Clusters of Galaxies

Introduction

Clusters of galaxies, the largest virialized systems known, are key tracers of the large-scale structure in the Universe and are critical tools in constraining cosmological models. By studying their mass, distribution and evolution, we obtain vital insights into the formation history of the underlying mass field: a fundamental goal of cosmology. Theoretically, clusters provide an opportunity to study the universe in the linear clustering regime, thereby enabling an analytical approach to the calculation of structure evolution (e.g. Mann et al. 1993). Observationally, galaxy clusters can be identified and classified to large lookback times ( z~1 ), thus allowing comparison studies with the nearby Universe. At present, some of the evidence for a rapidly evolving universe comes from clusters of galaxies (e.g. Henry et al. 1992).

There is little doubt that the SDSS survey will revolutionize cluster research. It will provide much needed advances in systematic studies of clusters, currently limited by the unavailability of modern, accurate, complete, and objectively selected catalogs, and by the limited photometric and redshift information for those that do exist. The three main catalogs of rich clusters, those of Abell (1958), Zwicky et al. (1961-1968), and Abell et al. (1989) were obtained from visual inspection of the Palomar Sky Survey and the ESO Survey. Even with this inaccurate procedure, and with only a relatively small number of cluster redshifts (cf. Struble and Rood 1991), the catalogs have contributed significantly to our understanding of large scale structure (Bahcall and Soneira 1983, Postman et al. 1992). Recent advances include wide-angle catalogs selected by objective algorithms from digitized photographic plates (Dodd and MacGillivray 1986, Picard 1991, Dalton et al. 1992, Lumsden et al. 1992) and from deep CCD images (Postman et al. 1995). These efforts have raised some questions about the integrity of the previous visually-constructed catalogues, but the new catalogues are not yet extensive, and we cannot make conclusive statements about completeness (Nichol et al. 1992, Nichol and Connolly 1995). The SDSS photometric survey will solve this by covering, in total, an area 3 times greater that the APM (Dalton et al. 1992) survey and being nearly as deep as the Postman et al. (1995) CCD survey (which covered only five square degrees). Further, it will be in five colors and be the foundation for over 10⁶ spectra! The SDSS will also allow the systematic study of the relationship between the X-ray emitting gas and cluster properties, and of the evolution of the cluster population, through the spectroscopy of ~ 10⁵ luminous red galaxies to z ~ 0.5. In the following section, we outline the expected scope of the SDSS with respect to clusters of galaxies.

The SDSS Cluster Catalogs

The digital photometric and redshift catalogs produced by the main SDSS survey will permit the selection of galaxy clusters using well-defined, automated algorithms. The SDSS can be thought of as producing four different, yet parallel, catalogs of overdensities from the ~10⁶ galaxy redshifts and 5 x 10⁷ galaxies in the photometric data. The digital nature, and the completeness, of the data will allow us to catalogue a continuous range of galaxy overdensities from rich clusters down to poor groups. We ask the reader to bear this in mind while reading the following sections even though we may neglect to explicitly discuss this tremendous dynamic range. Below, we summarize the four SDSS cluster catalogues:

a. 3D Selection

Using the ~10⁶ galaxies with measured redshifts, we will be able to construct a catalogue of overdensities seen in 3D. This will transcend all present concerns of projection effects in optical catalogues and allow us to confidently map the local large-scale structure in clusters. The number of clusters we expect to detect in 3D can be predicted using the Abell catalogue as our template. This catalog is generally thought to be complete for richness class R >= 1 clusters out to z ~ 0.08 (Huchra et al. 1990). In this volume, the SDSS spectroscopic galaxy survey will produce redshifts for galaxies a magnitude fainter than L_* in nearby clusters (Colless 1989), so there will be many galaxies ( >= 100 ) per R>= 1 cluster in the spectroscopic sample. The number of redshifts per cluster decreases with cluster redshift, down to a few redshifts per R >= 1 cluster at z ~ 0.2. We can thus expect this shallower SDSS catalog, identified in galaxy redshift space, to be nearly as deep as Abell's catalog. The surface density of rich clusters ( R >= 1 ) in the survey is expected to be ~ 0.1 per square degree for z <~ 0.2 . This translates to ~1000 clusters some of which have >=100 redshift measurements per cluster, which will be an unparalleled data set of great scientific merit.

A potential problem concerning the 3D selection of clusters is the 55 arcsecond minimum spacing of the spectroscopic fibers (Chapter 11), which will allow only one galaxy of a pair closer than this to be spectroscopically observed per plug plate. However, as we discuss in Chapter 12, we propose to distribute the spectroscopic fields to allow more than one observation of dense regions, a scheme called `adaptive tiling', and in most cases, the second member of a close pair in a dense region, i.e. a cluster, could be targeted in the second pass.

b. 2D selection

The 5 color photometry is predicted to be complete for galaxies to a magnitude limit of r'=22 (Chapter 5). This is substantially deeper than current photographic plate surveys like the APM and EDSGC (Collins et al. 1992), and represents a wonderful opportunity to statistically catalogue galaxy overdensities out to intermediate redshifts ( z<0.4 ). Much current research shows that this may be a dynamic epoch of cluster, and galaxy, evolution.

We will implement automated cluster-finding algorithms to identify clusters in the photometric survey. The wide wavelength coverage of the SDSS photometric data will allow us to search for overdensities in position-magnitude-color space. This will greatly improve the efficiency of cluster identification by reducing foreground and background contamination, which is especially important at high redshift. Furthermore, we can tie these photometric cluster catalogs to those identified solely from the spectroscopic data, thus providing a valuable check on the validity of the photometric identification procedures.

