2.3 Conformal diagrams 51ηχi^0 πi−−πη = constχ= constI−̃̃Fig. 2.6.The conformal diagrams show explicitly that flat and open de Sitter universes
are geodesically incomplete. For instance, following a geodesic for a photon, which
arrives atχ= 0 ,into its past, we find that this geodesic leaves first the open and
then the flat de Sitter region.
Finally, we note that the hypersurfaces of constant time in all coordinate systems
become increasingly flat and similar forχπ/2asη→ 0 −. In this limit, the
scale factor is inversely proportional to conformal time or, equivalently, increases
exponentially with physical time.
The reader may naturally wonder why we need to study the same de Sitter
spacetime in three different coordinate systems. As mentioned previously, the de
Sitter spacetime is useful in a practical sense because it can be viewed as the leading
order approximation to a universe undergoing inflationary expansion. In realistic
inflationary models, time-translational invariance is broken and the energy density
varies slightly with time. The hypersurface along which inflation ends is usually the
hypersurface of constant energy density and the geometry of the future Friedmann
universe depends on its shape. It can, in principle, be the surface of constant time
in closed, flat or open de Sitter coordinates and, as a result of a graceful exit from
inflation, one obtains a closed, flat or open Friedmann universe respectively.
The full cosmic history can be represented by gluing together the pieces of
conformal diagrams describing different phases of the universe’s evolution. When
gluing these pieces, however, one should not forget that every point of the diagram