Introduction 109(and indeed, in consequence of this remark) many of the ways of estimating the
model parameters depend on models of structure formation. Thus the previous
questions and this one interact in a number of ways.
This review will look at the first two questions in some depth, and only
briefly consider the third (which is covered in depth in Peacock’s chapter). To
examine these questions, we need to consider the family of cosmological solutions
with observational properties like those of the real universe at some stage of their
histories. Thus we are interested in thefull state space of solutions, allowing
us to see how realistic (lumpy) models are related to each other and to higher
symmetry models, including, in particular, the FL models. This chapter develops
general techniques for examining this family of models, and describes some
specific models of interest. The first part looks at exact general relations valid in
all cosmological models, the second part examines exact cosmological solutions
of the field equations and the third part looks at the observational properties of
these models and then returns to considering the previous questions. The chapter
concludes by emphasizing some of the fundamental issues that make it difficult
to obtain definitive answers if one tries to pursue the chain of cause and effect to
extremely early times.
3.1.1 Spacetime
We will make the standard assumption that on large scales, physics is dominated
by gravity, which is well described by general relativity (see, e.g. d’Inverno
[19], Wald [129], Hawking and Ellis [68] or Stephani [117]), with gravitational
effects resulting from spacetime curvature. The starting point for describing
a spacetime is an atlas of local coordinates{xi}covering the four-dimensional
spacetime manifoldM, and a Lorentzian metric tensorgij(xk)at each point of
M, representing the spacetime geometry near the point on a particular scale. This
then determines the connection componentsijk(xs), and, hence, the spacetime
curvature tensorRijkl, at that scale. The curvature tensor can be decomposed into
its trace-free part (the Weyl tensorCijkl:Cijil= 0 )and its trace (the Ricci tensor
Rik≡Rsisk)by the relation
Rijkl=Cijkl−^12 (Rikgjl+Rjlgik−Rilgjk−Rjkgil)+^16 R(gikgjl−gilgjk),(3.1)whereR≡Raais the Ricci scalar. The coordinates may be chosen arbitrarily in
each neighbourhood inM. To be useful in an explanatory role, a cosmological
model must be easy to describe—this means they have symmetries or special
properties of some kind or other.
3.1.2 Field equations
The metric tensor is determined, at the relevant averaging scale, by theEinstein
gravitational field equations(‘EFEs’)
(Rij−^12 Rgij)+λgij=κTij⇔Rij=λgij+κ(Tij−^12 Tgij) (3.2)