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

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4. Comparison of the spatial distribution of density and potential

The theory of gravitational instability as the origin of large scale structure is based on the close connection of the gravitational potential and the matter density field. However, this general statement has to be refined to discriminate between the influence of the potential on the density evolution and vice versa. Very roughly, we expect a leading role of density perturbations on small scales, and a leading role of potential perturbations on large scales. For example, the potential perturbations are responsible for the matter distribution in richer filaments (LSS) and in wall like structures (SLSS). As a rule, these structures map the potential wells. On the other hand, the thickness of these structure elements is due to dynamical effects in the evolution of the density field, and a theoretical model has to be used to trace it back to the characteristics of the potential field.

While the spatial distribution of the potential is driven by the large scale part of the power spectrum and, hence, evolves relatively slowly, the density distribution is mainly influenced by the small scale part of the power spectrum, and it evolves much faster. However, it can be shown that the distribution of the small scale density inhomogeneities is closely connected to the large scale spatial potential distribution.

This relation becomes evident considering the density field smoothed on a large scale. Then the evolution of both fields is controlled by the same part of the power spectrum. But the gravitational potential leads to a large scale modulation of the matter distribution, i.e. the much smoother potential provides a large scale imprint on the density field. Hence, it is natural to investigate this type of interaction avoiding any additional smoothing and to consider the density field given on the simulation grid in comparison with the potential. In this manner we can test more clearly the emergence of a natural bias or of resulting large typical scales in the matter density, and possibly in the spatial galaxy distribution leading to the wall-like SLSS. Here, we want to describe this link qualitatively and quantitatively using numerical simulations, a more detailes theoretical description should be given elsewhere.

4.1. Qualitative description

Contrary to the large scale potential field, the density perturbations evolve strongly both in the linear and the non-linear regime, cp. Fig. 9 for one simulation (for the same where the potential slice is shown in Fig. 2). We can clearly see an evolution towards more pronounced structures. This evolution looks like strings of clumped matter points moving toward the biggest clumps (clusters). These big clumps are pulling the smaller clumps along the inward pointing strings or filaments. Further we see richer filaments at redshift [FORMULA] than at [FORMULA]. They seem to become less well defined with time, while at the same time large areas are more or less empty of matter, so that small voids appear.

[FIGURE] Fig. 9. The evolution of the density field with time from [FORMULA] to [FORMULA] in comoving coordinates. The contour levels are the same for all three redshifts

In Fig. 10 we show the high density regions of the same CDM75 slice as in Fig. 9 at [FORMULA] for different contour levels. The slice corresponds to the gravitational potential field shown in Fig. 2b. The contour levels include 50% of the total matter in the slice in (a), 25% in (b) and 10% in (c). While the contour levels of the dark matter shown in 9c have a clear filamentary structure, they disappear gradually if looking at the high density regions (Fig. 10a and b). Fig. 10c contains only a few dense clumps, which can be interpreted as clusters, or at least as dark matter halos. From Fig. 10 we conclude that at [FORMULA] more than [FORMULA] of all matter in this slice is concentrated in a relatively low number of high density peaks in accordance with the estimates of Doroshkevich et al. (1997a). This is a typical feature of the considered late evolutionary stage. From Fig. 10 we also conclude that the spatially biggest clusters are also the heaviest.

[FIGURE] Fig. 10. A density slice for the CDM75 simulation, taken at [FORMULA]. It is the same slice as in Fig. 9 but there the high density regions at [FORMULA] correspond to a fraction 75% of the total mass in the slice, here we show 50% in a, 25% in b, and 10% in c.

Let us now compare Figs. 2, 9 and 10. In Fig. 9 we see the motion of smaller clumps and their merging with bigger clumps to the double cluster at the left, another cluster in the bottom and a third one near the middle. These big clusters have their counterparts in Fig. 2b, where 3 very deep potential wells confine the clusters formed by [FORMULA]. It is worth noting again that the evolution of the density field changes mostly the local potential distribution, but there are no global changes of the [FORMULA] contour levels with time. Also not much matter is flowing across these levels.

Comparing the gravitational potential in Fig. 2b with the contours of the density field in Fig. 10, we see that the weakest objects disappear with increasing contour levels, and, as expected, only the heaviest remain in the deepest potential wells. The highest density peaks are clearly correlated with the ODR. Thus the largest clusters trace the ODR.

The strong link between the spatial distribution of the density perturbations and the potential field is illustrated in Fig. 11, where a concentration of density peaks in regions of negative potential becomes obvious.

[FIGURE] Fig. 11. The evolution of a slice from [FORMULA] to [FORMULA] of the BSI200 and CDM200 models, respectively. The density is shown as grey tones, where white corresponds to empty space and the darkest spots to the highest density peaks. The solid lines correspond to [FORMULA]. The UDR are situated in the corners. The ODR at [FORMULA] are clearly seen as it accumulates the main fraction of high dense clumps.

