Red Sequence Luminosity Function

The Red Sequence Luminosity Function in Massive Intermediate Redshift Galaxy Clusters

S. M. Crawford,11affiliation: South African Astronomical Observatory, Observatory, 7935 Cape Town, South Africa; crawford@saao.ac.za M. A. Bershady,22affiliation: Washburn Observatories, U. Wisconsin - Madison, 475 N. Charter St., Madison, WI 53706; mab@astro.wisc.edu and J. G. Hoessel
Abstract

We measure the rest-frame B-band luminosity function of red-sequence galaxies (RSLF) of five intermediate-redshift (), high-mass ( km s) clusters. Cluster galaxies are identified through photometric redshifts based on imaging in seven bands (five broad, and two narrow) using the WIYN 3.5m telescope. The luminosity functions are well-fit down to for all of the clusters out to a radius of . For comparison, the luminosity functions for a sample of 59 low redshift clusters selected from the SDSS are measured as well. There is a brightening trend ( increases by 0.7 mags by z=0.75) with redshift comparable to what is seen in the field for similarly defined galaxies, although there is a hint that the cluster red-sequence brightening is more rapid in the past (), and relatively shallow at more recent times. Contrary to other claims, we find little evidence for evolution of the faint end slope. Previous indications of evolution may be due to limitations in measurement technique, bias in the sample selection, and cluster to cluster variation. As seen in both the low and high redshift sample, a significant amount of variation in luminosity functions parameters and exists between individual clusters.

galaxies:cluster: general — galaxies: evolution — galaxies: luminosity function

1 Introduction

In the nearby Universe, galaxy clusters are dominated by early type galaxies (Dressler 1980) along the so-called “red sequence,” the name for which derives from the tight color-magnitude relation observed for these galaxies (Visvanathan & Sandage 1977). The most luminous cluster galaxies, located at the tip of the red sequence, appear to have relatively little star formation since (Bower et al. 1992; Ellis et al. 1997; Stanford et al. 1998; Kelson et al. 2001). Observations of high redshift () clusters are consistent with this inference, indicating at least the bright end of the color-magnitude relationship has only passively evolved since (Stanford et al. 1997, Mullis et al. 2005, Stanford et al. 2005). However, evidence for recent ( Gyr) bursts of star formation in today’s lower-mass cluster galaxies (Poggianti et al. 2001, Conselice et al. 2003) indicates not all cluster galaxies have the same homogeneous star formation histories. While the luminous red sequence in clusters may have long been in place, the extension of the red sequence to lower luminosity may be a relatively recent phenomenon.

Recent, deep imaging programs have led to the claim of a deficit of red sequence at low luminosities in high redshift clusters (Nakata et al. 2001, de Lucia et al. 2004, Goto et al. 2005, Tanaka et al. 2005). This would suggest many of the galaxies seen along today’s red sequence may be late additions, perhaps culled from a quenched and faded blue population. This fits neatly with a notion that the red sequence is not monolithic, as evinced by the seminal studies of the Virgo cluster luminosity function (Sandage, Binggeli & Tammann 1985). However, the deficit may not be ubiquitous (e.g., Andreon 2006). Differences in the analysis methods, clusters samples, and the inherent limitations in different data sets (i.e., the depth of available data) have left the observational evidence for a deficit inconclusive.

Differences between studies are also complicated by the possible existence of evolutionary phenomenon that is a function of cluster mass. Substantial differences exist between the cluster and field red sequences (de Propris et al. 2003, Croton et al. 2005), and there is evidence for differences in the luminosity function with cluster mass, both locally (Hansen et al. 2005, Hilton et al. 2005) and at intermediate redshift (Koyama et al. 2007, Gilbank et al. 2007). However, evidence to the contrary has also appeared in the literature (Barkhouse et al. 2007, Andreon 2007). Given these disparate claims, it is difficult to assemble a clear picture of the red-sequence evolution – a challenge again exacerbated by differences in cluster samples and measurement techniques between studies.

A common theme repeated in many of the studies is the large cluster to cluster variation seen in any of the parameterizations of the cluster populations, particularly the luminosity function. Estimates of the dwarf-to-giant ratio (DGR, de Lucia et al. 2006) and Schechter-function (Schechter 1976) fits to the luminosity function (Barkhouse et al. 2007, Andreon 2007) both show significant cluster-to-cluster variation. Unique features also have long been identified in the shape of luminosity functions of local clusters (Biviano et al. 1995, Yagi et al. 2002), indicative of a multi-component population even among the red-galaxy population. Traditionally, deep studies of the so-called red-sequence luminosity function (RSLF) in local clusters have fit two functions (either double Schechter functions or a Gaussian and Schechter function) to the distribution of galaxies (Sandage et al. 1985, Jerjen & Tammann 1997, Popesso et al. 2005). The bright end of the luminosity function is well described by a functional form with a sharp turnover at lower luminosities, while the fainter luminosity-function component is generally found to be rapidly rising at lower luminosities. Variations in the relative amplitude of these components may well drive overall variations in observed cluster luminosity functions. Studies of intermediate redshift clusters are dominated by this brighter component, due to depth limitations. While such studies rarely reach deep enough to well-characterize the rapidly-rising faint end, its modulation may significantly impact the observed count-parametrization.

In this paper, we explore the red-sequence population in five high-density regions containing massive, bona fide, and well-studied clusters between . Our study is based on deep, multi-band imaging data from the WIYN 3.5m telescope and extant spectroscopic data. The depth of our WIYN data is comparable to other cluster studies at similar redshifts, and allows an independent assessment of the cluster RSLF significantly below the knee in the bright end of the luminosity function. The extent to which this, or any other extant study at intermediate redshift, can address the faint-end of the cluster RSLF is a point we address. For comparison, we include measurements of the luminosity function for a large sample of low redshift clusters from the Sloan Digital Sky Survey measured in the same manner as our intermediate redshift sample.

In §2 we summarize the observation data and highlight our measurement techniques. In §3, we present the luminosity functions measured for each of the clusters as well as the tests to confirm and assess the reliability of the measured luminosity functions. In §3.3, the low-redshift cluster sample is described. Finally, we discuss the observed evolution in our luminosity functions (§4), compare to results in the literature, and discuss the implications of our results in §5. Throughout this work, we adopt H km s Mpc, , and ; all absolute magnitudes are in the rest-frame (Johnson) B-band in the Vega system.

2 Observations and Analysis

Observations were obtained between 1999 October and 2004 June, with the WIYN111The WIYN Observatory is a joint facility of the University of Wisconsin-Madison, Indiana University, Yale University, and the National Optical Astronomy Observatories. 3.5m telescope’s Mini-Mosaic Camera ( per pixel and field of view, FWHM median seeing) as part of the survey described by Crawford et al. (2006) and Crawford (2006). Cluster images were taken in the Harris UBRI, Gunn z (Schneider, Gunn, & Hoessel 1983), and two narrow band filters. The narrow band filters () were specifically designed to detect [OII] and adjacent continuum at each cluster redshift. Table 1 lists the five clusters observed, and their salient attributes.

Data analysis proceeded on deep mosaic images created from reduced Mini-Mo frames, flat to of their initial sky values. The deep mosaics were photometrically calibrated through a variety of independent methods yielding uncertainties below . Object detection was performed on the R-band (typically deepest) images using SExtractor (Bertin & Arnouts 1996) with the criterion that objects contained contiguous pixels above of the sky noise. This corresponds to a minimum signal-to-noise (S/N) of 13.4 in an equivalent circular aperture of 0.35 arcsec radius (just slightly smaller than the typical seeing disk). The S/N at the 50% detection limit (see below) will be higher since most objects intrinsically at the specified limiting size and surface-brightness will have some fraction of the pixellated signal noise-aberrated below the detection threshold. As the S/N depends on the aperture size and tends to peak around the the half-light radius, it is essential to define the S/N used in any analysis to avoid confusion.

Two types of magnitudes were used for all calculations. Seeing-matched222All images were degraded to worst-case seeing for each cluster for the purpose of this photometric measurement. aperture magnitudes with a radius corresponding to 7.5 kpc at each cluster redshift were used for all colors and photometric redshifts. This is equivalent to aperture diameters between 1.93 and 2.45 arcsec, or roughly stellar FWHM on average. This aperture defines the S/N value for the R-band that we use in subsequent analysis, particularly in the next two sub-sections concerning detection completeness and photometric-redshift errors. Tailored magnitudes (Crawford et al. 2006) based on the curve of growth, concentration index, and Petrosian ratio are used for total apparent and absolute magnitudes. Absolute magnitudes and colors were calculated for all objects using photometric (see below) or spectroscopic redshifts. K-correction calculations adopt method 4 of Bershady (1995).

2.1 Detection Completeness

We determine the completeness limit of our data through Monte Carlo simulations similar to what is described in Bershady, Lowenthal & Koo (1995). In each detection image, we extract a sample of bright, representative objects of different sizes. Objects are dimmed, and then re-inserted into the image. For each half-magnitude interval, we insert 50 objects randomly across the usable field of view, repeat the detection procedure, and repeat the process 10 times, and for a minimum of three different object sizes. For small, unresolved objects, the completeness magnitude is typically around magnitudes. For objects with lower surface brightness, our completeness limits can be significantly ( mag) more shallow. The R-band completeness curves are presented in Figure 1. For guidance as to the depth of these data, we mark in the Figure the location of M* at the appropriate cluster redshift based on the evolving RSLF estimated for the field (Willmer et al. 2006). When correcting for incompleteness, we calculate the correction relative to the measured size of each object. S/N in seeing-matched apertures at the 50% and 90% detection completeness are roughly 20 and 25, respectively, which is well above the limiting S/N for detection.

For luminous red galaxies, the completeness limit is relatively shallow due to the diffuse profile shape and large size of such galaxies. Although a significant portion of the light is concentrated into the central regions, most of the light is in the low surface brightness wings of the galaxy. However, these systems are not at the detection-limit of our survey at the cluster redshifts. Instead, intrinsically faint (low luminosity), red galaxies are sampled near our detection limits, for which the above detrimental properties are not in play. Three natural phenomenon are working in our favor in terms of detecting these lower-luminosity galaxies. First, the light profiles of lower-luminosity red-sequence galaxies are closer to an exponential distribution than an (Graham & Guzmán 2003), thereby enhancing detectability. Second, according to the luminosity-size distribution of red galaxies (Shen et al. 2003, McIntosh et al. 2005), we expect the faint, red galaxies to be unresolved in our data set, and hence our deepest detection limits are relevant. We return to, and illustrate these two points later in §3. Finally, lower-luminosity galaxies are expected to be slightly bluer than their luminous red-sequence counterparts due to a slope in the color-magnitude relationship. Hence, the effect of the k-correction for these galaxies at the R-band detection limit is less severe. On this basis, we compute the depth of our survey relative to the completeness limit for unresolved objects in that field. These depths are listed in Table 1, and relative to the field RSLF, they range between 2.6 and 3.6 mag below .

