by Jaan Einasto
Now let us now compare the evolution of the locations of high-density peaks of wavelets of various scale. Fig. 8.4 shows that at all redshifts high-density peaks of wavelets of medium and large scales almost coincide. In other words, density perturbations of medium and large scales have a tendency of phase coupling or synchronisation at peak positions. Figure 8.4 shows that the synchronisation of medium and large scales applies also to underdense regions.
Fig. 8.4 The high-resolution density field of the full model M256 (where density perturbations of all scales are present) is shown in the left column, at k = 153 coordinate. The second, third, and fourth columns shows the wavelet w6, w5, and w4 decompositions at the same k, respectively. The upper row gives data for present epoch, z = 0, the second row for redshift z = 1, the third row for redshift z = 5, and the last row for redshift z = 10. Densities are expressed in linear scale (Einasto et al., 2011a).
This analysis suggests that the synchronisation of peak positions of wavelets of various scales represents a general property of the evolution of the density field of the Universe. How this occurs will be discussed in the next section.
As shown by Kofman & Shandarin (1988), the skeleton of the supercluster–void network is created already at a very early stage of the evolution of the Universe, in the post-inflation phase. Thereafter the structure is frozen as respective waves move out of the horizon. Scales larger than the sound horizon at recombination, ≈146 Mpc, were outside the horizon most of the time. This scale (105 h−1 Mpc for the presently accepted Hubble constant h = 0.72) is surprisingly close to the characteristic scale of the supercluster–void network (Einasto et al., 1997a, 2001), and to the scale of the baryon acoustic oscillations peak, seen in the correlation function of main galaxies of the Sloan survey by Eisenstein et al. (2005) and of LRG-galaxies by Hütsi (2006), as well as in the distribution of galaxies as demonstrated by Arnalte-Mur et al. (2012).
8.3.4 The fine structure of the cosmic web
So far we looked at how the general pattern of the cosmic web forms. Our next question is: How to explain the evolution of the fine structure of the web, in particular the formation of the filamentary web? It is well known that the structure of the cosmic web is hierarchical. Supervoids are not empty, but are crossed by filaments of galaxies and groups of various strength (Jõeveer & Einasto, 1978; Einasto et al., 1980a; Einasto & Einasto, 1989; van de Weygaert & van Kampen, 1993; Lindner et al., 1995). We have to understand how all this evolves in a natural way. To follow the evolution of the fine structure of the cosmic web we use our high-resolution models of the series L100; for more details see Einasto et al. (2011b).
To follow the evolution of the fine structure of the density field we compared high-resolution density fields of models of the L100 series at various time-steps and wavelength cuts. Fig. 8.5 shows slices at the coordinate k = 51 of the full model L100.100, and of the strongly cut model L100.016 with λcut = 16 h−1 Mpc, at epochs z = 0 and z = 2. The k coordinate is chosen so that the slice of the model L100.100 crosses a large under-dense region between a rich supercluster and several rich clusters. To see the differences between the density fields of models L100.100 and L100.016 better we show in Fig. 8.5 only the zoom-in of the central 50 × 50 h−1 Mpc (256 × 256 pixels) region. To compare the present field with the initial density field, we use the smallest scale wavelet w1 for both models at the redshift z = 10, shown as a zoom-in plot in Fig. 8.6, which is similar to the plot for the present epoch in Fig. 8.5.
The power spectrum of density perturbations has the highest power at small scales. Accordingly, the influence of small-scale perturbations relative to large- scale perturbations is strongest in the early period of structure evolution. For this reason the density fields and wavelets w1 at early epochs are almost identical for the full model L100.100, and for the model cut at small scales, λt = 8 h−1 Mpc, L100.008, as seen in Fig. 8.6. Eventually, perturbations of larger scale start to affect the evolution. These perturbations amplify small-scale perturbations near maxima and suppress small-scale perturbations near minima. In this way the growth of small-scale perturbations becomes non-linear. Thereafter still larger perturbations amplify smaller perturbations near their maxima, and suppress smaller perturbations near their minima, and so on. The largest amplification (non-linearity) occurs in regions where the maxima of perturbations of all scales happen to coincide. In such a way the synchronisation of phases of waves of different scales occurs as a natural process.
