Proc. Natl. Acad. Sci. USA Vol. 90, pp. 4853-4858, June 1993 Colloquium Paper

This paper was presented at a coUoquium entitled "Physical Cosmology," organized by a committee chaired by David N. Schramm, held March 27 and 28, 1992, at the National Academy of Sciences, Irvine, CA.

Redshift survey with multiple pencil beams at the galactic poles A. S. SZALAY*t, T. J. BROADHURSTt, N. ELLMAN§, D. C. Koo§, AND R. S. ELLIS¶ *Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218; tDepartment of Physics, E6tv6s University, Budapest, H-1088 Hungary; tRoyal Observatory, Edinburgh, EH9 3HJ, United Kingdom; §University of California Observatories, Lick Observatory, University of California, Santa Cruz, CA 95064; and 1Department of Physics, University of Durham, DH1 3LE, United Kingdom

ABSTRACT Observations of the large-scale structure of Pisces supercluster extends to large distances in the trans- the universe suggest inhomogeneities on scales between 100h-1 verse directions. Also, the angular two-point correlation and 150h-1 Mpc (where h 0.5-1 is the Hubble constant in function of the automated plate measuring (APM) survey units of 100 kmns-l-Mpc'1; 1 pc = 3.09 x 1016 m). A deep indicates a statistically significant deviation from CDM (9). redshift survey with a "pencil-beam" geometry of galaxies at The advent ofefficient multi-object spectrographs on large the galactic poles indicated strong clustering, with a provoca- optical telescopes has considerably accelerated the progress tive regularity at 128h1- Mpc [Broadhurst, T. J., Ellis, R. S., toward completing redshift surveys of faint galaxies. The Koo, D. C. & Szalay, A. S. (1990) Nature (London) 343, instrumentation used in such surveys typically sample 10-50 726-7281. Using newly acquired data, we demonstrate how galaxies within fields of 10- to 40-arcmin diameter, and thus multiple deep probes overcome most of the statistical problems "pencil-beam" geometries are produced which are charac- associated with single pencil beams. Our results from cross teristically different from strategies used to map the galaxy correlations of multiple pencil beams, containing over 1200 distribution locally. With 40 fields on 4-m telescopes, the galaxies, indicate that the strong peak in the power spectrum diameter of the survey at the median redshift of about z 0.3 results from structures of large transverse size, in agreement is 6h-1 Mpc; such geometries are close to optimal for with our original conjecture. We also discuss the sensitivity of detecting wall-like topologies on scales comparable to those pencil-beam surveys to the topology of large-scale structures revealed in the CfA surveys. and compare them with sparsely sampled wide-angle local Broadhurst et al. (10), hereafter BEKS, provided evidence surveys. from a combined sample of galaxies in the north and south galactic poles for structures on scales > lOOh-1 Mpc with a provocative regularity. The BEKS survey consisted of two 1. Introduction deep surveys spanning 2000h-1 Mpc-the deepest so far- and two previous brighter surveys by others (5, 11), which To model the large-scale structure of the universe, cosmol- together cover a volume well approximated by a cylinder of ogists consider a variety of initial conditions and follow the a constant comoving radius. The two northern fields lie subsequent evolution with combined analytic and numerical within the CfA slice, and the Great Wall is readily detected. techniques, making certain additional assumptions. The pre- Surprisingly, however, at large radial distances most galaxies dictions are compared with the distribution of galaxies and lie in a few discrete "spikes" separated typically by 130h-1 fluctuation limits for the cosmic background radiation (CBR). Mpc. This is revealed by using the one-dimensional pair The most popular galaxy formation theory is the cold dark counts and the one-dimensional power spectrum, which has matter (CDM) model (1), in which most of the mass density a very sharp peak at the wavenumber corresponding to is in the form of noninteracting dark particles. Together with 130h-1 Mpc. a scale-invariant Zeldovich-Harrison spectrum for the initial In this paper we present additional data collected in the density fluctuations, the theory satisfies many observational same direction, but using a slightly different strategy, con- constraints on small (< lOOh-1 Mpc) scales (2). However, to firming our original results. In Section 2 we begin by con- explain the absence of CBR fluctuations, its proponents trasting two rather different observing strategies, the sparse- invoked the concept of "biasing," whereby galaxies form sampled local survey and the deep pencil-beam surveys, only at high peaks of the mass fluctuations. Much stronger demonstrating the unique features of each. In Section 3 we correlations are predicted in the distribution of visible gal- discuss the statistical consequences of the one-dimensional axies than for the underlying mass (3, 4). nature of the pencil beams and discuss the effect of using The standard biased CDM scenario predicts a universe multiple pencil beams. In Section 4 we present a more relatively homogeneous on scales above 30-40h-1 Mpc. detailed statistical analysis of data acquired recently, and in Observationally, it becomes more and more apparent that Section 5 we summarize our results. there is stronger large-scale clustering than the CDM predic- tion. Almost a decade ago, Kirshner et al. (5) found a 2. Strategies for Mapping Large-Scale Structure 60h-1-Mpc sphere with a large underdensity in the galaxy distribution. deLapparent et al. (6) delineated the "Great First we discuss the question of the best strategy for delin- Wall" in their Center for Astrophysics (CfA) surveys-a eating the large-scale distribution of galaxies, given that its structure connecting several known Abell clusters over a precise topology remains unclear. With the exception of spatial extent of lOOh-1 by 50h-1 Mpc. Chincarini et al. (7) dedicated telescopes, a typical redshift survey can measure and Giovanelli et al. (8) had earlier shown that the Perseus- only a few thousand redshifts in a few years of observation. Indeed, there is probably no single strategy optimal in all The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" Abbreviations: APM, automated plate measuring; BEKS, Broad- in accordance with 18 U.S.C. §1734 solely to indicate this fact. hurst et al.; CDM, cold dark matter. 4853 Downloaded by guest on September 26, 2021 4854 Colloquium Paper: Szalay et al. Proc. Natl. Acad Sci. USA 90 (1993) situations, since the choice is strongly influenced by what Voronoi foam, R= 1.6, smoothed original presumptions hold about the statistical properties and the topology of the large-scale galaxy distribution. There is a fundamental difference between the strategies and goals of the sparse-sampled wide-angle surveys and those ofthe deep pencil-beam surveys. Their correlation functions measure different statistical aspects of the same distribution. In gen- eral, answers depend on what questions were asked. When discussing large-scale structure, one should first specify what these words mean. For some, large-scale structure is equal to the large-scale behavior of the galaxy two-point correlation function. Others like to draw maps ofthe galaxy distribution and associate that with the large-scale structure. These data sets contain different information, and this represents the fundamental difficulty in obtaining a single coherent view of the universe. We will elaborate on these differences below and focus on finding large coherent structures, our working definition of large-scale structure. If fluctuations in the universe are strictly Gaussian, their full statistical description is contained in the two-point cor- relation function or in