## 2. A generic third-order solutionLet us recall the class of third-order solutions on which we base our models. We require that, initially, the peculiar-velocity to be proportional to the peculiar-acceleration : where we have defined the fields as usual (compare Peebles 1980, Buchert 1992). Henceforth, we denote the peculiar-velocity potential at the initial time by S, , and the peculiar-gravitational potential at by , , where denotes the nabla operator with respect to the Lagrangian coordinates . 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
as with: 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). An important remark relevant to any realization of the solution (2)
concerns the possibility of setting 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 The scalar potential and the vector
potential 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
only, neglecting the transverse part
altogether. However, as will be demonstrated, keeping only the
generating function 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 © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |