© 1998 RAS

Mon. Not. R. Astron. Soc. 297, 128–142 (1998)

The K-band Hubble diagram for brightest cluster galaxies in X-rayclusters

C. A. Collins1 and R. G. Mann2,3

1Astrophysics Research Institute, School of Engineering, Liverpool John Moores University, Byrom Street, Liverpool L3 3AF2Astronomy Unit, Queen Mary and Westfield College, Mile End Road, London E1 4NS3Astrophysics Group, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2AZ

Accepted 1998 January 15. Received 1998 January 15; in original form 1997 March 13

A B STR ACTThis paper concerns the K-band Hubble diagram for the brightest cluster galaxies(BCGs) in a sample of X-ray clusters covering the redshift range 0.05szs0.8. Weshow that BCGs in clusters of high X-ray luminosity are excellent standard candles:the intrinsic dispersion in the raw K-band absolute magnitudes of BCGs in clusterswith LXa2.3Å1044 erg sÐ1 (in the 0.3–3.5 keV band) is no more than 0.22 mag, andis not significantly reduced by correcting for the BCG structure parameter, a, or forX-ray luminosity. This is the smallest scatter in the absolute magnitudes of any singleclass of galaxy, and demonstrates the homogeneity of BCGs in high-LX clusters. Bycontrast, we find that the brightest members of low-LX systems display a widerdispersion (10.5 mag) in absolute magnitude than commonly seen in previousstudies, which arises from the inclusion, in X-ray flux-limited samples, of poorclusters and groups which are usually omitted from low-redshift studies of BCGs inoptically rich clusters.

Spectral synthesis models reveal the insensitivity of K-band light to galaxyevolution, and this insensitivity, coupled with the tightness of its Hubble relation,and the lack of evidence of significant growth by merging (shown by the absence ofa correlation between BCG structure parameter, a, and redshift), makes our sampleof BCGs in high-LX clusters ideal for estimating the cosmological parameters WM

and WL , free from many of the problems that have bedevilled previous attemptsusing BCGs. The BCGs in our high-LX clusters yield a value of WM\0.28¹0.24 ifthe cosmological constant L\0. For a flat Universe we find WM\0.55+0.14

Ð0.15 with a 95per cent confidence upper limit to the cosmological constant corresponding toWLs0.73. These results are discussed in the context of other methods used toconstrain the density of the Universe, such as Type Ia supernovae.

Key words: galaxies: clusters: general – galaxies: elliptical and lenticular, cD –galaxies: evolution – cosmology: observations – infrared: galaxies.

1 INTRODUCTION

The Hubble (redshift–magnitude) diagram for brightestcluster galaxies (BCGs) is a classic cosmological tool(recently reviewed by Sandage 1995). Sandage and col-laborators (e.g. Sandage 1972a,b; Sandage & Hardy 1973;Sandage, Kristian & Westphal 1976) used the Hubble dia-gram for BCGs to verify Hubble’s linear redshift–distancelaw out to a distance 100 times greater than that probed bythe initial bright galaxy sample of Hubble (1929), and

attempted to measure the deceleration parameter from itsdeviation from the linear Hubble law. More recently, theBCG redshift–magnitude relation has appeared in studiesof the formation and evoluton of elliptical galaxies (Aragon-Salamanca et al. 1993) and of large-scale bulk flows in theUniverse (Lauer & Postman 1994; Postman & Lauer 1995;Hudson & Ebeling 1997).

Underlying all this work is the fact that BCGs appear tobe good standard candles after correction for systematiceffects and the dependence of absolute magnitude on galaxy

structure and environment, with their aperture luminositiesat low redshift showing a dispersion of only 10.3 mag (San-dage 1988). This small dispersion is all the more remarkablesince BCGs do not appear to be a particularly homogeneousclass of objects at first sight: some exhibit the large,extended envelopes that make them cD galaxies; many (upto half according to Hoessel & Schneider 1985) have muti-ple nuclei; and their immediate local environments varyfrom relative isolation in the cluster core to interaction andmerging, in the case of dumbbell galaxies. BCGs tend todominate their host clusters visually, but little is known oftheir place in the formation and evolution of the clusters, orof their own origins.

Answers to some of these questions may result fromextending the Hubble diagram for BCGs to higher redshifts,but there are several problems with that. First, as one movesto higher redshifts, optical wavebands become increasinglysensitive to the effects of star formation, and K-correctionsbecome large and uncertain. This problem may be cirum-vented, as discussed in detail below, by working in the Kband, which continues principally to sample the maturestellar population out to high redshifts: the K-band Hubblediagram for a small sample of BCGs was presented by Ara-gon-Salamanca et al. (1993).

A more serious problem is that of cluster selection. In thepast, cluster samples have been selected from optical sur-veys, on the basis of identifying enhancements in the surfacedensity of galaxies above a fluctuating background. Such aprocedure suffers from the well-known projection effectscaused by chance alignments of galaxies along the line ofsight (van Haarlem, Frenk & White 1997, and referencestherein). This contamination will be more severe at faintermagnitudes, thus strongly discouraging the use of thismethod for detecting high-redshift clusters. A solution tothis problem is provided by X-ray cluster selection. Mostbona fide clusters contain hot gas in their cores, emitting acopious flux of X-rays and making them visible to largedistances. The intensity of the thermal bremsstrahlung X-ray emission from a cluster is directly related to the depth ofits gravitational potential well, unlike its optical galaxy rich-ness, while the compactness of the X-ray-emitting region,compared to the extent of the galaxy distribution, alsomeans that projection effects are minimal in comparison tothose arising in optical cluster selection (Romer et al. 1994;van Haarlem et al. 1997).

These concerns motivated the current study, the long-term goal of which is to investigate the formation and evolu-tion of BCGs in host clusters with known X-rayluminosities, by studying their physical properties over awide span of cosmological time. In this paper we concen-trate on extending the infrared Hubble diagram for BCGsto higher redshifts than studied before, using K-band imagesof a large sample of BCGs in X-ray clusters with zx0.8. Theplan of the remainder of this paper is as follows. In Section2 we briefly describe our cluster sample, the observationsmade and their reduction, leading to the presentation of theuncorrected K–z diagram for our sample in Section 3,followed by a discussion of K-correction models for ourBCGs in Section 4. We consider the physical properties ofthe BCGs and their host cluster in Section 5, and look forcorrelations between them in Section 6. These correlationsare used to produce a corrected Hubble diagram in Section

7, which is used to derive constraints on the cosmologicalparameters WM and WL in Section 8. In Section 9 we discussthe results of this paper, and present the conclusions wedraw from them. Appendix A tabulates some statisticalresults omitted from the main body of the text.

2 THE DATA

2.1 The X-ray cluster sample

The X-ray clusters for this work were chosen principallyfrom the Einstein Medium Sensitivity Survey (EMSS)(Gioia et al. 1990; Stocke et al. 1991) catalogue of 104 X-ray-selected clusters (Gioia & Luppino 1994), but supple-mented by two clusters from the ROSAT All-Sky Survey(Voges 1992; Trumper 1993). Fig. 1 shows the X-ray lumi-nosities listed by Gioia & Luppino for all the EMSS clusters,as a function of redshift, compared with the sample studiedhere: the X-ray luminosities were calculated assuming thatWM\1, WL\0, and the Hubble constant H0\50 km sÐ1

MpcÐ1. No attempt was made to observe a statistically com-plete subsample of EMSS clusters, although it is clear fromFig. 1 that those selected are broadly representative of therange of X-ray luminosities for clusters within the parentsurvey: a two-sided Kolmogorov–Smirnov (K–S) test yieldsa probability of 0.26 that the cumulative X-ray luminositydistributions of the EMSS clusters imaged and not imagedwould differ by more than observed if both subsamples weredrawn from the same parent distribution. The correspond-ing K–S test on the redshift distributions of EMSS clustersubsamples imaged and not imaged yields a probability ofonly 0.04, reflecting the fact that we preferentially selectedEMSS clusters at higher redshifts, where the luminositydistance varies appreciably with cosmology and, thus, con-straints on WM and WL may be derived from the BCGHubble diagram.

