An Optical Observational Cluster Mass Function at z\sim1 with the ORELSE Survey
D. Hung, B. C. Lemaux, R. R. Gal, A. R. Tomczak, L. M. Lubin, O. Cucciati, D. Pelliccia, L. Shen, O. Le Fèvre, G. Zamorani, P-F. Wu, D. D. Kocevski, C. D. Fassnacht, G. K. Squires
MMNRAS , 1–13 (2020) Preprint 8 January 2021 Compiled using MNRAS L A TEX style file v3.0
An Optical Observational Cluster Mass Function at 𝑧 ∼ with theORELSE Survey D. Hung , B. C. Lemaux , R. R. Gal , A. R. Tomczak , L. M. Lubin , O. Cucciati ,D. Pelliccia , , L. Shen , O. Le Fèvre , G. Zamorani , P-F. Wu , D. D. Kocevski ,C. D. Fassnacht , G. K. Squires University of Hawai’i, Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822, USA Department of Physics & Astronomy, University of California, Davis, One Shields Ave., Davis, CA 95616, USA INAF - Osservatorio di Astrofisica e Scienza dello Spazio diBologna, via Gobetti 93/3 - 40129 Bologna - Italy UCO/Lick Observatory, Department of Astronomy & Astrophysics, UCSC, 1156 High Street, Santa Cruz, CA, 95064, USA Aix-Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388 Marseille, France National Astronomical Observatory of Japan, Osawa 2-21-1, Mitaka, Tokyo 181-8588, Japan Department of Physics and Astronomy, Colby College, Waterville, ME 04961, USA Spitzer Science Center, California Institute of Technology, M/S 220-6, 1200 E. California Blvd., Pasadena, CA 91125, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present a new mass function of galaxy clusters and groups using optical/near-infrared wavelength spectroscopic andphotometric data from the Observations of Redshift Evolution in Large-Scale Environments (ORELSE) survey. At 𝑧 ∼
1, clustermass function studies are rare regardless of wavelength and have never been attempted from an optical/near-infrared perspective.This work serves as a proof of concept that 𝑧 ∼ ∼
240 galaxy overdensity candidates in theredshift range 0 . < 𝑧 < .
37 at a mass range of 13 . < log ( 𝑀 / 𝑀 (cid:12) ) < .
8. This mass range is comparable to existing opticalcluster mass function studies for the local universe. Our candidate numbers vary based on the choice of multiple input parametersrelated to detection and characterization in our cluster finding algorithm, which we incorporated into the mass function analysisthrough a Monte-Carlo scheme. We find cosmological constraints on the matter density, Ω 𝑚 , and the amplitude of fluctuations, 𝜎 , of Ω 𝑚 = . + . − . and 𝜎 = . + . − . . While our Ω 𝑚 value is close to concordance, our 𝜎 value is ∼ 𝜎 higher becauseof the inflated observed number densities compared to theoretical mass function models owing to how our survey targetedoverdense regions. With Euclid and several other large, unbiased optical surveys on the horizon, VMC mapping will enableoptical/NIR cluster cosmology at redshifts much higher than what has been possible before. Key words: galaxies: clusters — galaxies: groups — cosmology: large-scale structure of Universe — cosmology: cosmologicalparameters — techniques: spectroscopic — techniques: photometric
Cosmological models seek in part to explain the growth and distri-bution of large-scale structure in the universe. One such quantifyingmetric is the cluster mass function, which describes the number den-sity of galaxy clusters as a function of their mass. How the massfunction evolves over time will depend on cosmological parameters,and thus measuring the mass function over wide redshift ranges of-fers the power of greater statistical leverage (see, e.g., Allen et al.2011). Different cosmologies in theoretical mass functions show non-negligible discrepancies in the predicted number counts of clusters(e.g., Pacaud et al. 2018), motivating the need for comparisons withobservational data.Constraints on cosmological parameters can be obtained throughfitting a number of independent probes, such as the cosmic mi- crowave background (CMB) anisotropy (e.g., Planck Collaborationet al. 2016; Hinshaw et al. 2013), the brightness/redshift relation fortype Ia supernovae (SNe; e.g., Riess et al. 1998; Perlmutter et al.1999), and baryon acoustic oscillations (BAO) data (e.g., Eisensteinet al. 2005). The cluster mass function can be used to constrain thematter density, Ω 𝑚 , and the amplitude of fluctuations on the scaleof 8 ℎ − Mpc, 𝜎 , by fitting the predicted halo abundance, the halomass function. 𝜎 shows a strong degeneracy with Ω 𝑚 when deter-mined from cluster abundance data. However, the confidence levelsof the Ω 𝑚 - 𝜎 likelihoods found by the cluster mass function areadvantageously almost orthogonal to those found by the CMB (e.g.,Rozo et al. 2010). Combining the two probes therefore helps breakthe degeneracy between Ω 𝑚 and 𝜎 and reduce their uncertainties,while BAO and SNe studies can constrain Ω 𝑚 independent of 𝜎 (e.g., Vikhlinin et al. 2009; Abdullah et al. 2020). © a r X i v : . [ a s t r o - ph . C O ] J a n Hung et al.