Fig. 8.5 Zoom-ins to the high-resolution density fields of the models L100.100 and L100.016, left and right columns, respectively. Zoom factor is 2; central 50 × 50 h−1 Mpc (256 × 256 pixels) of all models are shown. Upper panels are for the present epoch z = 0, lower panels for the epoch z = 2. All panels are at the k = 51 coordinate. Cross sections (beams) at coordinates j = 222 and k = 51 for both models are shown in Fig. 8.7 for three redshifts to see the evolution of the density field and its wavelets. In the upper left corner of the Figure there is a rich supercluster in the model L100.100, absent in the model L100.016. Both models have at the right edge of the Figure a rich cluster. This cluster is well seen in Fig. 8.7 at the i = 390 coordinate. Densities are expressed in the logarithmic scale, identical lower and upper limits for plotting with the SAO DS9 package are used. The border between the light blue and the dark green colours corresponds to the critical density Dloc = 1.6, which separates low-density halos and halos collapsed during the Hubble time (Kaiser, 1984; Bardeen et al., 1986). Note that in both models and simulation epochs the majority of filaments in voids have densities below the critical density (Einasto et al., 2011b).
Fig. 8.6 Wavelets w1 of models L100.100 and L100.008 at redshift z =10 are shown in the left and right panels, respectively, at coordinate k = 51. Densities are expressed on a linear scale; only over-dense regions are shown. As in Fig. 8.5, the central 256 × 256 pixels of the full models are shown. The characteristic scale of density perturbations corresponding to this wavelet is 0.4 h−1 Mpc. Note the weakening of peak densities of the model L100.100 in the region of the future large under-dense region, seen in Fig. 8.5 (Einasto et al., 2011b).
The differences between the models L100.100 and L100.016 at the present epoch z = 0 are very well seen in Fig. 8.5. In the model L100.100, between the supercluster at the left corner and the cluster at the right edge there is a large low-density region. This region is crossed near the centre by a filament having several knots in the green and red colour. In the model L100.016 there are no rich superclusters; the whole region is covered by a web of small-scale filaments. In other words, large-scale perturbations, present in the model L100.100, have suppressed the growth of the density of filaments in void regions.
There exists a low-density smooth background, seen in the Fig. 8.5 in deep dark-blue colour. The density of this background, D ≈ 0.1, is lower at the present epoch z = 0, i.e. the density of the smooth background decreases with time. Regions of very low density have much larger sizes in the model L100.100 than in the model L100.016.
Further evolution of wavelets can be followed using the density field along i-axis beams at fixed j, k coordinates of the model L100, shown in Fig. 8.7. The largest wavelet w5 of the strongly cut model L100.016 has for all redshifts approximately the same amplitude and a sinusoidal shape, suggesting that density perturbations on this scale are in the linear growth regime. The shape of the next wavelet w4 is very different from a sinusoid. In regions of maxima of the wavelet w5 the wavelet w4 has very strong maxima. An example is the region near i ≈ 370. In this region all wavelets of lower scale also have strong maxima, and wavelets of all scales up to w4 are very well synchronised. The overall shape of the density profile is determined by the wavelet w3 that has a characteristic scale 3.1 h−1 Mpc. Most peaks seen in the density profile are due to the maxima of this wavelet. In most of these peaks wavelets of smaller scale also have maxima, i.e. near the peaks small-scale wavelets are synchronised.
Fig. 8.7 The evolution of the local density and wavelets of the mod
els L100.100 (left panels) and L100.016 (right panels) in beams along the i-coordinate at j = 220, k = 51. The same k-coordinate was used in plotting the density field in Fig. 8.5. Data are shown for epochs z = 0, 2 in the upper and lower panels, respectively. To see better details only the region 144 ≤ i ≤ 400 of length 50 h−1 Mpc is shown. The characteristic scale of the wavelet w5 is 12.5 h−1 Mpc. Wavelets are divided by the growth factor f ∝ (1 + z)−1 (Einasto et al., 2011b).