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Figure 1. The luminosities of the EMSS clusters (in the 0.3–3.5keV passband) against redshift. The filled circles are the clustersimaged at K and discussed in this paper, while the empty circles arethe rest of the EMSS cluster sample of Gioia & Luppino (1994).The two triangles are the ROSAT clusters. Note that there is nosingle, sharp flux limit to the EMSS survey – cluster detectionswere based on pointed observations whose limiting sensitivityranges from 5Å10Ð14 to 3Å10Ð12 erg cmÐ2 sÐ1.

2.2 Observations

Observations were made on 1994 November 8–10 and 1995April 20–22 using the IRCAM3 infrared camera on the3.8-m United Kingdom Infrared Telescope (UKIRT):IRCAM3 contains a 256Å256 InSb array, with a pixel sizeof 0.286 arcsec. Two observing procedures were followed.For distant clusters, a series of 100 s (10Å10 s co-added)exposures were taken, with the galaxy offset in a five-pointjitter pattern within the initial field of view of the chip. Fornearby clusters, where the BCG covered too large an area ofthe chip to allow accurate sky level determination by thismethod, sky exposures were obtained after each galaxyexposure, allowing the jitter pattern for the sky frames(four- or eight-point) to be built-up simultaneously withthat of the jitter pattern for the galaxy frames (five- or nine-point). Typical total on-source integration times used were11000 s for zx0.2, 1500–3000 s for 0.2xzx0.4, and 4500 sabove z\0.5. Dark frames were taken regularly throughoutthe nights, as were standard star observations, using sourcesfrom the UKIRT Faint Standards list (Casali & Hawarden1992).

2.3 Data reduction

The data were reduced using standard IRAF1 routines and

the following procedure. Appropriate dark frames weresubtracted from each galaxy or sky image, and the resultingframes were multiplied by a mask, flagging known bad pixelsin the array to be ignored subsequently, before normalizingthe frames to unit median: we found the median to be amore stable measure of the sky level than the mode. Flat-field frames were constructed, by median filtering the galaxyframes (for distant clusters) or sky frames (for nearby clus-ters), and these flat-field frames were normalized to unitmedian. The galaxy frames were then divided by the appro-priate flat-field frames, and final mosaic images were pro-duced by cross-correlating the individual flat-fielded galaxyframes to determine their relative offsets. A similar pro-cedure was used to reduce the standard star frames.

A total of 52 BCGs were observed. Two of these (those inMS 1401.9+0437 and 1520.1+3002) did not yield photo-metric quality data and were dropped from our sample, aswere two others (MS 1209.0+3917 and 1333.3+1725)which have dubious redshift identifications in Gioia &Luppino (1994). A fifth BCG, in MS 0440.5+0204, wassubsequently omitted, because it is close to a bright starwhich falls within our chosen photometric aperture. Thisleft us with a sample of 47 BCGs: note that Stocke et al.(1991) and Gioia & Luppino (1994) suggest that the twoEMSS sources MS 2215.7Ð0404 and 2216.0Ð0401 may bepart of the extended X-ray emission from a single cluster,but we have treated them as two distinct objects. We assumecosmological matter and vacuum enegy densities of WM\1and WL\0, respectively, and h\0.5 (where H0\100 h kmsÐ1 MpcÐ1 is the Hubble constant) throughout this paper,unless stated otherwise.

3 THE R AW K–z DI AGR A M

We obtained K-band aperture magnitudes for our BCGsusing the IRAF package APPHOT, with a fixed metric apertureof diameter 50 kpc: in the few cases where there were twosimilarly bright galaxies in the cluster, the BCG was taken tobe the brighter of the two as judged by this magnitude.Airmass corrections were made using the standard medianMauna Kea K-band extinction value of 0.088 mag air-massÐ1.

The choice of aperture was made on the basis of twoconsiderations. First, it allows direct comparison with theresults of Aragon-Salamanca et al. (1993), who studiedBCGs in 19 optically selected clusters. Secondly, as dis-cussed in Section 5.1, we require a sufficiently large metricaperture to allow accurate estimation of the BCG structureparameter, a, for our more distant clusters, given the seeingconditions obtaining during our runs.

From comparing frames of the same object taken at dif-ferent times on the same night, on different nights of thesame run, and between the two runs, we obtain formalstatistical errors of 0.04 to 0.07 mag on our 50-kpc magni-tudes, except for 11 BCGs observed on one particular nightfor which greater uncertainty in our photometric calibrationproduced uncertainties of 0.1 mag. For the one object (MS0015.9+1609=Cl 0016+16) we have in common with Ara-gon-Salamanca et al. (1993), they measure an aperture mag-nitude that agrees with ours to well within these estimatederrors (15.56 mag compared to our 15.58 mag). In additionto these statistical errors, there are likely to be systematicerrors resulting, for example, from contamination fromother cluster galaxies falling within the photometric aper-ture: we make no attempt to estimate these (necessarily veryuncertain) corrections. The aperture magnitude data set isgiven below, and the resultant raw K–z diagram is shown inFig. 2. It is clear from Fig. 2 that there is a large scatter inthe raw K–z diagram for the BCGs in our X-ray clustersample, and that most of the scatter comes from the lower zhalf of the sample: before expressing that scatter in terms ofthe rms dispersion in the absolute magnitudes of thegalaxies, and investigating its origin, we must discuss how toK-correct the galaxy magnitudes, which we do in the nextsection.

4 INTERPR ETING THE B CG K-BA ND M AGNITUDES

There is a consensus developing that cluster ellipticals areold. For example, Charlot & Silk (1994) model the evolu-tion of spectral indicators of E/S0 galaxies in low-redshift(zs0.4) clusters, and suggest that 12 per cent at most oftheir stars have been made in the past 2.5 Gyr; Bender,Ziegler & Bruzual (1996) use Mgb–s data for 16 clusterellipticals in three clusters at z30.37 to show that the major-ity of their stars must have formed at za2, and that themost luminous galaxies may have formed at za4; and Elliset al. (1997) use rest frame UV–optical photometry of ellip-ticals in z10.5 clusters obtained with the Hubble SpaceTelescope to show that the bulk of the star formation in theellipticals in dense clusters was completed before z13 inconventional cosmologies. If we assume that the same goesfor our BCGs as for cluster ellipticals in general, then our

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1IRAF is distributed by National Optical Astronomy Observatories,

which is operated by the Association of Universities for Researchin Astronomy, Inc., under contract to the National Science Foun-dation.

photometric data are easy to interpret, since, by choosingthe K band, we are primarily sampling the passively evolvingmature stellar population over our full redshift range, andthe resultant K-correction will be insensitive to the exact ageof the galaxy: we take the K-correction, K (z), as encompas-ing the bandwidth, bandshifting and evolution terms (e.g.Sandage 1995), defining it so that the absolute magnitude,M, of a galaxy observed to have an apparent magnitude m atredshift z is given by M\m+5Ð5 log10(dL)ÐK (z), wheredL’(z, WM, WL , H0) is the luminosity distance (in pc), andwhere we have neglected the effect of Galactic extinction.