2.2 Photometric Redshift Method, Accuracy and Precision

We have derived photometric redshifts for all objects with the highest precision possible. The photometric redshifts were determined through a hybrid of the template and training-set methods similar to Csabai et al. (2003). A template grid of model spectral energy distributions (Bruzual & Charlot 2003), and their corresponding broad- and narrow-band fluxes, was created. The spectra cover a range of star formation rates, ages, and redshifts. Using objects with spectroscopic redshifts and our observed multi-band fluxes, the grid was constrained (more specifically, morphed or adjusted in flux space) to match the measured data. Over 500 objects with existing spectroscopic redshifts in our combined survey fields (Dressler et al. 1997, Ellingson et al. 1998, Postman et al. 1998, van Dokkum et al. 1999, Moran et al. 2007, Tran et al. 2007) trained our set of templates in this way. The method was tested with the following procedure. A single spectroscopic object was removed from the catalog. The grid was trained using the remaining objects and then applied to the removed objects. The procedure was repeated so that an independent measure of the photometric redshift (and its error) was made for every spectroscopic object. In Figure 2 and 3, we compare the spectroscopic and photometric redshifts measured via this procedure. High quality photometric redshifts can only be obtained with large values of signal to noise as shown in Figure 4. Photometric redshifts are calculated with a precision and an accuracy of for red objects with in the R-band. Blue objects (defined below in §2.3) performed slightly worse with and with . Both types of objects have much larger scatter for galaxies with . At the detection threshold, the typical scatter for red objects is expected to be approximately .

2.3 Impact on Luminosity Function Calculations

Extensive simulations of the effect of photometric errors on the distribution of measured photometric-redshift errors, as well as the subsequent effect on the cluster-galaxy luminosity function were performed. The latter comes about because we define cluster membership based on photometric redshift (§3) when spectroscopic redshifts are not available. Photometric redshift errors at high signal to noise () are small, and have a Gaussian distribution. At low signal to noise, however, the error distribution is a combination of a Gaussian core around the fiducial redshift plus a catastrophic error component due to near-degeneracies in multi-color space. Combined with our definition of cluster membership, the photometric-redshift errors cause a significant portion () of the cluster population above our detection limit to be identified as field galaxies beyond our redshift selection window. Because of large cluster over-densities, contamination from the field into the cluster sample is minimal. However, the opposite is true when we construct the field-population estimate for the RSLF, as we show below in §2.3. Consequently, the nature of the correction is to account for a net loss, or incompleteness in the bona-fide number of cluster galaxies, and vice-versa for the field population. These corrections are independent of our detection completeness except in so far as both are correlated with apparent magnitude (or S/N). Statistical corrections, based on our simulations, are applied to our luminosity functions as a function of apparent magnitude. At the completeness limit in the images, the average correction is with a maximum of for CL0016. For the field-galaxy luminosity function that we construct below as a quality check, we push these corrections farther to illustrate the quality of these corrections. In §3.2.1 we discuss tests of the above issues of completeness and contamination in the specific context of our cluster selection function.

2.4 Quality Check: The Field RSLF at Intermediate Redshifts

To demonstrate our methodology for selecting and computing the statistical distribution of red cluster galaxies, we first construct the field RSLF luminosity function, derived from our data, in two redshift bins that should be uncontaminated by rich, over-densities, i.e., foreground and background regions in the cluster fields (Figure 5). The low redshift bin () contains the imaging from fields MS1054, Cl1322, and Cl1604, whereas the high redshift bin () is calculated based on data from Cl0016 and MS0451. In addition, one empty “field” image centered at 13:24:50.1h +30:11:18.5, which contains no evidence of any massive over-density, was included in both bins. Red galaxies were identified following the same prescription as Willmer et al. (2006):

(1)

This definition, used throughout the paper, is based on a fit to the color-magnitude relation plus a -0.25 mag shift in zeropoint.

Corrections for detection incompleteness and contamination due to photometric redshift errors (as described in the previous section) have been applied. The data are only presented to a magnitude limit where the completeness is , but in some cases the contamination corrections are larger. At low S/N (faint magnitudes), cluster sources with large photometric-redshift errors can contribute substantially to the field counts. Akin to the Eddington effect (Eddington 1913) where fainter, more plentiful objects have a more significant influence on brighter bins with less objects, the majority of field counts at faint magnitudes can be due to contamination from cluster over-densities. For this reason, and to keep contamination corrections manageable, we calculate the luminosity function after excluding redshift bins where, for a given luminosity, the contribution from the cluster is expected to be greater than . For example, at () in the low-z sample, the error on the photometric redshift is expected to scatter up to of the galaxies from MS1054 into the redshift bin of . Hence, in this magnitude bin, we would exclude galaxies with photometric redshifts between in the MS1054 field from the low-z luminosity function while still including galaxies with lower redshifts. An appropriate volume would be calculated for this magnitude bin according the corresponding redshift window.

In Figure 5, we compare our raw and corrected field RSLF to DEEP2’s Schechter-function parameterization of the same (Willmer et al. 2006). A reduced- statistic of this parameterization with respect to our data-set indicates a reasonable match between results, despite our smaller survey volume and use of photometric redshifts. We reach a similar conclusion comparing our field blue-galaxy luminosity function with corresponding results Willmer et al. (2006). This is interesting because in the case of the blue-galaxy luminosity, the field sample we construct has smaller contamination errors. On balance, these results indicate we are able to control our corrections for incompleteness and photometric errors reliably down to and below the level we will use in our cluster luminosity function analysis.

3 Cluster Red-Sequence Luminosity Functions

3.1 WIYN Intermediate Redshift Sample

Red-sequence cluster galaxies were identified with (i) the same color-luminosity relation (Eq. 1) as for the field, and (ii) as having redshifts within of the cluster redshift. Both of these selection criteria for our data can be seen in Figure 6. Most of the clusters do exhibit a strong red sequence even when data are included from the entire field of view, which is much larger than the cluster core-radius. All fields show an over-density at the redshift of the cluster. For those two clusters (Cl1322 at z=0.75 and Cl1604 at z=0.9) which do not show a strong red excess in the color-redshift plot due to contrast (Figure 6, right panel), they still show a well-populated red-sequence in the color-magnitude diagram (Figure 6, left panel). The raw histogram in absolute magnitude of red-sequence galaxies, normalized by the selection volume, are displayed in Figure 7 (left panel) for a selection radius of 333 is calculated based on the definition in Finn et al. (2005). (The volume was estimated for each cluster from the selection in redshift and radius.) Since the cluster is expected to cover a much narrower range in redshift, the volume, while well-defined, is an over-estimate. Aside from this volume normalization, this histogram represents the uncorrected measurement of the red-sequence luminosity function in terms of shape and luminosity normalization.

Several corrections were applied to the data to account for observational deficiencies:

(a) The first correction was for detection completeness, computed from the Monte Carlo simulations (§2.1). The correction is calculated for each galaxy based on its apparent magnitude and size.

(b) Because we sample a much larger volume than the cluster, a portion of the counts are expected to be from the field around the cluster. We therefore subtract an appropriately volume-scaled field sample based on the RSLF of Willmer et al. (2006), which we note is always a small fraction of the total uncorrected cluster counts (e.g., within Mpc).

(c) Finally, the incompleteness due to photometric redshift errors was applied (§2.2), again based on simulations. Specifically, we computed the fraction of galaxies as a function of apparent magnitude that will be missed (scattered out of our redshift selection window). We can ignore as negligible the fraction scattered into our redshift selection window from the field, because to first order this cancels with the number of field galaxies in our volume scattered out, and further we subtract a field component from our volume, as noted above in (b). By assuming a single spectral energy distribution for the red galaxies, this correction can then be applied to the binned counts as a function of absolute magnitude.

Absolute magnitude bins containing galaxies beyond our specified completeness limits were excluded from our calculations, and our primary results are quoted down to 50% detection-completeness limits. The final, corrected data within for these limits are presented as solid points in Figure 7 (left-hand panel). We also use 90% detection-completeness limits in some cases to further demonstrate the robust nature of our statistical results. These are indicated by the vertical dotted lines in this Figure.

For each cluster, we calculate the best-fit Schechter-function to the luminosity distribution by varying , , and to find the smallest reduced . Because of the nature of our volume normalization, we focus here on and , not , with our primary scientific emphasis being on . We plot (Figure 7, right) the error ellipses calculated from measurements based on the best fit model. Again, these are for data within projected about the cluster center. The luminosity function also was calculated over several different selection radii: Mpc, , and . These results and those for are shown in Figure 8 and listed in Table 2.

3.1.1 Trends with selection radius

Substantial differences in the shape of the luminosity function are seen at smaller selection radii, consistent with local clusters (Lobo et al. 1997, Popesso et al. 2006), and presumably due to a morphology-density relation in the dwarf-giant ratio. This effect is illustrated in our study in Figure 8. Our clusters show a general trend of a flatter () luminosity function with increasing cluster radius, albeit with significant scattered especially for Cl1322 and Cl1604, where our errors are largest. In the literature, Lobo et al. (1997) find a steeper faint-end slope in the central regions of Coma as compared to groups around the outskirts. Popesso et al. (2006) found that the brightness of the faint-end luminosity function increased with increasing cluster radius in the context of double Schechter-function fits. For a single Schechter-function fit, this would result in a flatter () fit to the luminosity function for larger selection radius, which is seen in our cluster sample. RSLF shapes are far more uniform with a selection, but field contamination becomes much greater especially for lower-density clusters. For direct comparison with other intermediate redshift work this result indicates it is critical to make comparisons within the same selection radius, preferably relative to for each cluster. This is a point we return to later. In general, however, we are obliged to use the Mpc aperture. For our clusters, this is between 0.40-0.66 .

3.1.2 Comparison to literature

A comparison can be made between our luminosity-function measurements and four previous measurements for three clusters.

Cl0016. Tanaka et al. (2005) find (assuming ) and . for Cl0016 in a selection radius that is close to . Relative to our measurement, this is mag fainter in (a modestly significant difference), but only shallower in . In actuality, however, Tanaka et al. use a local density criteria for their selection. Given the well-known elongation of the central, high-density region of Cl0016 (e.g., their Figure 6), this results in an effective selection radius which is substantially (%) smaller. Indeed, comparing to our selection yields very similar values (see Figure 8), with their only mag fainter, and their only steeper. These are well within the 1 measurement errors, and small in an absolute sense as well.

MS1054. Our measurements for MS1054 agree closely with those of Andreon (2006) for his 1 Mpc selection radius: and , which is different in and different in .

However, Goto et al. (2005) find a value of 444The value from Goto et al. (2005) is converted from AB magnitudes and for objects with within a selection radius of Mpc from a spectroscopic sample. Our value for within (1.82 Mpc) is negligibly different than theirs, but we find . As Andreon (2006) points out, the spectroscopic completeness in Goto et al. at the faint end is only , which may lead to a severe turnover. Furthermore, they use a single color cut to identify red galaxies instead of a luminosity-dependent color-cut, as done in this work. Consequently, bright red sequence galaxies are likely to be included whereas faint red galaxies are likely to be preferentially excluded due to signal-to-noise considerations and the slope of the red sequence. If not properly accounted, this effect could be manifest as an apparent ’deficit’ of faint red galaxies compared to bright ones. Their color also does not span the Balmer Break at .

As a further check of the Goto et al. measurements, we note the luminosity function they derive for early-type galaxies is inconsistent with their measurement for red galaxies. Early-type galaxies are selected in their study via a visual (qualitative) morphological classification based on HST images. On the other hand, their value of for early-types galaxies is very comparable to our measurement. The early-type galaxies at this redshift are still overwhelmingly red (and dominate the counts of red galaxies in MS1054; van Dokkum et al. 1999). No population of bright, red, late-type galaxies exist in MS1054 that would bias the luminosity function in such a manner as might be inferred by the Goto et al. measurements. Based on the luminosity-function agreement between studies, we conclude their morphological selection is more reliable, in this case, than their color selection.

