Physical Foundations of Cosmology

66 Propagation of light and horizons

1

redshift z

1.5 2

apparent magnitude

m

bol

Ωm = 0

Ωm = 0.3

Ωm = 1

0.5

Fig. 2.13.

Problem 2.15In Euclidean space, the observed fluxFfrom an object of luminosity
Lat distancedisF=L/ 4 πd^2 and the angular size of an object of known length
lisθ=l/d. Based on these relations, cosmologists sometimesformallydefine
the luminosity distancedLand the angular diameter distancedAto an object in an
expanding universe as

dL≡

(

L

4 πF

) 1 / 2

, dA≡

l

θ

, (2.82)

respectively. CalculatedL(z)anddA(z)in a dust-dominated universe. How are they

related in general? Verify that the distancesdLanddAcoincide only to leading

order inzand at smallzrevert to the Euclidean distanced. In contrast withdA, the

luminosity distancedLincreases withzat large redshift, as common sense would

suggest. Both, however, are only formal forz>1 where the notion of invariant

physical distance does not exist.

2.5.3 Number counts

A further kinematic test is based on counting the number of cosmological objects

with a given redshift. Suppose the number of galaxies or clusters per unit volume at