In the light of these results, we take as our default K-correction model that resulting from a Bruzual & CharlotGISSEL (Bruzual & Charlot 1993) model (1995 version; seeCharlot, Worthey & Bressan 1996) in which our BCGs formin a 1-Gyr burst (with a Salpeter 1995 IMF, over the range0.1RM/M>R125) at a redshift zf\3, in a universe withWM\1, WL\0, and h\0.5. This can be approximated tobetter than 0.02 mag over the range 0RzR1 by a fifth-orderpolynomial in z, given by

K (z)\Ð2.476zÐ0.5809z 2+14.04z 3Ð21.10z 4+9.147z 5.

(1)

This is computed by approximating the K-band filter by atop-hat between 2.0 and 2.45 lm: the computed K-cor-rection changes by x0.02 mag over 0RzR1 if instead we fitthe UKIRT K-band filter (S. K. Leggett, private communi-cation) with a ninth-order polynomial, which accurately fol-lows variations in its transmission from 1.9 to 2.5 lm. Themodel in equation (1) agrees to better than 0.05 mag withthe same empirical K-band K-correction derived byBershady (1995) from the large sample of zx0.3 fieldgalaxies constructed by Bershady et al. (1994), which is veryuniform, having an rms dispersion of 10.05 mag over thefull range of field galaxy classes.

To illustrate the insensitivity of K-band K-corrections tostar formation history, we show in Fig. 3 the differencebetween the default model in equation (1) and the K (z)

curves resulting from (a) varying zf , and (b) varying the IMFand form of the burst of star formation. It is clear that theage of the stellar population is the most important factor indetermining the K-correction, but the variation of the K-correction with age becomes noticeable only above z10.4,and is pushed to higher redshifts as zf increases: as long asthe bulk of a galaxy’s stars were formed before zf\1.0, thenits K-correction is known to 0.05 mag or better if its redshiftzR0.6 (we only have one BCG more distant than that), andis known to that accuracy out to z10.8 if zfa1.5. Fig. 3(b)

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Figure 2. The raw K–z diagram. The filled circles show theapparent K-band magnitudes of the 47 BCGs in our X-ray sample,while the empty circles are the BCGs from the optical sample ofAragon-Salamanca et al. (1993), excluding the one BCG in com-mon between the two samples. The magnitudes are corrected forGalactic extinction, but have not been K-corrected.

Figure 3. The difference between the default K-band K-correctionmodel of equation (1) and other models, generated using Bruzual& Charlot’s (1995) GISSEL library (see Charlot et al. 1996), allusing a Salpeter (1995) IMF and mass range 0.1RM/M>R125unless otherwise stated: the oscillating form of these DK (z) curvesis caused by the use of Chebyshev polynomials to produce the fitgiven in equation (1). In (a) we consider bursts of star formation of1-Gyr duration at zf\1 (dotted line), 1.25 (dashed line), 1.5 (dot–dashed line), 2 (dot–dot–dot–dashed line), 3 (solid lines) and 4(long dashed lines): the K (z) curve does not change significantly forzfa4. In (b) we show bursts at zf\3 with (i) 1-Gyr duration [solidline, as in (a)], (ii) exponential decaying star formation rate witht\1 Gyr (dotted line), (iii) instantaneous burst (dashed line), (iv)instantaneous burst with mass range 0.1RM/M>R2.5 (long-dashed line), (v) instantaneous burst with Scalo (1986) IMF (dot–dashed line), and (vi) instantaneous burst with Miller & Scalo(1979) IMF (dot–dot–dot–dashed line). We assume WM\1, WL\0and h\0.5 in all cases. The pronounced dips in K (z) at z\0.7–1.0for zf\1.0–1.5 are caused by AGB stars: their appearance is short-lived, and pushed to za1 if zfE1.5. We have only one BCG atza0.6, so their effect on our results will be negligible.

shows that K (z) is very insensitive to the IMF and durationof the burst of star formation.

If we convert the apparent magnitudes in Fig. 2 toabsolute magnitudes, MK , using the K-correction model inequation (1), then the K-corrected mean absolute magni-tude of the full 47 galaxy sample is MK\Ð26.40 mag, andthe rms dispersion about that mean is s\0.47 mag. If weconsider the higher redshift half of the sample (i.e., the 23galaxies with za0.227), then these figures becomeMK\Ð26.53 mag and s\0.30: this reduction in the rmsdispersion in MK is significant at the 98.6 per cent level, sinceonly 1399 out of 105 23-galaxy random subsamples of the fullBCG sample give s values less than 0.30. Rather thanassuming that the BCGs were formed at a single redshift, zf ,we could follow Glazebrook et al. (1995) and assume thatthe stellar populations of our BCGs are of a constant age,irrespective of the redshift at which they are observed: thatprocedure leads to very similar results as those of the K-correction in equation (1) as the age of the BCGs is variedover the range 11–10 Gyr (Glazebrook et al. 1995). Theassumptions of constant zf and constant age both representcrude treatments of the star formation history of our BCGs,but the essential point here is that the very weak depend-ence of K (z) to the exact formation redshift or exact ageillustrates how insensitive near-infrared K-corrections areto the details of the star formation history of galaxies; thisshows the power of K-band observations of early-typegalaxies with a narrow intrinsic spread in absolute magni-tude to act as standard candles.

It might be objected that the passive evolution modelsdiscussed in this section are not appropriate for BCGs, sincethey might have experienced more recent star formation,either due to cooling flow activity or due to merging withother cluster galaxies. Allen (1995) studied six coolingflow clusters, finding large UV/blue continua and strongemission lines, such as Ha and [O II], which can be inter-preted as emission from O stars. In order to test forongoing star formation, we have examined the difference inabsolute magnitudes between the EMSS BCGs in oursample which show some evidence for star-forming activityin their nucleus and the rest. From the EMSS cluster cata-logue (Gioia & Luppino 1994) the BCGs in MS 0302.7,0419.0, 0537.1, 0839.8, 0955.7, 1004.2, 1125.3, 1224.7 and1455.0 all have [O II], while MS 0015.9 has many galaxieswith ‘E+A’-type spectra. Despite this, all 10 lie very closeto the mean Hubble relation shown in Fig. 2. A two-sidedK–S test for BCGs in our sample with and without strongemission lines yields a probability of 0.17 that the differencein absolute magnitudes between the two populations wouldarise if the two subsamples are drawn from the same parentpopulation. This result suggests that the effect of anyongoing star formation in the centres of BCGs on the K-band magnitudes integrated over the 50-kpc diameter aper-ture is small. Such an interpretation of the spatial extent ofany star formation in these systems is consistent with anumber of studies which have concluded that optical colourgradients, indicative of active star formation, are eithercompletely absent in BCGs (Andreon et al. 1992) or areconfined to their inner regions (McNamara & O’Connell1992).

Some BCGs are radio galaxies, so another possible con-cern is that some of the K-band light from our BCGs comes

from AGN, rather than from stars (Eales et al. 1998, andreferences therein). Gioia & Luppino (1994) list radiodetections for only 10 of the BCGs in our sample, and onlysix of these are in the high-X-ray luminosity clusters inwhich we shall be particularly interested: a K–S test for theabsolute magnitude distributions of our BCGs in high-X-rayluminosity clusters yields a probability of 0.78 that BCGswith and without radio detections are drawn from the sameparent population.

