This is the generalization of (5, 6, 7).

A positive -term increases the acceleration of the expansion, and gives rise to the possibility that the two terms on the right hand side of (14) can at some time during the evolution balance:

This "zero-acceleration" time can easily be shown to occur at a redshift z given by

At a time before the discovery of the cosmic expansion by Hubble, Einstein proposed that the Universe could be static if there was such a term, and that the value of would determine the cosmic density. The later discovery of the cosmic expansion did not, however, cause people to drop the term from the equations.

We can integrate (14) once. Using the conservation of matter expressed as = a^-3 (equation 7), we get

The version of equation (13) follows by evaluating the integration constant k (the curvature constant) at the present epoch:

(15)

to give

(16)

A universe with k = 0 is said to be "flat". If = 0, then the flat universe is a = 1 universe (the Einstein de Sitter model). There is a considerable body of opinion in favour of k = 0, but until the idea of an early "inflationary" phase of expansion was introduced by Guth, the reasons for favouring such a model were largely aesthetic. Inflation generally demands k = 0. (There was an argument that large scale structures would have to form very early (z > ₀^-1) if ₀ were small. Since we see the quasar population growing to a maximum more recently than a redshift of 3, this would suggest a relatively recent formation epoch for galaxy clusters if we could think of some argument relating QSO activity to the origin of clusters! We have in fact no direct evidence as to when the first large scale structures formed.)

1.3.2. The parameter

Note that we can introduce a dimensionless measure of :

(17)

Then the curvature constant, k becomes

The k = 0 flat universe such as implied by inflationary theories therefore has

If we argued that the dark matter was all baryonic, contributing ₀ = 0.2, then we would need = 0.8 for consistency with standard inflationary scenarios.

The coasting redshift in terms of is

(18)

For values of such as those described above for a flat universe we see a coasting period at relatively recent redshifts z 1 - 2.

1.3.3. Why introduce ?

Current thinking on the issue of whether should be there or not varies over a short timescale of a few years. There is certainly no observational evidence for including the term in the equations. From the point of view of our limited understanding of the status of the Einstein Field Equations in Quantum Field theory, there is every reason to want it to be exactly zero. However, it brings an extra parameter into the cosmological model and an extra degree of freedom with which to fit the observations. Thus we often see it being brought in at a time when there appear to be difficulties in explaining a set of observations.

A notable example of "wheeling in the cosmological constant" was in the case of explaining why there were so many more Quasars having a redshift near to 2 than might have been expected. This could be explained by putting the coasting period at a redshift of 2 (Petrosian, Salpeter and Szekeres, 1967). More recently, it has been noticed that certain standard models for galaxy formation do not have enough clustering on the largest scales. This might be explained in part by the dynamical influence could have on the formation of large scale structure (Efstathiou, Sutherland and Maddox, 1990; Lahav et al. 1991).