We have confirmed that the Abell cluster sample contains a considerable fraction of clusters located in rich superclusters which form a cubic lattice with a period about 130 h-1 Mpc. This structure is best seen in the clusters populating high density regions, but it is present already in the whole Abell cluster sample. This supports the results obtained in Papers I and II by using other methods.
The cubic lattice we see in the distribution of clusters defines specific directions in space. A clear periodicity is observed only along a few alignments, while the best signatures of periodicity come from a cubic tiling that is nearly aligned with supergalactic coordinates.
Observationally, this result is not so surprising as it may seem at the first glance. It confirms earlier results on the presence of a high concentration of clusters and superclusters towards both the Supergalactic Plane (Tully et al. 1992), and towards the Dominant Supercluster Plane, which are at right angles with respect to each other (Paper I). Tully et al. also noticed the rectangular character of the distribution of rich clusters. Thin deep slices, such as slices of the LCRS and the Century Survey, also show a weak periodicity signal (Landy et al. 1996; Geller et al. 1997). The scale length found in these studies is of the same order as obtained in the present paper.
The supergalactic Y axis is very close to the direction of Galactic poles, thus it is natural to expect a well-defined periodicity along Galactic poles as indeed observed by Broadhurst et al. (1990). As noted already by Bahcall (1991), the nearest peaks of the Broadhurst et al. survey coincide in position and redshift with nearby rich superclusters. We note, though, that the periodicity in the distribution of rich superclusters over the full sky is much less pronounced. This is in agreement with observational periodicity studies along beams oriented in other directions (Guzzo et al. 1992; Willmer et al. 1994; Ettori et al. 1997).
Kerscher (1998) used a combination of the nearest neighbour distribution and the void probability function to measure the regularity of the structure in the Abell cluster sample. His method is more universal, and detects any regularity, but it is isotropic and cannot detect specific structures. He found that the rich supercluster sample shows clear signs of regularity, in agreement with Paper I. He also found that the regularity can be partly due to the friends-of-friends (FOF) procedure used to select the rich supercluster population. His analysis extends to scales about 60 h-1 Mpc and the neighbourhood distance 24 h-1 Mpc used for FOF can have an effect at these scales, but certainly not much at scales of 130 h-1 Mpc and larger.
The regularity of the large-scale structure of the Universe has been studied also using the distribution of centers of superclusters (Kalinkov et al. 1998). These authors looked for high-order clustering of superclusters using correlation analysis, and found that superclusters are not clustered. Our analysis in Papers I and II and that presented here suggests that the regularity is completely different. The supercluster-void regularity is seen in the distribution of clusters themselves, not in that of supercluster centers.
Theoretically, this result contradicts almost everything we know about the formation of structure in the Universe. It is not clear at all if it may be generated by an isotropic Gaussian random field of initial perturbations, even with bumps and wells in its power spectrum, since a rather large fluctuation would be necessary to generate a locally anisotropic realization required to explain the data. In particular, Eisenstein et al. (1998) speculate that it could be a domination of a few Fourier modes in our location, but the chance of that seems vanishingly small. If this were true, it would be easy to detect from future deeper surveys: the phase alignment observed in our neighbourhood will soon be lost at larger distances, and deeper surveys should not reveal any regularity.
Thus, if the regularity is real, then most probably it means a non-isotropic non-Gaussian admixture to a Gaussian density field in the region of k= (0.03-0.06) h Mpc-1 where we tentatively see non-smooth features in P(k). Of course, in this case P(k) is not sufficient for description of stochastic perturbations, a whole hierarchy of higher moments or probability distributions is required. Still, from a theoretical point of view, local non-Gaussian features superimposed on a Gaussian field of initial perturbations with an approximately flat power spectrum are fairly possible. The most natural mechanism of their generation is essentially the same as the mechanism for generation of local non-smooth features in the power spectrum - a fast phase transition in physical fields other than the inflaton field during an inflationary stage in the early Universe (occurring about 50 e-folds before its end), see, e.g., the discussion in Starobinsky 1998 - only a different kind of the phase transition is required. In particular (though not necessarily), cosmic strings or more complicated topological defects may be generated during it. However, we clearly know too little yet about the properties of the large-scale quasi-regularity to discuss specific scenarios.
A conclusive confirmation of large-scale regular patterns will demand deeper samples that we can presently use. In this respect, the future complete data release of the 2dF survey, and, especially, the BRG part of the SDSS survey will certainly help to clarify the situation.
The present paper is of an exploratory nature, so if traces of regularity will be seen in the future, there are many obvious ways to improve the methods used here. The notion of a "best periodogram'' is not too clear yet; one should consider weighing of points again to speed up calculations etc. We have not touched the problem of false periods at all; for people who study time series this is one of the main problems. The most time-consuming part of the work at present is the calculation of the "Euler cube'' to find starting values for the best period search. As the 3-D Fourier transform of density has directions as arguments, one could use that to find initial Euler values, looking for pairs of perpendicular wave vectors of maximum total Fourier amplitude (maximum change along the axes of the cube).
In conclusion, we have shown that in our neighbourhood up to a distance of 350 h-1 Mpc from us the the Universe is not fully homogeneous and isotropic on scales of the order of hundreds of Mpc. High-density regions, delineated by galaxy clusters in rich superclusters, form a quasi-cubic lattice in this volume of space. This lattice is roughly aligned with the supergalactic coordinates, the regularity is well pronounced and has a period of 120-140 h-1 Mpc. This inhomogeneity does not contradict the isotropy and homogeneity of the Universe on larger scales, which follows from the data on the spatial distribution of galaxies and clusters, from the data on the angular anisotropies of the CMB, and from the observed degree of isotropy of the X-ray background in the Universe (emitted by baryons in the recent epoch).
Acknowledgements
We thank Jaan Pelt and Vicent Martínez for discussion and suggestions. Our special thanks are to our referee, Rien van de Weygaert, for stimulating suggestions and for the program to calculate the Voronoi model. The present study was supported by Estonian Science Foundation grants 2625 and 2882. A.S. was partially supported by the grant of the Russian Foundation for Basic Research No. 99-02-16224, and by the Russian Research Project "Cosmomicrophysics''. H.A. thanks CONACyT for financial support under grant 27602-E, and E.S. thanks the University of Valencia where this paper was finished.
Copyright ESO 2002