by Jaan Einasto
8.3 The formation and evolution of the cosmic web
Numerical simulations of the evolution of the Universe applying the Zeldovich approach to set initial conditions reproduce the present structure of the cosmic web well. I was interested in understanding how the evolution actually works — How one gets from a random field of density perturbations of various scale the cosmic web with filaments, clusters, superclusters and voids? In other words, What processes are involved in this transformation from a random field to a quasi-regular cosmic web? One of our goals was to understand what causes the formation of superclusters, in particular the quasi-regularity of the location of very rich clusters and superclusters, discussed in the previous Chapter (Einasto et al., 1997a).
First I assumed that the observed quasi-regularity may be caused by a strong feature in the power spectrum of density perturbations. During a visit to Fermilab in 2000 I calculated a number of N-body models with an artificial strong peak at the wavelength 120 h−1 Mpc, as found from observations by Einasto et al. (1997a). This attempt failed. Even a narrow peak at this scale, exceeding tenfold the amplitude of the mean spectrum around the peak, did not create the observed quasi-regularity of rich clusters and superclusters. Evidently the reason of this phenomenon must lie somewhere else.
8.3.1 The luminosity density field of the SDSS
I discussed the problem with Enn Saar, who had applied various methods to study the density field. We came to the conclusion that we had to start the study with a more detailed analysis of the real luminosity density field. As an example of the real density field we chose the equatorial slice of the Sloan Digital Sky Survey (SDSS). The density field is usually described by the amplitudes of Fourier components of the field, expressed by the power spectrum of density perturbations. However, the spatial structure of the field depends both on the amplitudes and on the phases of the density field. Thus both aspects of the field must be investigated.
To study the role of phase information in more detail, we extracted a 2-dimensional rectangular region of size 512 Mpc, calculated for the Hubble constant h = 0.8, from the density field of the SDSS Northern equatorial wedge of 2.5 degrees thickness up to redshift z = 0.2. This region is shown in the upper left panel of Fig. 8.1. The observer is located at the lower left corner. The colour-coded density levels used in plotting are in the interval from 0 to 10 in mean density units with white corresponding to the highest value using the SAO Image program DS9 colour palette SLS. Then we Fourier transformed the 2D density field, randomised phases of all Fourier components, and thereafter Fourier transformed it back to see the resulting density field.
The modified field has the same amplitudes of all wave-numbers k as the original field; only the phases of waves are different. The results are shown in the upper right panel of the Fig. 8.1. With shifting phases of density waves some densities become negative, thus in this case colour codes are in the density interval ±3.5. We see that the whole structure of superclusters, filaments and voids has disappeared; the field is fully covered by tiny randomly spaced density enhancements. There are even no clusters of galaxies in this picture, comparable in luminosity to real clusters of galaxies.
This simple test shows the importance of phase information in the understanding of the structure of the web. Coles & Chiang (2000) came to the same conclusion by randomising phases of a simulated filamentary network.
To study the influence of waves of different scales Enn developed programs where only waves in a specified scale interval are randomised. I tried this program using again the 2D slice of the SDSS Northern equatorial wedge. To our dissatisfaction, we did not find clear evidence of the influence of randomising phases of waves of different scales.
Next we tried the wavelet method. It is well known that the Fourier space is not sensitive to the location of particular high-density features in real space, such as filaments, clusters, and superclusters. To have a better understanding of the texture of the cosmic web, the web must be studied in real space. For this purpose wavelet analysis can be applied, which analyses properties of waves of various scales in real space (Jones, 2009).
In the wavelet analysis the density field is decomposed into several frequency bands as follows. The high-resolution (zero level) density field is calculated with the kernel of width, equal to the size of one cell of the field. Every next field is calculated with the twice larger kernel. Wavelets are found by subtracting higher level density fields from the previous level fields. In such a way each wavelet band contains waves twice longer the previous band, in the range times the mean (central) wave (Martínez & Saar, 2002).
