4. High-resolution studies
We present high-resolution studies of the density field as predicted by the Lagrangian schemes for both models. This is done by collecting trajectories into a (comoving) Eulerian grid of cells for Model I ( into a grid of for Model II) (for the method see Buchert & Bartelmann 1991).
Fig. 1 displays three evolution stages of the density field predicted by Model I for the first-, second-, and third-order perturbation solutions. (Initial data were given at ; .) A manifest feature is the delay of the collapse time for perturbation solutions at different orders; higher-order corrections significantly accelerate the collapse. This result was already stated by Moutarde et al. (1991). To compare the spatial patterns for the different orders, we can roughly compare the density fields "diagonally" in Fig. 1 (this way of comparison will be discussed quantitatively in a forthcoming paper: Karakatsanis et al. 1996): while the first-order solution (the "Zel'dovich approximation") carries mainly kinematical information beyond the epoch of shell-crossing, the second-order solution modifies the shape of the first mass-shell after crossing and generates a second mass-shell as well as secondary sheets and filaments inside the first structures (in agreement with the previous study of the trajectory field by Buchert & Ehlers 1993); the third-order correction redistributes mass inside the two mass-shells as well as in sheets and filaments.
Fig. 2 displays the third-order density field splitted into different parts of the third-order corrections in the solution (2): we infer that the transverse part of the "interaction term" (2j,k,l) is not of crucial importance here and might be neglected, it merely deconcentrates the inner mass-shell more from the center (which can be seen by comparing full third-order with or without transverse part, or "truncated third-order" with or without transverse part). However, to neglect the "interaction terms" altogether results in a pattern which is further away from the second-order approximation than the full third-order approximation. The outer caustic is even absent. This indicates that the third-order approximation without "interaction terms" is not useful, the "main body" of the perturbation sequence is not a good model as was speculated in (Buchert 1994).
We continue by looking at Fig. 3 which presents the density field of Model II for the different orders at a late evolution stage, late, because we then are able to separate the different structures visually which appear much earlier in the evolution. Again the features quoted above are visible, the collapse is delayed by a huge factor in the first-order ("Zel'dovich") approximation. Also, the similarity between second- and third-order is striking, while the first-order model lacks some internal structures, which can be attributed to secondary shell-crossing events (a second-order effect).
An interesting aspect of these high-resolution studies relates to a new interpretation of the longstanding "fragmentation problem" in classical pancake theory: we appreciate small "fragments" sitting at the intersections of caustics (see Figs. 1-3). Since a finite resolution brings the density to a finite value, these "fragments" show up as almost spherical blobs with potential wells that have about 2 times more height than the potential of the mass-shells. In realistic situations, physical processes at the location of caustics and velocity dispersion in a dark collisionless component will do a similar job. We may interprete this phenomenon as "gravitational fragmentation": although the initial fluctuation is coherent like in pancake models, the collapse process forms fragments on a substantially smaller scale. This interpretation is appropriate, if gravity is the dominating interaction related to the existence of a mass dominating dark matter component in the Universe. It has far reaching consequences in a self-gravitating medium, since we expect the phenomenon of multiple shell-crossing (termed "non-dissipative gravitational turbulence" by Gurevich & Zybin 1988a,b) to continue down to smaller and smaller scales yielding a hierarchy of nested caustics. This has been demonstrated in a two-dimensional simulation by Doroshkevich et al. (1980). Since further and further mass-shells are generated in the center of a cluster, more and more caustics are simultaneously present and consequently create a huge number of "fragments" as an internal organization of mass-shells. This way, a cluster naturally creates a gravitational potential which is distinctly rippled and thereby prepares the sites for galaxy formation: we expect the baryons to preferentially drop into these "fragments".
Although this consideration has to remain premature at this stage, we think that "gravitational fragmentation" as we describe it is a generic effect in gravitational clustering and should be taken seriously as soon as a dark matter component dominates the matter density. The fact that we need high-spatial resolution studies to uncover these fragments explains their absence in the literature. It is interesting to note here that another type of fragments appeared in a two-dimensional numerical simulation at high resolution (Melott & Shandarin 1990), which results from a redistribution of mass inside filaments (compare their plot with the filaments in the third-order approximation in Fig. 1).
The detailed study of caustic metamorphoses begun by Arnol'd et al. (1982) for the "Zel'dovich-approximation" in two spatial dimensions will provide the necessary insight to further understand this phenomenon. We have continued this study in three spatial dimensions (Buchert et al. 1996a,b); for an overview see (Buchert 1995).
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998