Cl1604. Andreon (2008) measured the luminosity function within the central 0.45 Mpc of the cluster from two-band HST imaging. He derived a value of for the slope of the faint end. Our value of within an aperture of R=0.37 Mpc agrees within the limits.

3.2 Reliability of the Intermediate-Redshift RSLF Determination

To further verify the accuracy of our results, we performed a number of tests of our data, reported here, including four tests of the measurement reliability given the known amplitude of random errors, two tests investigating the systematic impact of the selection function for red galaxies, and a final test of the surface brightness bias present in our sample. Each of these tests validate a different aspect of our cluster measurement and reveal the quality of our data and analysis, and the robustness of our results on the RSLF measurement.

3.2.1 Random-Error Effects

The first test we conducted was to measure the completeness and reliability of detecting red galaxies within the cluster volumes selected. We calculated the colors of a red galaxy (from synthetic spectra) at the redshift of each cluster, and at a redshift 0.1 and 0.15 about the cluster redshift, i.e., at 2 and 3 times the distance in redshift as our nominal selection cut ( 0.05 about the cluster redshift). We then simulated the appropriate multi-band magnitudes and errors for the object for to 26, in steps of 0.5 mags. For every magnitude interval and redshift bin, the simulated galaxy’s photometry was realized 100-500 times as a perturbation about the nominal colors, drawing statistically on the error distribution for each band to determine the perturbation. For each realization the photometric redshift was calculated. To determine completeness we simply counted the fraction of the galaxies simulated at the cluster redshift that retained photometric redshifts within 0.05 of the cluster redshift. To determine the reliability, we calculated the percentage of galaxies simulated at 0.1 and 0.15 away from the cluster redshift had photometric redshifts estimated to lie within 0.05 of the cluster redshift. Such galaxies would be selected in our scheme as red-sequence cluster galaxies. This percentage was then normalized by the ratio of the number of galaxies in the cluster vs. the number of galaxies in the volume from , as estimated in the real data. The contribution from the more distant shell was vanishingly small compared to the nearer shell, indicating our simulation should be quite accurate.

The results of this simulation are shown in Figure 9. From this it is clear that contamination is always 10%, and typically only a few percent even at the 50% source-detection limit. The selection completeness is % at the 50% detection limit, and % at the 90% detection limit. This is roughly what one would expect given the S/N and associated photometric-redshift errors at these limits. If anything, the results are somewhat optimistic, which can be understood in terms of the idealized nature of the simulation, e.g., the simulated galaxies are drawn from the same set that are used to detect and derive photometric redshifts. Overall, however, the simulation shows we are able to recover close to the expected number of cluster sources in a controlled situation resembling the actual data under analysis.

The remaining three tests directly probe the derived luminosity function itself.

In the second test, we exclude galaxies from our RSLF calculation where the detection-completeness was less than (instead of the ). While we expect using corrected data down to the completeness-limit is reliable (due to the extensive completeness simulations performed and the corrections derived therefrom), the robustness of our results is most directly shown by examining the truncated data set. We follow the same steps to calculate the luminosity function parameters as described in §3 but with fewer points. As can be seen by comparing the open to filled diamonds in Figure 7 (right), none of the clusters show a significant change in the parameterization of the luminosity function. At completeness, all of our data extends beyond .

The third test was to create 100 realizations of each cluster RSLF from the measured errors in the corrected counts of galaxies identified as red-sequence cluster members. For each cluster, the observed luminosity function was convolved with the errors at each magnitude. Then, the parameterization of the luminosity function was measured for each realization, and the averages for the Monte-Carlo simulation were computed. The values found for and are plotted as gray-scale in the left-hand plots in Figure 7. For all the clusters, the averages are well within the error measurements for the parameterization of the cluster luminosity function and the variance is of the same order as well.

The fourth and final test was a more extensive test of the entire process of measuring the cluster luminosity function. Following the procedure of Toft et al. (2004), we produced ten realizations of our photometric catalogs for each cluster. In these realizations, the measured aperture photometry was smeared, in a statistical fashion, drawing from the photometric error distribution for each flux measurement. This means that the error distribution for each realization is roughly larger than the initial (measured) catalog, although we did not update the effective error distribution for these realizations. With the smeared photometry, we recalculated the photometric redshifts. These new redshifts were then applied to the calculation of rest-frame properties and the selection of cluster galaxies. The luminosity functions were built using the same corrections and procedures as previously described. Finally the parameterization of the luminosity function was measured for all of the clusters. The results of each of the realizations are presented in Figure 7. For all but one of the clusters, the realizations produce results which show no significant difference in the mean from the measurement of the luminosity function. The dispersion in the luminosity function parameters, however, is larger, as expected from the additional noise introduced in the simulation process. For one of the smallest cluster in our sample (Cl1322), the results indicate a flatter () luminosity function then measured in the single data alone. However, the original measurement is contained within the spread of slopes that are found, and is not statistically significant.

3.2.2 Systematics with Color-Selection

To test what impact the specific selection of red cluster galaxies has on the derived RSLF, we selected galaxies from the ten realizations of each cluster catalog by applying perturbations to the color, magnitude, redshift, and radial selection functions.

Changes to the magnitude zero-point of the color-magnitude relation made no changes to the selection of red galaxies due to the steep nature of the relationship. Large changes ( mag) to the color zero-point did result in a small shift in the faint end slope of the luminosity function, with a tendency to find steeper downturns (more positive by ) with a redward shift in the cutoff, and flatter slopes (more negative by ) with a blueward shift. This is qualitatively consistent with the canonical picture that the red-galaxy population and blue-galaxy population are characterized, respectively, by shallow and steep faint-end slopes to their luminosity functions. A similar trend was observed with changing the slope of the color-magnitude relationship. However, neither change to the color selection-function would result in a measurement outside of the confidence limits for the value found for the original color selection. This result is in qualitative agreement with previous analysis (Andreon et al. 2006, de Lucia et al. 2007, Barkhouse et al. 2007).

3.2.3 Systematics with Redshift Window and Radius

Variation in the redshift window do result in a small systematic change in the luminosity function. The magnitude of the change, however, is much smaller than the errors on the measurement of an individual cluster: for a factor of two change in the redshift window. When the redshift window is increased, the luminosity function becomes steeper (closer to a flat faint-end with ), and becomes shallower (closer to ) when the window is decreased.

The systematic change in the luminosity function with the change in selection window may indicate an underestimation of the photometric-redshift error-correction or contamination from field sources from the larger volume being investigated. Without extensive spectroscopic redshifts at faint magnitudes, it is difficult to conclude the source of this bias. Alternatively, this may be similar to the general trend seen in the change of slope with selection radius, albeit washed out by errors in photometric redshift, i.e., a steepening of the slope as the core is more preferentially sampled. Regardless, the small magnitude of the change provides confidence in our measurement and the robustness of our corrections.

Small changes to the radial selection function do not result in significant changes to the luminosity function. Trends with larger variations in the radial selection, already noted, will be discussed again later.

3.2.4 Surface-Brigtness Selection Effects

Finally, we check that the depth of our observations – in terms of surface-brightness sensitivity – are sufficient to detect cluster members at these redshifts. As can be seen in Figure 10, the size-magnitude locus of the red sequence sources for most of the clusters is well within our surface-brightness detection limits. Only for the highest redshift cluster, Cl1604, are we truly in danger of missing some of the objects.

However, if we are missing a significant amount of objects at the faint end, this will have the effect of causing the slope to fall more steeply than it should ( too large), which would mimic the astrophysical effect claimed by others to be an evolutionary phenomenon. We do not see this effect in Cl1604 or MS1054, which both have slopes on par with our other clusters, and in fact Cl1604 has nominally the most negative in our sample.

3.3 The RSLF at

A number of studies have measured the luminosity function from a variety of different sources using a wide range of techniques. To provide a single, simple comparison to our body of work, we have measured the RSLF for a large sample of Abell clusters (Abell, Corwin, & Olowin 1989) based on SDSS imaging data and spectroscopy (Adelman-McCarthy et al. 2008) using a similar procedure as for our clusters. Clusters are selected from the Abell catalog that also appear in the SDSS DR6 imaging and spectroscopic catalogs. For all the clusters that do appear in the SDSS data, we measure the velocity dispersions of the clusters based on the spectroscopic data. From these measurements, we only include clusters with a minimum of 30 spectroscopically confirmed members. Clusters with extremely high velocity dispersions ( km s) or with significant difference in richness and are examined individually to confirm that they are not close superpositions of two smaller cluster. Superpositions are eliminated from the sample. Finally, cluster velocity dispersions are compared to values found in the literature from Struble & Rood (1991), Wu, Xue, & Fang (1999), Miller et al. (2005), or Popesso et al. (2005). Clusters with large disparities between their literature value and that measured by the SDSS data were removed as well. Our final sample of 59 clusters have redshifts between and km s with average values of and km s. Twenty-six of the clusters have km s. The selected clusters are listed in Table 3.

Using the SDSS photometric data, we have identified and analyzed the data following the same procedure as our cluster sample. The primary difference is that we only use the five SDSS bands that are available and no narrow band data. For magnitudes, we used corrected Petrosian magnitudes following the recipe in Graham et al. (2005), and for colors, we use the SDSS fiber magnitudes, which are analogous to our aperture magnitudes. We have employed the same photometric redshift technique with a training sample created from the SDSS spectroscopic sample, calculated absolute magnitudes and rest-frame colors, and selected red sequence galaxies using the same selection function except shifted to z=0.08 to account for luminosity evolution. Cluster galaxies were selected by having photometric redshifts within of the cluster redshift. The luminosity function was calculated in the same manner as for our clusters with corrections applied for photometric redshift errors and field subtraction. However, since the magnitude limit of the SDSS at low redshift probes much deeper down the cluster luminosity function, no correction for incompleteness need be applied for measuring the luminosity function to . The luminosity function for each cluster is reported in Table 3 within a selection radius of .

A large dispersion in the value of and is present for the clusters. Averaging all of the clusters together, we find and within , which is a significantly larger dispersion than if we had first summed the clusters together and then measured the luminosity function (see Appendix A for further issues with ensemble-averaged luminosity functions) or from the measurement error associated with an individual cluster (typically around ). For Mpc, we find values of and . These values are very comparable to similar studies once converted to our magnitude system. Figure 12 and Table 4 contain values and references to other measurements of and for low redshift clusters.

4 Evolution of the RSLF

The central question of this is work is determining changes in the cluster RSLF shape with redshift. To this end, the luminosity function parameters, and , are plotted in Figure 12 for individual clusters in our low- and intermediate-redshift samples, their mean values binned in redshift, and other clusters’ values published in the literature. The latter have been transformed into our magnitude system. Measurements of the luminosity function based on isolating the red-sequence via morphology, single-function fits to deep luminosity functions (; see §3.3 and 5.2), and in galaxy groups have been excluded. To provide the closest comparison to other studies, we plot the data for a cluster radius of 1 Mpc. Values for the parameterizations within instead of 1 Mpc generally are closer to with a brighter for both the low and intermediate redshift clusters in our sample, but give qualitatively the same trends with redshift.

Our clusters exhibit an increase in with redshift, as also seen for the field RSLF (Willmer et al. 2006). However, the cluster RSLF is 0.5-1 mag brighter in the field value at , but 0.5 mag fainter at –i.e., brightening with redshift appears steeper in clusters for In contrast, the value of cluster RSLF is relatively flat between , considering in concert local values from our low redshift sample or the literature. Overall, the brightening of rest-frame -band in clusters and the field agree well with the findings by de Propris et al. (1999) and Lin et al. (2006) in the near-infrared for the fading of simple stellar populations formed between .

