MODERN COSMOLOGY

Bianchi universes (s= 3 ) 135

theσαβ). This also simplifies other quantities (for example the choice of a shear
eigenframe will result in the tensorσαβbeing represented by two diagonal terms).
One hence obtains a reduced set of variables, consisting ofHand the remaining
commutation functions, which we denote symbolically byx=(γabc|reduced).
The physical state of the model is thus described by the vector(H,x). The details
of this reduction differ for classes A and B in the latter case, there is an algebraic
constraint of the formg(x)=0, wheregis a homogeneous polynomial.
The idea is now to normalizexwith the Hubble parameterH. Denoting the
resulting variables by a vectory∈Rn, we write

y=

x

H

. (3.88)

These new variables aredimensionless, and will be referred to asexpansion-
normalized variables. It is clear that each dimensionless stateydetermines a
one-parameter family of physical states(x,H). The evolution equations for the
γabclead to evolution equations forHandxand hence fory. In order that the
evolution equations define a flow, it is necessary, in conjunction with the rescaling
of the variables, to introduce adimensionless time variableτaccording to

S=S 0 eτ, (3.89)

whereS 0 is the value of the scale factor at some arbitrary reference time. Since
Sassumesvalues0<S<+∞in an ever-expanding model,τassumes all real
values, withτ →−∞at the initial singularity andτ →+∞at late times. It
follows that
dt
dτ

=

1

H

(3.90)

and the evolution equation forHcan be written

dH

dτ

=−( 1 +q)H, (3.91)

where thedeceleration parameter qis defined byq=−SS ̈ /S ̇^2 , and is related to
H ̇byH ̇=−( 1 +q)H^2. Since the right-hand side of the evolution equations for
theγabcare homogeneous of degree 2 in theγabc, the change (3.90) of the time
variable results inHcancelling out of the evolution equation fory, yielding an
autonomous differential equation (DE):

dy

dτ

=f(y), y∈Rn. (3.92)

The constraintg(x)=0 translates into a constraint

g(y)= 0 , (3.93)

which is preserved by the DE. The functionsf:Rn→Rnandg:Rn→Rare
polynomial functions iny. An essential feature of this process is that the evolution