To summarize, there is no evidence to suggest we shouldnot interpret the K-band photometry of our BCGs as arisingfrom the passive evolution of a mature stellar populationformed at zz3 similar to that advocated by Ellis et al.(1997) and others for luminous cluster ellipticals in general.We shall therefore adopt equation (1) as our preferred K-correction, but will return, in Section 8, to discuss the effectthat varying this assumed K(z) relation has on the con-straints on WM and WL we deduce from the BCG Hubblediagram.

5 PROPERTIES OF B CGs A ND THEIR HOST CLUSTERS

The scatter in the raw Hubble diagram shown in Fig. 2should come as no surprise, since it is well known that thereexist strong correlations between the optical luminosities ofBCGs and properties both of themselves, and of their hostclusters, and that it is only after correction for these thatBCGs are revealed to be good standard candles. In thissection we investigate some of the physical properties of theBCGs and their host clusters likely to be correlated withBCG luminosity.

5.1 The BCG structure parameter

Sandage (1972a,b) found that the luminosities of BCGscorrelate with the richnesses and Bautz–Morgan (Bautz &Morgan 1970) type of their host clusters, and Hoessel(1980) showed how this could be expressed as a correlationbetween Lm, the BCG luminosity within a metric apertureof radius rm, and a=[d log(Lm)/d log(r)]! r\rm

, the logarith-mic slope of the BCG growth curve. Correction for thecorrelation with the structure parameter, a, greatly reducesthe dispersion in BCG absolute magnitudes, and the Lm–arelation forms the basis of the recent use of BCGs asstreaming velocity probes (Lauer & Postman 1994; Postman& Lauer 1995; Hudson & Ebeling 1997).

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To measure the structure parameters of our BCGs, wecomputed the growth curve for each BCG from luminositymeasurements in a set of apertures with radii increasing insteps of 1 pixel out to 100 pixels or 30 arcsec. Each curve wasthen modelled using a Hermite polynomial, which was fittedover the full range of the growth curve. The value of a wasthen determined for each BCG by evaluating the logarith-mic derivative of the polynomial fit to its growth curve at anangle ym, which corresponds to a radius rm\25 kpc, and isgiven by

ym (arcsec)\0.43 (1+z)

[1Ð(1+z)Ð1/2], (2)

for our assumed cosmology, with WM\1, WL\0 and h\0.5.There are three principal sources of potential systematicerror in our a estimate.

(i) Incorrect background subtraction. In all cases, ourgalaxies are small enough that an adequate portion ofuncontaminated image was available to estimate the skylevel. We estimated what effect residual variations in the skybackground have on the estimated structure parameter byrecalculating the sky level using more than one uncontami-nated part of each image. These tests were carried out on arandom subset of 28 BCGs in our sample. The results indi-cated that uncertainties in sky subtraction give rise to typical1s errors of about 0.06, which is 110 per cent on average;this dominates the uncertainty in the estimation of thestructure parameter.

(ii) The presence of contaminating sources within themetric aperture. For seven sources a star or second galaxywas found to lie within the circle corresponding to a pro-jected distance of 25 kpc from the centre of the BCG. Inthes cases the contaminating sources were masked out, andthe values of the pixels in the masked region (which, in allcases, were close to 25 kpc away from the BCG) werereplaced with the value of the pixel located at the samedistance from the BCG but diametrically opposite the con-taminating source. In tests with non-contaminated images,this process introduced very little additional error.

(iii) Profile broadening due to seeing. In the inner regionof the BCG growth curves, the broadening effect of seeingcauses the a values to be systematically overestimated. Overthe course of our observations the seeing varied between 0.9and 1.4 arcsec, with a median seeing of 1.1 arcsec: a seeingdisc of this size is equivalent to a physical size of 5 kpc atz\0.2, and 8 kpc at z\0.5. To investigate the size of thiseffect, we smoothed our images by convolving them with aGaussian filter with a FWHM of 1.1 arcsec, and remeasuredthe structure parameters. The mean increase in the mea-sured a parameter computed at rm\8 and 25 hÐ1 kpc inredshift bins is shown in Table 1. It is clear from Table 1 thatthe estimated corrections to the a parameter measured in a50-kpc diameter aperture are small compared to otheruncertainties, and that there is little variation with redshift,so it is not necessary to correct our data for the effects ofseeing. The advantages of using a larger aperture can alsobe seen, since the systematic corrections are significantlysmaller for a 50-kpc aperture compared with those for a16-kpc aperture. The adopted a values for our BCGs aretabulated in Table 2.

5.2 Host cluster X-ray luminosities

Hudson & Ebeling (1997) have recently argued that theresiduals in the Lm–a relation correlate with the X-ray lumi-

nosity of the host cluster, so that the Lm–a relation does notfully remove the environmental dependence on BCG lumin-osity: the potential use of the cluster X-ray luminosity and/or temperature in reducing the scatted in the BCG Hubblediagram was also discussed by Edge (1991). In Table 2 welist the X-ray luminosities of the host clusters of our BCGsample. The EMSS X-ray luminosities for the 45 EMSSclusters are taken from Gioia & Luppino (1994), and arequoted for the energy passband of the Imaging ProportionalCounter (0.3–3.5 keV). We supplement that informationwith the luminosities for two ROSAT clusters (R84155 andR843053) which were discovered as part of an investigationof large-scale structure in the southern hemisphere (seeRomer et al. 1994). The X-ray luminosities of these addi-tional clusters have been corrected for extinction by usingH I column densities interpolated from Stark et al. (1992),then K-corrected and transformed from the ROSAT Posi-tion Sensitive Proportional Counter energy passband(0.1–2.4 keV) to the EMSS energy range assuming a 6-keVthermal bremsstrahlung spectrum: the passband correctionis of the form LX(0.3–3.5 keV)\1.08ÅLX(0.1–2.4 keV).

5.3 Near-neighbour analysis

To test how well the correlations with a and LX can expressthe environmental dependences of the BCG absolute mag-nitude, we wanted also to study residual dependences on therichness of the host cluster. Since our principal goal was tomeasure BCG magnitudes, we did not image more of thehost clusters than fell into the field of view of the chip whileimaging the BCG itself. In the most extreme case, thismeans we can only count near neighbours out to a distanceof 100 kpc from the BCG: this means that we are looking atthe central core of the cluster, rather than determining therichness of the cluster as a whole, for which one would wantto count neighbours to some appreciable fraction of anAbell (1958) radius.

We used the Starlink PISA package (Draper & Eaton1996) to detect near neighbours within a projected distanceof 100 kpc of the BCG centre, by searching for sets of atleast 10 contiguous pixels lying at least 2s above the skylevel, and used the IRAF APPHOT package to estimate theirmagnitudes: we removed ‘obvious’ bright stars and fore-ground galaxies by eye, although few BCGs had one of thesefalling within the counting circle. From the sky noise level ineach frame we were able to deduce its limiting isophote,which we converted into a limiting magnitude. We thencomputed, for each frame, the magnitude differencebetween the 50-kpc BCG magnitude and the limiting mag-nitude for that frame, finding that all frames went at least4.66 mag deeper than the BCG magnitude. Our raw near-neighbour count for each BCG, N, was then taken to be thenumber of objects lying within a projected distance of 100kpc of the BCG and with a magnitude less than 4.5 magfainter than the 50-kpc aperture magnitude of the BCG.

