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Astron. Astrophys. 329, 1-13 (1998)

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5. Conclusions

Here we have described the evolution of the large scale density perturbations using the characteristics of the potential field. We have also investigated the relation of the density perturbations to the potential. The main results can be summarized as follows:

1. For all simulations the measured values of the typical separation between the zero points of potential, [FORMULA], are in agreement with the theoretically predictions. For the largest boxes (CDM200 and BSI200 models) an approximately Poisson distribution has been found for all redshifts (with the exception of the large scale tail in the abundance distribution where the influence of the limited number of harmonics becomes important). The strong dependence of the distribution on the box size and the realized power spectrum became obvious. In case of smaller box sizes the different separations are almost equally distributed. This is more typical for the scale free models rather than for the models with the broad band power spectra.

These results illustrate the properties of the scale [FORMULA] as a characteristic of the spatial potential distribution. The very broad Poisson like distribution function found for the separation between zero crossings of the potential demonstrates the complicated spatial structure of the potential distribution where both narrow deep potential wells and very extended regions of positive (or negative) potential exist.

2. Our analysis demonstrates (see Figs. 4, 12, 13 and Table 2) that during the evolution the matter is slowly concentrating within [FORMULA] regions which coincide roughly with ODR. The initial spatial potential distribution is mapped into the observed large and superlarge scale structure. Thus the initial potential distribution may be used to predict properties of the large scale matter distribution.

3. The potential distribution forms a smooth and coherent field. On small scales (for scales up to 8 and 60 [FORMULA] Mpc for CDM25 and CDM200, respectively) the autocorrelation of the potential is positive. This behaviour agrees with the theoretically prediction (see Fig. 8), but at larger scales the correlation vanishes. This is due to the finite box size which strongly distorts the large scale properties of simulations.

4. A complex evolution has been found for the correlation coefficient of potential and density distribution (Fig. 14). While intuitively we expect a monotonous growing correlation due to the successive matter concentration in massive clumps, the dimensionless correlation coefficient firstly decreases to [FORMULA] from its initial amount before it starts to increase slightly.

It is expected that the potential distribution in the universe is traced by the observed galaxy distribution. Indeed, the most promising way for explaining the concentration of galaxies in SLSS elements is to consider the impact of the gravitational potential on structure formation. Also the potential distribution has the required scale range. Thus, we hope to explain the SLSS formation as matter infall to the potential wells, see the more detailed discussion in Demianski  &  Doroshkevich, 1997. In order to test this hypothesis, we have investigated the main properties of the potential distribution and its time evolution in numerical simulations with different box sizes thereby using power spectra truncated at different scales.

The next step was the comparison of the matter clustering with the potential distribution. As it has been recently shown (Doroshkevich et al. 1997a) for broad band CDM like power spectra there are three evolutionary stages. During the first stage, anisotropic gravitational collapse leads to a fast increase in the number of structure elements with exponentially decreasing mean separation between them. This period is described well by the nonlinear Zeldovich theory. During the second evolutionary stage, the intersection and merging of structure elements dominates, and the slow evolution of the large scale structure is driven by the velocity field. This period can be described by a modified Zeldovich theory tested by Doroshkevich et al. 1997a. During this period, there begins the formation of more massive structure elements. This process dominates the third evolutionary stage when the structure elements disrupt successively to the system of separate massive clumps. Clearly the distinction between these stages is a bit schematic since it depends on the considered scale, and for Gaussian initial conditions the different processes overlap. But it can be expected that the large-scale potential distribution studied in this paper is responsible mainly for the first stage, and all the detailed relaxation processes are occurring within the initial potential distribution. Let us note however, that for scale free power spectra with negative spectral index typical for the galaxy scale, the second evolutionary stage may become very short, and the relaxation of matter elements in deep potential wells will occur immediately after the pancake formation.

This sequence of processes is obvious in the evolution of the potential distribution. The successive intersection and merging of the structure elements results in the collapse of matter in regions of high potential gradients leading to a small number of narrow deep wells and to a smoothing of the overall potential distribution. The appearance of very massive sheetlike structure elements with a large separation between them has been recently found in one simulation. They remind us in many aspects on the SLSS walls found in observed galaxy distribution, but the simulation cannot reproduce it completely (see Doroshkevich et al. 1997b, and Demianski  &  Doroshkevich 1997, for a more detailed discussion). Perhaps, the difference is explained by the truncated power spectrum used in simulations. On the other hand it may also result from a large scale bias between the dark matter and the galaxies.

Considering the gravitational potential perturbations we must necessarily improve also the definition of the scale of homogeneity in the matter distribution in the universe. Usually this scale is assumed to be a few times larger than the scale of nonlinearity as defined e.g. by Efstathiou et al. (1988). However, the latter characterizes only the regions of strong matter clustering. Thus, the scale of nonlinearity is similar to the typical scale of the LSS which arises by the merging of structure elements due to their peculiar motion. The spatial distribution of SLSS elements can be understood by the typical scales of the potential distribution. This scale is neither linked directly with matter displacement nor with the scale of nonlinearity. It characterizes the spatial structure of the primordial perturbations.

The gravitational potential perturbations have a strong influence on the matter distribution of the largest scales that leads probably to the appearance of SLSS elements which comprises a significant fraction of all galaxies.

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© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997