The first analytical expression of the halo mass function was de-rived by Press & Schechter (1974), followed by Bond et al. (1991);Lee & Shandarin (1998); Sheth et al. (2001). However, the rise ofN-body simulations helped reveal limitations in the existing models,and the most recent halo mass functions have been calibrated usingnumerical results. These models are chiefly distinguished betweentwo widely used halo definitions. Haloes may be defined using thespherical overdensity (SO; Lacey & Cole 1994) algorithm, wherespherical apertures are placed around isolated density peaks, suchthat the mean interior density is some set multiple relative to thebackground or critical density. Haloes may also be defined with theFriends-of-Friends (FoF; Davis et al. 1985) algorithm, where a par-ticle is matched with neighbors within a given linking length, andthose neighbors are matched with other neighbors until no more arefound. The final group of particles then represents an isodensity con-tour in space. SO and FoF masses are strongly correlated for relaxed,isolated haloes (White 2001; Tinker et al. 2008), but irregular haloescan cause significant disagreement. Most theoretical models havefollowed the convention of Jenkins et al. (2001) and used FoF haloesin order to obtain a more universal halo mass function independentof redshift or cosmology. However, SO haloes tend to be preferredfor comparisons to observational studies, due to the more direct linkwith how virialized structures are defined in spherical apertures.Determining a cluster mass function from observational data re-quires a cluster sample where cluster masses have been estimatedeither directly (as in, e.g., weak gravitational lensing) or by using anobservational proxy. Typically, the cluster sample is X-ray selected,and the masses are derived through more indirect proxies such as X-ray luminosity or optical cluster richness (e.g., Reiprich & Böhringer2002; Mantz et al. 2008; Vikhlinin et al. 2009; Wen et al. 2010;Pacaud et al. 2018; Costanzi et al. 2019). However, the resultingcluster mass function can have large uncertainties due to factors suchas the scatter in the mass scaling relations as well as incompletenessin the cluster sample due to selection biases or other observational ef-fects. These issues are especially a concern at redshift 𝑧 > ∼ .
5, wherethe intracluster medium (ICM) begins to be underdeveloped, partic-ularly for intermediate or low-mass clusters. Many clusters have beenfound to be X-ray underluminous, compared to what was suggestedby their dynamics (e.g., Rumbaugh et al. 2018) or the luminosity-mass relation with weak lensing masses (e.g., Giles et al. 2015).Cluster samples selected by other means often see a sizable portionwith no detected X-ray counterpart at 𝑧 ∼ 𝑀 (cid:12) .In contrast, X-ray and optical/near-infrared (NIR) surveys are moreeffective at finding low-mass clusters, particularly at lower redshifts. More recently, the growing scale of photometric and spectroscopicsurveys at optical and NIR wavelengths have enabled cluster searchesindependent of any X-ray data. Such searches identify clusters byusing galaxies to trace mass overdensities (e.g., Abell 1958; Okeet al. 1998; Rykoff et al. 2016). Though there have been severalsuccessful cluster searches done at optical wavelengths, cluster massfunction studies at optical and NIR wavelengths have been scarce andso far limited to the local universe. Such studies use optically selectedcatalogs with masses derived from supplemental weak lensing or X-ray data (e.g., Rozo et al. 2010; Costanzi et al. 2019; Kirby et al.2019) or through the virial mass theorem (e.g., Abdullah et al. 2020).Beyond constraining cosmology, contrasting cluster mass functionswith X-ray and optically selected samples at different redshifts couldyield key insights on structure formation and development of theICM. While at least some attempts have been made at X-ray and SZwavelengths at redshifts up to 𝑧 ∼ − . from an optical/NIR perspective for the first time outsideof the relatively local universe. In Hung et al. (2020), we foundgalaxy clusters using a powerful new technique known as Voronoitessellation Monte-Carlo (VMC) mapping and apply it to opticaland NIR photometric and spectroscopic data over the redshift range0 . < 𝑧 < .
37. Unlike other cluster search algorithms, VMC map-ping makes no assumptions about cluster geometry or morphology.With VMC mapping, we count all galaxies irrespective of color to alimit of stellar masses > ∼ 𝑀 (cid:12) to trace overdensities, independentof the ICM emission.In searches of clusters with X-ray observations, there is a possi-bility of observing a decreasing number of systems at a given X-rayluminosity with increasing redshift. In such a case, an ambiguitywould exist in the interpretation of the trend as this behavior couldeither be attributed to intermediate- to high-redshift structures of agiven mass having an underdeveloped ICM relative to the local coun-terparts (as is true for at least some ORELSE systems, see Rumbaughet al. 2018), a true lack of structure at higher redshift, or some combi-nation of the two. The same ambiguity does not exist in optical/NIRcluster searches with spectroscopically confirmed redshifts as galax-ies will presumably always trace clusters. In our search, becausewe indiscriminately count galaxies without constraining ourselves toany particular subpopulations such as the red sequence, we shouldbe able to detect a cluster so long as it is galaxy-rich with any typeof galaxies.In Hung et al. (2020), we demonstrated VMC mapping’s sensi-tivity to detecting unprecedentedly low mass structures, quantifyingpurity and completeness estimates down to total masses of 10 . 𝑀 (cid:12) .Our search recovered 51 previously known structures and found 402new overdensity candidates, with estimated masses between 10.2 < log ( 𝑀 / 𝑀 (cid:12) ) < and a spectroscopic redshift fraction of atleast 5%. In this paper, we seek to derive a cluster mass functiondrawn from this sample. Our sample includes overdensity candidates with masses as small as ∼ . 𝑀 (cid:12) , which fall below the typically defined mass limits of galaxyclusters and instead would traditionally be regarded as groups. Though thesestructures are all included in the mass function, we use the term “cluster massfunction” in this paper for the sake of brevity. As we can only correct for purity and completeness down to masses of10 . 𝑀 (cid:12) , it is possible that many of the overdensity candidates with smallermasses are spurious detections. We refer the reader to §6.1.1 in Hung et al.(2020) for a more in-depth discussion.MNRAS , 1–13 (2020) RELSE Mass Function This paper is organized as follows: In §2, we briefly review thephotometric and spectroscopic data we used and our overdensitycandidate detection method. In §3, we go over several parameters thataffect the overdensity candidate sample. In §4, we describe how wetransform these parameters and their varying overdensity candidatesamples into one mass function. In §5, we compare our observationalmass function with a theoretical model to fit for Ω 𝑚 and 𝜎 . In §6, wediscuss the implications of our findings as well as a few other clustermass function studies and highlight where our methodolgy could beuseful with data from future surveys. Finally, we present a summaryof this work in §7. Unless otherwise noted, we use a flat Λ CDMcosmology throughout this paper, with 𝐻 = 70 km s − Mpc − , Ω 𝑚 = 0.27, and Ω Λ = 0.73. All reported distances are in proper units. Our previous work in Hung et al. (2020) searched for serendipi-tous cluster candidates in the Observations of Redshift Evolutionin Large-Scale Environments (ORELSE; Lubin et al. 2009) survey,a large multi-wavelength photometric and spectroscopic campaigntargeted at several known large-scale structures over redshifts of0 . < 𝑧 < .
