26 An introduction to the physics of cosmology
Figure 2.2.This plot shows the different possibilities for the cosmological expansion as
a function of matter density and vacuum energy. Models with total>1arealways
spatially closed (open for<1), although closed models can still expand to infinity if
v=0. If the cosmological constant is negative, recollapse always occurs; recollapse is
also possible with a positivevifmv.Ifv>1andmis small, there is the
possibility of a ‘loitering’ solution with some maximum redshift and infinite age (top left);
for even larger values of vacuum energy, there is no big bang singularity.
One further way of presenting the model’s dependence on time is via the
density. Following this, it is easy to show that
t=√
1
6 πGρ(matter domination)t=√
3
32 πGρ(radiation domination).An alternativek=0 model of greater observational interest has a significant
cosmological constant, so thatm+v =1 (radiation being neglected for
simplicity). The advantage of this model is that it is the only way of retaining the
theoretical attractiveness ofk=0 while changing the age of the universe from
the relationH 0 t 0 = 2 /3, which characterizes the Einstein–de Sitter model. Since
much observational evidence indicates thatH 0 t 0 1, this model has received
a good deal of interest in recent years. To keep things simple we shall neglect
radiation, so that the Friedmann equation is
a ̇^2 =H 02 [ma−^1 +( 1 −m)a^2 ],