132 Cosmological models
FL models can occur even ifk =0ork =− 1 ,for example with a toroidal
topology [27]. These universes can be closed on a small enough spatial scale that
we could have seen all the matter in the universe already, and indeed could have
seen it many times over; see the discussion on ‘small universes’ later.
3.6.3 Growth of inhomogeneity
This is studied by looking at linear perturbations of the FL models, as well as
by examining inhomogeneous models. The geometry and dynamics of perturbed
FL models is described in detail in other talks, so I will again just make a few
remarks. In dealing with perturbed FL models, one runs into thegauge issue:
the background model is not uniquely defined by a realistic (lumpy) model and
the definition of the perturbations depends on the choice of background model
(the gauge chosen). Consequently it is advisable to use gauge-invariant variables,
either coordinate-based [2] or covariant [39]. When dealing with multiple matter
components, it is important to take carefully into account the separate velocities
needed for each matter component, and their associated conservation equations.
The CBR can best be described by kinetic theory, which again can be presented
in a covariant and gauge invariant way [10, 59].
3.7 Bianchi universes (s=3)
These are the models in which there is a group of isometriesG 3 simply transitive
on spacelike surfaces, so they are spatially homogeneous. There is only one
essential dynamical coordinate (the timet) and the EFE reduce to ordinary
differential equations, because the inhomogeneous degrees of freedom have been
‘frozen out’. They are thus quite special in geometrical terms; nevertheless, they
form a rich set of models where one can study the exact dynamics of the full
nonlinear field equations. The solutions to the EFE will depend on the matter in
the spacetime. In the case of a fluid (with uniquely defined flow lines), we have
two different kinds of models:
(1)Orthogonal models, with the fluid flow lines orthogonal to the surfaces
of homogeneity (Ellis and MacCallum [45], see also [128]). In this case the fluid
4-velocityuaisparallelto the normal vectorsnaso the matter variables will be
just the fluid density and pressure. The fluid flow is necessarily irrotational and
geodesic.
(2)Tilted models, with the fluid flow lines not orthogonal to the surfaces of
homogeneity. Thus the fluid 4-velocity isnot parallelto the normals, and the
components of the fluid peculiar velocity enter as further variables (King and
Ellis [15, 77]). They determine the fluid energy–momentum tensor components
relative to the normal vectors (a perfect fluid will appear as an imperfect fluid in
that frame). Rotating modelsmustbe tilted, and are much more complex than
non-rotating models.