its Fourier transform, the power spectrum. The phases of the individual Fourier components, the "plane waves" are random for such a process. In this case the sparse-sampled survey is indeed the best way to Voronoi foam, R= 1.6, random phases obtain this information. If the universe, however, contains some very sharp large-scale features, such as the Great Wall, a sparsely sampled survey may fail to identify those. The presence of such structures also means that higher-order correlations are present, or equivalently, that the phases of the fluctuations are correlated. The result can be very dif- ferent from a homogeneous isotropic Gaussian random field with the same second-order statistical properties. We demonstrate this with a simple graphic example. Fig. 1 Upper is a two-dimensional Voronoi foam, generated by the median surfaces between Poisson "seeds" at the mean separation of lOOh-1 Mpc. In this simple toy model, galaxies reside only on the walls of the foam, smoothed, so the walls have a finite thickness. The structure has a well-defined second-order statistic but also has well-correlated phases. By using an image-processing program developed by George Djorgovski, this picture has been Fourier-transformed, all the phases randomized, and then transformed back again. The result (Fig. 1 Lower) is another two-dimensional density plot, with the same second-order properties, but with a Gaussian distribution. It is obvious how different the two distributions are. If we use a two-point correlation function estimator with sparse sampling, neither the statistic nor the distribution of FIG. 1. Simulation of a Gaussian (Lower) and a non-Gaussian the sparsely sampled points will tell us the difference. How- (Upper) process containing strong phase correlations, with identical ever, if we observe both pictures with well-sampled pencil- second-order statistical properties in two dimensions (done with the beam surveys, in the Voronoi image every beam would go help of George Djorgovski). through a sequence of walls, yielding very strong features in each one-dimensional correlation function. An average over determined by the number of redshifts obtained, and the an infinite number of pencil beams would provide a signal "clustering noise" is from the observed small-scale correla- similar for both images. The large variance around the mean tion function ofgalaxies; the latter depends on the shape and from one pencil beam to the next is the signature of a very size of the survey volume. For a given geometry, and a non-Gaussian behavior. By taking multiple pencil beams at second-order statistic (i.e., correlation function or power small angular separations, the spikes detected in the individ- spectrum), there is an optimum number of galaxies to ob- ual beams will be strongly correlated. serve, at which point the sampling noise equals the clustering The pencil beams cannot be arbitrarily thin, due to the noise. Beyond this critical number, the signal-to-noise ratio well-known small-scale clustering of galaxies, for large-scale correlations is not significantly improved. -1.8 Such an argument has been used, for example, to determine Ir the optimum sparse sampling rate for wide-angle or all-sky \ro -, [1] surveys (13). An optimum geometry can be defined by using I-I) exactly the same considerations. For example, if the large- where ro = 5h-1 Mpc is the correlation length (12). Any linear scale structure is dominated by walls with a well-defined dimension chosen to be less than r0 will cause noise from surface density, separated by large voids, how can we best small scales to dominate the signal from large-scale structure. determine whether such structures are common? Evidently, To be more specific, there are two competing sources of the sparse-sampled local surveys are farfrom optimum in this statistical noise in redshift surveys. The "sampling noise" is case. Downloaded by guest on September 26, 2021 Colloquium Paper: Szalay et al. Proc. Natl. Acad. Sci. USA 90 (1993) 4855 However, the argument above is only the sampling crite- In this latter case there is another implication: the one- rion for second-order methods. If we are interested in the dimensional power spectrum originating from a three- detection oflocalized high overdensities, the problem is quite dimensional homogeneous isotropic random process (not complicated, involving higher-order statistics, and the detec- necessarily Gaussian) is always monotonically decreasing. tion probability becomes a very sensitive function of the As the simplest example, plane waves with their wave sampling rate (I. Szapudi and A.S.S., unpublished work). vectors perpendicular to the line of sight will contribute to This has also been investigated experimentally by deLappar- fluctuations in the dc level of the one-dimensional power. ent et al. (14), and Ramella et al. (15). They considered the As one can see in the form of Eq. 5, ifthere is a large-scale Great Wall data set and placed the wall at different redshifts. (low-frequency) cutoff in P3 at the wave number ko, then P1 Since the wall does not have a uniform surface density- is constant from 0 to ko: it is a "white noise," with the level rather it is quite "patchy"-they inferred that the BEKS P1(0). The noise level can be physically interpreted as the pencil beams would typically detect more than five galaxies three-dimensional variance o3 = (18(X)12) multiplied with the in 50% of the walls in the redshift range of 0.1-0.4. This expectation value (1/e)/r, the inverse wave number where number is indeed very sensitive to the diameter of the pencil the maximum contribution to P1 comes from for a diameter of 3h-1 Mpc the probability is 20%o, beam, 1 00 whereas for 10h-1 Mpc it is 90%. For multiple pencil beams = de e the probability of missing an existing wall becomes increas- P10O Iw PAOt ingly smaller as the number of the beams increases. 1 K1e 1 de? 3. Statistics of One-Dimensional Projections e2 P3(e)]. [6] 2i As is well known in statistics, one-dimensional subsets or projections of three-dimensional random fields have rather The values of A(k) at different k are not independent; they different statistical properties from the original one. If we correlate due to the convolution with the window function. have a density fluctuation field 8(x) and an observational For realistic survey geometries we have to calculate the selection function W(x), we measure A(x) = 6(x)W(x), their appropriate window function, and with a basic model for product, excluding sampling noise, etc. If 8(x) is a random P3(k), such as CDM or a minimal truncated power-law field with a known power spectrum P3(k), the power spec- correlation function, we can predict the one-dimensional trum of the observed field A can be written by using the power spectrum. If we approximate the survey as a cylinder convolution theorem of fixed radius R and effective depth D, there are two separable window functions, a radial and a transverse one. For more complicated geometries generally the transverse (1A(k')12) = (I6 * w12) = J. d3k P3(k)jw(k'- k)l2, [2] window function depends on the z coordinate; for a fixed solid angle the z dependence is relatively simple. There is a natural coherence scale in k-space due to the large D, roughly where w(k) is the window function, the Fourier transform of k = 1D. The projection in k-space takes place then in disks the selection function W(x). In obtaining the above relation, of this thickness, and three-dimensional power is projected we have used only the translational invariance of the ensem- only from within a transverse window of conjugate radius of ble average, requiring that approximately 2ir/R. For a cylindrical survey the transverse window function is independent of the z coordinate (8(k')8*(k)) = (2r)3j(3)(k' - k) P3(k), [3]

J1(4tR)J(R where 5(3) is the three-dimensional Dirac delta. For simplicity wR(e) = 2 [7] let us consider an extreme window, an infinitely thin pencil along the z axis, extending to infinity. The window function in this case is w(kl, k2, k3) = 21r6(1)(k3). It is easy to show that where e is the magnitude of the wave vector in the plane of in this case the ensemble averaged one-dimensional power the projection. The physical importance of these two length spectrum Pl(k) becomes scales can be easily understood; the depth D will enter through the number of independent cells along the line of 1 sight, while the transverse scaleR determines the level ofthe P1(k3) = (2-)2 dkldk2 P3(k1, k2, k3). [4] small-scale clustering noise. Kaiser and Peacock (16) claimed that the BEKS result can be an aliasing effect: the high peak is entirely due to a chance fluctuation in a strong clustering The integral describes a projection: ifin a given point k3 along noise. Here we show how their hypothesis can be tested with the survey axis in Fourier space we project all power in the a new data set and how the aliased small-scale clustering (k1, k2) plane, perpendicular to the axis, intersecting it at k3 noise can be reduced at will by using more than one pencil into a single value Pl(k3), a lot of power is aliased from small beam. to large scales. Now we can consider various options for the If the survey consists of multiple pencil beams, sparsely three-dimensional power spectrum, from the most general sampling a larger surface area with a radius B, the important case to the homogeneous isotropic process. Until now, our scale will be this larger B rather than the small R. Let us treatment is fully general. If 8(x) is not a homogeneous consider two such transverse windows (both two-dimension- random process, like having well-defined "wall"-like fea- al), the first describing the individual pencil beams with tures comparable to the survey size, generally P3(k) can have radius R, WR(X), and the wider one with the scale B, describ- sharp spikes that can show up in P1(k3) as well. On the other ing the of N sparsely distributed probes at positions hand, if 5(x) is a homogeneous and isotropic random process, pattern the power spectrum P3(k) will depend on k = lkl only, and rn: Pl(k) can be written as 1 N = - W(X) N [8] 1 0o 1 fk WR(X-irn) Pl(k) = 2 i de i P3(e) = P(O)- - d e P3(eN). [5] In the k-space the Fourier transform becomes Downloaded by guest on September 26, 2021 4856 Colloquium Paper: Szalay et al. Proc. Natl. Acad. Sci. USA 90 (1993)

N Our strategy and the underlying reasons are described in the w(k) = - wRR(k)e'k'n. [9] sections above. We took approximately 10 pencil beams each N n=1 at both galactic poles, distributed over a 6° by 60 area. The size of the individual pencil beams is more or less determined The relevant quantity for correlations and power spectra is by the design of the telescope and the spectrograph. These do not go to such a faint magnitude as the original deep probes, 112 ~~12l 2 E ik(r,-r.)] [ Iw(k)12 = IWR() 2- E. eikrrr [10] but the total number more than compensates for this effect. In the United States, D.C.K., N.E., and A.S.S. have used the Kitt Peak 4-m Mayall telescope with the NESSIE multiobject The 1/N term comes from the autocorrelations of the N spectrograph. They obtained 292 new redshifts in 8 pencil pencil beams, and the second term describes the N(N - 1)/2 beams over a 60 by 60 area around the north galactic pole. The cross correlations. For the aliased clustering noise this win- typical separation of the beams has been 2-3°. T.J.B. and power spectrum: the noise expected from dow multiplies the R.S.E. took spectra with the Anglo-Australian Telescope; a single pencil beam will thus be attenuated by 1/N. The cross-correlation term is in general not isotropic; still, taking nine highly complete fields have been selected to cover a range of angular separations from the original south-galactic- an average over the direction of k we get a sum over Bessel pole beam covering four Schmidt plates, or 100 by 100 in total, functions, - rml). Ink-space this function will be much Jo(klr0 average beam separation of 20. Three hundred narrower than wR; thus it filters out most of the small-scale with the thirty clustering noise. This can be understood very intuitively: in new redshifts were obtained, with an 85% completeness in the cross correlations the minimum distance between pairs of redshift, and the sample was strictly magnitude-limited to 17 galaxies is the beam-beam separation, typically much larger < Bj < 20.5. In every pencil beam we measured 25-50 spectra than the beam size. The noise in the cross-correlation term over a 30-arcmin field. The total number of galaxies within a will be determined by the value of f on this scale. We can cylinder of 50h-1 Mpc radius in our new data set is 1182, conclude that the main source of aliased noise is still coming found in 21 separate pencil beams, compared with 396 in the from the scale of a single pencil beam, although attenuated by BEKS study. This includes further data from Kirshner et al. 1/N. (5) as well. However, this happens only if we stack all of our data Using this data set, we have estimated both A(k) and B(k). together and estimate its power spectrum. If we keep the data Fig. 2 Lower shows A(k). The scale on the horizontal axis is in the different beams separate and calculate only the cross- wavenumber, in units of 8000/h-1 Mpc, as in BEKS. The correlated power spectrum, we can entirely eliminate the largest peak is again at 128h-1 Mpc, and the second largest single-beam aliasing noise. We define the Fourier transform is its octave. In order to assess the probability that this is just of the nth pencil beam as the sum of Dirac deltas at each a chance occurrence in a Gaussian random noise, we fol- galaxy, lowed the procedure described in ref. 17. For a Gaussian Nn noise, the distribution of A is exponential, characterized by fn(k)= E e2,1fi, [11] a single "noise level" Ao. This can be estimated from the j=1 cumulative distribution of the amplitudes of A(k), and the value of the signal-to-noise ratio x = A/Ao determines the and the cross-correlated power spectrum as probability of finding a peak of that height.

