30 Kinematics and dynamics of an expanding universe
(ten parameters in the four-dimensional case). In this book, we pay special attention
to the de Sitter universe because it plays a central role in understanding the basic
properties of inflation. In fact, in most scenarios, inflation is nothing more than a
de Sitter stage with slightly broken time-translational symmetry.
To find the metric of the de Sitter universe, we use three different approaches
which illustrate different mathematical aspects of this spacetime. First, we obtain the
de Sitter metric in a way similar to that discussed in Section 1.3.1, namely, as a result
of embedding a constant curvature surface in a higher-dimensional flat spacetime.
For the sake of simplicity, we perform all calculations for two-dimensional surfaces.
The generalization to higher dimensions is straightforward. As a second approach,
we analytically continue metric (1.39), describing a homogeneous, isotropic three-
dimensional space of constant positive curvature with Euclidean signature, to obtain
a constant curvature space with Lorentzian signature. Finally, we obtain de Sitter
spacetime as a solution to the Friedmann equations with positive cosmological
constant.
De Sitter universe as a constant curvature surface embedded in Minkowski space-
time (two-dimensional case)Let us consider a hyperboloid
−z^2 +x^2 +y^2 =H−^2 , (1.90)embedded in three-dimensional Minkowski space with the metric
ds^2 =dz^2 −dx^2 −dy^2. (1.91)This hyperboloid has positive curvature and lies entirely outside the light cone
(Figure 1.8). Therefore, the induced metric has Lorentzian signature. (We noted
in Problem 1.8 that Lobachevski space can also be embedded in a space with
Lorentzian signature. However, Lobachevski space corresponds to a hyperbolic
surface lying within the light cone and has an induced metric with Euclidean
signature.) To parameterize the surface of the hyperboloid, we can usexandy
coordinates. The metric of the hyperboloid can then be written as
ds^2 =(xdx+ydy)^2
x^2 +y^2 −H−^2−dx^2 −dy^2 , (1.92)wherex^2 +y^2 >H−^2 .This is the metric of a two-dimensional de Sitterspace-
timeinx,ycoordinates. As with the cases considered in Section 1.3.1, it is more
convenient to use coordinates in which the symmetries of the spacetime are more
explicit. The first choice ist,χcoordinates related tox,yvia
x=H−^1 cosh(Ht) cosχ, y=H−^1 cosh(Ht) sinχ. (1.93)