Observations and horizons 139Bianchi evolution theorem 2.In each set of Bianchi models of a type admitting
intermediate isotropization, there will be spatially homogeneous models that are
linearizations of these Bianchi models about FLRW models. These perturbation
modes will occur in any almost-FLRW model that is generic rather than fine-
tuned; however, the exact models approximated by these linearizations will be
quite unlike FLRW models at very early and very late times.
Proof is by linearizing the previous equations (see the following section)
to obtain the Bianchi equations linearized about the FLRW models that occur
at the saddle point leading to the intermediate isotropisation. These modes will
be the solutions in a small neighbourhood about the saddle point permitted by
the linearized equations (given existence of solutions to the nonlinear equations,
linearization will not prevent corresponding linearized solutions existing).
The point is that these modes can exist as linearizations of the FLRW model;
if they do not occur, then the initial data have been chosen to set these modes
precisely to zero (rather than being made very small), which requires very special
initial conditions. Thus these modes will occur in almost all almost-FLRW
universes. Hence, if one believes in generality arguments, they will occur in the
real universe. When they occur, they will, at early and late times grow until the
model is very far from a FLRW geometry (while being arbitrarily close to an
FLRW model for a very long time, as per the previous theorem).
3.8 Observations and horizons
The basic observational problem is that, because of the enormous scale of the
universe, we can effectively only see it from one spacetime point, ‘here and
now’ [26, 29]. Consequently what we are able to see is a projection onto a 2-
sphere (‘the sky’) of all the objects in the universe, and our fundamental problem
is determining the distances of the various objects we see in the images we
obtain. In the standard universe models, redshift is a reliable zero-order distance
indicator, but is unreliable at first order because of local velocity perturbations.
Thus we need the array of other distance indicators (Tully–Fisher for example).
Furthermore, to test cosmological models we need at least two reliable measurable
properties of the objects we see, that we can plot against each other (magnitude
and redshift, for example), and most of them are unreliable both because of
intrinsic variation in source properties, and because of evolutionary effects
associated with the inevitable lookback-time involved when we observe distant
objects.
3.8.1 Observational variables and relations: FL models
The basic variables underlying direct observations of objects in the spatially
homegenous and isotropic FL models are: