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Astron. Astrophys. 318, 680-686 (1997)

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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:

[EQUATION]

where [FORMULA] and [FORMULA] 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

[EQUATION]

and thus

[EQUATION]

and

[EQUATION]

Furthermore, since [FORMULA] (Eq. (10)), we obtain, using Eq. (25),

[EQUATION]

From the definition of [FORMULA] (Eq. (4)) and matter conservation (Eq. (8)) we obtain

[EQUATION]

If we now insert Eqs. (26), (28), (29) and (30) into Eq. (24), sort the terms appropriately and cancel [FORMULA], which appears in all terms, we obtain

[EQUATION]

where a prime denotes a derivative with respect to redshift and from Eq. (12) follows

[EQUATION]

From the definition of the angular size distance (Eq. (13)) it is obvious that it follows the same differential equation as l:

[EQUATION]

with special boundary conditions at the redshift [FORMULA] where the two considered light rays intersect. The first boundary condition is trivially

[EQUATION]

and the second boundary condition follows from the Euclidean approximation for small distances, i.e.

[EQUATION]

hence

[EQUATION]

where the sign has been chosen such that D is always [FORMULA] locally. We denote these special solutions of Eq. (33) with [FORMULA], and, following the definition (Eq. (13)), the angular size distance of an object at redshift [FORMULA] is then given as

[EQUATION]

Fig. 1 shows the influence of z, [FORMULA] and [FORMULA] on the angular size distance, calculated using Eq. (33) with our numerical implementation.

[FIGURE] Fig. 1. The angular size distance from the observer ([FORMULA]) and from [FORMULA] (lower right) as a function of the redshift [FORMULA] for different cosmological models. Thin curves are for [FORMULA], thick for [FORMULA]. The upper curves near [FORMULA] ([FORMULA] at lower right) are for [FORMULA], the lower for [FORMULA]. [FORMULA] for all curves. The angular size distance D is given in units of [FORMULA]

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 [FORMULA] (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 [FORMULA], [FORMULA], [FORMULA] and [FORMULA], where 1(2) und 3(4) refer to [FORMULA] ([FORMULA]) during the expanding (contracting) phase, then symmetry considerations dictate that [FORMULA] and [FORMULA] as long as the dependence of [FORMULA] 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.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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