From the histograms in Figs. 5 and 6 we concluded that from all our simulations CDM200 had the greatest number of intersections with potential levels [FORMULA] along a random straight line, and we should therefore expect to find the highest number of SLSS elements in this model. In order to get a visual impression of the SLSS in the simulations, 3 redshift realizations for the two [FORMULA] Mpc models have been plotted (Fig. 11). The first obvious property of the sequence of slices is the static nature of the levels [FORMULA] (the solid lines), they practically do not change their positions in space from [FORMULA] to [FORMULA], and so they are unaffected by the large-scale motion of matter. We also see that the evolution of the density field with time in [FORMULA] -regions is faster than in [FORMULA] -regions.

The difference between the two cosmological models is likewise evident. This is not surprising, as the CDM model has more power than the BSI model on all scales covered by the simulation (see Fig. 1). Indeed, we see from Fig. 11 that the CDM model at [FORMULA] behaves approximately like BSI at [FORMULA]. Not only the density peaks grow faster in the CDM model, but also the LSS forms earlier. Moreover, the LSS elements seem to break up already at [FORMULA]. In the BSI model all evolution is later and slower. On larger scales the two models become more alike, which is also expected from the behaviour of the power spectrum. The placement of the potential levels [FORMULA] in the two models are practically the same, because the same random realization was used at [FORMULA]. The stronger evolution of the CDM model is also seen by the larger number of clusters than in the BSI model. In the CDM model well defined clusters are already present at [FORMULA], but they do just first appear at [FORMULA] in the BSI model.

It is interesting that one heavy cluster has developed in the [FORMULA] -region (lower left corner in Fig. 11) that shows that several dark matter clumps may arise also in [FORMULA] -regions. This is a natural consequence of the random character of the perturbations.

Using the periodicity of the simulation box, we can identify a connected (i.e. a percolating) structure which seems to lie approximately in the centre of the [FORMULA] -region in the CDM model at [FORMULA]. Especially the highest density peaks (the darkest spots) seem to trace this structure. As mentioned earlier, we would expect to see about 1 to 2 walls along a straight line, however, the evolution of the CDM model on small scales already breaks up this wall-like structures.

Voids of different sizes are present. The mean void size grows with time while the number of voids decreases. It looks as though there are two kind of voids in Fig. 11. One type is bounded by the weaker structure, and the other bounded by the strongest density peaks. The latter defines a kind of super-voids inhabited by relatively weak filamentary structures.

The comparison of Figs. 2, 9, 10 and 11point to the growth of the correlation between matter density and the gravitational potential due to the matter concentration into high dense clumps. At the same time the strong difference in the evolutionary rate of potential and density perturbations can be clearly seen. During the period under consideration the potential perturbations are changed on small scales, while on larger scales the potential distribution is very stable.

4.2. Quantitative description of the potential-density interaction

The detailed quantitative characteristics of the potential-density interaction requires a theoretical derivation based on some approximation of the non-linear dynamics. As a first step, we only provide a description and general characterisation of this interaction.

Fig. 12 illustrates this interaction for the models CDM200. The other models were investigated as well and they show similar results. The functions [FORMULA] and [FORMULA] are defined as the cumulative fraction of volume and mass, respectively, lying below the potential level [FORMULA] whereas [FORMULA] is the mean density in this region. Fig. 12a shows the [FORMULA] versus [FORMULA] relation. The lower value of the fraction of volume [FORMULA], occupying the regions below the potential level [FORMULA], than the value of the corresponding fraction of mass [FORMULA] shows the formation of ODR and UDR, the same behaviour is seen in Fig. 4. The matter concentrates in negative potential regions during the evolution. When more and more matter is gathered, and it forms deep potential wells, then the curves in Fig. 12a turn towards the right low corner. In Fig. 12b we show that the density increases by 3 orders of magnitude basically in the negative potential region. This is accompanied by the decrease of the global minimum of the potential.

[FIGURE] Fig. 12. Density-potential evolution of CDM200: [FORMULA], [FORMULA] and [FORMULA] are the fraction of space, the fraction of mass and the particle density, respectively, below the level [FORMULA] in the gravitational potential. [FORMULA] and [FORMULA] are the differential fraction of space and of mass per potential interval (in units of [FORMULA] km/s). The thin lines in a, b and c correspond to [FORMULA], the thick lines to [FORMULA]. The dotted line in a and b correspond to a homogeneous distribution. In c the solid lines correspond to the space fraction [FORMULA] and the dashes lines to the mass fraction [FORMULA].

Fig. 12c shows the evolution of the differential distribution of the matter and space fraction, [FORMULA] and [FORMULA], respectively, at potential value [FORMULA] for two redshifts. At [FORMULA] both curves are symmetrical with respect to the zero point, while at [FORMULA] only [FORMULA] remains almost symmetrical and [FORMULA] becomes significantly asymmetric. The typical wings of [FORMULA] into the [FORMULA] regions arise from the matter infall into the potential wells while the shift of the median of the curve describes the slow mass flow through the [FORMULA] level.

