Chapter 3

Cosmological models

George F R Ellis

Mathematics Department, University of Cape Town, South Africa

3.1 Introduction

The current standard models of the universe are the Friedmann–Lemaˆıtre (FL)
family of models, based on the Robertson–Walker (RW) spatially homogeneous
and isotropic geometries but with a much more complex set of matter constituents
than originally envisaged by Friedmann and Lemaˆıtre. It is appropriate then to
ask whether the universe is indeed well described by an RW geometry. There
is reasonable evidence supporting these models on the largest observable scales,
but at smaller scales they are clearly a bad description. Thus a better form of
the question is:On what scales and in what domains is the universe’s geometry
nearly RW? What are the best-fit RW parameters in the observable domain?
Given that the universe is apparently well described by the RW geometry
on the largest scales in the observable domain, the next question is:Why is it
RW? How did the universe come to have such an improbable geometry? The
predominant answer to this question at present is that it results from a very early
epoch when inflation took place (a period of accelerating expansion through many
e-folds of the scale of the universe). It is important to consider how good an
answer this is. One can only do so by considering alternatives to RW geometries,
as well as the models based on those geometries.
The third question is: How did astronomical structure come to exist on
smaller scales? Given a smooth structure on the largest scales, how was
that smoothness broken on smaller scales? Again, inflationary theory applied
to perturbed FL models gives a general answer to that question: quantum
fluctuations in the very early universe formed the seeds of inhomogeneities that
could then grow, on scales bigger than the (time-dependent) Jeans’ scale, by
gravitational attraction. It is important to note, however, that not only do structure-
formation effects depend in important ways on the background model, but also

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