Any cluster-finding algorithm is of necessity biased towards galaxy distributions of a certain shape and profile, and we are working on developing techniques that can efficiently find clusters with strong deviations from circular symmetry and/or pronounced substructure, properties which low-density, "unrelaxed" clusters like Hercules often exhibit. We will certainly want to apply a variety of cluster-finding algorithms to the photometric data. At present, we are investigating two well-motivated algorithms that could be run simultaneously, thereby reducing the overall biases in the final selection. These are the matched-filter approach of Postman et al. (1995) and Kepner et al. (1996), and a wavelet-based method (Slezak et al. 1990). The former involves convolving the data with a filter constructed from a physical cluster model, with the angular galaxy distribution fit to a King profile and the magnitudes fit to a redshifted Schechter luminosity function. The wavelet algorithm entails smoothing the binned galaxy data with a mexican-hat function, which can be thought of as a series of local background-subtracting bandpass filters. We are presently testing and comparing results from both of these using large simulations. These methods will of course also be used to find overdensities in the spectroscopic sample.

The use of photometric redshifts (Connolly et al. 1995) gives us the tantalizing opportunity to carry out a full pseudo-3D selection of clusters from the entire SDSS photometric survey. To facilitate finding clusters, we will construct from the photometric data, as they are reduced, a compressed, coarsely-binned data `cube' describing the structure of the photometric galaxy survey, consisting of the position of each galaxy, its r' magnitude and its photometric redshift (see Chapter 14) which can be easily searched for the density enhancements corresponding to clusters. Methods for searching for clusters in pseudo-3D space are also under development by Kepner et al. (1996).

Once more, we can use existing cluster catalogues to gauge the size of this 2D catalogue. The EDSGC finds clusters and groups by looking for overdensities in the galaxy distribution; they find ~0.5 per square degree (Lumsden et al. 1992) to a limiting magnitude of the tenth brightest member of m₁₀(b_j)=18.5 . Because we need the galaxy distribution ~ 2 mags fainter than m₁₀ in order to identify clusters, we will be able to generate a complete catalog of clusters to m₁₀(r')~20.5 . This yields a surface density of ~5 clusters and groups per square degree, producing a catalogue of ~ 10⁴ rich clusters, an order of magnitude larger than any other catalogue in existence. The smaller groups will be harder to detect at high redshift. A further option will be to co-add the imaging data in the individual SDSS bandpasses and search for angular galaxy overdensities complete to over half a magnitude fainter than this.

There is an alternative approach to cluster finding, pioneered by Dalcanton (1996) and Zaritsky et al. (1996). They use the drift-scan data of Schneider, Schmidt, and Gunn (1994) to identify extended low-surface-brightness objects; some of these are associated with low-surface brightness galaxies (Section 3.4.5), while others are associated with clusters too faint for the individual galaxies to be identified. Zaritsky et al. estimate that there are two high-redshift (z~1!) clusters per square degree in the Schneider et al. data. The Northern SDSS survey is somewhat deeper than these data, and of course will be in five colors, and the Southern survey will be appreciably deeper, so we can look forward to several tens of thousands of high-redshift cluster candidates for follow-up on large telescopes.

c. Southern Survey

A major objective of the SDSS is to carry out a deep photometric and spectroscopic survey in the southern hemisphere. This survey is discussed in detail in Section 5.5. These data will be analyzed in the same fashion as the main SDSS data and will push our cluster catalogues to even higher redshift. The state-of-the-art for high-redshift cluster searches is the deep CCD survey of Postman et al. (1995), carried out on the Palomar 5-m telescope. This survey, a major observational undertaking with current instruments, goes to about 24th magnitude in two bands and covers five square degrees. The SDSS southern survey will go a full magnitude deeper than this, in five bands, and will cover about 300 square degrees. The resulting cluster catalog will be a superb data set for the study of cluster evolution, extending the redshift baseline out to z~1 .

d. Spectroscopy of a Distant, Volume Limited Sample of High Luminosity Elliptical Galaxies

The SDSS plans to obtain redshifts for a sample of the ~ 10⁵ most luminous red galaxies (hereafter BRGs) between r' ~ 18^m - 19.5^m , a depth ~ 1.5^m fainter than that of the main galaxy sample. These objects are frequently located at the centers of high-density regions (clusters), and can be used to measure the volume -limited cluster distribution to z ~ 0.5, greatly extending the scale on which large-scale structure can be measured and providing direct information on evolution for many thousands of clusters of galaxies. BRGs are metal-rich, strong-lined elliptical galaxies, and the SDSS spectra should easily be able to provide redshifts to the limit r' = 19.5^m (see Section 3.4.4).

BRGs have intrinsically high luminosities, and therefore sample a much greater volume than does the main galaxy survey. In the main galaxy sample, a limit of r' ~ 18^m corresponds to a redshift of ~0.17 for L^* galaxies; however, BRGs one magnitude fainter than this are at a redshift of ~ 0.45 at their mean luminosity. The BRG sample thus measures a comoving volume about four times larger than that measured by BRGs in the main galaxy sample. The sampling statistics are such that the power spectrum can be well determined to scales which overlap significantly those measured by COBE and expected to be measured by MAP (see Figure 3.1.9).

This sample of luminous red elliptical galaxies includes the brightest galaxies in clusters but is not limited to them. It has been known for many years that galaxies like BRGs are found in small groups and poor clusters as well as in the richest, highest density clusters. Recent work on the optical identification of ROSAT sources has turned up objects like these in large numbers (see, for example, Figure 3.2.1). The X-ray luminosity of the extended X-ray source shown in this figure, RX J0018.8+1602, is 1.3 x 10⁴⁴ erg s^-1 , similar to typical values for Abell-type optically rich clusters. The bright galaxy near the X-ray centroid is almost certainly associated with the X-ray emission; the nearby optical point source shown in Figure 3.2.1 is unlikely to be the X-ray emitter since a spectrum taken with the ARC 3.5 meter telescope shows it to be an r = 19.5^m G or K star.

The ARC images show that the giant elliptical galaxy at the position of RX J0018.8+1602 has a cD-type radial profile, an r magnitude of 19.2 and a color g-r=3.0+-0.4 (giving a rest-frame color of g-r=0.9, Frei and Gunn 1994). The nearby radio source 54w084 - also at z = 0.54 - has a similar color.

Both RX J0018.8+1602 and 54w084 are likely candidates for the SDSS BRG sample, and illustrate the power of this sample for finding clusters of galaxies which otherwise would have been missed by standard cluster finding algorithms. This example also shows that the SDSS sample is ideal for finding distant X-ray clusters and probing cluster evolution to significant redshifts.

These luminous red galaxies are rare, but not nearly as rare as the great clusters; estimates of their number indicate that they are plentiful enough to provide powerful leverage for large scale structure but are distinct and rare enough that they will not overwhelm the main galaxy sample. The objects in this extended sample are well represented in the main galaxy sample as well, and there will be more than enough information to determine their bias relative to the bulk of the galaxies in the main sample.

Figure 3.2.1

ARC 3.5 m telescope data for the X-ray source RX J0018.8+1602. This source was serendipitously detected in the deep ROSAT PSPC pointing towards the (in)famous cluster of galaxies CL0016+16 at z = 0.546. The X-ray source is coincident with a single giant elliptical galaxy which is at the same redshift as CL0016+16 and is surrounded by a host of much fainter galaxies (Connolly et al. 1996).

The key to the definition of this extended sample is the introduction of the photometric redshift as a selection criterion, without which it would be difficult or impossible to reject lower luminosity objects which would overwhelm the sample. Connolly et al. (1995) have shown that accurate colors obtained from photometry in several bands, plus the apparent magnitude, can measure the redshift to an accuracy of 0.03 or better for most galaxies (see Section 3.1.4.2). It is a happy coincidence that it is precisely for these very red objects that the photometric redshifts are best determined, and redshift errors of at worst 0.02 should be possible from photometry alone. With a photometric redshift, we can estimate the absolute magnitude of the galaxy, and target only the highest luminosity galaxies at a given apparent magnitude. We further cut on color, as we describe below.

The exact size of the sample selected this way depends on how tightly it is defined and on some as yet very poorly known astronomical parameters. We can estimate this number as follows. The luminosity functions of both early- and late-type galaxies are similar and are described roughly by a Schechter (1976) function with an alpha of about -1. (Note, however, that real brightest cluster galaxies in very rich clusters are known not to be part of the general Schechter function, but their numbers are small enough that they are unlikely to affect our estimate very much.) The numbers in the visible bands grow roughly like 10^0.4m , or inversely as the flux density. Thus the distribution of absolute magnitudes at a given apparent magnitude is the luminosity function weighted by one power of the flux, or, roughly, a pure exponential. Therefore, at a given absolute magnitude, about 1/e = 37% of the sample is brighter than L^*. The mean magnitude of the brightest cluster galaxies is 1.8 magnitude brighter than L^*, and they have a dispersion of about 0.35 magnitudes (and the dispersion is somewhat smaller for metric magnitudes, Lauer and Postman 1994; Postman and Lauer 1995). Since we want to be sure to include nearly all the brightest cluster galaxies, we want to include objects with luminosities brighter than about 1.1 magnitudes above L^*; these represent about 12% of the sample. We also make a cut in color, and restrict the sample to the reddest galaxies. In order not to bias the sample, one needs to be relatively generous, but the dispersion in the intrinsic colors is small, about 0.05 or less in B-V (cf., Bower et al. 1992). With generous errors in colors introduced by the uncertainty in the photometric redshift, it would appear that accepting the reddest 30% at a given photometric redshift is adequate to ensure that we have no worse than a 2-sigma cut in the color of the bright ellipticals. Thus at a given apparent magnitude, the sample of bright ellipticals is about 3.5% of the total. We wish, however, to go fainter. To reach a BRG with the mean luminosity of these objects at z=0.5 we need to reach a brightness level of r'=20.4 in the 3 arcsecond fiber, about 1.5 magnitudes fainter than the typical main sample galaxy. There are four times as many objects in this sample than at the cutoff of the main sample, but one-fourth of those are already in the main sample, so we have three times as many additional objects, or 10% of the main sample. Simulations indicate that redshift-quality spectra for bright ellipticals at r'(3")=20.4 are readily obtainable, and that, indeed, one could even go significantly fainter: simulations for a z = 0.5 galaxy 0.7 magnitudes fainter still produce spectra which would yield good redshifts (see Section 3.4.4).

This sample will be accommodated by reducing the main galaxy sample by about 10%, or about 0.1^m in limiting magnitude. What do we gain in return? The volume of the new sample is four times that of the main sample, and indeed the sample is very nearly volume-limited, so that it does not thin out nearly so severely as the main sample does at large distances. Estimates of the shot-noise contribution to the power spectrum P(k) for this sample indicate that the measurements become shot-noise limited only for scales larger than one Gigaparsec h^-1 , and at that scale the shot noise and cosmic variance together contribute an error of about 20% to P(k) (see Figure 3.1.9). This scale is well beyond the peak in the power spectrum predicted by CDM models and overlaps well into the range explored by COBE. The window function of the main sample restricts us to scales more than a factor of two smaller. Indeed, these are scales on which the cluster correlation function (to say nothing of the properties of the clusters themselves) is expected to evolve appreciably, and quantitative measurement of this evolution has the potential to distinguish strongly between various cosmological models (e.g., Peacock 1996). The bias of the BRG sample, which is known to exist and be large, can be adequately explored within the main sample. The redshift to which the BRG sample reaches encompasses the range in which there are indications that there are strong evolutionary effects in cluster properties, and some of these may be seen within the sample. It will also allow a comparison of the evolution of optical and X-ray properties of clusters; there are indications that there may be little evolution in X-ray properties of clusters to z = 0.5, even though significant evolution in the optical properties is seen over this redshift range (Nichol et al. 1996). The sample will be an important bridge to the cluster survey in the south, which will reach objects with (photometric) redshifts in excess of unity, and to the scales probed by the quasar absorption lines (see Section 3.3.4.4).

The sample can also be used to study the evolution of stellar populations in these red galaxies. Combining the spectra in bins of redshift will give spectra with very high signal-to-noise ratios up to z = 0.5 or so, and detailed population synthesis models can be fit. There will be some 20,000 spectra of nearly identical systems between redshifts 0.4 and 0.5; the composite of these, even divided into several different categories, should be breathtaking.

Finally, since this sample is a safe superset of the brightest cluster galaxies, it greatly simplifies the cluster target selection process; these objects are targeted on the basis of their photometric properties alone (and the photometric redshifts are calculated from a simple quadratic expression in the colors, so their determination does not place a burden on the target selection software) and do not depend on finding associated density maxima. The clusters working group has identified a way of finding density enhancements using a simple catalog structure constructed by the target selection pipeline, as described above; we still wish to construct such catalogs for use in other investigations, but it would not be necessary to run a separate piece of code between target selection and tiling in order to calculate the density maxima.

The Combination of SDSS and X-ray Data

With advances in satellite instrumentation, X-ray studies of clusters of galaxies have recently provided important new information on the general cluster population. For example, ROSAT has proved to be one of the most successful X-ray satellites for cluster studies both in its pointing and survey phases. Deep ROSAT PSPC cluster pointings have shown that a large fraction of clusters have substructure and complicated internal dynamics, indicative of recent evolution. In the optical, such studies have been hampered by the need for large numbers of cluster redshift measurements. Also, the ROSAT pointings have shown that the X-ray emitting gas comprises the largest fraction of the baryonic mass in clusters (greater than or equal to the sum of all the galaxies) and that, in total, there is more baryonic matter than expected from nucleosynthesis if Omega = 1 . In parallel, the ROSAT All-Sky Survey (RASS) is now producing objective catalogues of X-ray clusters that are challenging the present optical catalogues in terms of volume sampled and reduced systematics like projection effects. Romer et al. (1994) recently presented the first results from such a catalogue showing that their large-scale distribution was quantitatively similar to that of the optical clusters.

It is clear to us that a marriage of the X-ray and optical data on clusters of galaxies will be extremely beneficial to both parties. On a practical front, optical clusters suffer from projection effects where the alignment of galaxies along the line-of-sight can give rise to a phantom cluster. At low redshift, this will not be a problem for the SDSS due to the spectroscopic survey; however, above z~0.1 this problem will start to surface again. Projections are less severe in X-rays and the coincidence of an X-ray source (especially if extended) with an optical overdensity is a powerful indication of a true mass concentration.

For X-ray cluster searches the major problem is mis-classification. Clusters represent only ~10% of all X-ray sources detected, and at faint flux limits clusters are indistinguishable from the dominant AGN/star populations (Section 2.1). Previous optical follow-up of X-ray sources (Gioia et al. 1990) has been laborious and painful, usually requiring both deep imaging and high-quality spectroscopy. The severity of these problems cannot be overstated, especially at intermediate to high redshifts ( z>0.15 ) - a small error in the identification of stars/AGNs as clusters/groups will give rise to large contamination in the subsequent X-ray cluster candidate catalogues. Therefore, having a full optical database, with 5 colors and substantial spectrophotometry for each X-ray source, will be an incredible advantage and will, in almost all cases, provide an unambiguous candidate for the X-ray emission.

Figure 3.2.2

Right ascension cone diagram of X-ray clusters. The plot shows the redshift distribution of 154 clusters selected from the ROSAT All-Sky Survey (Romer 1994). The declination direction has been compressed and ranges from 0° to -50°. The Sculptor Supercluster (Guzzo et al. 1992) is easily visible at cz~35000 km/s and dominates the nearby large-scale structure. This supercluster is probably one of the most massive structures presently known. The degree of statistical clustering seen in this sample agrees very well with previous optical results (Nichol et al. 1992). The incompleteness beyond cz~40000 km/s is due not to the X-ray data, but to the incompleteness of optical identifications and redshift measurements, since the existing photographic catalogues do not go deep enough.

These points are vividly illustrated by two on-going X-ray cluster surveys. First, Romer (1994) has presented details on her work in constructing an objectively-selected cluster catalogue from the RASS. In her thesis, she defines a sample of 486 possible X-ray clusters in a region of sky centered on the South Galactic Pole. This catalogue is a combination of the available digitized optical photographic plates and the ROSAT survey data. It has already taken her and collaborators over 3 years, with substantial amounts of 4-meter telescope time, to securely identify 129 of these candidates: 15% of them are AGNs, stars and normal galaxies. Furthermore, an additional 10% of the candidates have resolved X-ray emission (a good indication of cluster X-ray emission) but do not coincide with an optical galaxy overdensity on the photographic plates. Deep imaging of a subset of these has uncovered several rich clusters at z>=0.3 (Romer; private communication). Finally, the cluster candidate list was drawn from the superset of ~5000 X-ray sources detected in total over her region. Presently, little is known about the number of clusters lost in this larger X-ray source list. The redshift distribution of these clusters is shown in Figure 3.2.2. The distribution is very incomplete beyond cz~40,000 km/s because the existing optical cluster catalogues, based almost entirely on photographic plates, are not deep enough. As an example of the advances we can expect to make with the SDSS, Figure 3.2.3 shows an r' band image at the position of an extended ROSAT source observed with the ARC 3.5 m telescope (Nichol et al. 1996). A faint cluster, well below the limit of the available photographic plate surveys, is clearly visible.

Figure 3.2.3

Discovery of an X-ray cluster. The gray-scale image shows a cluster of galaxies at the position of an extended ROSAT source imaged at r' band using the Digital Imaging Spectrograph on the ARC 3.5 m telescope. The X-ray contours are from ROSAT observations, and the limiting magnitude of the optical image is about 22.5^m (from the Serendipitous High Redshift Archival ROSAT Cluster Survey; Nichol et al. 1996).

Second, several groups are trying to improve upon the work of the Einstein Medium Sensitivity Survey (EMSS; Gioia et al. 1990) and search for clusters serendipitously detected in the ROSAT PSPC pointing archive. The ROSAT International X-ray and Optical Survey (RIXOS) has been the first to announce their results, finding an order of magnitude fewer clusters detected than the EMSS (Castander et al. 1995). This is remarkable considering that RIXOS samples nearly 3 times more volume than the EMSS. More recent results from the other groups suggest that this discrepancy may be due to object mis-classifications in both of these surveys (Rosati et al. 1995, Nichol et al. 1996).

Clearly, the optical follow-up of all X-ray sources is ultimately the only way to construct a complete and robust subsample of X-ray clusters. However, using conventional methods, this could take decades. For instance, the WGACAT catalogue produced by N. White at GSFC, which is a database of all point sources detected in over 3000 PSPC ROSAT pointing observations, contains ~68,000 sources. To date, only 10% have been identified, and the rate is a strong function of X-ray flux. This catalogue, even though it is biased towards X-ray point sources, could contain ~5000 X-ray clusters to z<1 . The SDSS will provide the identification and classification of tens of thousands of ROSAT sources, redshifts for thousands of X-ray clusters, a search for extended X-ray halos without optical clusters, optical clusters without X-ray halos, the dependence of X-ray emission on structure, redshift, richness, galaxy morphology mix, - - - the list goes on. The structure and evolution of clusters, superclusters and large-scale structure made possible by the combination of SDSS, ROSAT and WGACAT is certain to provide major progress in this field.

SDSS Cluster Science

In this final section, we explore the scientific advances in cluster research expected from the whole SDSS dataset. We will highlight areas that we, as a collaboration, will expect to pursue over the course of the SDSS. Obviously, when the data become available to all, it is certain that much more and more ingenious cluster research will be extracted from this huge database. We will also address some of the scientific rewards of combining the SDSS with other databases like the ROSAT catalogues. This will undoubtedly lead to a fuller understanding of the cluster population since the galaxies and the intracluster gas constitute most of the baryonic mass in clusters.

a. Cosmological Implications

Clusters of galaxies present an efficient method of mapping the distribution of matter on the largest possible scales (Bahcall 1988). Present large cluster redshift surveys still sample larger volumes of space than galaxy redshift surveys and provide the only observations at, or near, the expected turnover in the underlying density power spectrum ( ~100h^-1 Mpc; cf., Figure 3.2.4). An example of the present status of cluster redshift surveys is shown in Figure 3.2.2. The SDSS will put such studies on a much firmer basis because of its larger volume (possibly two orders of magnitude more cluster redshifts), its objective identification methods, and the use of cluster-finding algorithms that are less subject to projection contamination than Abell-type algorithms. These conclusions will be further strengthened with the combination of the optical and X-ray data (cf., Figure 3.2.2).

Figure 3.2.4

The correlation functions derived from three large cluster redshift surveys. The data are from Romer et al. 1994 (X-ray; Sample 6), Postman et al. 1992 (Abell clusters) and Dalton et al. 1992 (APM clusters). This demonstrates the power of cluster correlations for probing the large-scale structure on scales near the turnover in the power spectrum of the underlying density field at ~100h^-1 Mpc.

The high amplitude of the cluster correlation function xi_cc(r) was one of the earliest pieces of statistical evidence for strong clustering on very large scales (Bahcall and Soneira 1983 and Figure 3.2.4) and still remains in conflict with most, though not all, popular models of galaxy formation (Bahcall and Cen 1993, Croft and Efstathiou 1994). Kaiser (1984) suggested that the amplification of xi_cc relative to the galaxy correlation function reflects the tendency of clusters to form near high peaks of the primordial density fluctuations; this idea later became the inspiration for theories of biased galaxy formation. With the SDSS we will be able to measure xi_cc on large scales free from the systematic effects which bedevil existing samples (cf. Olivier et al. 1990). For example, the "shallow" SDSS catalog, identified solely in redshift space, will itself be large enough to yield a robust measure of xi_cc at nearby redshifts ( ~1000 clusters). We will also have a much larger catalog of clusters, identified in the galaxy photometric data with cluster redshifts obtained in the spectroscopic survey, which will yield very accurate correlation measurements and allow us to examine the dependence of xi_cc on cluster properties such as richness, velocity dispersion, morphology, and X-ray luminosity (cf. Bahcall and West 1992). Beyond this, the BRG sample will have the required redshift baseline to search for evolution in the clustering of clusters out to z~0.5 . This latter observation may give us our clearest view yet of the true underlying large-scale structure.

On larger scales, catalogs of nearby superclusters, constructed using Abell clusters with known redshifts, have revealed the largest structures presently known in the universe; ~ 150h^-1 Mpc (Bahcall and Soneira 1984). There are also claims for still larger systems, ~ 300h^-1 Mpc (Tully 1987), but these are still controversial (Postman et al. 1989). Existing galaxy redshift surveys probe a much smaller volume than do the cluster catalogs, but to the extent they can be compared, the superclusters identified from the cluster distribution match up well with structures in the galaxy distribution; the Perseus-Pisces supercluster (Haynes and Giovanelli 1988), and the Great Wall (Geller and Huchra 1989) are both clearly seen in the distribution of clusters and superclusters. The SDSS redshift survey will clarify the relationship between the cluster distribution and the full galaxy distribution. The larger cluster sample should yield definitive answers about structure at very large scales. Cluster peculiar velocities, obtained directly from D_n-sigma measurements (Lynden-Bell et al. 1988) and indirectly from the anisotropy of the cluster correlation function in redshift space (Chapter 3.1), will provide information about the internal dynamics of superclusters.

Finally, it has been known for a long time that brightest cluster galaxies are good standard candles (Sandage 1961; cf., Section 3.2.2.4). This was more recently confirmed in a very detailed study of the photometry of 100 of these objects by Lauer and Postman (1994), showing that the dispersion is as small as 0^m.25 and can probably be reduced significantly by the use of velocity dispersion data. The Hubble diagram given by the few thousand BRGs obtained from the SDSS should demonstrate the uniformity of or find significant nonuniformity in the Hubble flow out to redshifts approaching z=0.3 (cf., Lauer and Postman 1992). Distances to the nearby sample of clusters can be obtained by the Tully-Fisher (Tully and Fisher 1977) and D_n - sigma (Lynden-Bell et al. 1988) techniques (cf., Section 3.1.4), thus allowing us to establish the zero point of the BRG Hubble diagram and obtain a global value of the Hubble constant. We will, parenthetically, be able to confirm or deny the reality of the very large-scale flow found by Lauer and Postman (1994) in their study, and extend their techniques to a very much larger sample.

b. Cluster Properties

The complete cluster survey will allow a detailed investigation of intrinsic cluster properties. From the 2-D and 3-D information, we will be able to determine accurately such properties as cluster richness, morphology, density and density profile, core radius, velocity dispersion profile, optical luminosity, and galaxy content (morphological fractions and cD galaxies), and to look for correlations between these properties. These studies can be supplemented with the vast array of data from other passbands, for example, the FIRST 1.4 GHz survey, which covers the same area as the SDSS with 5 arcsecond resolution and a completeness limit of ~1 mJy. This survey will provide an exquisite catalogue of radio sources in nearby clusters, as well as aiding in the detection of X-ray bright, distant clusters through the presence of a wide-angle radio source (Nichol et al. 1994).

