Lectures on the Theory of Cosmological Perturbations

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.