136 Cosmological models
equation forH, namely (3.91), decouples from the remaining equations (3.92)
and (3.93). Thus the DE (3.92) describes the evolution of the non-tilted Bianchi
cosmologies, the transformation of variables essentially scaling away the effects
of the overall expansion. An important consequence is that the new variables are
bounded near the initial singularity.
3.7.2.2 Equations and orbits
Sinceτ assumes all real values (for models which expand indefinitely), the
solutions of (3.92) are defined for allτand hence define aflow{φτ}onRn.The
evolution of the cosmological models can thus be analysed by studying the orbits
of this flow in the physical region of state space, which is a subset ofRndefined
by the requirement that the matter energy densityμbe non-negative, i.e.
(y)=κμ
3 H^2≥ 0 , (3.94)
where the density parameteris a dimensionless measure ofμ.
Thevacuum boundary,definedby(y)=0, describes the evolution of
vacuum Bianchi models, and is an invariant set which plays an important role
in the qualitative analysis because vacuum models can be asymptotic states for
perfect fluid models near the big bang or at late times. There are other invariant
sets which are also specified by simple restrictions onywhich play a special
role: the subsets representing each Bianchi type (table 3.2), and the subsets
representing higher-symmetry models, specifically the FLRW models and the
LRS Bianchi models (table 3.1).
It is desirable that the dimensionless state spaceDinRnis acompact
set. In this case each orbit will have non-empty future and past limit sets, and
hence there will exist a past attractor and a future attractor in state space. When
using expansion-normalized variables, compactness of the state space has a direct
physical meaning for ever-expanding models: if the state space is compact, then
at the big bang no physical or geometrical quantity diverges more rapidly than
the appropriate power ofH, and at late times no such quantity tends to zero less
rapidly than the appropriate power ofH. This will happen for many models;
however, the state space for Bianchi type VII 0 and type VIII models is non-
compact. This lack of compactness manifests itself in the behaviour of the Weyl
tensor at late times.
3.7.2.3 Equilibrium points and self-similar cosmologies
Each ordinary orbit in the dimensionless state space corresponds to a one-
parameter family of physical universes, which are conformally related by a
constant rescaling of the metric. However, for an equilibrium pointy∗of the
DE (3.92), which satisfiesf(y∗)= 0 , the deceleration parameterqis a constant,
i.e.q(y∗)=q∗,andwefind
H(τ)=H 0 e(^1 +q∗)τ
.