60 Propagation of light and horizons
Note that asz→∞,χ(z)approaches the particle horizon. Hence, the redshift
parameter measures distance only within the particle horizon.
Finally, let us find the explicit expressions fort(z)andχ(z)in a dust-dominated
universe. In this case,ε(z)=ε 0 ( 1 +z)^3 and
H(z)=H 0 ( 1 +z)√
1 +
0 z.For a flat universe (
0 =1),the integrals in (2.63) and (2.65) are straightforward
and we find
t(z)=2
3 H 0
1
( 1 +z)^3 /^2,χ(z)=2
a 0 H 0(
1 −
1
√
1 +z)
. (2.66)
Problem 2.14Verify that in both open and closed dust-dominated universes
(χ(z))=2
√
|
0 − 1 |
20 ( 1 +z)[
0 z+(
0 − 2 )(√
1 +
0 z− 1)]
, (2.67)
where the function is defined in (2.3). Note that if
0 z1, then (χ(z))→
(
χp
)
, given in (2.9). Derive the explicit expressions fort(z).2.5Kinematictests
For an object at a cosmological redshift, it is desirable to measure its angular size
(the angle the object subtends on the sky) or its apparent luminosity. Given a class
of objects of the same size (standard rulers), we find that the corresponding angular
size changes with redshift in a specific way that depends on the values of the
cosmological parameters. The same is also true for the apparent luminosities of
objects with the same total brightness (standard candles). Therefore, if we know
the appropriate dependencies for particular classes of standard rulers or standard
candles, we can determine the cosmological parameters. Moreover, because the
measurements refer to earlier times when the universe was 1+ztimes smaller
than now, we can study its recent expansion history and distinguish among models
with different matter content.
2.5.1Angulardiameter–redshiftrelation
In a static, Euclidean space, the angle which an object with a given transverse
size subtends on the sky is inversely proportional to the distance to this object. In
an expanding universe, the relation between the distance and the angular size is
not so trivial. Let us consider some extended object of given transverse sizelat
comoving distanceχemfrom an observer (Figure 2.11). Without loss of generality,
we can setφ=const. Then, photons emitted by the endpoints of this object at time