1 The question we have to ask, how likely is this just due to B(k) = E [fn(k)f* (k) + f* (k)fm(k)]. [12] chance, or is it an upward fluctuation, arising from a simple Ng n

1 by chance is further complicated by the fact that the small- A(k) = E fn(k)f*n(k). [13] scale correlations increase the variance of the number of Ngfn galaxies. Thus, before we proceed any further, we specify our minimal null hypothesis for the small-scale clustering: (i) One should be aware that B(k) is no longer positive definite; galaxies have the well-known correlation function t(r) = in case of a Gaussian random noise it will fluctuate around (r/5h-1 Mpc)-l8 and (ii) they are uncorrelated on larger zero. The comparison ofA(k) with B(k) for the same data set scales (> 30h-1 Mpc). As we have seen above, this null tell us signal-if any-is due to will whether the detected hypothesis implies a white-noise power spectrum on scales aliasing, amplified by the small radius of the survey, or coming from coherent structures of much larger transverse above the truncation of the small-scale 6r). We have to be extent. If only chance distribution of small groups or poor careful to distinguish between ensemble averages and single clusters is the cause of the peak in the BEKS power spec- realizations. While ensemble averages will be monotonically trum, then the power in the cross correlations should rapidly decreasing, or remain white noise, the individual realizations disappear for wider angular separations. Also, the level ofthe can have peaks in various places. Also, if our survey is not aliased noise in the full power spectrum of the stacked data deep enough to be statistically fair, maybe the hypothesis of set should be considerably weaker, so the probability of a a homogeneous isotropic process is not valid (e.g., we sample high peak decreases rapidly. In the next section we test these only the few nearest cells). two predictions explicitly. Under our null hypothesis, the power-spectrum amplitudes have an exponential distribution, since the real and complex 4. New Results with Multiple Beams parts of the Fourier transform are distributed as a Gaussian. Even if parts of the signal in r-space are very non-Gaussian, Since the BEKS study (10) was published we have collected the power-spectrum amplitudes still closely resemble a Gaus- a considerable number of new redshifts in the same two sian process. In this case the probability distribution is general areas of the sky, the north and south galactic poles. described by the noise level Ao: Downloaded by guest on September 26, 2021 Colloquium Paper: Szalay et al. Proc. Natl. Acad. Sci. USA 90 (1993) 4857

