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

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3. Spatial distribution of gravitational potential

As the first step of our consideration we need to reveal and to describe the properties of the simulated potential distribution. We need also to introduce and to test some quantitative characteristics of the spatial distribution of the potential field.

The spatial distribution of random field can be conveniently characterized by various mean values that are combinations of moments of the power spectrum (some examples of such characteristics were given by Bardeen et al. 1986 and Sahni & Coles 1995). It is important to specify whether they are given for untruncated power spectra which is important for theoretical considerations and comparison with observations, or together with various window functions what is important for comparisons with simulations which operate with truncated power spectra.

3.1. Qualitative description

The main properties of the potential distribution can be demonstrated by the evolution of the spatial distribution of the potential in a slice plotted in Fig. 2 for CDM75 at [FORMULA] and at [FORMULA]. The larger features of the potential are unchanged while on small scales the structure of potential is modified strongly and all the wrinkles in the equipotential lines are smoothed out with time. The lines come from the intersection of the plane under consideration with the 2-dimensional equipotential surfaces in 3-dimensional space.

[FIGURE] Fig. 2. Slices of the gravitational potential from CDM75 (left at [FORMULA], right at [FORMULA] ). The contour numbers are the values of the potential in units of [FORMULA] (km/s)2

The matter concentrations in denser clumps influences locally the potential. This can be clearly seen in the most negative regions of the potential in Fig. 2, where some very deep potential wells evolve with time. The deep hole in the centre of Fig. 2b is especially prominent.

The isolines of potential distribution in Fig. 2 demonstrate also the well known fact, that the shapes of these isolines are more or less elliptical for high potential peaks and deep wells, and they become more and more complicated near the zero level. These properties are typical for random Gaussian fields which we now describe quantitatively.

3.2. Quantitative characteristics of the spatial potential distribution

Let us consider the potential along a random straight line. Thus we transform the complicated three-dimensional problem to the much simpler one-dimensional one. A 1D potential distribution is plotted in Fig. 3 for CDM75 and CDM200. The CDM75 profiles in Fig. 3a correspond to the cuts [FORMULA] Mpc of Fig. 2, so that the "blob" at [FORMULA] Mpc of Fig. 2a is included. This feature is seen in Fig. 3a as a tangent to [FORMULA]. Fig. 3 demonstrates that in general, excluding the high dense clumps, the potential becomes smoother with time. Indeed, here the potential contains only one large wavelength giving raise to one maximum and one minimum. It is interesting that both the CDM75 and the CDM200 simulation contain just one basic wavelength of the potential perturbations which is an characteristic of the realized range of the CDM potential perturbations.

[FIGURE] Fig. 3. One-dimensional potential distribution in CDM75 & CDM200

In the following we call regions with positive potential ( [FORMULA] ) [FORMULA] -regions. During cosmic evolution the density in these regions becomes lower than the mean density (under dense region - UDR). In [FORMULA] -regions ( [FORMULA] ) the density becomes higher than the mean density (over dense region - ODR). This can be clearly seen in Fig. 4 where the mean value of the density contrast in both regions is shown separately. Note, that the density contrast increases in [FORMULA] -regions only by a factor of about 1.4 for CDM200. In the more evolved CDM25 simulation, we find over/underdensities of 2.2 and 0.17 for [FORMULA] and [FORMULA], respectively. More detailed properties of [FORMULA] and [FORMULA] regions are given later on (Table 2). There we will show that the density contrast arises from a small mass flow through the boundary between [FORMULA] and [FORMULA] regions, whereas the respective volumes change only by a very small amount.

