**1.3. The Cosmological Constant
**

**
The cosmological constant has been recently reviewed by
Peebles (1988)
and by Weinberg (1989).
(See also
Klapdor and Grotz, 1986).
The wealth cosmological models that can arise through simply introducing
is
discussed in the great ancient book by
Tolman (1934).
**

**
Einstein presented a version of his famous field equations containing
an additional constant of nature, the Cosmological Constant,
. The
consequence of introducing this ad hoc term into the equations can be
seen by studying the dynamical equations with the
-term. In the
simplest case of zero pressure (which approximates the present day
circumstances):
**

**
**

(14) |

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.)

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.

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).