On a sufficiently large scale, the universe we live in appears to be both spatially homogeneous and isotropic (that is, on an appropriate spatial section its matter content is uniformly distributed on average, and it looks qualitatively the same in all directions). Space-times with these properties were systematically investigated from different points of view in the pioneering work particularly of Friedmann, Lemaître, Robertson and Walker. The solutions they developed underlie the foundation of modern cosmology. They provide a wide range of possible dynamical models of the universe, among which cosmologists can identify that which most closely resembles our own on appropriately large scales. In particular, they have lead to the prediction of an initial cosmological singularity known as the big bang.

In this chapter, we will describe such a family of spatially homogeneous and isotropic space-times. These are considered as idealised cosmological models containing a perfect fluid satisfying some equation of state. As such, they represent various possible types of uniformly distributed matter, including the most important special cases of dust and radiation. They also admit a non-trivial cosmological constant. Like the vacuum space-times described in the previous three chapters, they are also conformally flat. The geometrical reason for this is that their natural three-dimensional spatial subspaces have constant curvature. This curvature can be positive, zero or negative, giving rise to different models of closed or open universes whose dynamics are uniquely determined by the specific matter content.