The evolution of the largest wavelet w5 of the full model L100.100 is almost linear up to the epoch z ≥ 1. The shape of the wavelet w5 for different epochs is almost sinusoidal, and the heights of the maxima are approximately equal. The next wavelet w4 behaves as the first overtone of the wavelet w5 — near the minima of w5 there are maxima of w4, which have much lower heights than the maxima near the maxima of w5. This phenomenon is well seen also in the wavelet analysis of the Sloan Digital Sky Survey, see Fig. 8.1. Near the joint maxima of w5 and w4 there are very strong maxima of all wavelets of smaller order; this is very well seen at locations i ≈ 290 and i ≈ 380.
The density and wavelet distributions of the model L100.100 for the present epoch z = 0 are completely different from the distributions at higher redshifts. Here the dominant feature is the presence of a large under-dense region in the interval 120 < i < 380. This large under-dense region is due to the influence of large-scale density perturbations not shown as wavelets in this Figure. These large-scale density waves have in this region their minima and have suppressed the amplitudes of all density waves of smaller scales, including w4 and w5. The amplitudes of wavelets w3 and lower orders are almost zero. Near the density maxima seen at higher redshifts at i ≈ 210 and i ≈ 290 there are very weak density peaks with maxima below the mean density level. These maxima are seen in the density field as weak filaments in Fig. 8.5.
The most remarkable feature of the density field of the model L100.100 at the present epoch is the presence of a large under-dense region of very low density D ≈ 0.1, seen in Fig. 8.5 in deep-blue colour. At earlier epochs the density in this region was higher and there were numerous low-density peaks within it; for the present epoch most of these peaks are gone. This is caused by density perturbations of larger scales. The synchronisation of density waves of medium and large scales explains also the absence of even dwarf galaxies in voids, or the “void phenomenon”, as discussed by Peebles (2001).
Einasto et al. (2011b) concludes the wavelet analysis as follows: The wavelet analysis leads us to the conclusion that the properties of the large-scale cosmic web with filaments and voids depend on two connected properties of the evolution of density perturbations. The first property is the synchronisation of density waves of medium and large scales. Due to the synchronisation of density waves of different scales, positive amplitude regions of density waves add together to form rich systems of galaxies, and negative amplitude regions of density waves add together to decrease the mean overall density in voids. The amplification of density perturbations is another property of density evolution. Due to the addition of negative amplitudes of medium and large scale perturbations, there is no possibility for the growth of the initial small-scale positive density peaks in void regions. For this reason, small-scale protohaloes dissolve there. In the absence of medium and large-scale density perturbations, these peaks would contract to form haloes, which would also fill the void regions, i.e. there wouldbeno void phenomenon as observed.
The analysis by Suhhonenko et al. (2011) showed that density perturbations up to the scale ∼100 h−1 Mpc determine the scale of the cosmic web in terms of void sizes. In contrast, waves of larger wavelengths do not influence the scale of but only amplify the web, leaving its scale unaffected, see Fig. 8.2 above.
There are probably two effects which affect the evolution of density waves of large scales. As mentioned above, all waves of scale ≥140 Mpc were outside the horizon during the early evolution of the Universe and could not grow. For this reason the supercluster–void network has just this characteristic scale. On the other hand, after recombination the amplitudes of density perturbations increased until a certain epoch, which corresponds to redshift ∼0.7. Thereafter the value of the cosmological constant term exceeded the value of the density term, and the growth of the web was frozen, which hindered the formation of larger systems.
8.4 Dark energy
8.4.1 The discovery of dark energy
Observational evidence for the presence of a cosmological term in the mass/energy relation comes from the distant supernova experiments. Two teams, led by Riess et al. (1998, 2007) (High-Z Supernova Search Team) and Perlmutter et al. (1999) (Supernova Cosmology Project), initiated programs to detect distant type Ia supernovae in the early stage of their evolution, and to investigate with large telescopes their properties. These supernovae have an almost constant intrinsic brightness (depending slightly on their evolution). By comparing the luminosities and red- shifts of nearby and distant supernovae it is possible to calculate how fast the Universe is expanding at different times. The supernova observations give strong support to the cosmological model with the Λ term.
A more detailed discussion of the discovery of dark energy is outside the scope of this book. Here we discuss only some consequences of the presence of dark energy in the evolution of the Universe.