We then estimated, for each BCG, the number of fieldgalaxies expected to fall within the 100 kpc counting circleand the 4.5-mag strip. This was done using a fit to a compila-tion of K-band galaxy count data kindly provided by J. Gard-ner (private communication), which comprises data from anumber of authors: Mobasher, Ellis & Sharples (1986),Gardner et al. (1993), Glazebrook et al. (1994), Djorgovski

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Table 1. Seeing disc convolution results: the meanincrease in a for two values of rm, as a function of red-shift.

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Table 2. Data for the BCGs in the X-ray cluster sample.

The names, positions, redshifts and X-ray luminosities of the EMSS clusters (with prefix MS) are taken fromGioia & Luppino (1994): the RA and Dec. are given in B1950 coordinates and refer to the brightest clustermember in the V band; the X-ray luminosities, LX, are in units of 1044 erg sÐ1. mBCG is the BCG K-bandmagnitude within a 50-kpc diameter aperture, AK is the K-band extinction correction (computed using theNH I maps of Stark et al. 1992, RV\3.5 and the mean extinction law of Mathis 1990), a is the structureparameter at 50-kpc diameter, while N and N are, respectively, the raw and corrected BCG near-neighbournumbers. Clusters commonly known by another name are as follows: MS 0015.9+1609 (Cl 0016+16), MS0839.8+2938 (Zwicky 1883), MS 0904.5+1651 (Abell 744), MS 0906.5+1110 (Abell 750), MS1006.0+1202 (Zwicky 2933), MS 1127.7Ð1418 (Abell 1285), MS 1253.9+0456 (Zwicky 5587), MS1522.0+3003 (Abell 2069), MS 1531.2+3118 (Abell 2092), MS 1558.5+3321 (Abell 2145), MS1618.9+2552 (Abell 2177), MS 2255.7+2039 (Zwicky 8795), MS 2301.3+1506 (Zwicky 8822), MS2354.4Ð3502 (Abell 4059) and R84155 (Abell 2938).

et al. (1995), McLeod et al. (1995), Metcalfe et al. (1996),Huang et al. (1997) and Gardner et al. (1988). The data areshown in Fig. 4, together with the following fourth-orderpolynomial fit to them:

log10[,N (K).]\a0+a1K+a2K 2+a3K 3+a4K 4, (3)

over the range 11.25xKx23.5, where ,N (K). is theexpected galaxy count per mag per deg2 at magnitude K, andthe values of the coefficients in equation (3) are as follows:a0\26.509, a1\Ð8.3590, a2\0.86876, a3\Ð3.5824Å10Ð2 and a4\5.2630Å10Ð4. This fit has x 2\128 for 56degrees of freedom, and this is improved only marginally byincreasing the order of the fit to 10, say. In Table 2 wetabulate, for each BCG, the raw count N and the correctedcount, N, obtained from it by subtracting the model back-ground count. In six cases we find that the estimated con-tamination exceeds the raw count N: this is not unexpected,given the uncertainty in the background counts near thelimiting magnitudes of the frames (which are, typically,20xKx21) and our neglect of clustering in the backgroundpopulation, etc.

6 CORR EL ATING B CG A ND HOST CLUSTER PROPERTIES

The correlations between BCG and host cluster propertiesare graphically illustrated by Figs 5, 6 and 7, which show, onthe Hubble diagram of Fig. 2, the members of the quartilesof, respectively, a, LX and N. The similarity between thesethree figures shows how all these quantities are broad indi-cators of cluster richness and, in particular, how the fainterBCGs at zx0.25 tend to be low-a galaxies in poor systems,as defined both by the X-ray luminosity, LX, and by thenear-neighbour number, N. This suggests that there wouldbe redundancy in correcting the K–z relation for all threequantities, so, in this section, we perform a non-parametriccorrelation analysis (using the Spearman rank correlationcoefficient; e.g. Press et al. 1992) to study the relationship

between BCGs and their host clusters and, hence, to deter-mine the best way to correct the BCG Hubble diagram forthe effects of environment. We consider the five propertiestabulated in Table 2: (i) redshift, z ; (ii) absolute K-bandmagnitude, MK , computed using the K-correction model inequation (1); (iii) X-ray luminosity, LX; (iv) corrected near-neighbour number, N; (v) BCG structure parameter, a.

In Table 3 we present the results of this Spearman rankcorrelation analysis: in the upper half of the table we tabu-late the Spearman rank correlation coefficient, rAB, whilethe lower half gives PAB=Ðlog10[P (rAB)], where P (rAB) isthe probability that the absolute value of the correlationcoefficient would be as large as !rAB! for a sample of the samesize, under the hypothesis that the quantities A and B areuncorrelated. The P (rAB) values are computed under theassumption that the quantity rAB[(NÐ2)/(1Ðr 2

AB)]1/2, where

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Figure 4. K-band field galaxy count data. The solid line plots the fourth-order polynomial fit made to the count data, and used to predict thebackground count for each BCG. The symbols for the galaxy counts are as follows: diamonds, Mobasher et al. (1986); empty stars, Gardneret al. (1993); filled triangles, Glazebrook et al. (1994); crosses, Djorgovski et al. (1995); squares, McLeod et al. (1995); circles, Metcalfe etal. (1996); filled stars, Huang et al. (1997); and empty triangles, Gardner et al. (1998).

Figure 5. The K–z diagram from Fig. 2 showing the location ofmembers of quartiles in a: the first quartile is shown by the squares,the second by the diamonds, the third by the triangles, and thefourth by the circles.

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N is the number of galaxies in the sample, is distributed likeStudent’s t-statistic: from Monte Carlo simulations we judgethis approximation to be very good, typically estimating PAB

to an accuracy of 10.02. A very similar set of results toTable 3 is obtained if we use either the constant age K-correction of Glazebrook et al. (1995) or the empirical K-correction of Aragon-Salamanca et al. (1993), rather thanthe model fit in equation (1). Note that the use of a rankcorrelation coefficient underestimate the true level of corre-lation between MK and a to the extent that this relation isparabolic, as commonly assumed, rather than monotonic(see Section 7 below): none of the other relations tabulatedin Table 3 are so affected, justifying our use of a rankcorrelation analysis to quantify them.

A number of these correlations are worthy of comment.First, the K-band absolute magnitudes of our BCGs show noevidence of significant evolution with redshift, above thepassive evolution removed by the K-correction, but we seefrom Table 3 that MK is strongly correlated with LX, N anda. The strong correlation between z and LX simply reflectsthe flux-limited nature of EMSS cluster sample, but notethat the strong correlation between N and a (shown in Fig.8) is not due to near neighbours lying within rm of the BCG,since we explicitly excluded such contamination in calculat-ing a: instead, it illustrates how well a can express the rich-ness of the BCG’s local environment, as argued by Hoessel(1980).

These results could, of course, be due to correlations witha third quantity: for example, we know that N and a arestrongly correlated, so the correlation between N and MK ,say, could just result from that and a correlation between MK

and a. To test that we have computed, for our five physicalproperties, partial Spearman correlation coefficients (e.g.Yates, Miller & Peacock 1986) of the form rAB, C , where

rAB, C\rABÐrAC rBC

[(1Ðr 2AC)(1Ðr 2

BC)]1/2, (4)

where rAB, etc., are the Spearman rank correlation coeffi-cients, as before. The significance of the partial rank corre-

Figure 6. The K–z diagram from Fig. 2 showing the location ofmembers of quartiles in LX’: the first quartile is shown by thesquares, the second by the diamonds, the third by the triangles, andthe fourth by the circles.

Figure 7. The K–z diagram from Fig. 2 showing the location ofmembers of quartiles in N: the first quartile is shown by thesquares, the second by the diamonds, the third by the triangles, andthe fourth by the circles.

