[Paper Review] Lectures on the Theory of Cosmological Perturbations

Robert Brandenberger|Jun 9, 2003

Cosmology and Gravitation Theories49 references4 citations

TL;DR

This paper provides a comprehensive overview of the classical and quantum theory of cosmological perturbations, emphasizing their role in linking early-universe models like inflation to late-time observational data. It demonstrates that infrared back-reaction effects from super-Hubble modes on local observables—such as the Hubble expansion rate—vanish when physical time variables (e.g., scalar field value) are used, implying no un-suppressed back-reaction in single-field models, but potentially observable in multi-field scenarios.

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.

Motivation & Objective

• To establish the theoretical framework connecting quantum vacuum fluctuations in the early universe to observable large-scale structure and CMB anisotropies.
• To analyze the classical and quantum evolution of cosmological perturbations in Friedmann-Robertson-Walker spacetimes.
• To investigate the back-reaction of cosmological fluctuations on the background geometry, particularly focusing on infrared divergences and their physical observability.
• To clarify the conditions under which back-reaction effects are unambiguously observable, especially in multi-field models versus single-field models.

Proposed method

• Derives the relativistic theory of cosmological perturbations using the 3+1 decomposition of spacetime and covariant formalism, including the expansion rate Θ = u^α;α.
• Applies second-order perturbation theory to compute back-reaction effects on the local Hubble rate, using conformal time and metric perturbations φ.
• Transforms results from conformal time to physical time variables (e.g., proper time τ or scalar field φ) to assess physical observability of back-reaction.
• Analyzes the cancellation of leading infrared back-reaction terms in single-field models, showing Θ(φ) = √3√V(φ) at leading order.
• Compares results with existing literature on back-reaction in single- and multi-field models, including work by Abramo, Fabio, and Afshordi.
• Considers implications for parametric resonance and non-perturbative effects in reheating, especially in two-field models where back-reaction may remain unsuppressed.

Experimental results

Research questions

• RQ1 Under what conditions do infrared modes of cosmological perturbations lead to un-suppressed back-reaction on the local Hubble expansion rate?
• RQ2 Why do back-reaction effects vanish in single-field models when physical time variables are used, despite apparent quadratic divergences in conformal time?
• RQ3 How does the choice of physical clock (e.g., scalar field value) affect the interpretation of back-reaction in inhomogeneous cosmologies?
• RQ4 Can back-reaction effects from super-Hubble modes be physically observable in multi-field models, and if so, under what conditions?
• RQ5 What is the role of gauge invariance and physical observability in assessing the significance of back-reaction in inflationary and pre-big-bang scenarios?

Key findings

• In single-field models, the leading infrared back-reaction contributions to the local expansion rate Θ vanish when expressed in terms of the scalar field φ, yielding Θ(φ) = √3√V(φ).
• The apparent quadratic back-reaction terms in conformal time are unphysical and cancel upon transformation to physical time, indicating no un-suppressed infrared effects in single-field models.
• In two-field models, back-reaction effects on the local Hubble rate do not cancel, suggesting potentially observable gravitational back-reaction in such scenarios.
• The cancellation of back-reaction in single-field models is robust and consistent with results from Abramo and Afshordi, indicating that infrared modes do not alter local expansion dynamics in these cases.
• Parametric resonance during reheating, which is a gauge artifact in single-field models, becomes physical and unsuppressed in multi-field models, suggesting a pathway for observable back-reaction.
• The analysis implies that back-reaction effects are not generically present in single-component universes but may emerge in more complex, multi-field cosmological models.