**4.5. The Angular size - redshift relation**

Another potentially sensitive test of models is related to the fact
that the angular size
of an extended object *D* located at a redshift *z*,
depends rather sensitively on the properties of the cosmological
model in which it
is being measured. Knowing the absolute size of an object (e.g. galaxy or
radio source) and the angle subtended by a distribution of such objects
in the
universe, it may be possible (after correcting for projection and evolution
effects) to say something about the geometry of space and the matter content
of the universe.

It is easy to derive a relationship between *D* and
.
Consider an object of proper length *D* at a coordinate
distance *r*, and assume for simplicity that the object is aligned
along the axis so that
coordinates marking its `top' and `bottom' are respectively
(*r*, _{1}
+ _{1},
_{1}) and
(*r*,
_{1},
_{1}).
The observer is at *r* = 0.
The proper length of the object can be obtained by setting
*t* = constant in the FRW line-element (2) giving
[142]

(46) |

As a result we get the following expression for the angle subtended by the object at the location of the observer

(47) |

where *d*_{A} = *a*(*t*)*r* is the
`angular-size distance'. Since
1 + *z* = *a*_{0} / *a*(*t*) one gets
*d*_{A} = *d*_{L}(1 + *z*)^{-2},
where *d*_{L}(*z*) = *a*_{0}*r*(1 +
*z*) is the luminosity distance discussed in the previous section.
Accordingly (47) may be rewritten as

(48) |

In Fig 10 we have plotted the angular size -
redshift relation
for flat cosmological models with a cosmological constant.
(We have used expressions (28) for the luminosity distance
*d*_{L} and
(19) for the dimensionless Hubble parameter *h*(*z*), assuming
a flat universe
_{total}
= _{m} +
_{} = 1.)

We find that as the object is moved to higher redshifts its
angular size first decreases (as naively expected) but soon
begins to increase after passing through a minimum value.
The appearance of a minimum angular size at a given redshift
*z*_{min} is a generic feature
of cosmological models with
_{m} > 0.
Differentiating (48) with respect
to the redshift after substituting for *d*_{L} from (30),
and then setting
/
*z* = 0 gives
*z*_{min} = 1.25 for the flat matter dominated Einstein-de
Sitter universe.
From figure 10 we find that the location of
the minimum angular size moves to higher redshifts as
_{}
is increased, until in the limiting case
_{} = 1 there is no
minimum at all.
(Formally *z*_{min}
in de Sitter space,
indicating that
the angular size of an object decreases monotonically with redshift
without ever reaching a minimum value.)

The suggestion that angular sizes of galaxies could be used to discriminate
between cosmological models was first made in
[95].
Curiously the angular size of a typical galaxy at a redshift *z* ~ 1
is roughly one arc second which is close to the limiting value of the
angular resolution (`seeing') allowed by the Earth's atmosphere
Einstein is quoted as saying
[154].
Beyond *z* ~ 1 the angular size of an object increases, and if one is
confident that galaxies of a given class at higher redshifts are similar
in form to their
lower redshift counterparts, then this test can in principle provide a
powerful means of discriminating between world models especially with
the use of satellite data which can get around the `seeing' limit.
Other (larger) objects which can be used to probe the angular size-redshift
relation include clusters of galaxies
[142]
and both extended and compact radio galaxies
[105,
31].
Extended radio sources which include
the twin radio lobes surrounding a radio galaxy can have sizes ranging from
a few kpc to ~ 1000 kpc, consequently the typical angular size of such
objects is ~ 20 arc seconds which can easily be measured using
ground based techniques.

However, a word of caution must now be added,
both clusters and radio galaxies
are prone to strong evolutionary effects which could lead to a change in
size
over cosmological epoch. Thus a comprehensive understanding of physical
effects associated with both clusters (subclustering, virialization etc.)
and radio galaxies (evolution of radio lobes and the central engine etc.)
is necessary before the angular size-redshift relation
can be used to unambiguously determine cosmological parameters including
_{}.

Recently Kellerman (1993) and Gurvits et al.
(1998) have
studied the angular sizes of compact radio sources (QSO's and AGN's)
arguing that the central `engine' powering these objects is likely
to be controlled by a limited number of physical parameters (mass of
central black hole, accretion rate etc.) and may therefore be subject
to less evolutionary effects
than extended radio sources. On the basis of an analysis of a large number
of sources spanning a wide redshift range 0.01 < *z* < 4.73
these authors
claim that an increase in the angular size has been detected
which is consistent with
_{m} ~ 1.
(However working with the same data set as Kellerman (1993),
Kayser (1995) has shown that a significant
cannot be ruled out.)

An interesting feature of closed universes not present in the flat models considered in figure 10, or in open models, is the presence of antipodal points. The presence of antipodes can contribute to changing the angular size as well as to the lensing of a source galaxy or quasar [80] and therefore provide us with a good means to constrain closed cosmological models. A metric describing the closed FRW universe is

(49) |

where 0
,
, 0
2. If we assume that
the observer is located at
= 0 then the associated antipodal point is at
_{a} =
. Substituting
*ds*^{2} = 0 we obtain
=
*dt* / *a*,
where is the
conformal time coordinate. In a matter dominated universe the form for the
expansion factor is
*a*()
= *A*(1 -
cos),
*ct* =
*A*( -
sin),
and the Hubble parameter is given by
*H*
sin /
(1 - cos)
where 0
2. Thus a light ray from the
antipodal point _{a} =
reaches
= 0 at the time of maximum
expansion
=
(corresponding to *H* = 0). Consequently in a matter dominated
closed universe, light from an antipodal
point can *never* reach an observer during the expanding phase
(when *H* > 0). This situation changes when one considers a
closed universe
with a cosmological term. In this case the universe is not obliged to
recollapse and one can observe an antipodal point during the
expansion epoch.
The location of antipodal points can be derived from the following
considerations: from (28) we find for a closed universe

(50) |

where

(51) |

since *d*_{A} = *d*_{L}(1 +
*z*)^{-2} it follows that for

(52) |

and, from (47),
,
*i.e. *the angular size of
an object located close to one of the antipodal points (52)
can become very large. Consequently such an object will appear to us to
be extremely bright even if located at a high redshift !
The presence of `normal' galaxies
and quasars, as well as gravitationally lensed objects out to redshifts
4.92 set a lower limit
on the antipodal redshift
*z*_{a}(_{},
_{m}) >
4.92 which can be used to constrain the cosmological parameter pair
(_{},
_{m})
in a closed universe
[51,
124,
147].
(Multiple images of a source object located further away than
the antipodal redshift *z*_{a} are very difficult to form
[80].)
Since the supernovae analysis prima-facie appears to favour a closed
universe, antipodal constraints may be used to further narrow down the
allowed range in parameter space.