With these quantities in hand, we can study individual clusters in great detail. Measurements of galaxy density and morphology, velocity dispersion, and X-ray emission will shed light on the nature of the intracluster medium and its impact on the member galaxies, and on the relative distribution of baryonic and dark matter (cf. White et al. 1993a). High surface mass density clusters will be candidates for gravitational lenses; we can search the photometric data for systematically distorted background galaxies, especially in the south, and target these clusters for yet deeper imaging surveys with larger telescopes. These lensing studies will allow us to map out the dark matter distribution within a host of clusters (Tyson et al. 1990; Kaiser and Squires 1993).

Although the SDSS dataset will not break new ground in the detailed study of any one cluster, with a uniform database covering an enormous statistical sample of clusters, we can characterize the cluster population statistically with much greater accuracy than has been done before (cf. Bahcall 1977, 1997). This statistical description will give us new insights into a number of issues associated with structure formation and evolution:

• We will be able to calculate velocity dispersions for ~ 1000 clusters with z <~ 0.2 , compared to ~100 clusters with reliable dispersion measurements available now (Mazure et al. 1996). For somewhat nearer clusters we will have well-sampled galaxy density profiles and velocity dispersion profiles; these allow much more accurate cluster mass determinations than those provided by global virial mass estimators, which are the standard recourse when only a small number of galaxy redshifts is available. In addition, some clusters in this sample will undoubtedly possess high quality X-ray spectral information from EXOSAT, ASCA and, in the future, AXAF, to provide a further independent handle on the mass estimates. The distribution of nearby cluster masses puts tight constraints on the mass fluctuation amplitude on cluster mass scales, for any specified value of Omega (Bahcall and Cen 1993, White et al. 1993b). The mass and dispersion distributions can also be used to distinguish initially Gaussian and non-Gaussian primordial density fluctuations (Weinberg and Cole 1992; Bartlett et al. 1993). Beyond the SDSS, the next generation of large aperture telescopes can be used to observe a sample of SDSS distant clusters, thus measuring the distribution of velocity dispersions, and mass, at higher redshift. Due to the homogeneous nature of the SDSS data, we should be able to confidently compare these distant results to the z=0 universe and, for the first time, quantify any observed evolution in the distribution of cluster velocity dispersions.
• With optical luminosities, galaxy profiles, and velocity dispersion profiles, we can measure accurate mass-to-light ratios for a large sample of clusters. These provide constraints on Omega and the bias parameter of galaxies in clusters (cf., Carlberg et al. 1996). The infall patterns at larger separations yield another set of constraints on cluster masses and Omega from scales near the cluster turn-around radius (Kaiser 1987; Regös and Geller 1989). It will be especially interesting to compare these constraints from small and intermediate scales to the results obtained from the analyses of large scale clustering anisotropies (cf. Section 3.1.3); a complete theory of galaxy formation will need to explain any observed dependence of bias parameters on scale and environment.
• At present, there are over 400 specific PSPC pointings towards known clusters of galaxies (mostly Abell clusters). Such data can also be supplemented with Einstein observations. When combined with the SDSS data, this database will be crucial to studies of the relative distributions of luminous (baryonic) and dark matter, and hence of the nature of the dark matter itself. More powerful constraints will be obtained by using dark matter maps derived from lensing of background galaxies. The ratio of baryonic mass to dark mass can lead to a strong constraints on various cosmological parameters and test our understanding of nucleosynthesis (White et al. 1993a).
• The above combined sample of nearby X-ray/optical clusters will also shed light on the formation histories of nearby clusters, since mass elements that have collapsed into them, especially at recent epochs, will remain at higher densities. Such measurements are very hard to perform without the fully-sampled nature of the SDSS spectroscopic and photometric data; substructure may be lost outside a limited field-of-view. This is clearly demonstrated in recent work on the Coma cluster, where the combination of extensive X-ray and optical data highlighted the presence of a large subcluster falling in, or out, of the main overdensity. No longer can Coma be thought of as the archetypical relaxed system since it is obviously involved in recent evolution.

  The frequency of such subclustering and non-virial structures in clusters at low redshift (Geller 1990) tells us whether clusters are dynamically old or young. In a gravitational instability model with Omega ~ 1 , structure continues to form today, while in a low- Omega , open universe, clustering tends to freeze out at moderate redshift. Cluster profiles and substructure statistics thus provide a diagnostic for Omega that depends on the history of clustering (Richstone et al. 1992), an approach quite different from those described earlier, which depend on measuring the mean mass density.

