34 Kinematics and dynamics of an expanding universe
and hence the energy density stays constant during expansion. Substitutingε=
const into (1.66), we obtain
a ̈−H^2 a= 0 , (1.103)where
H=(8πGε/3)^1 /^2.A general solution of this equation is
a=C 1 exp(Ht)+C 2 exp(−Ht), (1.104)whereC 1 andC 2 are constants of integration. These constants are constrained by
Friedmann equation (1.67):
4 H^2 C 1 C 2 =k. (1.105)Hence, in a flat universe (k=0), one of the constants must be equal to zero. If
C 1 =0 andC 2 =0, then (1.104) describes a flat expanding de Sitter universe and
we can chooseC 1 =H−^1. If bothC 1 andC 2 are nonzero, the timet=0 can be
chosen so that|C 1 |=|C 2 |. For a closed universe (k=+1), we have
C 1 =C 2 =1
2 H
,
while for an open universe (k=−1),
C 1 =−C 2 =1
2 H
.
The three solutions can be summarized as
ds^2 =dt^2 −H−^2⎛
⎝
sinh^2 (Ht)
exp(2Ht)
cosh^2 (Ht)⎞
⎠
⎡
⎣dχ^2 +⎛
⎝
sinh^2 χ
χ^2
sin^2 χ⎞
⎠d
2⎤
⎦
k=−1;
k=0;
k=+ 1 ,(1.106)
where the radial coordinateχchanges from zero to infinity in flat and open uni-
verses. In contrast to a matter-dominated universe, where the spatial curvature is
determined by the energy density, here all three types of solutions exist for any given
value ofεV.They all describe the same physical spacetime in different coordinate
systems. One should not be surprised that it is possible to cover the same spacetime
using homogeneous and isotropic hypersurfaces with different curvatures, since de
Sitter spacetime is translational invariant in time.Anyspace-like hypersurface is a
constant density hypersurface.
The behavior of the scale factora(t), shown in Figure 1.11, depends on the
coordinate system. In a closed coordinate system, the scale factor first decreases,