4. The general differential equation for the angular size distance
In a series of papers Zeldovich (1964), Dashevskii and Zeldovich (1965) and Dashevskii and Slysh (1966) developed a general differential equation for the distance between two light rays on the boundary of a small light cone propagating far away from all clumps of matter in an inhomogeneous universe:
where and are functions of the time t (not the lookback time of Eq. 22). The first term can be interpreted as Ricci focusing due to the matter inside the light cone, and the second term is due to the expansion of space during the light propagation. We now have to transform this time dependent differential equation into a redshift dependent differential equation. From Eq. (11) we obtain 9
where a prime denotes a derivative with respect to redshift and from Eq. (12) follows
where the sign has been chosen such that D is always locally. We denote these special solutions of Eq. (33) with , and, following the definition (Eq. (13)), the angular size distance of an object at redshift is then given as
Fig. 1 shows the influence of z, and on the angular size distance, calculated using Eq. (33) with our numerical implementation.
For completeness we note that after the original derivation by Kayser (1985) an equivalent equation was derived by Linder (1988) which, however, is difficult to implement due to the cumbersome notation.
Special mention must be made of the so-called bounce models, which expand from a finite R after having contracted from (See, e.g., Feige (1992).) A glance at Eq. (10) shows that in these cosmological models there must be four distances for an (ordered) pair of redshifts. If we denote the distances by , , and , where 1(2) und 3(4) refer to () during the expanding (contracting) phase, then symmetry considerations dictate that and as long as the dependence of on z is the same during both phases. In this case, there are two independent distances per (ordered) pair of redshifts. If this is not the case, the degeneracy is no longer present and there are four independent distances per (ordered) pair of redshifts.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998