3. It was designed to look for surrounding large-scalestructure in each field, but it also probed the full dynamic range ofenvironments at all redshifts by targeting galaxies along the line-of-sight (Gal et al. 2008; Lubin et al. 2009). Over 15 fields, ORELSEhas a combined ∼ ∼ . 𝐵𝑉𝑟𝑖𝑧 ) imag-ing typically ranged from depths of 𝑚 𝐴𝐵 = 26.4 in the 𝐵 -band to 𝑚 𝐴𝐵 = 24.6 in the 𝑧 -bands. The NIR ( 𝐽𝐾 , Spitzer /IRAC) imaging reachedtypical depths of 𝑚 𝐴𝐵 = 21.9 and 21.7 respectively in the 𝐽 and 𝐾 / 𝐾 𝑠 bands (Tomczak et al. 2019). Its unprecedented spectroscopiccoverage includes ∼ ,
000 high quality spectroscopic objects andspectroscopic completeness of 25% to 80% among known structures(Lemaux et al. 2019). Additionally, the spectral member populationhas been found to be broadly representative of the underlying galaxypopulation (Shen et al. 2017; Lemaux et al. 2019). ORELSE’s ex-tensive dataset provides thousands of high-quality photometric andspectroscopic redshifts ideal for a cluster search.We identified galactic overdensities using a powerful new tech-nique, Voronoi tessellation Monte-Carlo (VMC) mapping, describedin detail in Lemaux et al. (2018) and applied to look specifically forstructure in ORELSE in Hung et al. (2020). A Voronoi tessellation isa density field estimator that splits a 2D plane by assigning a polyg-onal cell to every object in the plane whose area is the region closerto its host object than any other object. The cell size is thus inverselyproportional to the density at a given location. For each ORELSEfield, we separate our galaxy catalogs into redshift slices of approx-imately ± − in velocity space and apply the tessellationto each slice. The redshift slices are defined such that neighboringslices have 90% overlap to minimize chances of splitting individualstructures across slices.For each slice, we have galaxies with spectroscopic redshifts, 𝑧 𝑠 𝑝𝑒𝑐 , and galaxies with photometric redshifts, 𝑧 𝑝ℎ𝑜𝑡 . The pho-tometric redshifts have much higher uncertainties than the spectro-scopic redshifts, which we account for with our VMC technique. Foreach Monte-Carlo realization of a slice, we Gaussian sample the PDFof each galaxy’s 𝑧 𝑝ℎ𝑜𝑡 . As a result, some galaxies fall in or out of theredshift boundaries of the slice. We then perform the Voronoi tessel-lation on all the 𝑧 𝑠 𝑝𝑒𝑐 and 𝑧 𝑝ℎ𝑜𝑡 galaxies in the slice. We repeat this100 times, and the final VMC map of the slice is then computed bymedian combining the densities from all realizations. For full details on the VMC methodology within the context of ORELSE, see Hunget al. (2020). Overdensities are first found in the redshift slices (see§3.1), and then linked together across neighboring slices (see §3.2). How we find and catalog galaxy overdensity candidates depends onseveral independent parameters, ranging from how large an over-density must be for detection to peculiarities on how we translatethe overdensity we observe to a total mass. In Hung et al. (2020),our goal was to establish VMC mapping as a viable tool for findingoverdensities. We thus adopted a set of parameters best suited for thegeneral case of detecting any structure at all and left the specificsof fine-tuning the resulting overdensity candidate sample to futurework. We revisit our parameters in this work as we now require cru-cial informaiton such as the proper number of overdensities at eachmass threshold in order to build the cluster mass function. In thissection, we describe the effects of each revelant parameter, and in§4, we go over which values we use for our cluster mass function.We encourage the reader to refer to Hung et al. (2020) where theseparameters are described in greater detail.
We search for significant overdense regions in our VMC maps usingthe standard photometry software package Source Extractor (SEx-tractor; Bertin & Arnouts 1996). SExtractor’s DETECT_THRESHparameter sets how much higher the density floor must be for a validdetection relative to the RMS noise in the background. For example,a 4 𝜎 DETECT_THRESH stipulates that detections must be at leastfour times the background RMS. In Hung et al. (2020), we found thatDETECT_THRESH values of 4 and 5 𝜎 performed similarly in termsof purity and completeness, but we decided to use 4 𝜎 to maximizeour chances of detecting smaller overdensity candidates in that work.Often, galaxy clusters are located in close proximity to each other,and show up in SExtractor as single detections. Deblending in SEx-tractor refers to separating these detections out to their subcompo-nents so that we can identify the individual clusters. SExtractor hastwo parameters related to deblending: the number of deblendingsub-thresholds DEBLEND_NTHRESH and the minimum contrastDEBLEND_MINCONT. The deblending sub-thresholds refer to thenumber of exponentially spaced levels from the detection floor to thepeak of the detection. Substructure is identified with the minimumcontrast DEBLEND_MINCONT parameter, which is how large theoverdensity in a substructure must be compared to the total over-density in the entire structure to be counted as a separate detection.Of the two deblending parameters, we choose to focus on the DE-BLEND_MINCONT parameter as it is more sensitive to change. AsDEBLEND_MINCONT decreases, the more SExtractor splits aparta single structure (Fig. 1).Previously in Hung et al. (2020), we elected to adopt a DE-BLEND_NTHRESH of 32 and DEBLEND_MINCONT of 0.01. Wedeemed these parameters as acceptable as they were able to sepa-rate some known structures while also avoiding splitting others up.However, not every blended grouping of known structures was ableto be separated with these deblending parameters. For an unbiasedcluster mass function, we must be able to properly separate largerconglomerates of structure by way of carefully choosing the optimalset of deblending parameters. Not doing so would lead to an over-abundance of high mass overdensity candidates and a depletion oflow mass overdensity candidates. MNRAS , 1–13 (2020)
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Figure 1.
