All the well developed models of standard cosmology start with two basic
assumptions: (i) The distribution of matter in the
universe is homogeneous and isotropic at sufficiently large scales.
(ii) The large scale structure of the universe is essentially
determined by gravitational interactions and hence can be described
by Einstein's theory of gravity. The geometry of the universe can then
be determined via Einstein's equations with the stress tensor of matter
*T*^{i}_{k}(*t*,**x**)
acting as the source. (For a review of cosmology, see e.g.
[43,
44,
45,
46,
47]).
The first assumption determines the kinematics of the universe while the
second one determines the dynamics. We shall discuss the consequences of
these two assumptions in the next two subsections.

**2.1. Kinematics of the Friedmann model**

The assumption of isotropy and homogeneity implies that the large scale geometry can be described by a metric of the form

(11) |

in a suitable set of coordinates called comoving coordinates.
Here *a*(*t*) is an arbitrary function of time (called
*expansion factor*) and *k* = 0, ±1. Defining a new
coordinate through
= (*r*,
sin^{-1}*r*, sinh^{-1}*r*) for
*k* = (0, + 1, - 1) this line element becomes

(12) |

where *S*_{k}() =
(,
sin,
sinh) for *k* = (0, +
1, - 1). In any range of time during which *a*(*t*) is a
monotonic function of *t*, one can use
*a* itself as a time coordinate. It is also convenient to define a
quantity *z*, called the redshift, through the relation
*a*(*t*) = *a*_{0}[1 +
*z*(*t*)]^{-1} where *a*_{0} is
the current value of the expansion factor. The line
element in terms of [*a*, ,
,
] or
[*z*, ,
,
] is:

(13) |

where *H*(*a*) = ( /
*a*), called the *Hubble parameter*, measures the rate of
expansion of the universe.

This equation allows us to draw an important conclusion:
The only non trivial metric function in a Friedmann universe is the
function *H*(*a*) (and the numerical value of *k* which
is encoded in the spatial part of the line element.) Hence, any kind of
observation based on geometry of the spacetime, however complex it may
be, will not allow us to determine anything other than this single
function *H*(*a*). As we shall see, this alone is inadequate
to describe the material content of the universe and any attempt to do
so will require additional inputs.

Since the geometrical observations often rely on photons received from
distant sources, let us consider a photon traveling a distance
*r*_{em}(*z*) from the time of emission (corresponding
to the redshift *z*) till today.
Using the fact that *ds* = 0 for a light ray and the
second equality in equation (13) we find that the distance traveled by
light rays is related to the redshift by
*dx* = *H*^{-1}(*z*)*dz*.
Integrating this relation, we get

(14) |

All other geometrical distances can be expressed in terms of
*r*_{em}(*z*) (see eg.,
[44]).
For example, the flux of radiation *F* received from a source of
luminosity *L* can be expressed in the form *F* = *L* /
(4
*d*_{L}^{2}) where

(15) |

is called the *luminosity distance*.
Similarly, if *D* is the physical size of an object which subtends
an angle
to the
observer, then - for small
- we can define an
*angular diameter distance* *d*_{A} through
the relation
= *D* /
*d*_{A}. The angular diameter distance is given by

(16) |

with
*d*_{L} = (1 + *z*)^{2} *d*_{A}.

If we can identify some objects (or physical phenomena)
at a redshift of *z* having a characteristic transverse size
*D*, then measuring the angle
subtended by this
object we can determine *d*_{A}(*z*). Similarly,
if we have a series of luminous sources at different redshifts having known
luminosity *L*, then by observing the flux from these sources
*L*, one can determine the luminosity distance
*d*_{L}(*z*). Determining any of these functions
will allow us to use the relations (15) [or (16)] and (14) to
obtain *H*^{-1}(*z*). For example,
*H*^{-1}(*z*) is related to
*d*_{L}(*z*) through

(17) |

where second equality holds if the spatial sections of the universe are
flat, corresponding to *k* = 0; then *d*_{L}(*z*),
*d*_{A}(*z*), *r*_{em}(*z*) and
*H*^{-1}(*z*) all contain
the (same) maximal amount of information about the geometry.

The function *r*_{em}(*z*) also determines the proper
volume of the universe between the redshifts *z* and *z* +
*dz* subtending a solid angle
*d* in the sky.
The comoving volume element can be expressed in the form

(18) |

where the prime denotes derivative with respect to *z*.
Based on this, there has been a suggestion
[48]
that future observations of the number of dark matter halos
as a function of redshift and circular velocities can be used to determine
the comoving volume element
to within a few percent accuracy. If this becomes possible, then
it will provide an additional handle on constraining the cosmological
parameters.

The above discussion illustrates how cosmological observations
can be used to determine the metric of the spacetime, encoded by the
single function *H*^{-1}(*z*). This issue is trivial
in principle, though enormously complicated in practice because of
observational uncertainties in the determination of
*d*_{L}(*z*), *d*_{A}(*z*) etc.
We shall occasion to discuss these features in detail later on.