1 Introduction

The basic assumption in the standard Friedman-Robertson-Walker cosmology is that the Universe is homogeneous and isotropic on large scales (Peebles 1980). Clusters of galaxies and galaxy filaments form superclusters (Abell 1958); examples are the Local supercluster (de Vaucouleurs 1956; Einasto et al. 1984) and the Perseus-Pisces supercluster (Jõeveer et al. 1978; Giovanelli 1993). Superclusters and voids between them form a continuous network of high- and low-density regions which extends over the entire part of the Universe studied in sufficient detail up to a redshift of $z \approx 0.13$ (Einasto et al. 1994, 1997d, hereafter Paper I). This network has been called cellular (Jõeveer & Einasto 1978; Icke 1984), in which one cell is formed by a low-density region surrounded by superclusters and dense galaxy filaments. According to the conventional paradigm the location of superclusters and centres of voids in this cellular network is almost random, as density perturbations on all scales are believed to form a Gaussian random field (see, e.g., Peacock 1999).

A first hint that something may be wrong in our understanding of the formation of the large-scale structure of the Universe came when Broadhurst et al. (1990) measured redshifts of galaxies in a narrow beam along the direction of the northern and southern Galactic poles. They found that the distribution of galaxies along that beam is periodic: high-density regions alternate with low-density ones with a surprisingly constant interval of 128 h^-1 Mpc (here h is the Hubble constant in units of 100 km s^-1 Mpc^-1). An attempt to explain this distribution by a Voronoi foam model was made by van de Weygaert (1991). In the Voronoi model void centers are randomly distributed, clusters of galaxies occupy vertices of the network, and galaxies lie in sheets between voids. Quasi-periodicity is observed in approximately 10% of cases when a narrow beam is cut from the cellular network. Kaiser & Peacock (1991) showed that such a periodicity can occur even in a random distribution of galaxy clumps due to a specific geometry of the sample (very narrow beams). Thus the first reaction to the Broadhurst et al. result was that the observed quasi-periodicity can be explained in the framework of the conventional model of structure formation. However, more extensive simulations have shown recently that such regularity has a very small a priori probability, less than 10^-3 (Yoshida et al. 2001).

There exists growing evidence that the supercluster-void network is more regular than expected from the Voronoi foam and the random clump model. The three-dimensional distribution of clusters shows clear signs of regularity, as shown by Einasto et al. (1994) and (1997d, Paper I of this series). One method to characterize this regularity is correlation analysis. Kopylov et al. (1984, 1988), Fetisova et al. (1993), Mo et al. (1992), and Einasto & Gramann (1993) have found evidence for the presence of a secondary peak of the correlation function of clusters of galaxies at 125 h^-1 Mpc. This peak has been interpreted as the correlation of clusters in superclusters located at opposite sides of large voids. Such a secondary peak is expected also in the Voronoi foam model (see below). Recent studies show that the correlation function has not only one secondary maximum but a series of regularly spaced maxima and minima. In other words, it oscillates with a period equal to that of the periodicity of the supercluster-void network (Einasto et al. 1997b, hereafter Paper II). The existence of these oscillations has been confirmed using more recent data, although the statistical significance of the oscillations is not too high (Tago et al. 2000). An extremely low amplitude oscillation is seen also in the correlation function of LCRS galaxies (Tucker et al. 1997).

Regular oscillations of the correlation function correspond to a sharp peak in the power spectrum (Einasto et al. 1997c, hereafter Paper III). Baugh & Efstathiou (1993, 1994) and Gaztañaga & Baugh (1998) determined the three-dimensional power spectrum from the projected distribution of the APM galaxies; the Abell cluster power spectrum was derived by Einasto et al. (1997a) and Retzlaff et al. (1998), Tadros et al. (1998) calculated the power spectrum for the APM clusters. All these studies found the peak or turnover of the power spectrum near a wavelength of $\lambda_0$ = 120-130 h^-1 Mpc or a wavenumber $k_0=2\pi/\lambda_0=0.05$ h Mpc^-1.