Example SExtractor-generated segmentation maps of a blended structure in the SC1604 supercluster at 𝑧 ∼ . We have a measure of each overdensity candidate’s mass by wayof its isophotal flux 𝐹 , a measurement of density calculated by SEx-tractor, of the form:log ( 𝑀 / 𝑀 (cid:12) ) = 𝑎 + 𝑏𝐹 𝑐 𝑒 −( 𝐹 / 𝑑 ) (1)where 𝑎 , 𝑏 , 𝑐 , and 𝑑 are scalar constants. We fit this quantity withthe virial masses of the previously known structures in ORELSEto obtain a general flux to mass relation. The mass zero point wascalibrated with the virial masses of the most spectroscopically well-studied clusters and groups in ORELSE, which generally had spectralfractions of >
50% and an average of 24 spectroscopic members perstructure. The virial masses have been found to be comparable withinthe error bars of independent X-ray, lensing, and SZ measurementswhere available (see e.g., Clowe et al. 1998; Margoniner et al. 2004;Valtchanov et al. 2004; Jee et al. 2006; Maughan et al. 2006; Mu-chovej et al. 2007; Rzepecki et al. 2007; Stott et al. 2010; Piffarettiet al. 2011; Laganá et al. 2013; Pratt & Bregman 2020) as well asstatistically consistent with the masses we estimate from the over-density maps directly using the method described in Cucciati et al.(2018). We refer the reader to §6.1 of Hung et al. (2020) for furtherdiscussion of our mass calibration as well as comparisons with othermass estimation methods.Previously in Hung et al. (2020), we found best-fit values of 𝑎 = . ± . 𝑏 = − . ± . 𝑐 = − . ± . 𝑑 = . ± .
740 for equation 1. The exact mass fit willvary in this work as the detection floor set by the choice of DE-TECT_MINCONT will significantly change the total isophotal fluxvalues, but we find negligible differences with respect to the choicein deblending parameters. The overall mass, summed over all over-density candidates we find, remains relatively unchanged as we drop the DETECT_MINCONT parameter, while the fraction of high masscandidates significantly drops (Fig. 2).
SExtractor finds individual overdensities in each redshift slice ofa VMC map. We link these overdensities over successive redshiftslices in order to obtain a single overdensity candidate. We start witha given SExtractor detection in a redshift slice. Then, we look inthe neighboring redshift for any detections with a barycenter withinan RA-DEC distance equal to or less than the linking radius we set.The smaller the linking radius, the closer the detections must be inorder to be linked together. If we find a match, we take the isophotalflux-weighted barycenter between the two detections and continueour search into the next redshift slice. If there are multiple matches,we take both as separate linked chains and continue the search untilno more matches are found. Once we complete the search for everydetection in a redshift slice, we move on to the next redshift sliceand repeat the same search. For each linked chain of at least fiveSExtractor detections, we apply a Gaussian fit of the isophotal fluxesof the detections and accept the chain as an overdensity candidate ifthe Gaussian fit converges.Because we link every detection to every possible match, we havethe same detections assigned to more than one linked chain. Thus,we would overcount the number of overdensity candidates we hadif we treated each chain as a new unique candidate. In Hung et al.(2020), we resolved this issue by sorting the overdensity candidatesby greatest Gaussian fit amplitude and removing any other candidatesthat fell within 0.7 Mpc and Δ 𝑧 < . MNRAS , 1–13 (2020)
RELSE Mass Function Figure 2.
We plot the total mass of all overdensity candidates we find whilevarying the SExtractor deblending parameter DEBLEND_MINCONT. Thedeblending becomes finer for smaller values of DEBLEND_MINCONT. Fol-lowing our expectations, finer deblending yields more low-mass overdensitycandidates as more high-mass structures are broken up. Between the DE-BLEND_MINCONT extremes plotted, the total mass decreases by 19%,while the fraction of high mass candidates drops by 66%. concerned with establishing the VMC technique for finding overden-sities for the general case rather than precise optimization for ourparticular set of fields.For the purpose of constructing a cluster mass function, how-ever, including the smaller substructures we originally eliminated isparamount. For this work, we revised our linking scheme by em-ploying a goodness of fit test in order to remove only the duplicatedetections. We first remove all linked chains that are complete sub-sets of other larger chains. Then, we apply our Gaussian fit for eachlinked chain. We measure the goodness of fit with the coefficient ofdetermination, 𝑅 . The 𝑅 statistic ranges between 0 and 1, with thelatter indicating a perfect agreement between the model and data. Wesort our linked chains by their 𝑅 values from high to low. We acceptthe first linked chain as an overdensity candidate, and we remove allother linked chains that include any of the same SExtractor detec-tions as the accepted candidate. We repeat this iteratively with thenext highest 𝑅 linked chain until no SExtractor detections are sharedbetween any of the candidates. We emphasize that this removal pro-cess only eliminates duplicate detections of the same overdensitycandidates from our catalog; no real structure or substructure is lostas a result of this process.Ignoring the linked chains that were complete subsets of another,the removal process eliminated as few as 20 to over 1000 linked chains across all ORELSE fields depending on how sensitive we set thedetection and deblending parameters in §3.1. Despite this wide rangein removals, the number of linked chains remaining was fairly robust,typically being between a total of 300 to 400, so we consistently havearound the same number of overdensity candidates after removingall duplicate detections. More linked chains are removed by numberat lower redshifts due to a greater abundance of detections, thoughthe percentage of removals is not sensitive to redshift. We note thatthis methodolgy will disfavor irregularly shaped structures where thevelocity distribution deviates appreciably from a Gaussian, thoughthey are likely still picked up in many cases as our goodness of fittest is a relative measure. ORELSE was designed to target massive known clusters. Thanks tothe high levels of spectroscopy around these systems, we found a fewdozen more clusters and groups nearby in the fields on an initial,primarily spectroscopic search. In Hung et al. (2020), we optimizedour choice of SExtractor parameters in part based on how well wecould recover all known structures in the ORELSE fields, both thosethe survey was targeted at and those found with spectroscopy. Theinclusion of all these known structures will bias a mass function highrelative to structures found in a field survey. This is particularly true atthe high mass end as the known structures are among the highest massoverdensities we detect with our technique. In addition, there were asmall number of structures, such as clusters B and C in RXJ1716 (see§4.3.1 in Hung et al. 2020), that we were not able to separate no matterhow fine we set the deblending parameters. We would pick up theseblended structures as single overdensities and thus overestimate theirmasses. By excluding the previously known structures we recoverfrom our cluster mass function calculations, we can possibly avoidbiasing our data towards higher mass overdensities. Regardless ofthis removal, it is likely that the mass function will still be biasedhigh because of additional structures around the targeted structuresthat were missed by the original spectral search. 