We find no trend in the faint end slope, , with redshift within our sample. For the low redshift clusters, shows a large range of values. However, the average value for the low redshift clusters within Mpc is , which increases to if we only include clusters with km s. Using this number for the high mass clusters (which is probably most comparable to our cluster sample), only one cluster, MS0451, is significantly different at the level. Including this cluster, the intermediate redshift sample has an average of , which is not significantly different than the average value for the low-redshift cluster sample.

To further investigate the question of evolution, we plot as a function of cluster velocity dispersion for our full sample of low and intermediate redshift clusters in Figure 13. The data are calculated within a cluster radius of and . The mean of the two distributions is different but not at a significant level. From our sample of intermediate redshift clusters, we find no significant evidence for a deficit of galaxies occurring to as compared to low redshift clusters.

5 Discussion

5.1 Assessment of Results in the Literature

Recent literature contains claims and refutations of a deficit of faint, red galaxies in intermediate redshift clusters. A deficit would be significant because a change in the RSLF shape implies an ongoing or multi-epoch formation scenario, beyond passive evolution of a coeval population forming at high redshift. We suspect different conclusions regarding the deficit (or lack thereof) arise in part from differences in samples, analysis methods, or comparisons to local samples. We enumerate these points below. For comparison and reference purposes during this discussion, we list all of the relevant studies of the red sequence cluster luminosity function and their principal attributes in Table 4.

5.1.1 Analysis methods

A number of studies measure the luminosity function shape in terms of a dwarf-to-giant ratio (DGR). Typically, the DGR is defined as a ratio between the number of galaxies in two magnitude bins, and as such, it is a much simpler calculation than the luminosity function. However the DGR suffers from a number of problems rarely addressed in the literature. First and foremost, the DGR has had a range of definitions (cf. Ferguson & Sandage 1991, Secker & Harris 1996, Driver, Couch, & Phillipps 1998, Tanaka et al. 2005, de Lucia et al. 2006, Gilbank et al. 2007, Koyama et al. 2007, Stott et al. 2007 ), and consequently is not directly comparable between all studies. Often the DGR is based around observational (i.e., detection) limits such that the definitions of “dwarf” and “giants” do not necessarily reflect any natural division between galaxies and ignores traditional splits between such systems. A further weakness of the DGR is that its value is dependent on accurate measurements from each of two magnitude bins, whereas the luminosity function (as shown in Section 3.2) is relatively insensitive to variation in a single bin. As the faintest magnitude bin is likely to have the largest uncertainties due to incompleteness, photometric errors, redshift errors, or other sources, an accurate statistical calculation of the DGR has far greater errors than a similar value derived from the fit of the luminosity function.

One of the purported strengths of the DGR is that it does not suffer from the covariance between and that is a common complaint of Schechter-function fitting. However, without measuring the shape of the luminosity function, it is difficult to ascertain whether changes in the DGR are due to changes in the bright end (dominated by ) or changes in the faint end (dominated by ). While the DGR is strongly correlated to changes in , it is also weakly correlated with changes in . We illustrate this in Figure 11. Following the definition of de Lucia et al. (2006) for the DGR:

(2)

it can be shown the DGR will vary approximately as for constant and for constant . Variation in the DGR therefore may not only be due to evolution in . As will be discussed later (also see Appendix A for issues with ensemble-averaged luminosity functions) cluster-to-cluster variation and sample selection may affect the measurement of the DGR to bias the results and confuse variation in the bright-end normalization for evolution in the faint end slope. For these reasons we prefer a parametric estimate of the luminosity function via the Schechter fitting-function.

5.1.2 Survey quality

Significant variations exist between surveys in terms of the level of analysis and the quality of the data. For example, a significant difference between our study and most RSLF studies in the literature is the amount of simulations undertaken to understand the completeness and selection functions of the data. Few other studies have undertaken extensive analysis of their incompleteness, with several notable exceptions. For example, Barkhouse et al. (2007) undertook a number of simulations outlined in their appendix to understand the effects of projection, Eddington Bias, and color selection. Mercurio et al. (2006) performed extensive completeness simulations for analysis of a low redshift cluster, but then adopted conservative luminosity limits well above their detection- and selection-completeness limits. Andreon (2006) presented an in-depth description of the statistical method for calculating the luminosity function in the limits of a background population. However, of these examples, the first two focus on low-redshift clusters. Only Andreon’s (2006) study is at comparable intermediate-redshifts as our own, and, notably, finds similar results as our study.

Imaging depth also varies substantial from survey to survey. None of the intermediate redshift studies explore beyond , and many are substantially (1-2 mag) shallower, in contrast to many of the low-redshift studies, Shallow surveys are not able to probe the faint end of the luminosity function, and are really just probing the giant population.

Another salient difference between surveys is wavelength coverage, and the impact this coverage has on reliable selection of red-sequence galaxies. Our study, although smaller in the number of clusters than many of the other studies, has far greater wavelength coverage and information to constrain and identify the cluster populations. Many surveys use only two bands and background-subtraction to identify cluster sources. In contrast, it has been shown that multi-band data sufficient to construct precision photometric redshifts can be far more reliable in selecting cluster galaxies (Brunner & Lubin 2000, Rines & Geller 2008).

5.1.3 Band-pass effects

As has been shown by Goto et al. (2002), the slope of the RSLF is passband-dependent. We have avoided comparing our results to any of the papers that measure the K-band luminosity function, but the conversions between other optical bands may still suffer from issues beyond simple color transformations. Indeed, further complications between measuring the RSLF in different passbands was initially shown by Smail et al. (1998). When measuring the apparent I-band luminosity function for galaxies in clusters, a selection in apparent (U-B) resulted in a deficit of galaxies as compared to the same sample but selected in (B-I), which covered the 4000 break at that redshift. To maintain consistency, Andreon (2008) always used filters saddling the 4000 break regardless of cluster redshift. However, a number of studies (e.g., de Lucia et al. 2006, Gilbank et al. 2008) have not done this.

5.1.4 Radius effects mixed with sample selection

A further difficulty in comparing the luminosity functions from different studies is the selection radius adopted in analysis. The luminosity function does show variations with cluster radius (Popesso et al. 2006, Barkhouse et al. 2007, and §3.1.1 here); a selection of a small, fixed radius will bias results for clusters over a range of masses. This affect can be especially pronounced when comparing sources at different redshifts. The Stott et al. (2007) sample is an example of the possible problems that may arise when using a small selection radius of fixed metric size. The low redshift clusters in their sample have smaller mass (in their case, they use x-ray luminosity as a mass proxy) than the higher-redshift sample. However, they use a constant measurement radius of 0.6 Mpc for all of their clusters. For the intermediate redshift clusters, they are only measuring the core of the cluster where the decline in the faint-end of the luminosity function tends to be more pronounced, but for the low redshift clusters, they measure out to . Unsurprisingly, they measure a deficit for the higher-redshift clusters. Indeed, for the one low redshift cluster of similar x-ray luminosity as the intermediate redshift clusters (Abell 3555 in their sample with a ), they find a DGR similar to the intermediate redshift clusters. In contrast, Andreon (2008) also uses a nearly constant radius of 0.5 Mpc for the high redshift clusters in his sample, and he finds no trends with redshift, but fairly significant scatter between clusters.

5.1.5 Sample differences

Our selection of clusters samples the most massive structures in the Universe – clusters that are almost of factor of two more massive than the EDisCS sample (de Lucia et al. 2006), as shown in Figure 14. According to the simulations of Wechsler et al. (2002), the typical line of sight velocity dispersion of our clusters, today, should be around km s, placing their masses well above the Coma cluster. None of the local comparison surveys, or even the CNOC cluster sample (Yee et al. 1996) at lower redshift are a fair match in mass to our sample; the large sample of Popesso et al. (2004, 2005, 2006) with velocity dispersions estimated from their x-ray luminosity are also far less massive for the most part. The closest comparison is the Smail et al. (1998) sample with a few clusters overlapping our sample, but with a much lower average mass. The average mass of the Stott et al. (2007) intermediate mass sample is only slightly less than our sample, but their low redshift sample is far less massive. (We estimate the velocity dispersion for the Stott et al. sample either from their x-ray luminosities, or from the velocity-dispersion listed in Andreon (2008) for the overlapping subset.) In Andreon (2008), the very high redshift clusters are, on average, a factor of 30% less massive than the sample. The masses of clusters in the Barkhouse et al. (2007) and Gilbank et al. (2007) studies are estimated using the parameter, which is a measure of the cluster richness. As they mention, this estimator may have dependencies related to redshift. According to their conversions, most of their sample is far less massive than those studied here. Our low-redshift clusters do span a large mass range, but a third of the clusters do have km s, which is comparable to the predicted masses of our cluster sample. If luminosity function shape, or its evolution, depends on cluster mass, than cluster sample selection could be responsible for the different results found in the literature.

For illustration of possible sample effects, we plot in the bottom panel of Figure 12 the amount of evolution in (the faint-end slope) seen by Stott et al. (2007), assuming the value for of our low redshift clusters is correct. To make this comparison, we have converted their measure of the DGR evolution (a power-law dependence in ) to a function of (see Figure 11) using the relationship found between DGR and in §5.1.1 and assuming constant . As can be seen, this trend is inconsistent with our data, even when normalizing to our low-redshift data. Fitting to our own data, we find a much shallower trend which is consistent with no change. It is conceivable, we suggest, the difference in redshift-trends are due to differences in the two samples’ cluster masses, and systematic trends in cluster-mass with redshift in the Stott et al. sample.

In support of this argument, Koyama et al. (2007) find a trend between cluster mass and in the sense that more massive clusters have steeper turnovers. We see a similar trend in our intermediate redshift clusters (Figure 13). However, the strength of the trend appears sensitive to the radial selection, and our statistics are poor. In contrast, there is no apparent trend in our low redshift data. Andreon (2008) also sees no trend using a larger sample of clusters. On balance, while cluster-mass may play a role in explaining differences between survey results on the RSLF, it is more likely that cluster-mass differences within samples, particularly when correlated with redshift, plays a more substantial role in driving apparent evolutionary trends seen in some studies.

5.2 Challenges to Measuring Evolution

In the previous section our focus was on identifying survey differences that could explain, in part, the discrepant results found in the literature on RSLF evolution. Here we turn instead to the astrophysical nature of the RSLF, and how its complexity and variation presents fundamental challenges to its measurement.

In the low redshift Universe, an over-abundance of “true” dwarf, early-type galaxies (those with luminosities below ) are found in clusters compared to the field (Driver et al. 1994, de Propris et al. 2003) and lower density environments (Trentham et al. 2005), indicating some relationship between the environment and shape of the faint RSLF must exist. In fact, since the RSLF is a composite population (e.g., massive ellipticals, intermediate mass lenticulars, and low-mass spheroidals) a single Schechter function is likely an inadequate description of the RSLF; bumps and dips have long been noticed in cluster luminosity functions (Kashikawa et al. 1995, Jerjen & Tammann 1996). Since these different sub-populations of the red-sequence have different relative densities with environment, changes in the shape of the RSLF with environment is to be expected at all luminosities. This is likely what drives the observation that cluster RSLFs changes shape with selection radius, and perhaps also differences between clusters.

Due to the composite nature of the RSLF, a two-function fit (either two Schechter functions or a Gaussian for the luminous component plus Schechter) is commonly used for parameterizing low-redshift cluster luminosity-distributions (Biviano et al. 1995, Popesso et al. 2006). When only a single function is fit to the luminosity function, the faint end slope is strongly affected by the limiting magnitude. In intermediate-redshift cluster studies, single-function fits are made exclusively, yet as Table 4 shows, there is substantial variation in sampled depth.

Since the SDSS data has much greater depth then our intermediate-redshift sample, we can explore the amplitude of the effect of measuring the RSLF to different depths for a given selection radius. At the faint-end slope should be dominated by the dwarf population (Trentham & Tully 2002, Popesso et al. 2006). If we measure to instead of , we find a faint end slope of instead of -0.84, i.e., the luminosity function appears to be slightly rising instead of falling at these greater depths. (This value is possibly an underestimate as incompleteness will start to effect the number counts at in the SDSS data; no correction for incompleteness has been applied to these data.) In other words, as the sampled depth decreases relative to M (as we would expect and indeed has happened in higher redshift surveys), the apparent slope of the RSLF increases, i.e., an apparent deficit appears.

The reason for this effect is simply that the bright-end of the RSLF is typically characterized in shape as a ‘bump,’ i.e., having a maximum near M. This bright bump is dominated by giant galaxies. The apparent decline at lower luminosities is substantially modulated by the relative amplitude of the dwarf-galaxy luminosity function, but intermediate redshift surveys do not reach down to faint enough magnitudes to well characterize the luminosity function of this population itself. Our data for MS0451 is a good example. In this case there is some evidence for an up-turn in the RSLF at the faintest magnitudes and a dip occurring prior to this upturn about 2-2.5 mag fainter than M, indicative of a composite population. However, the faint-end slope of the luminosity function we measure down to M is dominated by the excess of bright galaxies.

Survey-depth alone is not a panacea. The above conclusions are modulated by variations seen between clusters RSLFs and possible correlations with mass. There are indications that the RSLF M is brighter in clusters than in the field (de Propris et al. 2003, but Hilton et al. 2005 notes a few exceptions), and brighter still in more massive clusters (Croton et al. 2005, Hansen et al. 2005). Similarly, for a sample of 10 clusters, Yagi et al. (2002) find the dip in the luminosity function around becomes much stronger in higher-mass systems.

Unfortunately, there is considerable scatter that washes out any clear correlation between the shape and normalization of the luminosity function and cluster mass for large samples (Biviano et al. 1995, Popesso et al. 2003, de Propris et al. 2003 and 2005, Barkhouse et al. 2007). Within our low redshift sample, does brighten with cluster mass within , but the correlation is not statistically significant. Variations in the complex luminosity-function shape are seen in the nearby clusters of A168 (Yang et al. 2004) and Shapely super cluster (Mercurio et al. 2005). In our survey, variations are most obvious in comparing two of our clusters. Even though both are at the same redshift and are of similar mass, MS0451 has a very pronounced bright end as compared to the relatively flat bright end of Cl0016.

Cluster to cluster variation is a significant effect and a detriment to measuring evolution from a small, diverse sample of clusters. The alternative, namely, to studying large samples of clusters to average over variations also presents a challenge, since each cluster must be observed to adequate depth to well characterize the luminosity-function, as discussed above.

Along these lines we close with one further caution pertaining to large samples. With significant variation in the luminosity function, the ensemble average luminosity distribution may not be the same as the average of the individual luminosity distributions. Simulations we have performed (see Appendix A) indicate systematic errors do result from ensemble averaging, and are particularly large for the DGR index. We conclude that in order to confirm the hypothesis of evolution of the red sequence, a far more significant sample of intermediate redshift clusters has to be observed to greater depth to constrain the true dwarf population. These observations must be analyzed in a manner consistent with analysis of low-redshift cluster samples, and should not be ensemble-averaged.

5.3 Evolution Scenarios

If we take the results of de Lucia et al. (2006) and Stott et al. (2007) at face value, in concert with ours, we conclude the red sequence is already in place at intermediate redshifts down to faint magnitudes () in the most massive clusters, whereas the build-up is still occurring in lower-mass clusters. Mass-dependent evolution would be in concordance with a general trend of down-sizing in hierarchical structure-formation scenarios, where the most massive structures form earliest. Here, this would have to occur on two scales: (i) The most massive clusters would be the ones to have their red sequences in place first. (ii) For any given cluster, the most massive galaxies would evolve to the red sequence first.

Another more subtle possibility may be occurring. While the hierarchical structure-formation scenario addresses directly the assembly and build-up of mass, there is the associated notion that the pace of star-formation is also set by this build up. Specifically we expect to see, and perhaps do see, a down-sizing in the co-moving star-formation rate such that it is dominated by lower-mass systems at later times. Star-formation, however, is stochastic on a galaxy scale. Further, small-mass bursts in large-mass systems can dramatically change the colors, providing a blue, if ephemeral, photometric icing. Consequently, while the mass build-up may indeed be hierarchical, the growth of the RSLF as a function of luminosity may be substantially modulated by on-going star-formation. If this is the case, the evolution of the shape of the RSLF (as distinguished from the normalization) may be a rather subtle phenomenon, as we observe within our own study.

By looking at the star-formation histories of galaxies currently on the red sequence, we can put some additional constraints on the RSLF evolution. Studies place the formation epoch of red-sequence galaxies, (particularly the massive ellipticals) at , with only passive evolution since that time (e.g., Holden et al. 2004). However, elliptical galaxy stellar populations are typically found to be far more uniform than S0 populations, yet S0’s are counted in the RSLF, and are a significant – if not dominant – component at bright absolute magnitudes in local samples. If a deficit of low-luminosity, passively evolving galaxies existed at intermediate redshifts, then we would expect to see some correlation between stellar ages and magnitude in local cluster red-galaxy populations. Perhaps S0’s are part of this picture.

Indeed, some S0 galaxies are relatively recent additions to the cluster environment. They are missing in intermediate clusters relative to local clusters (Dressler et al. 1997), and almost of those that are found at intermediate redshifts have spectra indicative of recent star formation (Tran et al. 2007). In some local clusters, S0’s show far greater spread in their stellar histories than elliptical populations (Kuntschner & Davies 1998), and the color-magnitude relationship is far tighter for ellipticals than S0 galaxies (Bower et al. 1992).

If an evolving S0 population is to be responsible for a deficit in the intermediate redshift RSLF at faint magnitudes, critical then is determining the luminosity range over which the S0’s dominate the RSLF. It’s well known that locally the S0 luminosity function is quite broad, and comparable to E’s (Binggeli, Sandage & Tamman 1988). Further evidence of their contribution to the RSLF can be found in the scatter in the color magnitude diagram. Down to , which is well below the magnitude limit of any of the intermediate redshift cluster surveys, the scatter in the CM relation for the red population in the nearby Perseus cluster is less than 0.07 mag (Conselice et al. 2002). The small amplitude of this scatter is typically interpreted to mean that there has been a very small range in star-forming histories in this red population. The lack of trend in the amplitude with luminosity is also indicative of galaxies having comparable range of formation histories. Greater scatter is seen at fainter magnitudes (Secker & Harris 1997) where the dE population begins to dominate, but these depths have not been probed at intermediate redshift. Similarly, Poggianti et al. (2001) find that of galaxies at all magnitudes have little to no evidence for any star formation since .

These lines of evidence indicate one of three possibilities: (1) the notion that S0’s are young is incorrect; (2) the scatter in the color-magnitude diagram is insensitive to the age-variations under consideration; or (3) the younger population is well-distributed in luminosity such that they contribute to scatter (increase it), but not in a way that drives a significant trend with magnitude. We suspect the latter scenario is most likely. An obvious next step is to repeat the RSLF experiment with adequate imaging or kinematic data to well-distinguish ellipticals from lenticulars in a large sample of clusters, probing to depths of at least M+3 and preferably to M+5.

6 Conclusion

In this paper, we have explored the red sequence luminosity function in five intermediate redshift galaxy clusters. The luminosity functions are measured from deep UBRIz plus narrow band imaging from the WIYN telescope. Red sequence galaxies are identified from their rest frame colors and photometric redshifts within selection radii of 1 Mpc, , , and . Extensive simulations are performed to assure the quality of the detection, photometric redshifts, and measurement of the luminosity function. The quality of the data is confirmed through the measurement of the field luminosity function from the off-cluster sample.

To provide a low redshift comparison sample, we also measured the red sequence luminosity function in a set of 59 high mass clusters with data from the SDSS. The same process for measuring the luminosity function for the higher redshift cluster was used here with similar definitions for cluster galaxies and the red sequence. For both sets of clusters, we find comparable luminosity functions to those fount in the literature for previously studied systems.

We have two primary conclusions concerning the RSLF evolution:

  • evolves in a similar manner as the field luminosity function and has faded by about 0.7 mags over the last 6.5 Gyrs. However, little evolution is seen between and for the massive clusters.

  • The faint end slope, , shows no indication of evolution between our low and intermediate redshift samples. In addition, we find no relationship between the cluster velocity dispersion, , and for the high mass clusters.

In an extensive comparison to measurements from the literature, we have two additional conclusions:

  • Selection effects can be critical to the determination of any signal of evolution. Clusters do show variations with the luminosity function in terms of mass and radius, which can lead to erroneous conclusions in terms of evolution, if not carefully accounted. Although small, this survey predominately measures the RSLF in massive system which seems to display different behaviors than low mass systems.

  • Significant cluster to cluster variations exists, even at a given mass. The dispersion in cluster luminosity-function parameters, measured for individual clusters, is typically an order of magnitude greater than the error estimate on those parameters from fitting to an ensemble-averaged luminosity function. Significantly more work needs to be done to better understand these cluster to cluster variations.

A clear picture of the evolution of the cluster RSLF remains elusive, yet such a picture is critical for understanding the processes that drive the growth and transformations of cluster-galaxies over cosmic time. It is conceivable that a sufficiently delineated map of the RSLF evolution with time, cluster-mass, and location within the local cluster environment can help confirm or refute predictions of models on time-scales relevant to the assembly of the most massive, virialized systems in the universe. While the observational challenge has yet to be met, the theoretical models also fall short on definitive predictions due to the complexity of the gas-physics (including star-formation) that channels the transformation of galaxies from the blue cloud onto the red sequence. For example, clean predictions of how and when the different sub-populations along the red sequence (e.g., E, S0, dE) are formed are yet to be had. As such, the red sequence, as interpreted simply as a mass sequence, will likely continue as a critical observational foil thrown up as a test of hierarchical models. Tracking the fate of the blue population of galaxies that cause the Butchler-Oemler effect is a key to understanding the physics behind the transformative processes in clusters; at least some of these systems are likely to be the progenitors of the red-sequence galaxies. The complexity of the astrophysics will likely stymie a definitive observational picture of the transformation process, however headway can be made by juxtaposing the blue and red populations in the context of environment. In future work we will investigate the cluster blue-galaxy population in this context.

We would like to thank Vy Tran and Chris Moran for providing access to their spectroscopic redshifts for MS1054 and MS0451, the anonymous referee for comments that dramatically expanded and improved this work, SAAO for support (SMC), and U. Toronto for hospitality while pursuing this work (MAB). Research was supported by STScI/AR-9917, NSF/AST-0307417, NSF/AST-0607516 and a Wisconsin Space Grant. We acknowledge use of the Sloan Digital Sky Survey (SDSS and SDSS-II; see //www.sdss.org/ for funding, management and participating institutions).

Appendix A Ensemble-average Luminosity Function

An important aspect of the RSLF brought forward by our work so far is the variation between clusters seen even at the same redshift and with similar masses. This phenomenon is fairly well documented for local cluster samples. A number of studies measure the luminosity function not for individual clusters but instead for the coadded luminosity distribution of an ensemble of clusters. Such studies generally either compute a straight average of the luminosity distribution, or a weighted average. The primary method for weighting is to normalize each cluster by the number of bright galaxies (e.g., Colless 1989).

Unfortunately there has been no investigation of whether systematics in the inferred luminosity-function parameters are introduced by the cluster-averaging described above, particularly relative to the measurement of individual clusters. Since there is a natural need to average over clusters at higher redshift where sources are fainter, it is important to understand if this averaging will lead to spurious trends with redshift.

To explore this possibility, we created a number of simulations using two mock catalogs of 10 and 100 clusters. We apply errors on the mock cluster-galaxy counts that are representative of those for our low and intermediate redshift data. For each of these two catalogs, we generate 100 realizations for each luminosity-function distribution we describe below. For each catalog realization, we measured the Schechter-function parameters (, M, and ) and DGR (following de Lucia et al. 2007) for each individual cluster, the average of all the clusters, and the weighted average of the clusters following the Colless (1989) prescription.

For an initial simulation test-set, we assume all clusters have the same luminosity function. Unsurprisingly, in this sample, all three methods return the same results for the luminosity function and DGR. For the remainder of the simulations, we assume a distribution of luminosity functions described by Gaussian distribution about and . Simulations are generated with distribution-widths in each of these parameters of 0.05, 0.10, and 0.25 (with appropriate units). The count-normalization of each luminosity function, is also assumed to have a Gaussian distribution with widths corresponding to variations of , , and about the mean (with the requirement that ). Simulations were carried out where each of the parameters (, , and ) was varied individually and also in combination with the other parameters. When multiple parameters are varied, the same distribution widths were used for and , with the corresponding low-, medium-, or high-percentage width for . For example, when = 0.10 for and , the distribution width is 40%. One further simulation (lowz) uses variations of , , and in , M, and , respectively, which closes matches the values found for the low redshift clusters.

The results for the simulations are presented in Table 5 for realizations including 10 clusters. The first column lists the parameter varied (, , or ); the second column lists the distribution width for that parameter. Subsequent columns list the average and standard deviation for , , and the DGR measured from the 100 realizations (a) for individual clusters (columns 3-8), (b) the composite luminosity function constructed by averaging the cluster-counts together (columns 9-14), and (c) the same for a composite luminosity function constructed through a weighted average of the luminosity functions following Colless (1989; columns 15-20 here). For each quantity, the standard deviation is the measured deviation about the mean measurement across the 100 simulations. Simulations including 100 clusters yielded identical results in the mean, but with smaller standard deviations about the mean for each value. This is due simply to a better sampling of the luminosity-function distribution. One exception is for the standard deviation of the DGR for individually-measured clusters, which are larger for the 100-cluster simulation (see below).

For measurements of individual clusters, we are able to recover the original Schechter function values of the parent distribution even for large spread in the parameterization. In contrast, the measured DGR does not behave in a similar manner as the parameterization. For a single cluster with the nominal luminosity function, the DGR would have a value of 4.71. However, as the variation in the luminosity function parameterization increases, the average value of the DGR tends to increase as well. The increase in the average value of the DGR is due to the Poissian-like distribution in the DGR values due to a Gaussian spread in . The median is a better statistic to measure the DGR from an ensemble of individual measurements than the average. For individual measurements of 100 clusters with , the median value of the DGR is 4.78, while the average is 5.00.

The results for the composite luminosity functions indicate that averaged luminosity functions tend to yield estimates of increasingly negative as the variation in increases, whereas the weighted luminosity yields the opposite effect. In both cases, we see a small increase in the measured mean value of , with the increase comparable to the measured dispersion. Neither case shows any change in parameterization with changes in . For both cases, the DGR is far better behaved (returning a DGR corresponding to the value measured for in each case), although the systematic deviation in the mean DGR value is no longer coupled with the underlying variation in luminosity function parameters. As expected, the composite luminosity function masks the intrinsic cluster to cluster variations.

For the lowz case, which represents the dispersion measured for our low redshift clusters, we find very different results from the individual, average, and weighted measurements. The individual cluster measurements do return the input luminosity function as expected, but with significant variation in and . The average DGR is large, but the median, once again, provides a much more accurate statistic. However, the average and composite luminosity functions both perform much worse. The average luminosity function measures a far steeper slope of rather than the input value of . The weighted luminosity function performs slightly better, but the measured also decreases to . Even worse, the measured increases to for both cases (nearly an 0.5 mag shift). Because of this shift in , the resultant DGR value is generally found to be lower than the nominal value.

These simulations indicate systematics effects are introduced by coadding data from an ensemble of clusters. These effects are minimized by avoiding the DGR formulation and measuring Schechter-function parameters. In the case where luminosity functions can be measured for individual clusters, this is clearly preferable. When this is not the case, we suspect a maximum likelihood approach that simultaneously fits the cluster ensemble assuming a distribution in luminosity function-parameters may be promising.

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Cluster RA DEC z aa Ref: Carlberg et al. 1998, Tran et al. 1999, Lubin et al. 2004, Gal & Lubin 2004. bb Ref: Donahue et al. 2003, Henry 2004, Lubin et al. 2002, Gioia et al. 2004
(J2000) (J2000) (km s) erg (Mpc)
MS0451 04:54:10.81 -03:00:56.85 0.54 1354 2.50 -17.12
Cl0016 00:18:33.52 +16:26:16.01 0.55 1230 2.25 -16.70
Cl1322 13:24:50.13 +30:11:18.49 0.75 1016 1.66 -17.59
MS1054 10:56:59.50 -03:37:28.39 0.83 1170 1.82 -18.45
Cl1604 16:04:18.27 +43:04:38.42 0.90 982 1.50 -18.95
Table 1: Intermediate-Redshift Cluster Properties
Cluster z R
(Mpc) ( Mpc)
MS0451 0.54 0.63 0.25 -0.00 -20.28 240.0 0.44
MS0451 0.54 1.00 0.40 -0.22 -20.54 130.9 0.69
MS0451 0.54 1.25 0.50 -0.14 -20.44 104.2 1.16
MS0451 0.54 2.50 1.00 -0.38 -20.72 33.6 1.22
Cl0016 0.55 0.56 0.25 -0.38 -19.84 250.0 0.76
Cl0016 0.55 1.00 0.44 -0.58 -20.34 110.1 0.98
Cl0016 0.55 1.12 0.50 -0.60 -20.36 99.4 1.14
Cl0016 0.55 2.25 1.00 -0.82 -21.36 25.5 1.23
Cl1322 0.75 0.42 0.25 -0.96 -22.80 40.0 0.35
Cl1322 0.75 0.83 0.50 -0.64 -21.78 27.4 0.56
Cl1322 0.75 1.00 0.60 -0.64 -21.16 12.3 0.37
Cl1322 0.75 1.66 1.00 -0.64 -21.54 14.5 1.18
MS1054 0.83 0.46 0.25 +0.42 -20.26 300.0 0.76
MS1054 0.83 0.91 0.50 -0.50 -21.20 110.6 1.52
MS1054 0.83 1.00 0.55 -0.76 -21.30 100.0 0.29
MS1054 0.83 1.82 1.00 -0.58 -21.16 35.8 2.29
Cl1604 0.90 0.37 0.25 -0.86 -23.50 30.0 0.01
Cl1604 0.90 0.75 0.50 -0.20 -21.62 46.1 0.16
Cl1604 0.90 1.00 0.67 -0.74 -22.28 16.4 0.54
Cl1604 0.90 1.50 1.00 -0.94 -22.50 6.5 0.40
Table 2: Intermediate-Redshift Cluster Red Sequence Luminosity Functions
Cluster RA DEC z
(Abell) (deg) (deg) (km s) ( Mpc)
0085 10.41 -9.34 0.0550 1266 51 -0.84 -20.52 194.5 1.46
0117 14.00 -10.03 0.0548 623 35 -0.66 -20.06 82.1 1.33
0160 18.21 15.51 0.0434 701 32 -0.88 -20.38 66.0 0.36
0168 18.79 0.25 0.0450 648 32 -0.70 -20.10 107.8 0.54
0279 29.09 1.06 0.0800 827 32 -1.00 -20.54 41.8 1.09
0659 126.02 19.40 0.0987 582 30 -0.76 -20.70 113.4 0.58
0671 127.12 30.42 0.0503 977 42 -1.38 -21.50 30.7 1.50
0724 134.58 38.57 0.0933 511 30 -1.14 -21.54 36.2 0.67
0795 141.01 14.17 0.1390 819 39 -0.82 -20.90 112.6 1.66
0933 151.92 0.52 0.0970 571 30 -0.16 -20.06 82.7 1.36
0957 153.49 -0.92 0.0444 766 41 -0.94 -20.34 111.8 1.13
1035 158.03 40.21 0.0725 1664 61 -0.94 -20.84 133.0 1.12
1066 159.85 5.17 0.0688 946 50 -0.94 -20.68 90.5 1.08
1185 167.70 28.68 0.0324 802 40 -1.12 -20.36 90.5 1.34
1190 167.94 40.84 0.0754 850 47 -0.22 -19.82 178.4 0.87
1203 168.49 40.29 0.0752 468 33 -0.24 -19.76 69.1 0.40
1205 168.34 2.51 0.0754 743 35 -0.78 -20.44 79.8 0.66
1213 169.12 29.26 0.0466 664 39 -0.06 -19.32 115.6 2.42
1291 173.02 56.02 0.0558 1272 33 -0.98 -20.72 93.1 0.90
1307 173.20 14.52 0.0810 927 45 -0.58 -20.20 144.8 0.74
1346 175.29 5.69 0.0984 732 36 -0.72 -20.46 84.9 0.71
1371 176.37 15.54 0.0689 611 33 -1.22 -21.44 16.5 1.80
1424 179.39 5.04 0.0768 573 30 -0.66 -20.32 52.6 0.73
1437 180.12 3.34 0.1344 1566 30 -1.04 -20.78 154.2 1.18
1516 184.74 5.24 0.0766 1068 30 -1.04 -21.36 59.8 0.75
1541 186.86 8.84 0.0903 835 45 -1.00 -21.18 57.6 1.41
1552 187.46 11.74 0.0856 921 47 -0.84 -20.42 124.7 0.87
1650 194.69 -1.75 0.0843 897 31 -0.74 -20.24 150.4 0.74
1663 195.69 -2.52 0.0834 791 41 -0.78 -20.68 76.5 0.61
1691 197.85 39.20 0.0722 923 34 -0.58 -20.16 127.0 0.59
1750 202.72 -1.84 0.0851 1062 41 -0.98 -20.74 128.5 1.01
1767 204.00 59.21 0.0710 888 40 -0.60 -20.24 99.5 1.49
1773 205.54 2.25 0.0777 998 41 -0.88 -20.76 84.3 1.04
1775 205.48 26.36 0.0742 1048 36 -1.04 -20.66 85.1 1.10
1795 207.25 26.59 0.0630 959 38 -0.50 -19.70 224.8 1.36
1809 208.33 5.15 0.0792 843 52 -1.00 -20.84 70.9 1.11
1890 214.39 8.19 0.0578 631 36 -1.00 -20.70 43.0 1.49
1904 215.53 48.56 0.0712 869 50 -0.96 -20.88 71.1 1.53
2026 227.14 -0.27 0.0906 751 32 0.06 -19.26 112.1 1.71
2030 227.82 -0.09 0.0914 534 36 -0.94 -21.54 31.1 0.21
2034 227.55 33.53 0.1137 1396 40 -1.08 -20.98 119.6 0.72
2040 228.19 7.43 0.0460 826 34 -1.20 -21.10 64.3 1.38
2048 228.82 4.38 0.0980 936 39 -0.96 -21.16 138.0 0.64
2051 229.19 -0.95 0.1195 948 32 -0.96 -20.54 298.3 1.91
2061 230.31 30.65 0.0773 772 64 -0.48 -20.02 154.5 1.24
2069 230.99 29.89 0.1137 1118 50 -0.70 -20.36 169.6 1.49
2094 234.15 -2.03 0.1448 735 30 -0.94 -20.78 80.1 0.62
2108 235.02 17.89 0.0907 795 32 -1.00 -21.20 74.6 0.75
2122 236.12 36.13 0.0656 815 33 -0.54 -20.18 107.1 0.89
2124 236.25 36.06 0.0664 1260 39 -0.78 -20.50 139.2 1.46
2142 239.57 27.22 0.0903 1217 90 -0.82 -20.56 547.7 1.81
2151 241.31 17.75 0.0344 661 56 -1.38 -21.32 36.2 2.61
2175 245.10 29.92 0.0960 994 34 -1.06 -21.08 145.0 0.62
2244 255.68 34.05 0.1000 1138 57 -1.16 -20.94 178.1 1.41
2245 255.69 33.53 0.0884 1276 55 -1.02 -21.14 216.3 1.28
2255 258.13 64.09 0.0803 1148 45 -0.50 -20.30 186.7 1.41
2399 329.39 -7.79 0.0584 592 43 -0.56 -20.46 102.3 1.13
2593 351.13 14.64 0.0409 800 36 -1.18 -20.20 49.4 1.63
2670 358.54 -10.41 0.0757 960 50 -0.72 -20.58 114.1 1.27
Table 3: Low Redshift Cluster RSLF within and
Reference zaaFor studies with a single cluster, gives the redshift of the cluster. For studies with multiple clusters, gives the range and median value of the redshifts. FOVbbThe Field of View (FOV) is in square arcminutes RccLimiting magnitude in the R band. Transformations are made to the R-band from other optical bands if necessary. Other magnitudes indicated when necessary. All magnitudes transformed to Vega. Filters Selection BackgroundddBackground refers to the method used for the seperation of field and cluster sources. The diferent techniques are: ’BGS’ is background subtraction, ’SZ’ is spectroscopic identification, ’PZ’ is photometric redshift identification, and ’Morph’ is using morphological information to identify cluster objects. CombineeeCombine refers to how the clusters were combined. The different methods are: ’Ind’ means the luminosity function was measured for each individual cluster, ’Ave’ means a single luminosity function fit to the average or sum of all of the clusters, and ’Wei’ means a single luminosity function to a weighted average of the clusters. MeasurementffType of measurement made to assess the density of objects. ’LF’ is luminosity function. ’DGR’ is for dwarf to galaxy ratio, but the definition differs between the different studies. Before each definition, the band (rest or observed) that the measurement is made in is labelled.
Andreon et al. 2006 0.0325 1 1176 1.3 Mpc 24.5 8 BVR B-V BGS Ind -LF
Tanaka et al. 2005 0-0.065(0.035) —- ggTanaka et al. use a density estimate to define clusters vs. field but the estimate corresponds to the viral radius, which is approximately equal to . Mercurio et al. also use a density estimate to differentiate between low and high-density regions. 23 3.5 ugriz hhSDSS magnitudes. SZ Ave -LF
Yagi et al. 2002 0-0.076(0.043) 10 4698 1 Mpc 20.0 6 R BGS Wei -LF
Mercurio et al. 2006 0.05 3+ 14400 —-  ggTanaka et al. use a density estimate to define clusters vs. field but the estimate corresponds to the viral radius, which is approximately equal to . Mercurio et al. also use a density estimate to differentiate between low and high-density regions. 22.0 7 BR B-R BGS Ave -LF
de Propris et al. 2003 0-0.11(0.064) 60 1.5 Mpc 22.5 5 b pca SZ Wei
Low-z clusters (This Study) 0-0.15 (0.077) 59 —- 23 3,5 ugriz hhSDSS magnitudes. PZ+BGS Ind -LF
Barkhouse et al. 2007 0-0.18(0.08) 57 529 23 6.0 BRI B-R BGS Ind/Ave -LF
Stott et al. 2007 0.08-0.15(0.12) 10 —- 0.6 Mpc 20.9 3.1 BR U-V BGS Ave -LF/DGR
Popesso et al. 2006 0-0.460(0.13) 59 —- 23 6.5 ugriz hhSDSS magnitudes. u-r BGS Ind/Wei g-LF
Goto et al. 2002 0.02-0.25(0.18) 204 0.75 Mpc 23 3.5 ugriz hhSDSS magnitudes. Morph PZ+BGS Wei
Smail et al. 1998 0.22-0.28(0.24) 10 90 0.7 Mpc 22.8 4 UBI B-I BGS Ave I-LF
Muzzin et al. 2006 0.35-0.47(0.296) 15 45-250 24 grK pca SZ Ave K-LF
Wilman et al. 2005 0.35-0.47(0.43) 26 30.5+ 0.5 Mpc 22.0 UBVRI EW SZ Ave DGR
Gilbank et al. 2007 0.40 57 —- 23.2 4 Rz R-z BGS Wei -LF
Barrientos and Lilly 2003 0.39-0.48(0.45) 8 213.2 1 Mpc 24.5 3.5 VIK V-I BGS All I-LF
Stott et al. 2007 0.47-0.68(0.54) 10 11.3 0.6 Mpc 25.3 3 Vi iiHST studies where V, i, and z are the F606W, F775W, and F850LP bands, respectively. For Stott et al. 2007, the HST bands for V and i are F555W and F814W. For Andreon 2008, there is only two HST bands that bracket the 4000 Åbreak. Filters used include F555W, F606W, F775W, F814W, and F850LP. U-V BGS Ave -LF/DGR
MS0451 (This study) 0.54 1 92.16 25.21 3.5 UBRIz+NB PZ+BGS Ind -LF
Cl0016 (This study) 0.55 1 92.16 25.62 4.6 UBRIz+NB PZ+BGS Ind -LF
de Lucia et al. 2006 0.40-0.80(0.59) 18 42.45 25.7 2.5 VRI V-I PZ+BGS Ind/Ave -DGR
Tanaka et al. 2005 0.55,0.83(0.69) 2 768 ggTanaka et al. use a density estimate to define clusters vs. field but the estimate corresponds to the viral radius, which is approximately equal to . Mercurio et al. also use a density estimate to differentiate between low and high-density regions. 26.0 3 BVRiz PZ+BGS Ind -LF
Gilbank et al. 2007 0.5-0.95 101 —- 23.2 1-3 Rz R-z BGS Wei -DGR
Cl1322 (This study) 0.75 1 92.16 25.56 3.9 UBRIz+NB PZ+BGS Ind -LF
Andreon 2008 0.50-1.27(0.76) 14 11.33  0.5 Mpc 25.5 3.5+ Vi iiHST studies where V, i, and z are the F606W, F775W, and F850LP bands, respectively. For Stott et al. 2007, the HST bands for V and i are F555W and F814W. For Andreon 2008, there is only two HST bands that bracket the 4000 Åbreak. Filters used include F555W, F606W, F775W, F814W, and F850LP. BGS Ind -LF
Koyama et al. 2007 0.81 1 768 26.0 2.0 VRiz R-z BGS Ind DGR
Goto et al. 2005 0.83 1 36.51 1 Mpc 24.5 2.5 Viz iiHST studies where V, i, and z are the F606W, F775W, and F850LP bands, respectively. For Stott et al. 2007, the HST bands for V and i are F555W and F814W. For Andreon 2008, there is only two HST bands that bracket the 4000 Åbreak. Filters used include F555W, F606W, F775W, F814W, and F850LP. i-z SZ Ind -LF
Andreon 2006 0.83 1 20.25  1 Mpc 26.5 3.5 ViK iiHST studies where V, i, and z are the F606W, F775W, and F850LP bands, respectively. For Stott et al. 2007, the HST bands for V and i are F555W and F814W. For Andreon 2008, there is only two HST bands that bracket the 4000 Åbreak. Filters used include F555W, F606W, F775W, F814W, and F850LP. V-I BGS Ind -LF
MS1054 (This study) 0.83 1 92.16 25.48 2.7 UBRIz+NB PZ+BGS Ind -LF
Cl1604 (This study) 0.90 1 92.16 25.42 3.5 UBRIz+NB PZ+BGS Ind -LF
Andreon et al. 2008 1.02 1 49.0 0.7 Mpc 24.5 2 Iz I-z BGS Ind -LF
Mei et al. 2006 jjMei et al. 2006, ApJ 639, 81 1.106 1 11.9 0.6 Mpc 25.5 izJK iiHST studies where V, i, and z are the F606W, F775W, and F850LP bands, respectively. For Stott et al. 2007, the HST bands for V and i are F555W and F814W. For Andreon 2008, there is only two HST bands that bracket the 4000 Åbreak. Filters used include F555W, F606W, F775W, F814W, and F850LP. i-z PZ Ind -LF
Stazzullo et al. 2006 1.11-1.27(1.17) 1 1.8-4.4 0.7-1.0 Mpc K TEM BSG Ind K-LF
Nakata et al. 2001 1.2 1 0.66 0.33 Mpc 26 VRIK PZ Ind none
Toft et al. 2004 1.237 1 16 1 Mpc 26 BVRIJK PZ Ind K-LF
Blakeslee et al. 2003 1.24 1 36.51 1 Mpc 26.5 1.3 iz iiHST studies where V, i, and z are the F606W, F775W, and F850LP bands, respectively. For Stott et al. 2007, the HST bands for V and i are F555W and F814W. For Andreon 2008, there is only two HST bands that bracket the 4000 Åbreak. Filters used include F555W, F606W, F775W, F814W, and F850LP. i-z Morph Ind none
Tanaka et al. 2007 1.24 1 25 0.5-1 Mpc 26.4 VrizK i-K none Ind K-LF