Figure 8.1 shows wavelets 7 to 4 of the Sloan Northern rectangular region. These wavelets characterise waves of length about 256, 128, 64, and 32 Mpc, respectively. In wavelet figures both under- and over-densities are shown. Extreme levels were chosen so that mean features of the structure are well visible.
The middle left panel of Fig. 8.1 shows the waves of length about 256 Mpc. In its highest density regions there are three very rich superclusters: N20 from the list by Einasto et al. (2003b), located in the upper part of the Figure, supercluster N13 (SCL126 from the list by Einasto et al. (2001) in the Sloan Great Wall) near the centre, and supercluster N02 (SCL82) in the lower right part of the panel. Supercluster numbers are shown in the middle right panel w6.
The next panel shows waves of about 128 Mpc. Here the most prominent features are superclusters N13 (SCL126) and N02 (SCL82). The supercluster N23 (SCL155) in the upper left part of the panel is also fairly strong, seen as weak density peak already in the previous panel. In addition we see the supercluster N15 just above N13 near the minimum of the wave of the 256 Mpc scale, and a number of poorer superclusters located mostly in voids defined by waves of larger size.
The lower left panel plots waves of scales about 64 Mpc. Here all superclusters seen on larger scales are also visible. A large fraction of density enhancements are either situated in the middle of low-density regions of the previous panel, or they divide massive superclusters into smaller subunits. This property is repeated in the next panel. Here the highest peaks are substructures of rich superclusters, and there are numerous smaller density enhancements (clusters) between the peaks of the previous panel.
Fig. 8.1 The 2-dimensional rectangular region of size 512 × 512 Mpc of the luminosity density field of the SDSS. Upper panels show the actual density field, and the field with phases randomly shifted. Middle and lower panels show the wavelets w7, w6, w5, and w4 of the field (Einasto et al., 2011a).
When we compare density waves of all scales, then we come to the conclusion that superclusters form in regions where large density waves of various scales combine in similar over-density phases. The larger the scale of the wave where this coincidence takes place, the richer the superclusters are.
Similarly voids form in regions where density waves of medium and large scales combine in similar under-density phases. In large voids medium-scale perturbations generate a web of filamentary structures with knots.
This simple analysis demonstrates very clearly the role of phase coupling (synchronisation) of density waves of different scales in the formation of the supercluster–void network.
8.3.2 The role of density waves of various scales
Our next goal was to investigate in more detail how structure has formed and what role is played by density perturbations of various scale. I discussed the problem again with Enn. After some thoughts and trial calculations we found that the best way to understand the role of density perturbations of different scales in the formation of the cosmic web is to use models with a varying large-scale cutoff of the power spectrum of initial perturbations. So far the influence of small-scale waves was studied by cutting small-scale perturbations; one such example is the HDM model where small-scale perturbations are absent. Now we decided to do it otherwise. Enn developed a variant of the generation of initial conditions of the N-body simulations, where the amplitude of density perturbations in a given scale interval is
put to zero.
Thus we performed several series of simulations. All simulations of a given series had identical initial conditions with ‘random’ initial positions and velocities of test particles, but the amplitudes of all perturbations on a scale exceeding a given scale were forced to be zero. In this way, we can follow how systems of galaxies grow under the influence of perturbations of various scales.
The first trial models were calculated by Enn already in 2003 with a rather small resolution, with 1283 particles and cells. As our computer park evolved we made new series with higher resolution. We also used simulations made in Potsdam in collaboration with Volker Muller, and in Tartu University computer center. The last simulations were made by our young collaborator Ivan Suhhonenko. Preliminary results of our calculations were reported at various conferences: “Bernard60” conference in Valencia 2006, void conference in Amsterdam 2006, Zeldovich memorial conference in Minsk 2009. The final results of this study were published only recently by Einasto et al. (2011b,a); Suhhonenko et al. (2011).
