## 1. IntroductionIt is commonly appreciated that Lagrangian perturbation solutions provide useful models of large-scale structure. Comparison with numerical simulations have put them into a strong position in the list of currently discussed analytical or semi-analytical approximations (see: Melott 1994 for a summary). Lagrangian perturbation schemes have been optimized by smoothing the high-frequency end of the power spectrum of density inhomogeneities such that they are capable of replacing N-body integrators above some scale close, but smaller than the non-linearity scale (i.e., where the r.m.s. density contrast is of order unity) (Coles et al. 1993, Melott et al. 1994, 1995, Bouchet et al. 1995, Sathyaprakash et al. 1995, Weiß et al. 1996). While their application to pancake models, i.e., models with a large high-frequency cutoff, demonstrates an excellent performance up to the epoch when shell-crossing singularities in the cosmic flow develop (Buchert et al. 1994), their application to later non-linear stages fails unless the initial data are smoothed to avoid substantial post-singularity evolution. This way the large-scale structure is restored, and small-scale features arise due to the collapse of the waves which were left in the initial data. In the present work we want to look in more detail at the collapsing structures around the epoch of shell-crossing on smaller scales by using high-resolution techniques described by Buchert & Bartelmann (1991) (however, here, we do not interpolate initial data). The present study can be viewed in line with previous high-resolution studies of pancakes (Buchert 1989a,b, Melott & Shandarin 1990, Beacom et al. 1991 (2D), and Buchert & Bartelmann 1991 (2D and 3D)). The applicability of the Lagrangian approximations has been tested in previous work on the basis of cross-correlation statistics of density fields in which the internal substructures are not resolved. We here address the question which substructures are predicted by these approximations and we shall single out first-, second-, and third-order effects in the evolution of caustics in the density field. The work by Alimi et al. (1990), Moutarde et al. (1991) and the comparison of Lagrangian perturbation solutions with the spherically symmetric solution by Munshi et al. (1994) comprise steps in this direction. We have taken care of the precision with which we realize the Lagrangian schemes. Thus far, these analytical models have to be realized numerically to set up the initial data in Fourier space. In particular, the third-order model provides a complication, since products of derivatives of perturbation potentials at first and second order (as solutions of Poisson equations) form the input for the third-order perturbation potentials, which describe interaction of perturbations (see the next subsection for details). It is therefore desirable to control this realization of initial data in an optimal way to minimize numerical uncertainties. We did this by calculating the perturbations fully analytically. We have also automated the process of finding the perturbation potentials from a single given initial velocity potential, or density field, respectively, by using algebraic manipulation systems. This procedure is suitable for spectra with not too much modes like in models with coherence length. Another advantage of this analytical procedure is the possibility of improving particle number, since we are not limited by storage as in the case of FFT realizations. We therefore can present realizations using particles. The structures shown can only be seen at resolutions higher than particles in the simulation box. We start with the derivation of a simple plane-wave model attempted earlier by Moutarde et al. (1991) (see also Alimi et al. 1990). Their third-order solution was derived just for this model. (However, this solution did not pass a test we did by inserting it into the Euler-Poisson system in Lagrangian form.) We, here, proceed differently. We start from the generic third-order solution given by Buchert (1994) and insert the plane-wave model as a special case. Since both the generic model and the special model have been checked to solve the original equations (by using algebraic manipulation systems), we are confident in our calculations. Besides automating algebraically the derivation of the potentials as mentioned above, we also exemplify the use of a set of local forms given by Buchert & Ehlers (1993) and Buchert (1994) in the case of the Moutarde et al. problem (see the appendix). We stick to that model first, since it is simple and already shows the principal features of the gravitational collapse we are interested in. Also in other work on related subjects this model is useful as an example (Mo & Buchert 1990, Matarrese et al. 1992), and can be used as a toy-model to compare different approximation schemes. We then move to generic initial data, i.e., data with no symmetry, but restricted to a small enough box to assure the resolution of patterns we are interested in. © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |