de Sitter Cosmology 935.2 de Sitter Cosmology
Let us now turn to another special case for which the Einstein equation can be solved.
Consider a homogeneous flat universe with the Robertson–Walker metric in which the
density of pressureless dust is constant,휌(푡)=휌 0. Friedmann’s Equation (5.17) for the
rate of expansion including the cosmological constant then takes the form
푎̇(푡)
푎(푡)=퐻, (5.57)
where퐻is now a constant:
퐻=√
8 휋
3
퐺휌 0 +
휆
3
. (5.58)
This is clearly true when푘=0 but is even true for푘≠0: since the density is constant
and푎increases without limit, the curvature term푘푐^2 ∕푅^2 will eventually be negligible.
The solution to Equation (5.57) is obviously an exponentially expanding universe:
푎(푡)∝eHt. (5.59)This is drawn as the de Sitter curve in Figure 5.1. Substituting this function into the
Robertson–Walker metric [Equation (2.32)] we obtain the de Sitter metric
d푠^2 =푐^2 d푡^2 −e^2 Ht(d푟^2 +푟^2 d휃^2 +푟^2 sin^2 휃d휙^2 ) (5.60)with푟replacing휎. In 1917 de Sitter published such a solution, setting휌=푝=0, thus
relating퐻directly to the cosmological constant휆. The same solution of course fol-
lows even with휆=0 if the density of dust휌is constant. Eddington characterized the
de Sitter universeas ‘motion without matter’, in contrast to the staticEinstein universe
that was ‘matter without motion’.
If one introduces two test particles into this empty de Sitter universe, they will
appear to recede from each other exponentially. The force driving the test particles
apart is very strange. Let us suppose that they are at spatial distancerafrom each
other, and that휆is positive. Then the equation of relative motion of the test particles
is given by Equation (5.5) including the휆term:
d^2 (ra)
d푡^2=휆
3
ra−4 휋
3
퐺(휌+ 3 푝푐−^2 )ra. (5.61)The second term on the right-hand side is the decelerating force due to the ordinary
gravitational interaction. The first term, however, is a force due to the vacuum-energy
density, proportional to the distance푟between the particles!
If휆is positive as in the Einstein universe, the force is repulsive, accelerating the
expansion. If휆is negative, the force is attractive, decelerating the expansion just like
ordinary gravitation. This is called ananti-de Sitteruniverse. Since휆is so small [see
Equation (5.22)] this force will only be of importance to systems with mass densities
of the order of the vacuum energy. The only known systems with such low densities
are the large-scale structures, or the full horizon volume of cosmic size. This is the rea-
son for the namecosmological constant. In a later chapter we shall meet inflationary
universes with exponential expansion.