Figure 8. Correlation of N and a. This relation is not simply theresult of contamination within a radius rm of the BCG, since theseobjects were removed before calculating a.

Table 3. Results of Spearman rank correlationanalysis. The upper half of the table gives theSpearman rank correlation coefficient, rAB,while the lower half gives PAB=Ðlog-10[P (rAB)], where P (rAB) is the probability thatthe absolute value of the correlation coeffi-cient would be as large as !rAB! for a sample ofthe same size, under the hypothesis that thequantities A and B are uncorrelated.

lation coefficients may be calculated using a similarapproximation as above – that rAB, C[(NÐ3)/(1Ðr 2

AB, C)]1/2 isdistributed as Student’s t-statistic – to compute the prob-ability, P (rAB, C), of obtaining a partial rank correlationcoefficient with absolute value as large as !rAB, C! under thenull hypothesis that the correlation between A and B resultssolely from correlations between A and C and between Band C: Monte Carlo simulations show that this approxima-tion works as well for P (rAB, C) as for P (rAB).

We relegate the full set of partial rank correlation resultsto Appendix A, and note only a few important points here.The most striking result is that once the dependence on X-ray luminosity has been removed, a becomes independentof z to a high degree: P (rza, LX

)\0.60. We find that the N–arelation is not due to separate correlations between thosetwo variables and a third: the lowest P (rAB, C) for A\N andB\a arises when C\MK , but, even then, there is a prob-ability of less than 4Å10Ð4 that the N–a correlation isentirely due to the MK–N and MK–a correlations. This leadsto high values of P (rAB, C) when A\N and C\a, or viceversa, since these variables are almost interchangeable.There is some reduction in the significance of the MK–LX

correlation when it is evaluated at constant a, but it remainsquite strong, consistent with the suggestion by Hudson &Ebeling (1997) that correction for LX can remove some ofthe residual environmental dependence of BCG magni-tudes left after the Lm–a correction has been made. Finally,it appears that none of the significant correlations betweenpairs of properties arise from separate evolutionary trendsof the properties with redshift. To summarize, it is clear thatMK is strongly correlated with both a and LX, and that nofurther correction for N need be made after the a-cor-rection, due to the strength of the correlation between Nand a.

7 THE CORR ECTED K–z R EL ATION

The relationship between absolute magnitude and a for ourBCG sample is shown in Fig. 9. The relationship divergesfrom a purely linear slope at large values of a, and is in thisrespect qualitatively similar to those of Hoessel (1980), whoused a sample of 108 Abell clusters, and Postman & Lauer(1995), who determined a for a sample of 119 BCGs atzR0.05 in the R band. In a similar way to Postman & Lauer(1995), we fit the correlation between structure parameterand K-band luminosity with a parabola, which gives a least-squares best fit of

MK\Ð23.84Ð(8.34¹1.14)a+(6.14¹1.00)a 2, (5)

which is also shown in Fig. 9. The vertical scatter about thisfit is s\0.29 mag, and the gradient in the linear part of therelation is significantly steeper for our sample than for theR-band sample of Postman & Lauer (1995). It is clear that aparabolic fit is a reasonable description of the data over therange 0.1RaR1.0, and Fig. 9 provides the first real evidencethat the Lm–a relation at least flattens at high a, rather thancontinuing linearly with increased scatter, as appears thecase for the Postman & Lauer (1995) data.

The filled squares in Fig. 9 are the BCGs in high-LX

clusters. Note that these are all to be found near the peak ofthe parabola, indicating how high-LX selection picks outBCGs with a narrow range of absolute magnitudes, and

suggesting that these magnitudes will be largely unaffectedby the correcting for the MK–a relation of equation (5). Thisis graphically demonstrated by Figs 10 and 11, which show,respectively, the relationship between LX and the raw anda-corrected MK values. Below host cluster X-ray luminosi-ties of LX32.3Å1044 erg sÐ1, there is a wide dispersion inraw BCG absolute magnitudes, which is largely, but notcompletely, removed by the MK–a correction, while thehigh-LX absolute magnitudes are barely changed. In whatfollows we use this threshold of LX\2.3Å1044 erg sÐ1 tomark the break between a homogeneous population ofBCGs in high-LX and a low-LX population which displays awider scatter in its properties. Intriguingly, the LX thresholdwe empirically found on the basis of BCG properties isalmost identical to that of Annis (1997), who has recentlyargued that it marks the dividing point in the X-ray proper-ties of EMSS clusters: above 2Å1044 erg sÐ1 the X-ray

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Figure 9. The Lm–a relation for our data set. The filled and emptysquares denote the BCGs in clusters with X-ray luminositiesgreater than and less than 2.3Å1044 erg sÐ1 respectively. The lineshows the best quadratic fit to the data.

Figure 10. The relationship between X-ray luminosity and rawBCG absolute magnitude. The empty squares denote the BCGs inclusters with LXs2.3Å1044 erg sÐ1, while the filled squares are theBCGs in clusters above that threshold. Note the much wider spreadof absolute magnitudes seen for the BCGs in the less X-ray-lumin-ous clusters.

properties are very homogeneous, while below that, Annisdetects two populations of clusters, with quite differingproperties, one of which he argues contains clusters wherethe X-ray emission is solely from the intracluster medium(the only sort of cluster found above 2Å1044 erg sÐ1), andthe other where a significant contribution comes from acooling flow. We note that the lack of strong correlationbetween MK and a for high-LX clusters found here is incontrast to the results found by Hudson & Ebeling (1997),who argue that the optical Lm–a relation actually steepensfor high-LX Abell clusters. However, their local Abellsample does not probe luminosities as high as the EMSSclusters – only two clusters in their sample haveLXE2.3Å1044 erg sÐ1 in the EMSS passband.

After correction for the MK–a relation of equation (5) wefind that the rms dispersion in absolute magnitudes fallsfrom s\0.47 to 0.29 mag, while the dispersion for the high-LX subsample (the 26 galaxies in clusters withLXa2.3Å1044 erg sÐ1) about its own best-fit MK–a relationis 0.22 mag, identical to the rms dispersion in their rawmagnitudes: the best-fitting relation for the high-LX sampleis MK\Ð26.18Ð(0.87¹2.34)aÐ(0.18¹1.89)a 2, indicat-ing the weakness of the dependence of MK on a shown inFig. 9 for these BCGs.

The residuals about the best MK–a fit for the high-LX

sample do not correlate with z, LX or N, while, for the fullsample, the only significant correlation is with LX , similar tothat reported by Hudson & Ebeling (1997). We also triedcorrecting our raw absolute magnitudes using the sameform of Lm–a–LX relation recently advocated by Hudson &Ebeling (1997) which, translated to the K band, reads

MK\c0+c1a+c2a 2+c3 log10(LX)+c4a log10(LX)

+c5a 2 log10(LX), (6)

where LX is the cluster X-ray luminosity in units of 1014 ergsÐ1. For the full sample, the best-fitting values of these

parameters are c0\Ð23.26, c1\Ð10.66¹1.58,c2\8.45¹1.49, c3\Ð3.63¹1.13, c4\12.23¹4.30 andc5\Ð10.37¹3.97, which means that the terms are in simi-lar proportion to those found by Hudson & Ebeling (1997)for the Lauer & Postman (1994) optical BCG sample. Withthis model, the rms dispersion of the full sample isdecreased slightly to 0.26 mag, while that for the high-LX

sample actually increases slightly to 0.23 mag, as the loss ofdegrees of freedom arising from the use of three moreparameters in the fit outweighs the marginal decrease in thescatter in absolute magnitudes brought about by correctionfor the residual dependence on LX left after correction fora.