• In addition to the properties of clusters themselves, the properties of galaxies in clusters are of great interest. It is well known that elliptical and S0 galaxies are more common in rich clusters than in the field, and that the fraction of early type galaxies depends on cluster density (Dressler 1984). It has recently been suggested, however, that a morphology-radius relation is a better description of existing data than a morphology-density relation (Whitmore et al. 1993). The SDSS will provide a powerful database for examining such issues, going beyond statistical characterization to detailed studies of the effects that might produce morphological segregation. Furthermore, for nearby clusters, galaxies will be classified spectroscopically as well as morphologically, and will have excellent colors. The interaction of individual galaxies with the X-ray emitting intracluster medium and tidal fields from their neighbors can be studied in detail using the excellent morphological, photometric, and spectroscopic data. To give just one example, measurements of the H alpha equivalent width of spirals as a function of velocity relative to the cluster rest frame and distance from its center may reveal the gas stripping and star formation histories of galaxies which are passing through the intracluster gas for the first time.
• One of the most important contributions the SDSS will make in cluster research will be in the study of cluster evolution. This is particularly important since it is potentially the most powerful diagnostic of the physical model causing structure formation. Current investigations are limited to a small numbers of well-known high-redshift clusters, but an even more serious obstacle is that high and low redshift samples of clusters are selected in very different ways. As a result, it is difficult to know whether high redshift concentrations represent fully-formed clusters, or are high-density cores on which more mass will accrete by the present. The first would be unexpected in an Omega ~ 1 model, but the second would be natural. The SDSS cluster sample will clearly remedy both of these shortfalls. The high and low redshift clusters will be selected from the same data using the same algorithms, so they can be compared directly to each other, and to numerical simulations. The number of clusters will be large; we will have redshifts for BCGs to z ~ 0.5 , and for more distant clusters we can obtain fairly accurate estimated redshifts from the photometry alone (using the colors and apparent luminosity function of member galaxies.) The southern survey will probe the cluster population out to redshifts of order unity (the i' filter will be most useful for this).

• The Butcher-Oemler (1984) effect is the strongest optical evidence for significant cluster evolution out to z~0.5 . However, as discussed by Kron (1995), these studies have been based on a rogue's gallery of well-studied clusters: "the clusters that formed the foundations of the Butcher-Oemler effect in 1984 are evidently very exceptional objects -- otherwise, objects discovered in the mid-1950's would not continue to appear so prominently in current research efforts". Using only the SDSS database, we will transcend such problems and quantify the Butcher-Oemler effect for thousands of clusters as a smooth function of galaxy richness, redshift and cluster morphology. By combining the SDSS with X-ray data, we can further investigate the relationship between the Butcher-Oemler effect and the density/temperature of the intracluster gas, which still remains relatively unknown. This combined dataset will directly test certain Butcher-Oemler hypotheses; for example, is the increased blue galaxy population due to ram-pressure enhanced star formation in infalling cluster galaxies (Gunn and Dressler 1988)? If so, one would expect a strong correlation between the X-ray properties of clusters and their corresponding galaxy populations as a function of cluster radius and velocity dispersion. At low redshift ( z<0.1 ), the high-quality SDSS data will allow us to move beyond the simplistic Butcher-Oemler definition of optical cluster color/morphology evolution by basing our studies solely on the spectrophotometric data. At higher redshifts, the five color photometry will be invaluable, since the K-corrections can be accurately computed.
• The observed space density of clusters as a function of redshift can place tight constraints on the evolution of the underlying density field. X-ray observations suggest a rapid decrease in the density of bright X-ray clusters (Henry et al. 1992), while at optical wavelengths, the space density appears to remain almost constant out to z=0.5 (Postman et al. 1995). At present, many authors simply wish to advocate the merits of one cluster selection method over the other, yet the combination of these two results may be telling us something fundamental about cluster evolution and the enrichment/heating of the intracluster gas. Using the SDSS, we will be able to examine such results and quantify any subtle selection bias at play. If such effects are ruled out, or quantified, we should then be able to de-couple the physics of cluster formation from the underlying cosmological density evolution. Again, this will only be possible via the unique marriage of the SDSS and X-ray data.
• Finally, at intermediate redshifts, there already appears to be a strong correlation between the optical morphology of clusters and their X-ray characteristics. This is illustrated by the work of Gioia and Luppino (1994) and Postman et al. (1995): over 90% of the z>0.3 EMSS clusters contain a dominant, central galaxy (cD), while in the optical, the fraction is only ~25% . Furthermore, the fact that a rich optical cluster is not an X-ray emitter is equally interesting and can only be determined through the combination of the objective X-ray and optical data. Such phenomena may be related to the above discrepancies in the cluster space densities and a direct indication of their contrasting evolutionary histories: relaxed, X-ray bright clusters may have a greater number of cluster crossing times, thereby enriching the intracluster medium and allowing the formation of a dominant galaxy via cannibalism. We will be able to carry out detailed cluster morphological studies (both photometrically and spectroscopically) testing such hypotheses. For example, do X-ray faint, optically rich systems have a significantly different E/S0 population compared to centrally condensed, X-ray bright clusters? If they do, this would be indicative of different formation epochs since these galaxies are dominated by old stellar systems.

References

Abell, G.O. 1958, ApJSuppl 3, 211.

Abell, G.O., Corwin, H.G., and Olowin, R.P. 1989, ApJSuppl 70, 1.

Bahcall, N. A. 1977, AnnRevAAp 15, 505.

Bahcall, N. A. 1988, AnnRevAAp 26, 631.

Bahcall, N. A. 1997, in Astrophysical Quantities, ed. A. Cox, AIP Press, in press.

Bahcall, N. A., and Cen, R. 1992, ApJL 398, 1.

Bahcall, N.A., and Cen, R. 1993, ApJL 407, 49.

Bahcall, N.A., and Soneira, R.M. 1983, ApJ 270, 20.

Bahcall, N.A., and Soneira, R.M. 1984, ApJ 227, 27.

Bahcall, N.A., and West, M.J. 1992, ApJ 392, 419.

Bartlett, J.G., Gooding, A.K., and Spergel, D.N. 1993, ApJ 403, 1.

Bower, R.G., Lucey, J.R., and Ellis, R.S. 1992, MNRAS 254, 601.

Butcher, H., and Oemler, A. 1984, ApJ 285, 426.

Carlberg, R.G., Yee, H.K.C., Ellingson, E., Abraham, R., Gravel, P., Morris, S., and Pritchet, C.J. 1996, ApJ 462, 32.

Castander, F.J. et al. 1995, Nature 377, 39.

Colless, M., 1989, MNRAS 237, 799.

Collins, C.A., Nichol, R.C., and Lumsden, S.L. 1992, MNRAS 254, 295.

Connolly, A.J., Csabai, I., Szalay, A.S., Koo, D.C., Kron, R.G., and Munn, J.A., 1995, AJ 110, 2655.