[FIGURE] Fig. 4. Evolution of the density contrast in the regions of positive ( [FORMULA] ) and negative ( [FORMULA] ) potential for CDM200

The 1D approach allows us to introduce a simple quantitative characteristic of the potential distribution, the mean separation (mean free path) between zero points of the potential, [FORMULA]. Demianski  &  Doroshkevich (1992) found a relation between [FORMULA] and the power spectrum [FORMULA],


where k is the comoving wave number. The value [FORMULA] is closely linked with the correlation length of the potential (see, e.g., Bardeen et al. 1986). Equation ( 1) can be used for simulations with truncated power spectra, however, it can obviously not be used for spectra with Harrison-Zeldovich asymptote [FORMULA] at [FORMULA], because the upper integral is divergent and leads to an infinite correlation length of the potential. Doroshkevich et al. (1997a) have derived a general relation which can be simplified for broad band power spectra to the equation


the solution of which gives the correct value [FORMULA] both for the truncated and the total power spectrum. For the standard CDM and BSI matter dominated power spectra with [FORMULA] we obtain


Note, that due to the scaling of the transfer function with [FORMULA] the resulting [FORMULA] of our models scales with [FORMULA].

The comparison of these results obtained for the theoretical spectra with similar estimates for simulations provides a simplest quantitative characteristic of the representativity of the simulations with respect to the large scale spatial potential distribution because both expressions (1) and (2) are sensitive to small k -values and to the size of the simulation box. Hence, it allows to estimate the impact of the computational box size on the matter evolution.

In simulations the mean separation of points [FORMULA] along a random straight line, [FORMULA], can be found directly. Thus, if [FORMULA] is the set of measured distances between levels [FORMULA] and N is their total number, then the mean separation and dispersion [FORMULA] are given by


In Table 1 the results are listed for CDM and BSI simulations at redshift [FORMULA], [FORMULA] and [FORMULA] together with the theoretical values of [FORMULA] calculated from Eq. (2) for the truncated spectra used for simulations. The upper and lower limits of the integral were taken from the box size and the cell size.


Table 1. Theoretical and measured values of the mean separation [FORMULA] and their dispersion [FORMULA] for potential levels [FORMULA]


Table 2. Parameters of the ODR and UDR, defined as regions where [FORMULA] and [FORMULA], respectively for redshifts [FORMULA] and [FORMULA]. The columns 25-0 denotes the comparison of the density field at [FORMULA] with respect to the the initial potential field at [FORMULA] (for [FORMULA] the columns 25 and 25-0 are identical). [FORMULA] is the fraction of mass, [FORMULA] is the fraction of space and [FORMULA] is the density (in units of the critical density)

The dispersion in Table 1 characterizes the actual distance distribution rather than the errors in the measurements of [FORMULA]. The reason for the large dispersion becomes clear from frequency distributions [FORMULA] of the measured [FORMULA] values (see Figs. 5 and 6 for CDM25 and CDM200, respectively). They are produced by 3963 random straight lines cutting the actual contour levels of the potential in the simulation box.

[FIGURE] Fig. 5. Frequency distribution [FORMULA] of measured distances [FORMULA] between potential levels [FORMULA] in the [FORMULA] Mpc simulations. The dashed lines are the mean [FORMULA] from Table 1, and the dotted lines are the theoretical values [FORMULA]

[FIGURE] Fig. 6. Frequency distribution [FORMULA] of measured distances [FORMULA] between potential levels [FORMULA] in the [FORMULA] Mpc simulations. The dashed lines are the mean [FORMULA] from Table 1, and the dotted lines are the theoretical values [FORMULA]. Fits to the histogram with the Poisson distribution are included as dotted lines.

In CDM25 and BSI25 simulations the different distances between levels [FORMULA] are almost equal abundant, but we see already a trend towards smaller distances. This becomes more evident in the bigger simulations. Actually, the histogram is well described by a Poisson distribution in case of the CDM200 and BSI200 simulations (see the dotted line in Fig. 6).

From a theoretical point of view, we expect a Poisson distribution of SLSS elements (and also of [FORMULA] ) along a random straight line for distances [FORMULA] where any correlation becomes negligible (White 1979, Buryak et al. 1991, Borgani 1996). However, we see that for the small box sizes the measured distribution is far from Poissonian. For distances ( [FORMULA] ) this is caused by the limitations of the simulations regarding SLSS elements because the number of long wave harmonics in the perturbations are strongly limited. However, in case of smaller scales the potential distribution is correlated and these correlations depend on the power spectrum. The CDM200 model shows the best indications of a Poissonian distribution because it contains the most SLSS elements in all the simulations. This can be easily seen from the values of [FORMULA] in CDM200, which shows an average of 3 to 4 intersections with levels [FORMULA] per box length, while on the other hand the [FORMULA] Mpc simulations have only about two.