In principle, galaxy merging in the centre of clustersshould increase the stellar mass of the BCGs with time.Hausman & Ostriker (1978) show that the a parameter is asensitive measure of the merging history of BCGs: we cantherefore determine the effect that the merging process hashad on our sample by using the a parameter as a tracer ofthe growth of the BCGs through merging. As the Spearmanresults of Section 6 for the full sample show, once thedependences of a on LX and of LX on z are removed, a isindependent of z: there is a 60 per cent probability that thecorrelation between a and z arises from the combination oftheir individual correlations with LX. This lack of a correla-tion between a and z is slightly more strongly confirmed forthe high-LX sample of 26 BCGs, as the Spearman test for acorrelation between a and z for these galaxies gives a corre-lation coefficient with probability of 0.63 of arising fromuncorrelated variables. We conclude, therefore, that theBCGs in the high-LX clusters sample form a very homo-geneous population, with an intrinsic dispersion in absolutemagnitude of no more than 0.22 mag, and that there hasbeen no significant evoluton due to galaxy merging since aredshift of z10.8. It follows that our high-LX BCG sampleconstitutes a set of standard candles ideal for use in classicalcosmological tests, such as the estimation of WM and WL .

8 ESTIM ATING XM A ND XK FROM THE B CG HUBBLE DI AGR A M

The Hubble diagram for a set of standard candles deter-mines the relationship between luminosity distance, dL, andredshift, which may be written (Carroll, Press & Turner1992; Perlmutter et al. 1997) in the general form

dL(z, WM, WL, H0)\c (1+z)

H0Z!k!(7)

ÅI 8Z!k! hz

0

dx [(1+x)2(1+WMx)Ðx (2+x)WL]Ð1/29,where (i) if WM+WLa1, then I (x)\sin(x) andk\1ÐWMÐWL; (ii) if WM+WLs1, then I (x)\sinh(x)and k\1ÐWMÐWL; and (iii) if WM+WL\1, then I (x)\xand k\1. As emphasized by Perlmutter et al. (1997), it isimportant to note that the form of equation (7) means thatdL depends on WM and WL independently, and not solelythrough the deceleration parameter, q0=WM/2ÐWL – i.e.,the confidence region in the (WM, WL) plane determinedfrom the Hubble diagram of a set of standard candles is not

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Figure 11. The relationship between X-ray luminosity and a-cor-rected BCG absolute magnitude. The empty squares denote theBCGs in clusters with LXs2.3Å1044 erg sÐ1, while the filledsquares are the BCGs in clusters above that threshold. Comparisonof this plot and Fig. 10 shows how the a-correction removes most ofthe scatter in the absolute magnitudes of the low-LX BCGs, whileleaving the high-LX ones virtually unchanged.

parallel to contours of constant q0, except at zss1. Weprefer, therefore, to solve for WM and WL using our Hubblediagram for BCGs in high-LX clusters, rather than for q0.

We do that by using the following procedure for eachcandidate cosmology.

(i) Select values for WM and WL; h\0.5 is assumedthroughout.

(ii) Using equation (7), compute the angle, ym, corre-sponding to a physical distance rm\25 kpc at the redshift ofeach of the 47 BCGs: ym\rm/dA, where dA\dL/(1+z)2 is theangular diameter distance.

(iii) Calculate the change in the apparent magnitudes ofthe BCGs, as DmBCG\a (dA, refÐdA)/dA, ref , where the avalues are those in Table 2, which assume WM\1 andWL\0, and dA, ref is the angular diameter distance in thatreference model. This first-order correction to the apparentmagnitude should be accurate so long as rm does not varymuch from the reference aperture with WM\1 and WL\0,and that condition holds true for all physically reasonablecosmologies.

(iv) Compute the ages of the galaxies in the cosmology,assuming their formation at a redshift zf\3, and computetheir K-corrections using the Bruzual & Charlot models, asdescribed in Section 4. Use the K-corrections for, and lumi-nosity distances of, the galaxies in the cosmology to derivethe BCG absolute magnitudes.

(v) Perform a quadratic fit to the new MK–a relation anduse it to correct the BCG magnitudes to a fiducial a value,which is taken to be the median a.

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(vi) Take the mean of the a-corrected MK as the estimatefor M0, the BCG K-band absolute magnitude, and deduceits standard error from the scatter of the MK values aboutM0.

(vii) For the redshifts of the 26 BCGs in high-LX

clusters, calculate the predicted apparent magnitude, givenM0 and the K (z) relation in the cosmology. Using the errorson the measured magnitudes of the 26 BCGs in high-LX

clusters, and the standard error on M0, compute x 2 for thecomparison of the 26 BCG apparent magnitudes and theapparent magnitudes predicted at their redshifts in thecosmology.

We report these results for two classes of cosmology, withand without a cosmological constant.

8.1 Friedmann models: XK 550

Much of the previous work trying to estimate cosmologicalparameters from the Hubble diagrams of distant galaxieshas assumed WL\0, in which case the deceleration param-eter, q0 , is given by q0\WM/2. Table 4 summarizes the q0

values obtained previously, from a variety of galaxy samples,using BCGs and radio galaxies, in both the optical and thenear-infrared. Several points are worthy of note, motivatingthe use of K-band photometry of BCGs in X-ray clusters toestimate q0. First, as discussed above, the use of K-bandphotometry is to be preferred to the use of optical photo-metry, due to the insensitivity of K-band light to the starformation history of the galaxy. This means that our esti-mated q0 value is insensitive to uncertainties in the age ofthe stellar population of the BCGs and in the stellar IMF:optical photometry would produce a greater uncertainty inq0 on both those scores. Secondly, X-ray cluster selection isto be preferred over optical cluster selection since, as wehave shown, it is very effective at selecting galaxies with avery narrow intrinsic dispersion in absolute magnitude.Thirdly, BCGs are to be preferred over radio galaxies, bothbecause the intrinsic dispersion in radio galaxy absolutemagnitudes appears larger (10.5 mag) and because biasmay be introduced by the significant non-stellar componentto the K-band light (Dunlop & Peacock 1993; Eales et al.1998).

If we assume WL\0, then we obtain WM\0.28¹0.24 ifzf\3. Consistent values are obtained if zf is varied betweenzf\1 (i.e., slightly higher than the redshift of our most dis-tant BCG) to zf\10, as shown in Fig. 12. These results areconsistent with the results of Perlmutter et al. (1997), whofind WM\0.88+0.69

Ð0.60 for a WL\0 cosmology from the Hubblediagram of the first seven high-z Type Ia supernovae dis-covered by The Supernova Cosmology Project, andWM\0.2¹0.4 when they include a z\0.83 supernova (Perl-mutter et al. 1998), and with those of Carlberg et al. (1996),who find WM30.24¹0.09 from CNOC cluster mass pro-files.

Table 4. Previous estimates of the deceleration parameter q0 from the Hubble diagram of BCGs andradio galaxies. The symbols in the waveband column denote whether the measurements were carriedout in the optical (O) or near-infrared (IR). Wherever possible, the q0 values listed are those quoted bythe authors after correcting for evolution; otherwise the raw values are given.