Mei et al. 2006 kkMei et al. 2006, ApJ 644, 759. Observations of a supercluster located at z=1.265. The two clusters are separated by 2 Mpc and have redshifts of z=1.26,1.27. 1.265 1 37 0.5 Mpc 26.5 iz iiHST studies where V, i, and z are the F606W, F775W, and F850LP bands, respectively. For Stott et al. 2007, the HST bands for V and i are F555W and F814W. For Andreon 2008, there is only two HST bands that bracket the 4000 Åbreak. Filters used include F555W, F606W, F775W, F814W, and F850LP. i-z SZ+morph Ind -LF
Table 4: Summary of Extant Cluster RSLF Studies
ParameteraaParameter refers to the luminosity function parameter which is being varied in each simulation. bb refers to the amount the parameter is being varied in the simulation. Individual Average Weighted
0.05 -0.90 0.02 -20.00 0.00 4.73 0.30 -0.91 0.02 -20.01 0.01 4.72 0.11 -0.90 0.02 -20.01 0.01 4.69 0.11
0.10 -0.90 0.03 -20.00 0.00 4.80 0.67 -0.92 0.03 -20.01 0.01 4.77 0.21 -0.89 0.03 -20.01 0.01 4.59 0.20
0.25 -0.89 0.07 -20.00 0.00 4.95 1.71 -0.97 0.08 -20.07 0.04 4.77 0.52 -0.82 0.07 -20.06 0.03 3.93 0.48
, 0.05 -0.90 0.02 -20.00 0.01 4.74 0.42 -0.91 0.02 -20.01 0.02 4.72 0.14 -0.90 0.02 -20.01 0.02 4.67 0.14
, 0.10 -0.90 0.03 -20.00 0.03 4.82 0.84 -0.92 0.03 -20.03 0.03 4.72 0.27 -0.90 0.03 -20.03 0.03 4.55 0.27
, 0.25 -0.89 0.08 -20.01 0.08 5.31 2.42 -1.02 0.09 -20.18 0.11 4.65 0.72 -0.86 0.09 -20.16 0.14 3.79 0.76
,, 0.05 -0.90 0.01 -20.00 0.02 4.73 0.40 -0.90 0.02 -20.01 0.02 4.70 0.15 -0.90 0.01 -20.00 0.02 4.66 0.14
,, 0.10 -0.90 0.03 -20.00 0.04 4.79 0.85 -0.92 0.04 -20.03 0.04 4.69 0.35 -0.89 0.03 -20.03 0.04 4.53 0.34
,, 0.25 -0.89 0.08 -20.01 0.08 5.30 2.40 -1.01 0.11 -20.18 0.13 4.67 0.81 -0.87 0.08 -20.18 0.14 3.77 0.70
, 0.05 -0.90 0.02 -20.00 0.00 4.73 0.33 -0.91 0.02 -20.01 0.01 4.72 0.10 -0.90 0.02 -20.01 0.01 4.68 0.11
, 0.10 -0.90 0.03 -20.00 0.00 4.73 0.66 -0.91 0.04 -20.01 0.01 4.71 0.25 -0.88 0.03 -20.01 0.01 4.54 0.22
, 0.25 -0.90 0.07 -20.00 0.00 5.07 1.73 -0.99 0.11 -20.08 0.05 4.94 0.71 -0.84 0.08 -20.06 0.03 3.98 0.54
0.05 -0.90 0.00 -20.00 0.02 4.74 0.26 -0.90 0.00 -20.00 0.02 4.72 0.09 -0.90 0.00 -20.00 0.02 4.72 0.09
0.10 -0.90 0.00 -20.00 0.03 4.78 0.54 -0.91 0.01 -20.01 0.03 4.71 0.18 -0.91 0.01 -20.01 0.03 4.71 0.18
0.25 -0.90 0.00 -19.98 0.08 5.15 1.55 -0.94 0.02 -20.09 0.09 4.63 0.42 -0.94 0.02 -20.08 0.09 4.64 0.42
, 0.05 -0.90 0.00 -20.00 0.02 4.72 0.25 -0.90 0.00 -20.00 0.02 4.70 0.08 -0.90 0.00 -20.00 0.02 4.70 0.08
, 0.10 -0.90 0.00 -20.00 0.03 4.78 0.51 -0.90 0.01 -20.00 0.03 4.72 0.17 -0.90 0.01 -20.01 0.03 4.71 0.16
, 0.25 -0.90 0.00 -19.99 0.08 5.11 1.47 -0.94 0.02 -20.08 0.10 4.65 0.52 -0.94 0.02 -20.09 0.09 4.61 0.43
20% -0.90 0.00 -20.00 0.00 4.71 0.00 -0.90 0.00 -20.00 0.00 4.71 0.00 -0.90 0.00 -20.00 0.00 4.71 0.00
40% -0.90 0.00 -20.00 0.00 4.71 0.00 -0.90 0.00 -20.00 0.00 4.71 0.00 -0.90 0.00 -20.00 0.00 4.71 0.00
100% -0.90 0.00 -20.00 0.00 4.71 0.00 -0.90 0.00 -20.00 0.00 4.71 0.00 -0.90 0.00 -20.00 0.00 4.71 0.00
lowzcc“lowz” is a simulation with variations matching our low-redshift cluster sample. The simulation has variations in , , and of , , and . -0.90 0.07 -20.03 0.14 7.10 6.60 -1.11 0.10 -20.48 0.24 4.16 0.77 -0.97 0.10 -20.48 0.27 3.37 0.79
Table 5: Simulations of Luminosity-Function Measurement Variations for a 10-Cluster Ensemble
Figure 1: R-band completeness profiles as function of object size for the six fields observed for this program. Curves (dotted, dashed, solid) are measured for real objects selected from the data with apparent half-light sizes of approximately 1.5’,1’, and 0.5’, respectively. For reference, the approximate location of M* for the evolving field RSLF (Willmer et al. 2006) is shown as a vertical line.
Figure 2: Photometric redshift vs. spectroscopic redshift for the full sample of galaxies with spectroscopic redshifts in our survey fields. Blue and red galaxies are defined in terms of rest-frame properties as specified in text.
Figure 3: Histogram of the difference between spectroscopic and photometric redshift as a function of z for the sample in Figure 1. The top plot is for all blue galaxies and the bottom plot is for red galaxies, as defined in text.
Figure 4: Photometric redshift error as a function of S/N for measurements in this study. S/N is defined within the seeing-matched apertures described in §2. For the sample in Figure 1, the squares represent red galaxies; triangles represent blue galaxies. Light gray objects are similar measurements made from the SDSS sample using our photometric redshift technique. Photometric redshift error begins to increase dramatically below S/N=10, but sources at or above the 50% detection limit have S/N 20.
Figure 5: Field RSLF at low (top panel; ) and high (bottom panel; ) redshift in our survey fields. Dashed and dotted vertical lines represent the estimated 50% and 90% detection-completeness limits for our data in each redshift bin. Light, open symbols represent raw (uncorrected) measurements in our images for redshift regimes free from over-densities (as defined in text). Dark, open symbols are corrected for detection completeness. Filled symbols are corrected for both detection completeness and cluster contamination due to photometric redshift efforts, as described in text. Unlike cluster luminosity-function derivations, here we extend the data beyond where contamination corrections are greater than for illustration purposes on the quality of the correction, which is based on simulations. For comparison, the field RSLF measured from the DEEP2 data set (Willmer et al. 2006) is plotted as a solid line. A reduced- statistic between our data and the DEEP2 Schecter-function (no degrees of freedom) is indicated in each panel.
Figure 6: (left) Rest-frame color-magnitude diagrams for each of the clusters. Only galaxies with photometric redshifts within of the cluster redshift and within are plotted. The redshift selections are illustrated in the right panels. Solid lines are the color-magnitude relationship from Willmer et al. 2006; dotted lines are the demarcation between red and blue galaxies (Equation 1 in text). (right) Apparent color vs redshift distribution for each cluster field. The solid vertical lines represent the redshift selection limits for each cluster, but all sources identified as galaxies in the field are plotted. The expected colors as a function of redshift for an non-evolving ellipitical galaxy (solid), Sb galaxy (dotted), and NGC4449 (dashed) are overplotted.
Figure 7: (left) RSLF within in each of our five clusters. Open squares represent raw counts; solid squares are the counts after all corrections (see text). The best-fit luminosity function are solid black lines, with confidence-limits as gray dotted lines. The best-fit and degrees of freedom is given in each frame. The 50% and 90% completeness limits are plotted as heavy dashed and dotted lines. (right) Error ellipses ( and ) are plotted for and based on the distribution. The best-fit value for data down to the 50% and 90% detection-completeness limit are plotted as solid and open circles, respectively. Best-fit values for ten realizations of the photometry with larger errors are plotted as plus symbols (see text). Gray-scale represents results from jack-knife estimates of the errors. We also plot the best-fit value from the low redshift cluster sample within .
Figure 8: (left) Red-Sequence cluster luminosity function for each intermediate-redshift cluster as measured by different selection radii. Points represent corrected counts within from Figure 7; the solid line is the best-fit Schechter-function luminosity function to those points. Other lines represent best-fits to data extracted with different selection radii: green dot-dashed (), red dashed (), blue dotted (1 Mpc). (right) Error ellipses ( C.L.) for the best-fit and parameters of the luminosity function for all selection radii are shown in the left-hand panels (same line-types and colors). Other measurements are shown for Cl0016 (Tanaka et al. 2005,square) and MS1054 (Goto et al. 2005, hexagon; Andreon 2006, triangle). Colors are only available in the electronic version.
Figure 9: Completeness and reliability of red cluster-galaxy selection due to effects from photometric redshift errors. Solid lines represent the fraction of simulated cluster galaxies that have photometric redshifts within the selection window for the cluster. Simulations are described in text. In the worse case, corrections for incompleteness due to photometric-redshift errors are 40% at the 50% detection-completeness limit, and typically are only 20-25%. At the 90% detection-completeness limit, the correction for selection incompleteness is typically below 15%. Dotted-lines represent the fraction of galaxies in the cluster luminosity function that are expected to be contamination from the field. Due to the much larger red galaxy luminosity function normalization, only a small fraction (5%) of field galaxies are expected to contribute even at the faintest magnitude bins.
Figure 10: Apparent size-magnitude locus for the red cluster objects is well offset from the (dashed line) and (solid line) completeness limit except for the highest redshift clusters (MS1054, and Cl1604). In these clusters, we may be missing a portion of the dimmest objects which would bias our luminosity functions towards a deficit. However, since our completeness corrections include corrections for object size and magnitude, this bias is diminished.
Figure 11: Dwarf-to-giant ratio (DGR) relationship with Schechter luminosity-function parameters (labels at top and x axis) and (labeled curves at right). The DGR is primarily a function of , but is also affected by the value of , and hence DGR measurement interpretation is degenerate between variations in both these quantities.
Figure 12: Evolution of RSLF (top) and slope (bottom) with redshift. Individual data from our survey is marked by filled, gray circles and diamonds, whereas the average values are marked by dark circles and diamonds (see key). Other studies are plotted as given in the key. All except two field studies (Madgwick et al. 2002, Willmer et al. 2006) are for cluster samples. Dashed curves in top panel show evolution in of a solar-metalicity, Salpter-IMF simple stellar population with formation epochs of (top to bottom), referenced to Madgwick et al. at . The dashed curve in the bottom panel represents the observed evolution in the DGR relationship found by Stott et al. (2007) normalized to our low-redshift measurement of (see text). rises much more smoothly with redshift in the field (Madgwick et al. 2002, Willmer et al. 2006) than in the cluster sample, whereas shows little to trend with redshift.
Figure 13: Faint end slope, , as a function of cluster velocity dispersion for low and intermediate redshift clusters. The low-redshift data is from measurements based on SDSS spectroscopic measurements of the Abell sample for magnitudes limits of . In the left panel all data are measured within a selection radius of 1 Mpc. The right panel shows the same, measured within a selection radius of R.
Figure 14: Cluster velocity dispersions (masses for constant size) as a function of redshift. Our smaller sample of clusters are far more massive than either the CNOC sample (Yee et al. 1996, Carlberg et al. 1997; open shields), the EDisCS samples (White et al. 2005, de Lucia et al. 2006; open diamonds), or even Coma (Colless & Dunn 1996; filled square). Other points are the same as in Figure 12. Predictions for the growth of halos from Wechsler et al. (2002) are over plotted as dashed line for halos with km s at .
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