𝑅 Cluster mass function studies typically compare to theoretical modelsthat calculate the dark matter halo mass function. Dark matter haloesare typically defined within spherical apertures of radii 𝑅 corre-sponding to an overdensity Δ = 𝑅 circ , of each overdensity candidate to their 𝑅 radii. Eachoverdensity candidate is made up of a series of SExtractor detectionswe linked together over several redshift slices. We obtain 𝑅 circ bytaking the largest SExtractor detection by isophotal area in an over-density candidate and finding its effective circularized radius. Wederive 𝑅 by treating the overdensity candidate’s estimated massfrom equation 1 in §3.1 as equal to 𝑀 with: 𝑅 = (cid:18) 𝐺 𝑀 𝐻 ( 𝑡 ) (cid:19) / (2)where 𝐻 ( 𝑡 ) is the Hubble parameter and 𝐺 is the gravitationalconstant.In Figure 3, we plot 𝑅 circ and 𝑅 values for four different DE-BLEND_MINCONT parameters using a DETECT_THRESH of 4 𝜎 .Above approximately 0.6 Mpc, 𝑅 circ predominantly outpaces 𝑅 . MNRAS000
We plot the total mass of all overdensity candidates we find whilevarying the SExtractor deblending parameter DEBLEND_MINCONT. Thedeblending becomes finer for smaller values of DEBLEND_MINCONT. Fol-lowing our expectations, finer deblending yields more low-mass overdensitycandidates as more high-mass structures are broken up. Between the DE-BLEND_MINCONT extremes plotted, the total mass decreases by 19%,while the fraction of high mass candidates drops by 66%. concerned with establishing the VMC technique for finding overden-sities for the general case rather than precise optimization for ourparticular set of fields.For the purpose of constructing a cluster mass function, how-ever, including the smaller substructures we originally eliminated isparamount. For this work, we revised our linking scheme by em-ploying a goodness of fit test in order to remove only the duplicatedetections. We first remove all linked chains that are complete sub-sets of other larger chains. Then, we apply our Gaussian fit for eachlinked chain. We measure the goodness of fit with the coefficient ofdetermination, 𝑅 . The 𝑅 statistic ranges between 0 and 1, with thelatter indicating a perfect agreement between the model and data. Wesort our linked chains by their 𝑅 values from high to low. We acceptthe first linked chain as an overdensity candidate, and we remove allother linked chains that include any of the same SExtractor detec-tions as the accepted candidate. We repeat this iteratively with thenext highest 𝑅 linked chain until no SExtractor detections are sharedbetween any of the candidates. We emphasize that this removal pro-cess only eliminates duplicate detections of the same overdensitycandidates from our catalog; no real structure or substructure is lostas a result of this process.Ignoring the linked chains that were complete subsets of another,the removal process eliminated as few as 20 to over 1000 linked chains across all ORELSE fields depending on how sensitive we set thedetection and deblending parameters in §3.1. Despite this wide rangein removals, the number of linked chains remaining was fairly robust,typically being between a total of 300 to 400, so we consistently havearound the same number of overdensity candidates after removingall duplicate detections. More linked chains are removed by numberat lower redshifts due to a greater abundance of detections, thoughthe percentage of removals is not sensitive to redshift. We note thatthis methodolgy will disfavor irregularly shaped structures where thevelocity distribution deviates appreciably from a Gaussian, thoughthey are likely still picked up in many cases as our goodness of fittest is a relative measure. ORELSE was designed to target massive known clusters. Thanks tothe high levels of spectroscopy around these systems, we found a fewdozen more clusters and groups nearby in the fields on an initial,primarily spectroscopic search. In Hung et al. (2020), we optimizedour choice of SExtractor parameters in part based on how well wecould recover all known structures in the ORELSE fields, both thosethe survey was targeted at and those found with spectroscopy. Theinclusion of all these known structures will bias a mass function highrelative to structures found in a field survey. This is particularly true atthe high mass end as the known structures are among the highest massoverdensities we detect with our technique. In addition, there were asmall number of structures, such as clusters B and C in RXJ1716 (see§4.3.1 in Hung et al. 2020), that we were not able to separate no matterhow fine we set the deblending parameters. We would pick up theseblended structures as single overdensities and thus overestimate theirmasses. By excluding the previously known structures we recoverfrom our cluster mass function calculations, we can possibly avoidbiasing our data towards higher mass overdensities. Regardless ofthis removal, it is likely that the mass function will still be biasedhigh because of additional structures around the targeted structuresthat were missed by the original spectral search. 𝑅 Cluster mass function studies typically compare to theoretical modelsthat calculate the dark matter halo mass function. Dark matter haloesare typically defined within spherical apertures of radii 𝑅 corre-sponding to an overdensity Δ = 𝑅 circ , of each overdensity candidate to their 𝑅 radii. Eachoverdensity candidate is made up of a series of SExtractor detectionswe linked together over several redshift slices. We obtain 𝑅 circ bytaking the largest SExtractor detection by isophotal area in an over-density candidate and finding its effective circularized radius. Wederive 𝑅 by treating the overdensity candidate’s estimated massfrom equation 1 in §3.1 as equal to 𝑀 with: 𝑅 = (cid:18) 𝐺 𝑀 𝐻 ( 𝑡 ) (cid:19) / (2)where 𝐻 ( 𝑡 ) is the Hubble parameter and 𝐺 is the gravitationalconstant.In Figure 3, we plot 𝑅 circ and 𝑅 values for four different DE-BLEND_MINCONT parameters using a DETECT_THRESH of 4 𝜎 .Above approximately 0.6 Mpc, 𝑅 circ predominantly outpaces 𝑅 . MNRAS000 , 1–13 (2020)
Hung et al.