8.2 Friedmann–Lemaıtre models: XK80

Fig. 13 shows the joint constraints on WM and WL we obtainif we allow a non-zero cosmological constant: this is verysimilar to the corresponding plots presented by Perlmutteret al. (1997, 1998). If we consider only spatially flat models,where WM+WL\1, we find WM\0.55+0.14

Ð0.15 , with a corre-sponding 95 per cent upper limit to WLof 0.73, for zf\3: asabove, these figures are very stable, varying marginally overthe full range z1R10. Perlmutter et al. (1997) findWM\0.94+0.34

Ð0.28 if they assume a spatially flat Universe, whichis again consistent with our results; their results translateinto a 95 per cent confidence level of WLs0.51, slightlytighter than ours. The addition of the z\0.83 supernova

(Perlmutter et al. 1998) reduces the best-fitting WM value toWM\0.6¹0.2. From the incidence of gravitational lensingof quasars, Kochanek (1996) derived a 95 per cent limit ofWLs0.66, while two recent studies have reported significantdetections of non-zero WL values: Henry (1997) findsWM\0.55¹0.17 from the evolution of the cluster X-raytemperature function, and Bender et al. (1998) deduceWM10.5, WL10.5 from cluster elliptical Fundamental Planerelations.

9 DISCUS SION A ND CONCLUSIONS

This paper has presented the K-band Hubble diagram for asample of BCGs in X-ray-selected clusters. Our principalresults are that the brightest X-ray clusters contain BCGswith a very narrow spread of absolute magnitudes, and thatthis, coupled with the insensitivity of K-band light to thedetails of the star formation processes in, and histories of,galaxies, makes samples such as ours ideal for performingclassical cosmological tests. We find that the BCGs in theclusters with high X-ray luminosities (LXa2.3Å1044 erg sÐ1

in the 0.3–3.5 keV passband) have an rms dispersion inabsolute K-band magnitude of no more than 0.22 mag, andthat leads us to an estimate of WM\0.28¹0.24 if L\0,while, for a spatially flat Universe, we find WM\0.55¹0.15and a 95 per cent limit of WLs0.73, with only a very weaksensitivity to the properties of the stellar population of theBCGs over a wide range of redshifts. The stability of theseresults demonstrates that BCGs are excellent standard can-dles, competitive with Type Ia supernovae (which also havean intrinsic magnitude dispersion of 10.2 mag; Perlmutteret al. 1997, 1998) in the estimation of cosmological param-eters WM and WL: the further refinement of these two tech-niques over the next few years holds out hope for thedetermination of these fundamental cosmological param-eters with high precision.

Finally, the lack of evolution we see in the K band, shownby the absence of a correlation between the BCG structureparameter, a, and redshift, adds to the growing body ofevidence that cluster ellipticals are old systems, possiblyundergoing only quiescent evolution of their stellar popula-tions (e.g. Bender et al. 1996; Dunlop et al. 1996; Ellis et al.1997). It is intriguing to note that a qualitatively similar lackof evolution is seen in the X-ray luminosity (Burke et al.1997; Collins et al. 1997) and temperature (Henry 1997;Mushotzky & Scharf 1997) functions. We shall return to thisquestion of the relationship between the formation andevolution of BCGs and their host clusters in a forthcomingpaper, incorporating recently acquired K-band imaging datafor BCGs in clusters from the deep Southern SHARCROSAT cluster survey (Collins et al. 1997): these clusterswere selected above a flux limit an order of magnitudefainter than that of the EMSS, enabling us to disentanglethe effects of redshift and X-ray luminosity in a mannerimpossible with the current data set.

ACKNOWLEDGMENTS

CAC acknowledges support from a PPARC AdvancedFellowship and RGM that from PPARC rolling grants atQMW and Imperial College. We thank Colin Aspin, JohnDavies, Dolores Walther and Thor Wold for their support

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Figure 12. The joint constraints on the value of WM and the BCGformation redshift zf deduced from the Hubble diagram for theBCGs in our high-LX clusters. The dotted line shows the locus ofthe best-fitting WM value for 100 values of zf in the range 1RzfR10,while the solid contours mark the 68, 95, 99 per cent confidencecontours for WM and zf : the small-amplitude random fluctuations inthe dotted line are numerical artefacts introduced by our methodof estimating WM, while the behaviour of all the curves in the range1.25RzR2.25 is caused by the transient presence of light fromAGB stars in the stellar synthesis model used to compute the BCGK-corrections, as discussed in Section 4.

Figure 13. The joint constraints on WM and WL obtained from theHubble diagram of the 26 BCGs in high-LX clusters. The contoursmark the 68, 90 and 95 per cent confidence levels for WM and WL

jointly, while the dashed and dotted lines mark the models withWL\0 and WM+WL\1 respelctively.

during the two UKIRT runs, Jon Gardner for making avail-able his compilation of K-band galaxy count data, and LanceMiller, Andy Fabian and Doug Burke for useful discussions;we are also grateful to the referee, Mike Hudson, for sug-gesting changes which improved the paper. The UnitedKingdom Infrared Telescope is operated by the Joint Astro-nomy Centre on behalf of the UK Particle Physics andAstronomy Research Council.

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APPENDI X A : PA RTI A L SPE A R M A N TEST R ESULTS

In this appendix we tabulate results from the partial Spear-man rank correlation analysis of Section 6 that it would betoo tedious to include in the main body of the text of thispaper. The partial Spearman rank correlation coefficientrAB, C (defined in equation 4 of Section 6) quantifies thestrength of the correlation between A and B, once the effectof their individual correlations with C has been removed.The significance of rAB, C may readily be computed. This isthe probability, PAB, C , that a correlation coefficient with

absolute value as high as !rAB, C! would arise if the correlationbetween A and B arose solely from their separate correla-tions with C: in common with Yates et al. (1986), we findthat Monte Carlo simulations indicate that this may be com-puted sufficiently accurately using the approximation thatrAB, C[(NÐ3)/(1Ðr 2

AB, C)]1/2 is distributed as Student’s t-statistic.

We tabulate here (in Tables A1–A5) the rAB, C values forthe five quantities studied in Section 5: (i) redshift, z; (ii)absolute K-band magnitude, MK , computed using the k-correction model in equation (1); (iii) X-ray luminosity, LX;(iv) corrected near-neighbour number, N; and (v) BCGstructure parameter, a. In each table, the upper half lists the

partial Spearman rank correlation coefficient, rAB, C, whilethe lower half gives PAB, C=Ðlog10[P (rAB, C)]. We considerthe effect of setting C equal to each of these in turn, bywhich we study the significance of correlations betweenpairs of variables A and B after removing the effect ofcorrelations between A and C and between B and C.

142 C. A. Collins and R. G. Mann

© 1998 RAS, MNRAS 297, 128–142

Table A1. Results of partial Spearman rankcorrelations analysis for constant redshift,i.e. C\z. The upper half of the table givesthe values of rABC, while the lower half givesPAB=Ðlog10[P (rAB, C)], where P (rAB, C) isthe probability that the absolute value ofthe correlation coefficient would be aslarge as !rAB, C! for a sample of the samesize, under the hypothesis that the correla-tions between A and B arise solely fromtheir individual correlations with C.

Table A2. Results of partial Spearman rankcorrelation analysis for constant absolutemagnitude, i.e., C\MK; format as forTable A1.

Table A3. Results of partial Spearman rankcorrelation analysis for constant X-rayluminosity, i.e., C\LX; format as for TableA1.

Table A4. Results of partial Spearman rankcorrelation analysis for constant correctednear neighbour number, i.e., C\N; formatas for Table A1.

Table A5. Results of partial Spearman rankcorrelation analysis for constant structureparameter, i.e., C\a; format as for TableA1.