MODERN COSMOLOGY

134 Cosmological models

Ta b l e 3. 2. Canonical structure constants for different Bianchi types. The parameter
h=a^2 /n 2 n 3.

Class Type n 1 n 2 n 3 a

A I 0000 Abelian

II +ve000

VI 0 0 +ve −ve 0

VII 0 0 +ve +ve 0

VIII −ve +ve +ve 0

IX +ve +ve +ve 0

BV000+ve

IV 0 0 +ve +ve

VIh 0 +ve −ve +ve h< 0

III 0 +ve −ve n 2 n 3 same as VI 1

VIIh 0 +ve +ve +ve h> 0

The set of tetrad equations (section 3.3) with restrictions (3.86) will
determine the evolution of all the commutation functions and matter variables and,
hence, determine the metric and also the evolution of the Weyl tensor. One can
relate these equations to variational principles and a Hamiltonian, thus expressing
them in terms of a potential formalism that gives an intuitive feel for what the
evolution will be like [92, 93]. They are also the basis of dynamical systems
analyses.

3.7.2 Dynamical systems approach

The most illuminating description of the evolution of families of Bianchi models
is a dynamical systems approach based on the use of orthonormal tetrads,
presented in detail in Wainwright and Ellis [128]. The main variables used
are essentially the commutation functions mentioned earlier, but rescaled by a
common time-dependent factor.

3.7.2.1 Reduced differential equations

The basic idea [12, 126] is to write the EFE in a way that enables one to study the
evolution of the various physical and geometrical quantitiesrelative to the overall
rate of expansion of the universe, as described by the rate of expansion scalar'
or, equivalently,the Hubble parameter H=^13 '. The remaining freedom in the
choice of orthonormal tetrad needs to be eliminated by specifying the variables

αimplicitly or explicitly (for example by specifying the basis as eigenvectors of