Fig. 8.2 Density fields for the models of the M256 series. The upper panels show the high-resolution fields forthe models M256.256, M256.064, and M256.032, the lower panels for the models M256.864, M256.016, and M256.008 (from left to right). The densities are shown for a layer of 6 h−1 Mpc thickness at the k = 75 coordinate. All fields correspond to the present epoch z = 0. The densities are expressed on a logarithmic scale to enhance the low-density regions; only the overdensity regions are shown (Suhhonenko et al., 2011).
To have both a high spatial resolution and the presence of density perturbations in a larger scale interval we used simulations in boxes of sizes varying from 64 h−1 Mpc to 768 h−1 Mpc, and resolutions and ; our basic simulations were made in boxes of sizes 100 and 256 h−1 Mpc (Einasto et al., 2011b,a). The notations for our models are: the first characters M and L designate models with resolutions of Ngrid = 256 and Ngrid = 512, respectively. The following number gives the size of the simulation box, L, in h−1 Mpc; the subsequent number indicates the maximum wavelength used in the simulation, also in h−1 Mpc. Some simulations were made in boxes of sizes 64,512, and 768 h−1 Mpc, to understand the role of the smallest and the largest density perturbations.
Fig. 8.2 shows the density fields of models of the series with cube length L = 256 h−1 Mpc, M256, at the present epoch z = 0. In the model M256.864 the amplitude of perturbations between wavelength 8 h−1 Mpc and 64 h−1 Mpc has been put to zero, to see the influence (actually the absence) of density perturbations of medium scales.
Fig. 8.3 The left panel show the cumulative mass functions of the density field clusters for the models with various cutoff scales. The right panel show the mean radii of voids, defined by the DF clusters for different threshold masses, Mth, and for various cutoff scales (Suhhonenko et al., 2011).
The Figure demonstrates very clearly the role of density perturbations of various scales in the formation of the cosmic web. In the absence of large-scale perturbations, systems larger than the cutoff scale do not form. The removing of only the medium-scale perturbations is of particular interest. In such a model there are no filaments, both within superclusters and between them. The distribution of small- scale systems is more or less random, and there are no compact systems of galaxies such as clusters — the compact systems are rather small. Superclusters are present, but they lack the fine structure. This simple exercise shows that for the formation of the cosmic web the combined influence of density waves of all scales is important.
To characterise quantitatively the dependence of the cosmic web on the scale of perturbations Suhhonenko et al. (2011) used two tests, the cumulative mass function of density field clusters and the void radius distribution, as determined by density field (DF) clusters of different mass. Density field clusters were defined as peaks of the high-resolution density field. In the search for maxima of the field we used a minimum density threshold, D0 = 2, and minimal mass, Dp = 5. The mass was calculated as a sum of density values within ±3 cells from the central one; the mass is given in Solar mass units.
The cumulative mass functions of the DF clusters for the models of the series L256 are shown in Fig. 8.3. We see that DF cluster masses strongly depend on the scale of the power spectrum cutoff. All models with a cutoff on the scale 8 h−1 Mpc have DF cluster mass distributions with rather sharp decreases on the high mass side. The maximum masses of the DF clusters in these models are 1.5 × 1013 M. With increasing spectrum cutoff scale, the mass distribution rapidly shifts to higher masses. This rapid increase in the maximum DF cluster mass continues up to the cutoff scale of 64 h−1 Mpc. This increase stops at the cutoff scale 128 h−1 Mpc. Analysis of models in larger boxes of sizes 512 and 768 h−1 Mpc confirms this result.
Next we studied the distribution of radii of voids — regions of space devoid of certain kinds of objects. Different types of objects define voids of different size. Large voids are determined by rich clusters and are crossed by filaments of faint galaxies. Almost all systems of galaxies contain outlying faint members (see Fig. 7.14 for the luminosity density field of a spherical shell of the Sloan Digital Sky Survey). Dwarf galaxies define much smaller voids than giant ones.
