MODERN COSMOLOGY

128 Cosmological models

Ta b l e 3. 1. Classification of cosmological models (with(μ+p)> 0 )by isotropy and
homogeneity.

Dim invariant variety

Dimension, s= 2 s= 3 s= 4

Isotropy Inhomogeneous Spatially Spacetime

group homogeneous homogeneous

q= 0 Generic metric form known. Bianchi: Osvath/Kerr

anisotropic Spatially self-similar, orthogonal,

AbelianG 2 on 2D tilted

spacelike surfaces,

non-AbelianG 2

q=1Lemaˆıtre–Tolman– Kantowski–Sachs, G ̈odel

LRS Bondi family LRS Bianchi

q= 3 None Friedmann Einstein

isotropic (cannot happen) static

Two non-ignorable One non-ignorable Algebraic EFE

coordinates coordinate (no redshift)

Dim invariant variety

Dimension s= 0 s= 1

Isotropy Inhomogeneous Inhomogeneous/

group no isotropy group

q= 0 Szekeres–Szafron, General metric

Stephani–Barnes, form independent

Oleson typeN of one coord;

KV h.s.o./not h.s.o.

The real universe!

3.5.2.1 Spacetime homogeneous models

These models withs=4 are unchanging in space and time, henceμis a constant,
so by the energy conservation equation (3.29) they cannot expand: ' = 0.
They cannot produce an almost isotropic redshift, and are not useful as models
of the real universe. Nevertheless they are of some interest for their geometric
properties.

Theisotropic case q=3(⇒r= 7 )is the Einstein static universe, the non-
expanding FL model that was the first relativistic cosmological model found. It is