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1.3. The Cosmological Constant Lambda

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 Lambda is discussed in the great ancient book by Tolman (1934).

1.3.1. Expansion with Lambda

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

Equation 14 (14)

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

A positive Lambda-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 zLambda 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 Lambda term, and that the value of Lambda would determine the cosmic density. The later discovery of the cosmic expansion did not, however, cause people to drop the Lambda term from the equations.

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


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

Equation 15 (15)

to give

Equation 16 (16)

A universe with k = 0 is said to be "flat". If Lambda = 0, then the flat universe is a Omega = 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 > Omega0-1) if Omega0 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 lambda parameter

Note that we can introduce a dimensionless measure of Lambda:

Equation 17 (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 Omega0 = 0.2, then we would need lambda = 0.8 for consistency with standard inflationary scenarios.

The coasting redshift in terms of lambda is

Equation 18 (18)

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

1.3.3. Why introduce Lambda?

Current thinking on the issue of whether Lambda should be there or not varies over a short timescale of a few years. There is certainly no observational evidence for including the Lambda 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 Lambda could have on the formation of large scale structure (Efstathiou, Sutherland and Maddox, 1990; Lahav et al. 1991).

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