In this respect the CDM25 and BSI25 models are similar to the scale free models with a power index between [FORMULA] and [FORMULA] (see Fig. 1). Only for the models CDM200 and BSI200 deviations from the power law becomes important, so that we have to deal with a real broad band power spectrum. As we could see, this is also reflected by the potential characteristics. Thus, both for [FORMULA] =75 [FORMULA] Mpc and [FORMULA] =25 [FORMULA] Mpc the values [FORMULA] are close to half the box size, 0.5 [FORMULA]. In all simulations besides CDM200, the realized values [FORMULA] are much smaller than the values [FORMULA] obtained for the untruncated power spectra. This means that only the CDM simulation in the larger computational boxes realistically reproduces the potential distribution, and therefore it can expected to describe the matter evolution on the SLSS scale, comparable with the observed matter distribution. In contrast, any simulations in smaller boxes have more or less methodical character.

A common feature of all histograms (see also Table 1) is the slow increase of [FORMULA] during evolution due to non-linear effects whereas [FORMULA] is defined by the initial power spectrum. However, [FORMULA] remains near to [FORMULA] during the whole evolution. The slightly faster evolution of CDM75 and CDM25 should be ascribed to the late evolutionary stage of these simulations. Indeed, some structure elements have been destroyed at [FORMULA], and the matter distribution starts to transform into a system of isolated clumps (see also the discussion in Doroshkevich et al. 1997a).

3.3. Potential-potential correlation

The simple but important quantitative characteristic of the potential distribution is the two-point autocorrelation function. It allows us to obtain the characteristics of potential distribution averaged over the entire simulation.

For the primordial Harrison-Zeldovich spectrum the `gravitational potential' cannot be used directly because the dispersion of the potential diverges in the limit [FORMULA]. Therefore, here one has to use the 'potential difference' between two points. This implies some modifications of the theoretical description (see, e.g., Demianski  &  Doroshkevich 1997). However, in case of simulations this problem disappears due to the finite box size. Thus, we can use here the `potential' without any restriction.

Thus, the standard correlation function of the potential in points [FORMULA] and [FORMULA] can be written as following:


where [FORMULA]. For CDM-like power spectra this relation can be fitted as following (Demianski  &  Doroshkevich, 1997)


where the value [FORMULA] is the mean separation of the zero potential along a random straight line defined by Eq. ( 2) and


is the dispersion of the potential perturbations.

In Fig. 7 the evolution of the power spectrum is depicted for CDM25 at redshifts [FORMULA]. At high redshifts the power spectrum has some unphysical fluctuations at short wavelengths, caused by the finite resolution of the simulation. As upper limit in the integration of the power spectrum, we take the Nyquist frequency. Later the evolution increases strongly the short wave amplitude relatively to the linear law. In our case the long wave spectral amplitude decreases slightly due to the special initial realization.

[FIGURE] Fig. 7. The evolution of the power spectrum with redshift for CDM25. The dashed lines are the corresponding power spectra calculated by linear theory. The horizontal bar indicates box size and Nyquist frequency.

The function [FORMULA] has been calculated for the CDM200 and CDM25 models using the power spectra reconstructed at several redshifts (z = 25, 2  &  0) from the simulation. The results are presented in Fig. 8 together with the theoretical values given by Eq. ( 5). The different behaviour of theoretical and measured values at large distances characterizes the different potential distribution for truncated power spectra realized in simulations and the theoretical CDM spectra, i.e. it is caused by the impact of the finite box size. This conclusion is consistent to the former discussion of the one-dimendional analysis of separations of zero-points of the potential since this separation concerns the length scales larger than half of the simulation box.

[FIGURE] Fig. 8. The normalized correlation function of potential vs. comoving distance for two models and two redshifts. The dotted lines present the simulation results and the dashed line the expectation of Eq. (5).
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© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997