A most interesting problem is the connection of the initial potential distribution with the current density distribution as it manifests the influence of initial perturbations to the later evolutionary stages. Such an influence lies on the basis of the adhesion model (see, e.g., Shandarin  &  Zeldovich 1989). The correlation between the potential at [FORMULA] and the density field at redshifts [FORMULA], 2 and 0 is shown in Fig. 13 for CDM25 and CDM200. Again one notices the slow flow of matter with time to the regions of negative initial potential. As it was shown above qualitatively, faster and more effective clustering occurs in regions of negative potential while the boundaries [FORMULA] remain practically at rest. Fig. 13 gives us the quantitative characteristic of this effect and shows that the matter flow across the surface [FORMULA] in CDM200 is small. On the other hand, a stronger clustering and strong matter concentration in ODR occurs in the smaller simulation CDM25 which is in the latest non-linear evolutionary period. A significant number of high peaks are present at [FORMULA]. They appear when the sheetlike and filamentary DM structures disrupt into a system of massive clumps (Doroshkevich et. al 1997a). Nevertheless, the strong correlation of the matter distribution with the initial potential distribution can be seen also in the small box. As expected, the correlation increases during the evolution.

[FIGURE] Fig. 13. The correlation between the potential at [FORMULA] and the density field at redshifts 25, 2 and 0, for CDM25 & CDM200. Shown are [FORMULA] at [FORMULA] (solid line) and [FORMULA] at redshifts [FORMULA] (long dashed line), [FORMULA] (dashed line) and [FORMULA] (dotted line)

In Table 2, the fraction of mass [FORMULA], the fraction of space [FORMULA] of ODR and UDR, and the mean density [FORMULA] in ODR and UDR, respectively, are shown for both the initial and final redshifts versus the initial and current potential distribution.

At [FORMULA], [FORMULA], [FORMULA] and [FORMULA] have almost identical values both in ODR and UDR (and in all simulations) because they characterize the (almost) homogeneous initial matter distribution. At redshift [FORMULA] this situation has changed. In all simulations, the mass fractions have increased in ODR and decreased in UDR, clearest in CDM25 where [FORMULA] of the mass have gathered in ODR. In the two small simulations the space fraction occupied by the ODR decreases also. This can be hardly seen in the two big simulations, where [FORMULA] is approximately equal to [FORMULA] from [FORMULA] to [FORMULA]. Due to the more significant change in [FORMULA] in comparison to [FORMULA], the density of UDR [FORMULA] decreases with time (for instance [FORMULA] in CDM200). These issues are in agreement with results of previous investigations (Doroshkevich et al., 1997a) using the correlation function and the core-sampling approach. They provide us with additional quantitative characteristics of the evolution of the models.

Generally speaking, Table 2 shows that during matter clustering only a moderate or negligible change takes place of [FORMULA] while the value [FORMULA] changes much more. Almost the same ratios are obtained if the ODR/UDR are defined with respect to the initial potential distribution. From Fig. 11 one could expect that the highest density peaks were significantly correlated with the potential. But this figure consist only of a small slice and might not be representative enough while Table 2 presents results for four complete simulations.

4.3. Potential-density correlations

Qualitatively, there is no doubt that the (negative) cross-correlation between the potential and the density increases with time because the matter concentration in the high dense clumps is always accompanied by the growth of (negative) potential. However, looking for a quantitative characterization, the correlation between the gravitational potential and density fields turns out to be more complicated.

The simplest characteristic of such a correlation is the correlation coefficient [FORMULA], given by

[EQUATION]

Here [FORMULA] denotes a volume average represented in Fourier space as an average over moments of the spectrum (see, e.g., Demianski  &  Doroshkevich 1992). The potential [FORMULA] and the density excess [FORMULA] must be taken at the same point. The dispersion of the potential [FORMULA] was defined in Eq. ( 6). The other parts of Eq. ( 7) are defined in the same manner:

[EQUATION]

[EQUATION]

Evidently, the correlation coefficient is independent of the normalisation of the power spectrum, but it is sensitive to the nonlinear evolution of the spectral shape. Here the correlation coefficient [FORMULA] was found using the simulated power spectra [FORMULA], compare, e.g., Fig. 7). This procedure is similar to the one used in Sect. 3.3 for the calculations of [FORMULA].

In Fig. 14 the evolution of the correlation coefficient [FORMULA] with redshift z is shown. At the start of the simulations it has about the value of linear theory, after which the magnitude decreases while in the later evolutionary stages ( [FORMULA] ) it stays almost constant or slowly increases. This unexpected behaviour of [FORMULA] requires more detailed investigations with a wider sample of simulations.

[FIGURE] Fig. 14. Evolution of the correlation coefficient [FORMULA] with redshift z for CDM25 (full circles) and for CDM200 (stars) vs. redshift z. The two horizontal lines show the correlation coefficient calculated by the linear power spectrum (CDM25 long-dashed line, CDM200 dashed line)
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© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997
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