The cosmological Λ term is presently interpreted as vacuum or dark energy. Dark energy has two important properties: its density ρυ is constant, i.e. the density does not depend not on time nor on location; and it acts as a repulsive force, i.e. it accelerates the expansion of the Universe.
Studies of the Hubble flow in nearby space, using observations of type Ia supernovae with the Hubble Space Telescope (HST), were carried out by several groups. The major goal of the study was to determine the value of the Hubble constant. As a by-product also the smoothness of the Hubble flow was investigated. In this project supernovae were found up to the redshift (expansion speed) 20 000 km s−1. This project (Sandage et al., 2006) confirmed earlier results that the Hubble flow is very quiet over a range of scales from our Local Supercluster to the most distant objects observed. This smoothness in spite of the inhomogeneous local mass distribution requires a special agent. Dark energy as the solution has been proposed by several authors (Chernin (2001); Baryshev et al. (2001) and others). Sandage emphasises that no viable alternative to dark energy is known at present, thus the quietness of the Hubble flow gives strong support for the existence of dark energy.
8.4.2 The role of dark energy in the evolution of the Universe
As noted above, dark energy has two important properties: its density depends neither on time nor on location; and it acts as a repulsive force or antigravity (for detailed discussions see Chernin (2003, 2008)).
The first property means that in an expanding universe in the earlier epoch the density of matter (ordinary + dark matter) exceeded the density of dark energy. As the universe expands the mean density of matter decreases and at a certain epoch the matter density and the absolute value of the dark energy effective gravitating density were equal. This happened at an epoch which corresponds to redshift z ≈ 0.7. Before this epoch the gravity of matter decelerated the expansion; after this epoch the antigravity of dark energy accelerated the expansion. This is a global phenomenon — it happened for the whole Universe at once.
Dark energy influences also the local dynamics of astronomical bodies. The local effect of dark energy on the dynamics of bodies has been studied by Karachentsev, Chernin, Tully and collaborators. Using the Hubble Space Telescope and large ground-based telescopes Karachentsev determined accurate distances and redshifts of satellite galaxies in the Local group and several nearby groups of galaxies (Karachentsev et al., 2002,2003,2007, 2009; Tully et al., 2008). This study shows that near the group centre up to distance R ~1.25 h−1 Mpc satellite galaxies have both positive and negative velocities with respect to the group centre; at larger distance all relative velocities are positive and follow the Hubble flow.
This test demonstrates that dark energy influences both the local and the global dynamics of astronomical systems. For rich clusters of galaxies the zero gravity distance is about 10 h−1 Mpc; for rich superclusters it is several tens of h−1 Mpc, which corresponds to the radius of cores of rich superclusters. The antigravity of dark energy also explains the absence of extremely large superclusters: even the richest superclusters have characteristic radii of about 50 h−1 Mpc.
8.4.3 Cosmological parameters
Tartu astronomers participated in the determination of cosmological parameters in only a few cases. The first of these cases was the estimation of the total density of matter, including dark matter, by Einasto et al. (1974b). Actually this density estimate concerns only clustered matter in systems of galaxies. By definition, half of the matter was initially in low-density regions; this fraction decreases due to the infall of void matter to systems of galaxies. Presently the fraction of matter in voids is about 25%, depending slightly on the cosmological model (Einasto et al., 1994a). Taking this into account we get for the whole matter density about 0.27 in units of the critical density.
Together with Fernando Atrio-Barandela I made in the late 1990’s a lot of calculations with the CMBFAST program (Seljak & Zaldarriaga, 1996) to calculate power spectra for various cosmological models, and to compare the results with observed power spectra of galaxies (Atrio-Barandela et al., 1997; Einasto et al., 1999a,b,c). Our results suggest that, if the neutrino contribution is negligible, then the matter density parameter is about 0.3; values as high as 0.4 and as low as 0.2 are definitely excluded. I had a chance to discuss these results with Mike Turner and Jerry Ostriker. We agreed that the mean value, 0.3, is favoured; it coincides with the density parameter of the ‘concordant model’ by Ostriker & Steinhardt (1995).
