So far we have only studied the evolution of perturbations assuming that the initial conditions are fixed and given once and for all. Now we want to study how classical perturbations are generated out of quantum fluctuations during a simple inflationary phase. The fact that inflation generates a nearly scale-invariant spectrum of scalar perturbations in good agreement with the observations of the cosmic microwave background is to be considered as its greatest success. The solution of the flatness and entropy problems with an inflationary phase are actually ‘post-dictions’ while the scale-invariant spectrum of scalar perturbations was first predicted in Mukhanov & Chibishov (1982) and Mukhanov et al. (1992), long before its discovery by the COBE satellite by Smoot et al. (1992). It represents therefore a real prediction of inflation. There are also other models for structure formation which predict a scaleinvariant spectrum of fluctuations but which disagree with the detailed spectrum of CMB fluctuations such as topological defects (Durrer et al., 2002).

In this chapter we first study perturbations in a FL universe filled with a scalar field. Next we discuss the generation of fluctuations during inflation. We especially determine the spectral index of scalar and tensor perturbations and the ratio of their amplitudes in the slow roll approximation. This will lead us to the well known consistency relation for slow roll inflation. We study in detail the simple case of one scalar field, the ‘inflaton’.