Introduction

In this chapter we develop in detail the theory of linear perturbations of a Friedmann–Lemaître universe. This theory is of utmost importance, since we assume that the observed structure in the Universe (galaxies, cluster voids, etc.) have grown out of small initial fluctuations. Their entire evolution from the generation of the fluctuations until the time when they become of order unity can be studied within linear perturbation theory. This is especially relevant for the fluctuations in the CMB which are still very small today. It is also one of the main reasons why CMB anisotropies are so important for observational cosmology: they can be calculated to very good accuracy within linear perturbation theory, which is simple and lends itself to highly accurate and fast computations.

The idea that the large-scale structure of our Universe might have grown out of small initial fluctuations via gravitational instability goes back to Newton (letter to Bentley, 1692 (Newton, 1958)). The first relativistic treatment of linear perturbations in a Friedmann–Lemaître universe was given by Lifshitz (1946). There he found that the gravitational potential cannot grow within linear perturbation theory and he concluded that galaxies have not been formed by gravitational instability.

Today we know that in order to form structures it is sufficient that matter density fluctuations can grow. Nevertheless, considerable initial fluctuations with amplitudes of the order of 10−5 are needed in order to reproduce the cosmic structures observed today.