Figure 3.
In an attempt to better match our cluster mass function to theoretical halo mass function models, we can scale our overdensity candidates circular radii 𝑅 circ to their equivalent 𝑅 radii, assuming their mass is equal to 𝑀 . We plot linear fits of 𝑅 circ and 𝑅 for our four different DEBLEND_MINCONTparameters using a fixed DETECT_THRESH of 4 𝜎 . We calculate 𝑅 according to Equation 2. We measure 𝑅 circ from the largest SExtractor detection byarea in each overdensity candidate. The scatter points represent all found overdensity candidates for each DEBLEND_MINCONT parameter. The filled scatterpoints represent the overdensity candidates with masses log ( 𝑀 tot / 𝑀 (cid:12) ) > .
5. The fits between the DEBLEND_MINCONT parameters do not significantlychange. 𝑅 circ and 𝑅 appear to be closest to equal (the dashed black line) below around 0.6 Mpc. In other words, it is likely we are estimating a mass for larger overden-sity candidates at an effective radius larger than 𝑅 . The disagree-ment between the two radius measures implies that our assumptionthat our mass estimate is equal to 𝑀 is incorrect, which means thecomparisons between our observed mass function and the theoreticalmass functions may also be off due to the latter using 𝑅 . Thisindicates a possible need to scale down the masses of such overden-sity candidates to the mass enclosed by their 𝑅 radii to match thecomparisons we make with the theoretical halo mass function in §5.Equation 37 of Coe (2010) gives the mass of a Navarro-Frenk- For transparency, we note that we also allow the theoretical value to varyto account for the imprecision in this process. More details can be found in§5.
White (NFW) dark matter halo within a sphere of radius 𝑟 = 𝑥𝑟 𝑠 as: 𝑀 ( 𝑟 ) = 𝜋𝜌 𝑠 𝑟 𝑠 (cid:16) ln ( + 𝑥 ) − 𝑥 + 𝑥 (cid:17) (3)where 𝜌 𝑠 is the scale density, 𝑟 𝑠 is the scale radius, and 𝑥 is amultiplicative factor. We use this equation to calculate the quotientof the mass enclosed at 𝑅 , 𝑀 , and the mass enclosed at somegeneralized radius. From equation 1 of Coe (2010), the scale radiusis equivalent to 𝑟 𝑠 = ( 𝐶 𝑣𝑖𝑟 / 𝑟 𝑣𝑖𝑟 ) − , where 𝐶 𝑣𝑖𝑟 is the concentrationat the virial radius, and 𝑟 𝑣𝑖𝑟 is the virial radius. We assume that 𝑟 𝑣𝑖𝑟 = 𝑅 / .
14 and estimate that 𝐶 𝑣𝑖𝑟 ≈ . 𝜅 of themasses enclosed in 𝑅 circ over 𝑅 reduces to: 𝜅 = ln ( ) − / (cid:16) ln ( + 𝑥 ) − 𝑥 + 𝑥 (cid:17) (4) MNRAS , 1–13 (2020)
RELSE Mass Function 𝑥 = 𝑅 circ 𝑅 = ( − 𝑏 / 𝑅 ) 𝑚 (5)where 𝑚 and 𝑏 are the slope and intercept of the 𝑅 circ and 𝑅 linear fit.For a given overdensity candidate, we would multiply its mass by 𝜅 to scale it back to 𝑀 . As overdensity candidates with 𝑅 ≤ . 𝑅 ≈ 𝑅 circ , weconsider this correction only for candidates with 𝑅 > . 𝜅 decreases with mass, giving typical values of 0.90, 0.71, and 0.64for masses of 10 𝑀 (cid:12) , 10 . 𝑀 (cid:12) , and 10 𝑀 (cid:12) respectively. We represent the cluster mass function as a cumulative distribution,where we plot the number density 𝑁 ( > 𝑀 ) for a given mass. InHung et al. (2020), we constructed several mock candidate catalogsto estimate our purity and completeness numbers. The mock catalogssampled slightly different mass ranges depending on the redshift, butthey inclusively covered log ( 𝑀 / 𝑀 (cid:12) ) = 13.64 to 14.81. We use 10equally logarithmically spaced points in the same mass range for ourmass function. The spacing between the mass bins do not affect themeasured number densities as long as they are wider than the averagemass uncertainties. Because we have purity and completeness esti-mates, and associated uncertainties on those purity and completenessvalues, as functions of redshift, mass, and spectroscopic function, wedo not need to rely on using a 𝑉 𝑚𝑎𝑥 method to limit our sample towhere we have high completeness. We assign the overdensity candidates to the mass bins using a Monte-Carlo method. For each overdensity candidate, we have an estimateof its mass and redshift and their associated uncertainties. We Gaus-sian sample each to obtain a new mass 𝑀 𝑖 and redshift 𝑧 𝑖 for aniteration. 𝑀 𝑖 and 𝑧 𝑖 are used to compute the purity and completenesscorrections, which also have associated uncertainties and are againGaussian sampled. The number density 𝑛 𝑖 of the candidate is then: 𝑛 𝑖 = 𝑃 𝑖 / 𝐶 𝑖 𝑉 (6)where 𝑃 𝑖 and 𝐶 𝑖 are the purity and completeness for the giveniteration, and 𝑉 is the comoving volume, which is 9 . × Mpc for our redshift range of 0.55 to 1.37 and effective transverse surveyarea of 1.4 square degrees. 𝑛 𝑖 is added to its mass bin, assigned by 𝑀 𝑖 . As more overdensity candidates are assigned to the same massbin, the larger the total number density in the bin grows. We repeatthis process 1000 times and then take the median of all iterations asthe final number densities for each mass bin, with the 16th and 84thpercentiles as approximate 1 𝜎 uncertainties. We tested for the presence of Eddington (1913) bias in our sample ofoverdensity candidates using a toy cluster mass function. We deviseda mock cluster mass function and sampled from it a population of ob-served synthetic clusters. Redshifts and spectroscopic fractions weregenerated for each synthetic cluster by uniform randomly samplingthe full range of these two parameters of the real overdensity candi-date sample. Our purity and completeness estimates were unchanged.