To find the distribution of void radii, we used a simple void finder suggested by Einasto et al. (1989). The mean void radii were found for a broad range of the DF cluster mass thresholds, Mth. This variation imitates the various luminosity limit in real galaxy and cluster samples. The highest threshold used was selected so that the volume density of the DF clusters in the sample is approximately equal to the volume density of the Abell clusters, about 25 × 10−6 (h−1 Mpc)3 (Einasto et al., 2006).
The results of our calculations are shown in Fig. 8.3. We see that for a low DF cluster mass threshold, the void radii are almost independent of the mass threshold Mth. This means that in this DF cluster mass interval the clusters are located in identical filaments. If the DF cluster mass threshold increases further, the void radii start to increase. This means that some filaments are fainter than the respective mass threshold limit, and do not contribute to the void definition. In the case of models L256.008 and L256.016 with a high mass threshold the void radii increase very rapidly until clusters disappear. This effect is due to the very sharp decrease in the number of DF clusters of a high mass. These rare clusters define very large voids which are not characteristic of the overall cosmic web pattern of the particular model.
Fig. 8.3 shows that the Rv versus Mth curves for the models M256.032, M256.064, M256.128, and M256.256 are almost identical. This means that the scale of the cosmic web is determined essentially by density perturbations of a scale up to 32 h−1 Mpc. Some small differences in the Rv versus Mth curves remain between the models M256.032 and M256.064. The higher cutoff models are practically identical in this void size test.
Thus, the void analysis confirms our results from the mass distribution of the DF clusters, that density perturbations of large scales have little effect on the pattern of the cosmic web as characterised by void sizes. This analysis suggests that the cosmic web with filamentary superclusters and voids is formed by the combined action of all perturbations up to the scale ∼100 h−1 Mpc. The largest perturbations in this range determine the scale of the supercluster-void network. Perturbations of the largest scales > 100 h−1 Mpc modulate the richness of galaxy systems from clusters to superclusters, and make voids emptier, but do not change the pattern of the web.
8.3.3 The phase coupling of density perturbations of various scale
Our next task was to look at how the phase coupling or synchronisation may arise. To clarify this problem numerical simulations are needed, where the role of density perturbations of various scales can be investigated. Ryden & Gramann (1991) studied the phase shifts in the evolving density field, and showed that small-scale phase information is lost during the evolution. Our aim was to find the evolution of the phases of large and medium scales which determine the basic elements of the cosmic web.
If the hy
pothesis of primordial Gaussianity is correct, then density waves of different scales began with random and uncorrelated spatial phases. As the density waves evolve, they interact with others in a non-linear way. This interaction leads to the generation of non-random and correlated phases which form a spatial pattern of the present cosmic web. How is this process in action?
To see this process we used again the wavelet decomposition of the density field. Figure 8.4 shows the high-resolution density fields of the full model M256 at four redshifts: z =10, 5, 1, 0. Wavelet decompositions at levels w6, w5, and w4 for the same redshifts are shown in the same Figure. Colour-coding of wavelets at different redshifts is chosen so that a certain colour corresponds approximately to the density level, corrected by the linear growth factor for that redshift. Blue wavelet colours correspond to under-dense regions of density waves, green colours to slightly over-dense regions, and red and white colours to highly over-dense regions.
We see that the pattern of the cosmic web on the scale of the wavelet w6 is almost identical at all redshifts; only the amplitude of the density waves is increasing approximately in proportion to the linear growth factor. This linear growth is expected for density waves of large scales, which are in the linear stage of growth. The pattern of the web of the wavelet w5 changes little, but the growth of the amplitude of density waves is more rapid. The pattern of the wavelet w4 changes much more during the evolution, and the amplitude of density waves increases more rapidly, but essential features remain unchanged, i.e. the locations of high-density peaks and low-density depressions are almost independent of the epoch.
