112 Cosmological models
The description of matter and radiation in a cosmological model must be
sufficiently complete to determine theobservational relationspredicted by the
model for both discrete sources and the background radiation, implying a well-
developed theory ofstructure growthfor very small and for very large physical
scales (i.e. for light atomic nuclei and for galaxies and clusters of galaxies), and
ofradiation absorbtion and emission. Clearly an essential requirement for a
viable cosmological model is that it should be able to reproduce current large-
scale astronomical observations accurately.
I will deal with both the 1+3 covariant approach [21, 26, 28, 91] and the
orthonormal tetrad approach, which serves as a completion to the 1+3covariant
approach [41].
3.2 1 +3 covariant description: variables
3.2.1 Average 4-velocity of matter
The preferred 4-velocity is
ua=dxa
dτ, uaua=− 1 , (3.10)whereτis the proper time measured along the fundamental world-lines. Given
ua, uniqueprojection tensorscan be defined:
Uab=−uaub⇒UacUcb=Uab,Uaa= 1 ,Uabub=ua,
hab=gab+uaub⇒hachcb=hab,haa= 3 ,habub= 0. (3.11)The first projects parallel to the velocity vectorua, and the second determines
the metric properties of the (orthogonal) instantaneous rest-spaces of observers
moving with 4-velocityua.Avolume elementfor the rest spaces:
ηabc=udηdabc⇒ηabc=η[abc],ηabcuc= 0 , (3.12)where√ ηabcdis the four-dimensional volume element (ηabcd=η[abcd],η 0123 =
|detgab|) is also defined.
Furthermore, two derivatives are defined: the covariant time derivative ‘ ̇’
along the fundamental world-lines, where for any tensorTabcd
T ̇abcd=ue∇eTabcd, (3.13)and the fully orthogonally projected covariant derivative∇ ̃where, for any tensor
Tabcd,
∇ ̃eTabcd=hafhbghpchqdhre∇rTfgpq, (3.14)
with total projection on all free indices. The tilde serves as a reminder that ifua
hasnon-zerovorticity,∇ ̃ isnota proper three-dimensional covariant derivative