1 s r W s s 1 .1 I I 1 I I I I I I I I .08 I .06 .04 .02 0 rA I AE 1-= A A g-DP O%Chc%4-., 11j~wr ~V- 1APVr; a'J - %J -P- - -.02 I I I I I I I I I I I I = 0 100 200 300 400 500

.1 .08 .06 .04 .02 0 -.02 0 100 200 300 400 500

FIG. 2. Power spectra of both the auto- and cross correlations of our sample, consisting of 1182 galaxies in 21 separate pencil beams. The power spectrum ofthe whole combined data set (Lower) and the power spectrum using only cross correlations between the pencil beams (Upper) are shown. Note that the power is almost unaffected. Abscissa, wavenumber in units of 8000/h-1 Mpc.

dAk ters of galaxies also detected considerable overdensities at dP(Ak) = exp(-Ak/Ao) -. [14] the redshifts of the original BEKS spikes, at angular sepa- Ao rations of several degrees (19-21). This leaves little doubt that the detected regularity is not a statistical artifact, as All models for the small-scale clustering or a more detailed suggested by Kaiser and Peacock (16), but a genuine feature survey enter this influence of the geometry through single ofthe galaxy distribution in that direction. The redshift spikes number. There is a way to obtain this number independently found by BEKS are real and are of considerable transverse of any hypotheses, using an internal estimator. We can calculate the cumulative distribution of the actual (numeri- extent. one is to ask: is the universe cally determined) power-spectrum amplitudes and fit an After this, again tempted exponential to this distribution. The slope of the exponential periodic? We should make it very clear that the observed at small amplitudes gives us the required noise level. The regularity does not mean a periodic universe. It indicates the most conservative signal-to-noise ratio for the new wide- presence of large-scale power, on scales of 100-150h-1 Mpc. angle data set becomes x > 18.0, considerably larger than the Pencil beams in other directions of the sky (D.C.K., R. G. value of 11.8 for the original BEKS data set! This means that Kron, J. Munn, and A.S.S., unpublished work) do not show the relative height of the largest peak, still at the comoving such a sharp regularity, but each direction studied has excess scale of 128h-1 Mpc, increased as we were opening the power on these very large scales. The most likely interpre- survey angle, in sharp contrast with the expectations based tation is that the three-dimensional power spectrum of the upon Kaiser and Peacock's (16) conclusion. The correspond- galaxy distribution is still not very smooth and isotropic over ing probability, that we find the largest peak with this relative very large volumes, it has some "hot spots" between 100 and height in an exponentially distributed noise, measured in -30 150h-1 Mpc. In pencil beams with orientations close to those independent spectral bands is P(xma, > 18.0130) = 7 x 1O-7, hot spots ofthe power spectrum, we may detect a sharp peak. suggesting a statistically significant clustering of galaxies on Shallower surveys with a much broader window function will very large scales. The other test we have made (only in smooth such sharp spikes, so we predict that such hot spots qualitative terms here) is the comparison between A(k) and will eventually be detected by other surveys as well, but with 2 one can see B(k), the latter shown in Fig. Upper. First, that a much broader peak. Such a nonisotropic power spectrum B fluctuates mostly around zero, except for a few regions does not necessarily violate our assumptions about the Gaus- around the peak and its octave. By eliminating the autocor- relations of the 21 individual pencil beams, the height of the sian nature of the initial conditions. The gravitational insta- can can create peak has hardly been affected; thus the large-scale power is bility form these spikes, nonlinearities large contained primarily in the cross correlations, where the coherent structures and correlate the phases ofthe individual some excess power aliasing noise is irrelevant. Fourier components, but is still needed over the standard CDM scenario. 5. Discussion Our strategy of using multiple pencil beams with separa- tions of a few degrees is very powerful; it allows us to control In summary, using a much larger data set in the same general the effects of the aliasing from small-scale clustering. The direction of the north and south galactic poles, we have comparison of the full power spectrum and the cross- confirmed the original BEKS detection ofthe very large peak correlated power spectrum shows that most ofthe large-scale in the one-dimensional power spectrum, corresponding to a signal comes from large extended structures, in agreement scale of 128h-1 Mpc (18). Independent observations of clus- with the original conclusions of BEKS. Downloaded by guest on September 26, 2021 4858 Colloquium Paper: Szalay et al. Proc. Natl. Acad. Sci. USA 90 (1993)

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