10
The standard model of cosmology
where the parameter h is measured to have the value
h = 0.72 ± O.OS.
(1.66)
Thus, the present age of the universe is
to '" 10 10 yr.
(1.67)
1.7 The cosmological constant
In 1917, attempting to apply his general theory of relativity (GR) to cosmology,
Einstein sought a static solution of the field equations for a universe filled with
dust of constant density and zero pressure. The general static solution of (1.34)
and (1.36) has
p= ~ (_A _ _ p)
(1.68)
3 41£GN
and
k
81£GN
A
- = - - p + -
(1.69)
R2
3
3·
With zero cosmological constant (A = 0), the only solution of these equations,
apart from an empty, flat universe, requires that either the energy density p or the
pressure p is negative. It was this unphysical result that led him to introduce the
cosmologicalterm. Then the solution for pressureless dust is
A
(1.70)
p = 41£GN
and
k
R2 =A.
(1.71)
Assuming that p is positive requires that A is positive, so that
k = +1
(1.72)
and
I
R = ../K.
( 1.73)
Hence, the universe is closed and has the geometry of S3 with volume V and mass
M given by
v =
1£
21£2 R3 = 21£2 A -3/2
( 1.74)
M = 2GN"/x.
A non-zero cosmological constant also allows non-trivial static (de Sitter)
solutions of the Einstein field equations with no matter (p = 0 = p) at all. It
was, therefore, a considerable relief in the 1920s when the redshifts of distant