14 Kinematics and dynamics of an expanding universe
velocities and having arbitrary pressure. We know that radiation, which has a pres-
sure equal to one third of its energy density, dominated the universe for the first
100 000 years after the big bang. Additionally, evidence suggests that most of the
energy density today has negative pressure. To understand these important epochs
in cosmic history, we are forced to go beyond Newtonian gravity and turn to a fully
relativistic theory. We begin by considering what kind of three-dimensional spaces
can be used to describe a homogeneous and isotropic universe.
1.3.1 Geometry of an homogeneous, isotropic space
The assumption that our universe is homogeneous and isotropic means that its evolu-
tion can be represented as a time-ordered sequence of three-dimensional space-like
hypersurfaces, each of which is homogeneous and isotropic. These hypersurfaces
are the natural choice for surfaces of constant time.
Homogeneity means that the physical conditions are the same at every point of
any given hypersurface. Isotropy means that the physical conditions are identical
in all directions when viewed from a given point on the hypersurface. Isotropy
at every pointautomatically enforces homogeneity. However, homogeneity does
not necessarily imply isotropy. One can imagine, for example, a homogeneous yet
anisotropic universe which contracts in one direction and expands in the other two
directions.
Homogeneous and isotropic spaces have the largest possible symmetry group;
in three dimensions there are three independent translations and three rotations.
These symmetries strongly restrict the admissible geometry for such spaces. There
exist only three types of homogeneous and isotropic spaces with simple topology:
(a) flat space, (b) a three-dimensional sphere of constant positive curvature, and
(c) a three-dimensional hyperbolic space of constant negative curvature.
To help visualize these spaces, we consider the analogous two-dimensional ho-
mogeneous, isotropic surfaces. The generalization to three dimensions is straight-
forward. Two well known cases of homogeneous, isotropic surfaces are the plane
and the 2-sphere. They both can be embedded in three-dimensional Euclidean space
with the usual Cartesian coordinatesx,y,z. The equation describing the embedding
of a two-dimensional sphere (Figure 1.4) is
x^2 +y^2 +z^2 =a^2 , (1.28)whereais the radius of the sphere. Differentiating this equation, we see that, for
two infinitesimally close points on the sphere,
dz=−xdx+ydy
z=±
xdx+ydy
√
a^2 −x^2 −y^2