8
The standard model of cosmology
Equation (t .50) may be understood as a constant number of massive particles
occupying a volume expanding as R3 (t) as the universe expands. Equation (t .49)
may be understood as the number density of photons (or other massless particles)
decreasing as R-3(t), as for massive matter but, in addition, the energy of each
photon decreasing as R-I(t) because of the redshifting of the photon energy
discussed in section 1.2. Another interesting case is w = -I, which gives
p = constant p= -po
(1.51 )
This may be interpreted as vacuum energy and allows us to incorporate the
cosmological constant into the discussion without introducing it explicitly. if we
wish.
1.5 Time dependence of the scale factor
It is easy to solve the Friedmann equation (t .34) in the case of zero cosmological
constant and k = 0, a spatially flat universe. Both of these assumptions are
always good approximations for sufficiently early times because, as discussed
in section 1.4, p ex R- 4 for radiation domination and p ex R-3 for matter
domination. Consequently, for a 'big-bang' universe with R -+ 0 as t -+ 0,
the J1rGNP tenn in (1.34) becomes more important than the k/R2 or A/3
tenns. With the energy density p given by (t .48), the solution of (t.34) (provided
w # -I) is
R(t) ex t-~(\+W).
(t .52)
In particular,
R ex tl/2
and
H = !t- I
for radiation domination
(1.53)
and
2-1
R ex t 2 / 3
and
H = Jt
for matter domination.
(1.54)
However, if at some stage in the history of the universe the cosmological constant
is (positive and) large enough to dominate over the energy density and curvature
tenns in (1.34), then the Friedmann equation has the solution
fA,
R(t) ex eV'I .
(1.55)
This is the de Sitter universe.
1.6 Age of the universe
We shall estimate the age of the universe in the case A = o. We shall also
assume a matter-dominated universe for the calculation. This is a reasonable