Table 1.
Overdensity Candidate ParametersParameter ValuesMass Fit (DETECT_THRESH 𝜎 ) Original (Hung et al. 2020), 4, 5DEBLEND_MINCONT 0.03, 0.01, 0.005, 0.003Linking Radius (Mpc) 1.0, 0.50, 0.25Using Known Structures Yes, NoUsing 𝑅 Correction Yes, NoEach parameter listed here will affect how many overdensity candidates arefound and at what mass, directly affecting whatever mass function weattempt to derive. As we do not know a priori the optimal set of parametersto use, we consider reasonable ranges of values for five total parameters,described in §3 and §4, giving 144 unique arrangements.
We assigned every synthetic cluster the same fixed mass uncertaintyand tested two cases: 0.05 dex, the typical uncertainty we see in ourreal overdensity candidate sample, and 0.15 dex, one of the largestuncertainties in the sample. We assigned the synthetic clusters intomass bins with the same Monte-Carlo method described in §4.1 tosee if we could recover our toy cluster mass function. Though Ed-dington bias was always clearly present, we found that our numberdensities only noticeably deviated from the toy mass function whenthe size of the mass bin was smaller than the clusters’ mass uncer-tainties. Any deviation was negligible otherwise. Given that we usea much larger mass bin of 0.38 dex for our real mass function, wecan consider the effects of Eddington bias as small compared to ourtypical mass error and purity and completeness corrections.
The overdensity candidates in our catalog will change depending onour choice of parameters, and we do not know a priori which choiceis optimal for building our cluster mass function. However, we candefine for each parameter a reasonable range of values from our priorrigorous testing on both real and mock data in §4 and 5 in Hung et al.(2020). With the values we choose below, we were able to recoverhigh fractions of previously known structures with similar redshiftand transverse position offsets from their fiducial coordinates. Like-wise, our estimated levels of completeness and purity largely fellwithin a 5% variation.We define our grid in Table 1 by the set of parameters describedin §3, and we plot the variations in our overdensity candidates innumber and redshift in Figure 4. The mass fit is dependent on theDETECT_THRESH 𝜎 used, as a higher 𝜎 decreases the sizes ofthe SExtractor detections. We use new mass fits drawn from us-ing DETECT_THRESH values of 4 and 5 𝜎 . Because we have fourunique values for DEBLEND_MINCONT, the mass fit will slightlydiffer for each one. In order to obtain a single mass fit for the sameDETECT_THRESH value, we compute a mass fit for each DE-BLEND_MINCONT value, leaving us four sets of best-fit terms forthe fitting function in Eq. 1. We then obtain an average mass fit bytaking the median for each term, and we treat the median absolute de-viation as the uncertainty in the term. We also use our original massfit from Hung et al. (2020), which used a DETECT_THRESH of4 𝜎 , as a means of testing another methodological approach divorcedfrom the choice of parameters described in this paper.We chose a reasonable range of our DEBLEND_MINCONT val-ues by examining by eye five pairs of known structures within 0.2to 3.5 Mpc in the transverse dimensions and 𝑧 < Δ .
02 in red-shift across different ORELSE fields. The deblending becomes finerfor smaller values of DEBLEND_MINCONT. Dropping the DE-
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MNRAS000 , 1–13 (2020)
Hung et al.
Figure 4.
Our sample of overdensity candidates will change based on the choice of input parameters in our detection algorithm. Here we show how the totalnumber (left) and redshifts (right) of overdensity candidates vary by averaging over the ranges of values we set in Table 1. The points denote the median numberin each mass bin, and the error bars show the 16th and 84th percentiles. For simplicity, we assume all individual candidate mass and redshift uncertainties as wellas purity and completeness correction uncertainties to be 0 in this plot. The median total number of candidates is 241 in the plotted mass range. The candidatenumbers fall just short of 0 for our highest mass bin, and we see the largest spreads in candidate numbers below 14.2 dex. The dashed red line gives the overallmean redshift of the sample at 𝑧 = .
94, which falls within 1 𝜎 of the redshift range in every mass bin. BLEND_MINCONT too far runs the risk of breaking apart indi-vidual overdensities, so we looked for the most conservative valueof DEBLEND_MINCONT that was able to deblend a given pair ofknown structures. For four of the pairs, we respectively found DE-BLEND_MINCONT values of 0.03, 0.01, 0.005, and 0.003. For theremaining fifth pair, we were not able to split the two substructureswithout breaking the conglomerate detection in SExtractor into morethan two components.We consider linking radii of 1.0, 0.50, and 0.25 Mpc, which wehad previously tested in Hung et al. (2020). Finally, we examinethe effects of including previously known structures or not, whichcan bias our cluster mass function to high mass overdensities, andusing the 𝑅 correction, which will shift our larger overdensitycandidates to lower masses.In total, we consider 144 unique sets of values over five indepen-dent parameters for the purposes of our cluster mass function. Foreach set, we use the Monte-Carlo method described in §4.1 to as-sign the candidates to each mass bin and compute a number densityby taking the median over 1000 iterations. We do this for each setof values, meaning we end up with 144 number densities. Becausewe do not expect any sample of overdensity candidates to be moreindicative of reality than another, we then take the median of thesenumber densities to give us our final mass function, with the 1 𝜎 upper and lower bounds defined by the 84th and 16th percentiles. For our analysis, we use the halo mass function by Tinker et al.(2010), derived from identifying dark matter haloes in N-body sim-ulations of flat Λ CDM cosmology. Using the spherical overdensity(SO) algorithm, haloes are identified as isolated density peaks. Thehalo mass is defined in spherical apertures enclosing overdensities Δ ,defined as the mean interior density relative to the background. Thehalo mass function is not the same as the cluster total mass function,but simulations suggest a tight correlation between halo mass andcluster mass proxies (e.g., Kravtsov et al. 2006; Nagai 2006). Wechose to use the Tinker et al. (2010) halo mass function as it andTinker et al. (2008) are highly cited as a point of comparison forobservational studies, and both models are very nearly equal for ourredshift and mass ranges. We note, however, that choosing other mod-ern halo mass functions (e.g., Crocce et al. 2010; Courtin et al. 2011;Bhattacharya et al. 2011; Angulo et al. 2012; Watson et al. 2013)does not significantly change our results as the differences betweenthe models are smaller than our typical number density uncertainties(Fig. 5).We generate the theoretical models with the Halo Mass Functioncalculator (HMFcalc; Murray et al. 2013) available through an onlineinterface as well as a Python package . The main cosmological pa-rameters that define the halo mass function are Ω 𝑚 and 𝜎 . The other Version 3.0.12; https://github.com/steven-murray/hmf
MNRAS , 1–13 (2020)
RELSE Mass Function Figure 5.
