62 Propagation of light and horizons
increases with distance and as it approaches the horizon its image covers the whole
sky. Of course, the apparent luminosity drops drastically with increasing distance,
otherwise remote objects would completely outshine nearby ones.
To understand this unusual behavior of the angular diameter, it is again useful to
turn to a low-dimensional analogy and consider how an observer on the north pole
of the Earth would see an object of a given size at various distances. In this analogy,
light propagates along meridians, which are geodesics on the Earth’s surface, and
we find that the angular size decreases with distance, but only if the object is north
of the equator. If the object is south of the equator, the angular size increases with
distance until, finally, an object at the south pole “covers the whole sky.” This
analogy, while illuminating, is not complete. The angular size of a very remote
object also grows in a flat universe because of the time dependence of the scale
factor; the 4-curvature of spacetime is responsible for the unusual behavior of the
angular diameter.
The angular sizeθ can be expressed as a function of redshift z.Since
a 0 /a(tem)= 1 +z,we can write (2.69) as
θ(z)=( 1 +z)l
a 0 (χem(z)), (2.70)
whereχem(z)is given by (2.65). In a flat universe filled with dust, the function
(χem)equalsχem, whose explicit dependence onzwas given in (2.66). Hence,
the angular diameter as a function ofzis
θ(z)=
lH 0
2( 1 +z)^3 /^2
( 1 +z)^1 /^2 − 1. (2.71)
At low redshifts (z1), the angular diameter decreases in inverse proportion to
z, reaches a minimum atz= 5 /4, and then scales aszforz1 (Figure 2.12).
The extension to more general cosmologies is straightforward. For example,
substituting (χem)from (2.67) into (2.70), we find that in a nonflat dust-dominat-
ed universe,
θ(z)=
lH 0
2
20 ( 1 +z)^2
0 z+(
0 −2)(( 1 +
0 z)^1 /^2 −1). (2.72)
In principle, having standard rulers distributed over a range of redshifts we could
use the measurements of angular diameter versus redshift to test cosmological
models. Unfortunately, the lack of reliable standard rulers has hampered progress
in this technique for many years.
One spectacular success, though, has been a single standard ruler extracted from
measurements of the cosmic microwave background. The temperature autocor-
relation function measures how the microwave background temperature in two