Connolly, A.J., Szalay, A.S., Koo, D.L., Romer, A.K., Holden, B., Nichol, R.C., and Miyaji, T. 1996, ApJL 473, 67.

Croft, R.A.C., and Efstathiou, G., 1994, MNRAS 267, 390.

Dalcanton, J.J. 1996, ApJ 466, 92.

Dalton, G.B., Efstathiou, G., Maddox, S.J., and Sutherland, W.J. 1992, ApJL 390, 1.

Dodd, R.J., and MacGillivray, H.T. 1986, AJ 92, 706.

Dressler, A. 1984, AnnRevAAp 22, 185.

Frei, Z., and Gunn, J.E. 1994, AJ 108, 1476.

Geller, M. 1990, in Clusters of Galaxies: Space Telescope Symposium Series # 4, W. Oegerle, M. Fitchett, and L. Danly, eds. (Cambridge: Cambridge University Press).

Geller, M., and Huchra, J.P. 1989, Science 246, 897.

Gioia, L., Henry, J. P., Maccacaro, T., Morris, S. L., Stocke, J., Wolter, A. 1990, ApJ 356, L35.

Gioia, L., and Luppino, G. 1994, ApJSuppl 94, 583.

Gunn, J. E., and Dressler, A. 1988, Large Scale Structure of the Universe, IAU Symposium 130, eds. Audouze et al.

Guzzo, L., Collins, C. A., Nichol, R. C. and Lumsden, S. L., 1992, ApJ 393, L5.

Gunn, J. E., and Dressler, A. 1988, Towards Understanding Galaxies at Large Redshift, eds. Kron and Renzini.

Haynes, M.P., and Giovanelli, R. 1988, in Large Scale Motions in the Universe: A Vatican Study Week, edited by V.C. Rubin and G.V. Coyne, S.J. (Princeton: Princeton University Press), p. 31.

Henry, J. P., Gioia, L., Maccacaro, T., Morris, S. L., Stocke, J., Wolter, A. 1992, ApJ 386, 408.

Huchra, J.P., Henry, J.P., Postman, M., Geller, M.J., 1990, ApJ 365, 66.

Kaiser, N. 1984, ApJL 284, 49.

Kaiser, N. 1987, MNRAS 227, 1.

Kaiser, N., and Squires, G. 1993, ApJ 404, 441.

Kepner, J.V., Fan, X., Bahcall, N.A., Gunn, J.E., and Lupton, R.H. 1996, in preparation.

Kron, R.G., 1995, "The Butcher-Oemler Effect: Lecture 8", Sante Fe Lecture Series.

Lauer, T. R. and Postman, M., 1992, ApJL 400, 47.

Lauer, T. R. and Postman, M., 1994, ApJ 425, 418.

Lumsden, S. L., Nichol, R. C., Collins, C. A., and Guzzo, L., 1992, MNRAS 258, 1.

Lynden-Bell, D., Faber, S. M., Burstein, D., Davies, R. L., Dressler, A., Terlevich, R. J., and Wegner, G. W., 1988, ApJ 326, 19.

Mann, R.G., Heavens, A., and Peacock, J.A., 1993, MNRAS 263, 798.

Mazure, A. et al. 1996, AstrAp 310, 31.

Nichol, R.C., Collins, C.A., Guzzo L., and Lumsden S.L., 1992, MNRAS 255, 21P.

Nichol, R.C., and Connolly, A.J., 1995, MNRAS 279, 521.

Nichol, R.C., Romer, K., Holden, B., Ulmer, M. Collins, C. and Burke, D. 1996, Ap. J. (in press).

Nichol, R.C., Ulmer M.P., Kron R.G., Wirth G.D., and Koo, D.C. 1994, ApJ 432, 464.

Olivier, S., Primack, J.R., Blumenthal, G.R., Dekel, A., and Stanhill, D. 1990, ApJ 356, 1.

Peacock, J.A. 1996, MNRAS, in press (astroph/9608151).

Picard, A. 1991, PhD. Thesis, California Institute of Technology.

Postman, M., Lubin, L.M., Gunn, J.E., Oke, J.B., Hoessel, J.G., Schneider, D.P., and Christensen, J.A. 1995, AJ 111, 615.

Postman, M., Huchra, J.P., and Geller, M. 1992, ApJ 384, 404.

Postman, M., and Lauer, T. 1995, ApJ 440, 28.

Regös, E., and Geller, M.J. 1989, AJ 98, 755.

Romer, A.K., 1994, PhD Thesis, Univ. of Edinburgh.

Romer, A.K., Collins, C.A., Cruddace, R.G., MacGillivray, H., Ebeling, H., Bohringer H., 1994, Nature 372, 75.

Rosati, P., Della Ceca, R., Burg, R., Norman, C., and Giacconi, R. 1995, ApJL 445, 11.

Sandage, A., 1961, ApJ 133, 355.

Schechter, P.L. 1976, ApJ 203, 297.

Schneider, D.P., Schmidt, M., and Gunn, J.E. 1994, AJ 107, 1245.

Slezak, E., Bijaoui, A., and Mars, G., 1990, AstrAp 227, 301.

Struble, M.F., and Rood, H.J. 1991, ApJSuppl 77, 363.

Tully, R. B. 1987, ApJ 323, 1.

Tully, R. B., and Fisher, J. R., 1977, AstrAp 54, 661.

Tyson, J.A., Wenk, R.A., and Valdes, F. 1990, ApJL 349, 1.

Weinberg, D.H., and Cole, S. 1993, MNRAS 259, 652.

White, S.D.M., Navarro, J.F., Evrard, A.E., and Frenk, C. S. 1993a, Nature 366, 429.

White, S.D.M., Efstathiou, G., and Frenk, C.S. 1993b, MNRAS 262, 1023.

Whitmore, B.C., Gilmore, D.M., and Jones, C. 1993, ApJ 407, 489.

Zaritsky, D., Nelson, A.E., Dalcanton, J.J., and Gonzalez, A.H. 1996, preprint (astro-ph/9612021)

Zwicky, F. et al. 1961-1968, Catalog of Galaxies and Clusters of Galaxies (Pasadena: California Institute of Technology).