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| © CAMBRIDGE UNIVERSITY PRESS 1999 |
1.2.1 Friedmann-Robertson-Walker Universes
For a homogeneous and isotropic fluid of density 
and pressure
p in a homogeneous universe with curvature k and cosmological
constant 
, Einstein's system
of partial differential
equations reduces to the two ordinary differential equations labeled
in Table 1.2 as FRW E(00) and
E(ii), for the diagonal time and spatial
components (see, e.g.,
Rindler 1977,
Section 9.9). Dividing E(00) by
H02, and subtracting E(00) from
E(ii) puts these equations into
more familiar forms. Dividing the latter by 2E(00) and evaluating all
expressions at the present epoch then gives the familiar expression
for the deceleration parameter q0 in terms of
0 and
.
Multiplying E(00) by a3, differentiating with respect
to a, and
comparing with E(ii) gives the equation of continuity. Given an
equation of state p = p (), this equation can be integrated to
determine
(a); then E(00) can be integrated to determine a (t).
Consider, for example, the case of vanishing pressure p = 0, which is
presumably an excellent approximation for the present universe since
the contribution of radiation and massless neutrinos (both having
p =
c2 / 3) to the mass-energy density is at the present epoch
much less than that of nonrelativistic matter (for which p is
negligible). The continuity equation reduces to
(4 
/ 3)
a3 = M
= constant, and E(00) yields Friedmann's equation
This gives an expression for the age of the universe
t0 which can
be integrated in general in terms of elliptic functions, and for
= 0 or k = 0 in terms
of elementary functions (cf. standard textbooks, e.g.
Peebles 1993,
Section 13, and
Felton & Isaacman
1986).
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Figure 1.1. (a) Evolution of the scale
factor a (t) plotted vs. the time
after the present (t - t0 in units of Hubble time
tH |
Figure 1.1 (a) plots the evolution of the
scale factor a for three
interesting examples: (0, =
(1,0), (0.3,0),
and (0.3,0.7). Figure 1.1 (b) shows how
t / tH depends on 0
both for = 0 (dashed) and
= 1 - 0 (solid).
Notice that for = 0,
t0 / tH is somewhat greater for
0 = 0.3 (0.81) than
for = 1 (2/3), while for
0 = 1- = 0.3 it is substantially greater: t0 /
tH
= 0.96. In the latter case, the competition between the attraction
of the matter and the repulsion of space by space represented by the
cosmological constant results in a slowing of the expansion at a ~
0.5; the cosmological constant subsequently dominates, resulting in
an accelerated expansion (negative deceleration q0 =
-0.55 at the present epoch), corresponding to an inflationary universe. In
addition to increasing t0, this
behavior has observational implications that we will explore in
Section 1.3.3.
H0-1 = 9.78 h-1 Gyr for
three different cosmologies:
Einstein-de Sitter (