5.6 “Menu” of scenarios 261of the initial conditions problem. We would like to point out, however, that the
initial conditions problem is posed to us by nature, while the other problems are, at
present, not more than internal problems of theories beyond the Standard Model.
By solving these extra problems, inflation opens the door to theories that would
otherwise be prohibited by cosmology. Depending on one’s attitude, this is either
a useful or damning achievement of inflation.
De Sitter solution and inflationThe last point we would like to make concerns the
role of a cosmological constant and a pure de Sitter solution for inflation. We have
already said that the pure de Sitter solution cannot provide us with a model with
a graceful exit. Even the notion of expansion is not unambiguously defined in de
Sitter space. We saw in Section 1.3.6 that this space has the same symmetry group
as Minkowski space. It is spatially homogeneous and time-translation-invariant.
Therefore any space-like surface is a hypersurface of constant energy. To charac-
terize an expansion we can use not onlyk= 0 ,±1 Friedmann coordinates but also,
for example, “static coordinates” (see Problem 2.7), which describe an expanding
space outside the event horizon. In all these cases the 3-geometries of constant time
hypersurfaces are very different. These differences, however, simply characterize
the different slicings of the perfectly symmetrical space and there is no obvious
preferable choice for the coordinates.
It is important therefore that inflation is never realized by a pure de Sitter so-
lution. There must be deviations from the vacuum equation of state, which finally
determine the “hypersurface” of transition to the hot universe. The de Sitter universe
is still, however, a very useful zeroth order approximation for nearly all inflationary
models. In fact, the effective equation of state must satisfy the conditionε+ 3 p< 0
for at least 75 e-folds. This is generally possible only if during most of the time
we havep≈−εto a rather high accuracy. Therefore, one can use the language
of constant time hypersurfaces defined in various coordinate systems in de Sit-
ter space. Our earlier considerations show that the transition from inflation to the
Friedmann universe occurs along a hypersurface of constant time in the expanding
isotropic a coordinates(η=const), but not along ar=const hypersurface in the
“static coordinates.” The next question is out of the three possible isotropic coor-
dinate systems (k= 0 ,±1), which must be used to match the de Sitter space to
the Friedmann universe? Depending on the answer to this question, we obtain flat,
open or closed Friedmann universes. It turns out, however, that this answer seems
not to be relevant for the observable domain of the universe. In fact, if inflation
lasts more than 75 e-folds, the observable part of the universe corresponds only to
a tiny piece of the matched global conformal diagrams for de Sitter and Friedmann
universes. This piece is located near the upper border of the conformal diagram for