This wavelength, however, corresponds to the case of the Einstein-de Sitter cosmological model ( $\Omega_{\rm m}=1$). Recent observational data have changed our conception about the form of the matter power spectrum in the wavelength interval involved. First, observations of type Ia supernova explosions at high redshifts (Perlmutter et al. 1998, 1999; Garnavich et al. 1998; Riess et al. 1999) strongly support the existence of a positive cosmological constant in the Universe. When combined with results of the most recent CMB experiments BOOMERANG (de Bernardis et al. 2000) and MAXIMA-1 (Hanany et al. 2000), these data favor the flat Universe with $\Omega_{\rm m}=1-\Omega_{\Lambda}=0.3\pm 0.1$. In the low density flat LCDM Universe with the flat ( $n_{\rm S}=1$) initial spectrum of adiabatic perturbations, the maximum of the present matter power spectrum should lie at $k\approx 0.1 \Gamma$ h Mpc^-1, where $\Gamma \approx \Omega_{\rm m} h$ is the shape parameter, i.e. at $k\sim 0.02$ h Mpc^-1. Thus, the additional peak at k=k₀=0.05 h Mpc^-1 discussed above does not coincide with the main peak of the smooth power spectrum P(k) (as would take place for $\Omega_{\rm m}=1$), but is located on its right slope. Moreover, if the empirical power spectrum P(k) obtained in Einasto et al. (1999, hereafter E99) for $k\ge 0.03$ h Mpc^-1 from galaxy and cluster catalogs is taken with the proper bias factor (see Einasto et al. 1999) and compared with the $n_{\rm S}=1$ power spectrum for the LCDM cosmological model, normalized to the COBE data on anisotropies of the cosmic microwave background (CMB) at very small k, the peak at k=k₀=0.05 h Mpc^-1 appears to be only a local bump above a general depression in the empirical spectrum occurring for k>0.03 h Mpc^-1. So, there is no (or practically no) excess power in this region, as compared to the case of a featureless, scale-free initial matter power spectrum. In addition, a pronounced well at $k\sim 0.035$ h Mpc^-1 arises. Therefore, now we suppose that a feature in the power spectrum corresponding to the large-scale quasi-regularity which we are considering probably has a more complicated structure than simply one bump (see Atrio-Barandela et al. 2001 for more discussion).

Figure 1: Observed magnitudes of a variable star (left panel), folded using two trial periods, and the resulting phase diagrams - for a period of 4 units (middle panel) and for a period of 2$\pi$ units (right panel).

The most recent data on the power spectrum P(k) from galaxy and cluster catalogs - the de-correlated power spectrum of the IRAS PSCz redshift survey (Hamilton et al. 2000) that supersedes earlier results by Sutherland et al. (1999), the cluster spectrum by Miller & Batuski (2001), and the spectrum of the REFLEX X-ray cluster survey (Schuecker et al. 2001) all show the continuing increase of P(k) to the direction of smaller kup to $k\sim 0.02$ h Mpc^-1, in excellent agreement with the expected location of the main maximum in P(k) given above. The situation with superimposed features that may reflect quasi-regularity is much more complicated. While the combined structure - the (relative) peak at k=0.05 h Mpc^-1 and the depression at $k\sim 0.035$ h Mpc^-1 is clearly seen in the de-correlated PSCz spectrum and even in the Miller-Batuski data, still it is (almost) inside $1\sigma$ error bars for each point since errors are rather large in this region. Moreover, no sharp features are seen in the REFLEX spectrum. Once more, its error bars are sufficiently large to hide all expected deviations from the n_S=1initial spectrum. The recent determinations of the power spectrum of the 2dF survey also do not resolve the problem of the existence of local features. While Percival et al. (2001) find oscillations, in a later paper Tegmark et al. (2001) do not detect significant wiggles in the power spectrum.

A more detailed analysis of this question will be presented elsewhere. However, it is clear that presently available power spectra do not give a definite answer to the question if some local features in P(k) over the range k=0.03-0.06 h Mpc^-1 related to the quasi-regularity exist or not. The same may be said about the present CMB data (see the discussion below, in Sect. 4).

The main reason why the recent precise results on P(k) and the CMB multipoles C_l are still unable to resolve this problem (and still a significantly better accuracy is required) is because all these quantities are obtained from the module of the Fourier transform of inhomogeneities and averaging over rotations in 3-space. Valdarnini (2001) proposed to describe the density distribution by order parameters, which are sensitive both to radial and angular separations, but these are also found by averaging over rotations. Thus, these descriptors are insensitive to directional and phase information, and can describe regularity only indirectly. So far, the only positive results on regularity of the large-scale three-dimensional distribution of high-density regions have been found on the basis of oscillatory features of the correlation function of Abell clusters of galaxies, and confirmation of this result by independent methods is needed.

In this paper we shall use a new geometric method to investigate the distribution of clusters of galaxies, sensitive to the regularity of the distribution. Our main goal is to obtain independent evidence on the regularity problem. In the next section we describe the new method and check it using various test samples. Then we apply the method to the sample of Abell clusters of galaxies. Finally, we discuss and summarize our results.