the background), the wavelengths of the fluctuations become much larger than the Hubble radius for a long period of cosmic evolution. Hence, both Quantum Mechanics and General Relativity are required in order to understand the generation and evolution of fluctuations. After a review of the Newtonian theory of perturbations, I discuss first the classical relativistic theory of fluctuations, and then their quantization. Briefly summarized are two new applications of the theory of cosmological fluctuations: the trans-Planckian ``problem'' of inflationary cosmology and the study of the back-reaction of cosmological fluctuations on the background space-time geometry.

Figures (3)

• Figure 1: Space-time diagram (sketch) showing the evolution of scales in inflationary cosmology. The vertical axis is time, and the period of inflation lasts between $t_i$ and $t_R$, and is followed by the radiation-dominated phase of standard big bang cosmology. During exponential inflation, the Hubble radius $H^{-1}$ is constant in physical spatial coordinates (the horizontal axis), whereas it increases linearly in time after $t_R$. The physical length corresponding to a fixed comoving length scale labelled by its wavenumber $k$ increases exponentially during inflation but increases less fast than the Hubble radius (namely as $t^{1/2}$), after inflation.
• Figure 2: Space-time diagram (sketch) showing the evolution of scales in a cosmology of PBB or Ekpyrotic type. The axes are as in Figure 1. Times earlier than $t_B$ correspond to the contracting phase, times after describe the post-bounce phase of expansion as described in standard cosmology. The Hubble radius decreases relative to a fixed comoving scale during the contracting phase, and increases faster in the expanding phase. Fluctuations of cosmological interest today are generated sub-Hubble but propagate super-Hubble for a long time interval.
• Figure 3: Space-time diagram (physical distance vs. time) showing the origin of the trans-Planckian problem of inflationary cosmology: at very early times, the wavelength is smaller than the Planck scale $\ell _{\rm Pl}$ (Phase I), at intermediate times it is larger than $\ell _{\rm Pl}$ but smaller than the Hubble radius $H^{-1}$ (Phase II), and at late times during inflation it is larger than the Hubble radius (Phase III). The line labeled a) is the physical wavelength associated with a fixed comoving scale $k$. The line b) is the Hubble radius or horizon in SBB cosmology. Curve c) shows the Hubble radius during inflation. The horizon in inflationary cosmology is shown in curve d).