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Astron. Astrophys. 318, 1-10 (1997)

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2. A generic third-order solution

Let us recall the class of third-order solutions on which we base our models. We require that, initially, the peculiar-velocity [FORMULA] to be proportional to the peculiar-acceleration [FORMULA]:

[EQUATION]

where we have defined the fields as usual (compare Peebles 1980, Buchert 1992). Henceforth, we denote the peculiar-velocity potential at the initial time [FORMULA] by S, [FORMULA], and the peculiar-gravitational potential at [FORMULA] by [FORMULA], [FORMULA], where [FORMULA] denotes the nabla operator with respect to the Lagrangian coordinates [FORMULA]. The restriction (1) has proved to be appropriate for the purpose of modeling large-scale structure, since the peculiar-velocity field tends to be parallel to the gravitational peculiar-field strength after some time, related to the existence of growing and decaying solutions in the linear regime (Buchert & Ehlers 1993, Buchert 1994).

With a superposition ansatz for Lagrangian perturbations of an Einstein-de Sitter background the following mapping [FORMULA] as irrotational solution of the Euler-Poisson system in Lagrangian form up to the third order in the perturbations from homogeneity has been obtained (Buchert 1994). [FORMULA] defines the displacement map from Lagrangian coordinates [FORMULA] to Eulerian coordinates [FORMULA] which are comoving with the unperturbed Hubble-flow; the general set of initial conditions [FORMULA] is restricted according to [FORMULA] (see equation (1)); [FORMULA]:

[EQUATION]

with:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION][EQUATION]

[EQUATION][EQUATION]

where the initial displacement vectors have to be constructed by solving seven elliptic boundary value problems (summation over repeated indices; i,j,k = 1,2,3 with cyclic ordering).

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

An important remark relevant to any realization of the solution (2) concerns the possibility of setting [FORMULA] without loss of generality, if initial data are spatially periodic (compare Buchert 1996, Ehlers & Buchert 1996 for details and proofs). With this setting, the first-order solution reduces to the well-known "Zel'dovich approximation" (Zel'dovich 1970, 1973; Buchert 1992), which then assumes its familiar local form. Also, the truncated third-order model (i.e., neglecting interaction terms) is then, although of course non-locally, expressible in terms of the initial data (compare eqs. (2f-h)).

The scalar potential [FORMULA] and the vector potential [FORMULA] generate interaction among the first- and second-order perturbations. The general interaction term is not purely longitudinal: inspite of the irrotationality of the flow in Eulerian space, vorticity is generated in Lagrangian space starting at the third order for this set of intial data. For more general initial data, this happens already at second order. As our analysis of the solution will show, it has sense to include the interaction term [FORMULA] only, neglecting the transverse part altogether. However, as will be demonstrated, keeping only the generating function [FORMULA] results in a density pattern, which is not an adequate generalization of the second-order approximation. This "truncated third-order" model has been proposed in (Buchert 1994) as the "main body" of the perturbation sequence in the early nonlinear regime, since all higher-order solutions are made up of interaction terms among the perturbation potentials. A closer look at the features presented in this work shows that the third-order model has to be run with the interaction term [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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