A standard topic in an introductory General Relativity (GR) course is the study of maximally symmetric solutions. These are flat (Minkowski) spacetime, de Sitter spacetime (obtained when the cosmological constant is positive) and Anti-de Sitter spacetime (when the cosmological constant is negative). While this last space has been of great interest in physics during the last fifteen years due to its central role in the correspondence between gauge theories and gravity, it is de Sitter space with which I'll be concerned here.
The idea of cosmological inflation is our best developed idea of how the physics of the early universe might lead to the observed universe today. This idea has been widely discussed in popular books and beyond, and in this context, many students have heard the loose description that inflation occurs when the universe is in an almost de Sitter state, and undergoes exponentially rapid expansion. There is nothing wrong with this explanation, but one consequence of accepting it before having a thorough grounding in GR is that it seems to imply that de Sitter space is a solution to GR that undergoes a rapid change over time. This leads to a few confused looks when I get to maximally symmetric spaces in my course.
You see, maximal symmetry means that you should be able to look at the space at different places and at different times and the metric should be just the same. So how are we to square that with the idea of an exponentially growing universe? Well, it all comes down to coordinate choices and the crucial existence of other matter in the universe.
Pure de Sitter space - the solution to the Einstein equations with a positive cosmological constant and no other matter sources - is, indeed, a maximally symmetric space. There exist a number of particularly useful coordinate choices for this space. In some cases, these consist of picking a useful time choice, and thus defining a family of spacelike surfaces (the spatial part of the spacetime at a constant value of this time choice). This is referred to as a slicing of the space, and it is, actually, possible to slice the space in three different ways that correspond to cosmologically expanding spaces with flat, positively-curved and negatively curved spatial parts, respectively. These are the ways of describing de Sitter space that are useful when considering inflation. However, there also exists a choice of coordinates in which the metric does not depend on time at all, and the mere existence of such a choice is enough to tell us that there is no fundamental sense in which this is an expanding cosmological spacetime. In fact, from what I just wrote, you might have a related question: even in the cosmological coordinates, what decides if the universe is flat, positively, or negatively curved?
In the case of pure de Sitter space there is no answer to these questions. All the coordinate choices are equally allowed of course, and so we might as well look at the static coordinates, and there is no cosmology here. However, importantly, in cosmology we are never interested in pure de Sitter space. We know that there is other matter in the universe. This may be either in the form of particles like us, or, in the case of inflation, the background field that causes inflation in the first place - the inflaton. These types of matter mean that the behavior of the metric is at best almost de Sitter - the difference from pure de Sitter being that, crucially, there are only certain coordinate systems in which the regular matter is homogeneous and isotropic, whereas for a cosmological constant this is true in all coordinate systems. Thus an almost de Sitter space has less symmetry than pure de Sitter. One is free to transform coordinates as much as one likes, but there will no longer be any choices in which the metric is static!
Of course, we find it most convenient to discuss cosmology in the (Friedmann, Robertson-Walker) coordinates that exploit the natural homogeneity and isotropy of the relevant matter sources. This picks out a slicing of the spacetime, and in this slicing, when the universe is almost de Sitter, the universe does expand almost exponentially rapidly - inflation! This also decides among the flat, positively and negatively curved options for the spatial part of the metric.
So it matters that inflation is "quasi-de Sitter". It is this that gives sense to statements about inflation beginning, ending, and even operating in the way we usually describe. de Sitter space is beautiful symmetric and rich, but out real universe is somewhat messier, even at its earliest times.
