Astron. Astrophys. 323, 295-304 (1997)

Appendix A: determination of

A.1. First-order model I

The first-order particle orbits are

where the perturbation potential is related to the initial gravitational potential through the relation

The inversion of (A1) is then:

and therefore is given by:

Thus, we obtain:

From this we derive the well-known result for the Zel'dovich approximation used in the Sachs-Wolfe effect that

since

A.2. Second-order model I

Eqs. (17)(18) in terms of the perturbation potentials read

where

We write the perturbation potentials in terms of and by inverting (A9):

where

Using (A14), we invert (5) in the form

where

Therefore, given the condition of parallelism(19), we have that (A19) reduces to (20) with

A.3. Second-order model II

Following the same procedure as in Appendix A.2, we write

where

Therefore, inversion of (A24) leads to:

where

Appendix B: Jacobian

The Jacobian is given by:

where

and the functions and are model-dependent. For instance, for the case of model I of Sect. 3,

where and are defined in Appendix A.2. The assumption of parallelism (19) adopted, is simplified by making the following substitutions in (B1):

where

Hence, all terms containing are set equal to zero, and thus (B1) simply reads

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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