Before going on to galaxy formation and evolution, it is worthwhile to
look at the connections between cosmological models and basic tenets of
general relativity for the commonly-considered instance of an
expanding Universe in the *Friedman model*.
The connection between the world model and
observables comes about from Weyl's postulate - that the wordlines of galaxies
form a 3-bundle of non-intersecting geodisics orthogonal to a series of
spacelike
hypersurfaces. This is what is now generally meant by a comoving coordinate
systems, since we appreciate that galaxies and clusters depart from this
ideal through having nonzero peculiar motions. An excellent source for
this material is Peebles' *Physical Cosmology* (Princeton 1971).
Pedantically, using the relativistic coordinates
*x*^{m
}, a spacelike
hypersurface satisfies *x*^{0}=constant, and a worldline which
follows
the Weyl condition has
*x*^{m
} constant for
m=1,2,3.
This constrains
the geometry of spacetime (i.e. the fact that these exist and
can be orthogonal) such that the connections *g* between coordinates
*g*_{ 0 m
}*x*^{0}
*x*^{m
} = 0 where we use the relativistic summation
convention for repeated indices. For the metric *ds*, this implies
that the governing equation

reduces to (for
m=1,2,3)
G
_{ 0 0 }^{
m} = 0.
This in turn means that the three spatial coordinates are equivalent,
so *g*_{00}=*g*_{00}*x*^{1},
and this is conventionally taken as 1.
This gives a metric - the expression connecting spacetime intervals with their
components in some coordinate system - the form

If we incorporate the cosmological principle that the result is
homogeneous and isotropic, we reach the *Robertson-Walker metric*

where we also transform for convenience into a spherical set of
spatial coordinates.
Here, *S* is a scale-factor (clearly time-dependent, but
location-independent by construction through invoking the
Cosmological Principle), and *k*
is determined by the large-scale curvature. This is conventionally
folded into the constant of integration in the denonimator of the
term in *dr*², although the choice is sometimes made of letting
this express the curvature radius *R* (positive, negative, or infinite)
such that *k*= 1/*R*².

Integration for the behavior of components of the interval between events in this metric yields

Including the energy equation to relate mass-energy density and curvature,
on can derive analytical solution for *k*=0, ± 1. For *k*=0
(equivalent to infinite radius of curvature) we have
the *Einstein-de Sitter* model, at critical density,
for which
*S* ~ *t*^{2/3} and
*t*_{0} = 2/3H as we've noted before.
Given enough justification, one may add a *cosmological constant*
L to
the expressions for
*R*_{m
} above, which gives the
*Lemaitre* model; depending on
L, the
Universe can hang for
a long time at nearly constant *R* before settling on an asymptotic
late-time solution of the familiar Hubble expansion.

Last changes: 11/2000 © 2000