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Astron. Astrophys. 329, 1-13 (1998)
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
than at
. 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]](img76.gif) |
Fig. 9. The evolution of the density field with time from
to
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
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
more than
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]](img79.gif) |
Fig. 10. A density slice for the CDM75 simulation, taken at
. It is the same slice as in Fig. 9 but there the high density regions at
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
. 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
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]](img82.gif) |
Fig. 11. The evolution of a slice from
to
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
. The UDR are situated in the corners. The ODR at
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
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
Mpc models have been plotted (Fig. 11). The
first obvious property of the sequence of slices is the static nature
of the levels
(the solid lines), they practically do not
change their positions in space from
to
, 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
-regions is faster than in
-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
behaves approximately like BSI at
. 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
. 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
in the two models are practically the same,
because the same random realization was used at
. 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
, but they do just first appear at
in the BSI model.
It is interesting that one heavy cluster has developed in the
-region (lower left corner in Fig. 11) that
shows that several dark matter clumps may arise also in
-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
-region in the CDM model at
. 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
and
are defined as the cumulative fraction of
volume and mass, respectively, lying below the potential level
whereas
is the mean density in this region. Fig. 12a
shows the
versus
relation. The lower value of the fraction of
volume
, occupying the regions below the potential
level
, than the value of the corresponding fraction
of mass
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]](img92.gif) |
Fig. 12. Density-potential evolution of CDM200:
,
and
are the fraction of space, the fraction of mass and the particle density, respectively, below the level
in the gravitational potential.
and
are the differential fraction of space and of mass per potential interval (in units of
km/s). The thin lines in a, b and c correspond to
, the thick lines to
. The dotted line in a and b correspond to a homogeneous distribution. In c the solid lines correspond to the space fraction
and the dashes lines to the mass fraction
.
|
Fig. 12c shows the evolution of the differential distribution of
the matter and space fraction,
and
, respectively, at potential value
for two redshifts. At
both curves are symmetrical with respect to the
zero point, while at
only
remains almost symmetrical and
becomes significantly asymmetric. The typical
wings of
into the
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
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
and the density field at redshifts
, 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
remain practically at rest. Fig. 13 gives us the
quantitative characteristic of this effect and shows that the matter
flow across the surface
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
. 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]](img96.gif) |
Fig. 13. The correlation between the potential at
and the density field at redshifts 25, 2 and 0, for CDM25 & CDM200. Shown are
at
(solid line) and
at redshifts
(long dashed line),
(dashed line) and
(dotted line)
|
In Table 2, the fraction of mass
, the fraction of space
of ODR and UDR, and the mean density
in ODR and UDR, respectively, are shown for
both the initial and final redshifts versus the initial and current
potential distribution.
At
,
,
and
have almost identical values both in ODR and
UDR (and in all simulations) because they characterize the (almost)
homogeneous initial matter distribution. At redshift
this situation has changed. In all simulations,
the mass fractions have increased in ODR and decreased in UDR,
clearest in CDM25 where
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
is approximately equal to
from
to
. Due to the more significant change in
in comparison to
, the density of UDR
decreases with time (for instance
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
while the value
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
, given by
![[EQUATION]](img106.gif)
Here
denotes a volume average represented in Fourier
space as an average over moments of the spectrum (see, e.g., Demianski
& Doroshkevich 1992). The potential
and the density excess
must be taken at the same point. The dispersion
of the potential
was defined in Eq. ( 6). The other parts of Eq.
( 7) are defined in the same manner:
![[EQUATION]](img110.gif)
![[EQUATION]](img111.gif)
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
was found using the simulated power spectra
, compare, e.g., Fig. 7). This procedure is
similar to the one used in Sect. 3.3 for the calculations of
.
In Fig. 14 the evolution of the correlation coefficient
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 (
) it stays almost constant or slowly
increases. This unexpected behaviour of
requires more detailed investigations with a
wider sample of simulations.
![[FIGURE]](img115.gif) |
Fig. 14. Evolution of the correlation coefficient
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
helpdesk.link@springer.de |