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

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Appendix A: analytical construction of displacement vectors for Model I

The first-order displacement vector

In this section we exemplify how special solutions to (2) can be constructed by using a set of local forms which provide first integrals of the seven Poisson equations (2f-l). Let us consider a simple plane-wave model for the initial peculiar-velocity potential [FORMULA], which was studied earlier by Moutarde et al. (1991):

[EQUATION]

The amplitude [FORMULA] plays the role of the perturbation parameter here and is related to the total amplitude [FORMULA] of the density contrast [FORMULA] as [FORMULA]. The amplitudes [FORMULA] allow for triaxial deformations of the model; one has to choose [FORMULA] in order to keep the r.m.s. amplitude of [FORMULA] the same. In this paper we shall use [FORMULA], since different amplitudes give no further information about internal structures of the model. Although the model (A.1a) is simple, it has no symmetries which destroy the generic feature of the singularites formed like plane or spherical symmetry would do. The structure of the cluster formed will only retain reflection and rotational symmetries manifest in the potential (A.1a) for our choice of amplitudes. As, e.g., demonstrated in (Buchert & Ehlers 1993) for a similar two-dimensional model, we have with models like (A.1a) the possibility of studying principal kinematical features of a generic collapse such as the formation of cusped caustics, interconnected network structures, infall of matter onto the cluster. Additionally, internal differentiation of a multi-stream system resulting in a hierarchy of shell-crossings, which are attributed to a generic feature of a gravitational collapse, can be demonstrated nicely with this model. The model has periodic boundary conditions which makes it accessible for numerical treatment.

From (A.1a) we have for the first order displacement vector:

[EQUATION]

We now scetch a procedure how to construct the higher-order potentials from this initial condition. The procedure is based on a list of local forms given by Buchert & Ehlers (1993) and Buchert (1994), which are first integrals of the quadratic and cubic source terms in the Poisson equations of the solution (2). These integrals only hold for special classes of initial data, although they might also be useful as approximations for generic initial data. For the potential (A.1a) it turns out that it belongs to the class of initial data which, for all orders, admits such first integrals.

The second-order displacement vector

According to corollary 1 proved in (Buchert & Ehlers 1993), a local form can be obtained for second-order displacements. It reads

[EQUATION]

The local form (A.2) is constructed such that its divergence agrees with the source term in (2g), its curl is, however, in general non-zero, it only vanishes if (A.2b,c,d) are statisfied. Inserting the potential (A.1a) we immediately obtain the second-order displacement vector:

[EQUATION]

The vector (A.2e) is curl-free as can be easily demonstrated, so it obeys the constraints (A.2b,c,d) necessary to admit a potential. This potential can now be guessed from (A.2e) to be of the form

[EQUATION]

The third-order displacement vector of the "truncated model"

Similarily, we can ask for a local vector form whose divergence agrees with the source term in equation (2h). An expression given in Buchert (1994, corollary 1) has the required property: The vector [FORMULA] with the components

[EQUATION]

has the property

[EQUATION]

where [FORMULA] are the subdeterminants of the tensor [FORMULA] (a comma always denotes partial derivative with respect to Lagrangian coordinates). The following constraints have to be satisfied in order that [FORMULA] be curl-free:

[EQUATION]

Inserting the potential (A.1a) into (A.3a) gives for the displacement vector

[EQUATION]

Again, the vector (A.3e) is found to be curl-free which renders the contraints (A.3b,c,d) satisfied. A potential generating this displacement is again easily found from (A.3e). It reads

[EQUATION]

The third-order displacement vector of the interaction term - longitudinal part

The source term in (2i) which describes the longitudinal part of the interaction of first- and second-order perturbations has a similar structure as the second-order source term (2g). We are able to construct a local form by analogy (Buchert & Ehlers 1993, corollary 1):

[EQUATION]

Here, the linear combination of the two possible integrals as a general integral has to be taken, where [FORMULA]. In order to satisfy the requirement that the vector (A.4a) be a solution of the Poisson equation (2i), we have to assure that it is curl-free which implies (Buchert & Ehlers 1993, corollary 1):

[EQUATION]

As can be seen from (A4), we have to determine the parameters [FORMULA] and [FORMULA] suitably in order to fulfil the constraints (A.4b,c,d). Although, we can find the two first integrals for the potential (A.1a), the resulting vectors are not curl-free. It is a matter of some algebra until one finds the correct linear combination of the two vectors, which is curl-free. This can be achieved by first guessing the form of the potential [FORMULA] from the two vectors. It is clear that, in general, we will not be successful. We obtain [FORMULA], [FORMULA]. Thus, the displacement vector reads

[EQUATION]

with

[EQUATION]

i.e.,

[EQUATION]

with the potential

[EQUATION]

The third-order displacement vector of the interaction term - transverse part

Finally, we ask for a first integral of the transverse part of the interaction vector (2j,k,l). In (Buchert 1994, corollary 2) the vector form needed has been given again as a linear combination of the two possible integrals. The vector

[EQUATION]

has the property

[EQUATION]

We have to assure [FORMULA]. In order to satisfy the requirement that the vector components (A.5a,b,c) be solutions of the Poisson equations (2j,k,l), we have to guarantee that the vector field [FORMULA] is source-free which implies after using well-known vector identities

[EQUATION]

Again, we find the two integrals after inserting the potential (A.1a) to be not source-free. We have to determine the correct linear combination of the two integrals. As in the longitudinal case we first guess the form of the vector potential [FORMULA] from the two integrals. We are successful with the parameters [FORMULA], and [FORMULA], but we have to add another function [FORMULA] such that the total displacement is source-free. The potential [FORMULA] is given by

[EQUATION]

(This is possible, since the local forms discussed above are only determined up to the gradient of some potential, see Buchert 1994). The vector displacement [FORMULA] reads

[EQUATION]

with the vector-potential

[EQUATION]

Remarks

For the following discussion we need the explicit expressions of the source terms in the solution (2) for the special model (A.1a). We derive

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

From the generating functions constructed above we infer the following property: except for the longitudinal part of the interaction term, the perturbation potentials obey equations which are typical for bound systems:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Recall that the condition (A.7a) implies [FORMULA] for an initially irrotational peculiar-velocity field [FORMULA] ; [FORMULA], i.e., the motion is initiated to follow the gradient of the density-contrast field. At the third order the evolution model shows that this property of the flow is lost.

(The algebraic program we used to compute the perturbation potentials for Model II also reproduces the perturbation potentials derived here.)

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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