We compare the Tinker et al. (2010) halo mass function with severalof the most recent models available in the Halo Mass Function calculator,fixing 𝑧 = . Δ = parameters do not strongly affect the halo mass function, so we keepthem fixed, adopting the nine-year
Wilkinson Microwave AnisotropyProbe (WMAP9; Hinshaw et al. 2013) parameters, which is availablein HMFcalc as a pre-defined cosmology. We plot our observationalpoints against several sets of Ω 𝑚 and 𝜎 in Figure 6. We fix theredshift to 𝑧 = .
94, which is the mean redshift of our overdensitycandidate sample, a mean which does not depend strongly on struc-ture mass (see Figure 4). When comparing 𝑅 circ and 𝑅 across oursample, we have seen the former be consistently larger. Though wetry to correct our masses to roughly 𝑀 , we plot Δ ranges of 100to 200 to compare with larger radii due to our ignorance in how ourmasses are constructed over the same spatial extents (Fig. 6).Our observational points consistently appear high compared toconcordance cosmology, which is likely a consequence of our surveybeing targeted around previously known clusters. Despite this, wecan still attempt to fit our observed points for Ω 𝑚 and 𝜎 to demon-strate the proof of concept that such constraints are possible from 𝑧 ∼ Ω 𝑚 and 𝜎 , wedefine a grid of values in 0.005 steps for 0 . < Ω 𝑚 < .
600 and0 . < 𝜎 < .
000 which we iterate over in the Tinker et al. (2010)halo mass function at 𝑧 = .
94, with
Δ = Ω 𝑚 is varied in HMFcalc such that the total densityparameter Ω tot remains flat. At each point in the grid, we measurethe 𝜒 difference between the cluster and halo mass functions usinga standard least squares method, which is transformed to a likeli-hood by 𝑒 − 𝜒 / . When fitting for Ω 𝑚 and 𝜎 , we split our observednumber densities into two groups: one that contains all overdensity candidates as described in §4 and one excluding all previously knownstructures (Fig. 7). Depending on the choice of SExtractor detectionand deblending parameters, we recover between 77 to 93% of the56 previously known structures in the ORELSE fields. The knownstructures constitute 13 to 20% of all overdensities in our sample bynumber. At masses greater than log ( 𝑀 / 𝑀 (cid:12) ) = .
5, however, theknown structures make up between 53 to 82% of the sample. Theknown structures are among the most massive in our sample, andthus we expect them to bias our observed high-mass densities.We find Ω 𝑚 = . + . − . and 𝜎 = . + . − . among our com-plete sample, and Ω 𝑚,𝑛𝑘 = . + . − . and 𝜎 ,𝑛𝑘 = . + . − . when the known structures are removed. We plot these fits againsttheir respective observed points in Figure 8. Under the Λ CDM model,WMAP9 gives Ω 𝑚 = . ± .
025 and 𝜎 = . ± . . Ourbest-fit Ω 𝑚 values agree with the concordance value within 1 𝜎 , while 𝜎 is discrepant at the ∼ ∼ 𝜎 levels when the known struc-tures are included and excluded, respectively. From the right panelin Figure 6, we see that our observed points closely follow a lineof fixed 𝜎 for Ω 𝑚 = .
27, which consequently shifts 𝜎 higher tocompensate. Likewise, the best-fit Ω 𝑚 of our sample with and with-out the known structures are very similar, but 𝜎 is slightly smallerfor the sample without the known structures. However, we note thata considerable fraction ( ∼ 𝜎 parameter. We present our Ω 𝑚 and 𝜎 fits as a proof of concept that cosmo-logical fitting can be done with optical/NIR data at 𝑧 ∼
1, which tothe best of our knowledge has not been done before outside of therelatively local universe. As a result of our consistently high numberdensities, especially at the high mass end (see discussion in §3.3 and5), while our best-fit Ω 𝑚 is consistent within 1 𝜎 with the concor-dance value, our best-fit 𝜎 is roughly 2 𝜎 higher than the equivalentconcordance value. However, the ORELSE survey was by designtargeted around known large-scale structures, so we would expect tosee more galaxy overdensities per volume than an equivalent fieldsurvey. Though other recent studies such as Abbott et al. (2020)have found a tension in their derived cosmological parameters dueto disagreements between different mass proxies, we do not sharesimilar concerns, at least at the high-mass end, since our dynamicalmass estimates are within the error bars of the X-ray, lensing, andSZ mass measurements found in other studies. However, the issuein Abbott et al. (2020) was primarily at the low-mass end. To checkif this is potentially an issue for our results, we excluded the twolowest mass bins in our observational mass function and re-derivedthe cosmological parameters. We found no meaningful difference inour results, with Ω 𝑚 and 𝜎 being entirely within the error bars ofthe values we found in §5.Optical-wavelength mass function studies conducted for the localuniverse have recently begun to emerge. Abdullah et al. (2020) de-rived a cluster mass function using 756 Sloan Digital Sky Survey(SDSS; Albareti et al. 2017) clusters with masses estimated from the https://lambda.gsfc.nasa.gov/product/map/dr5/params/lcdm_wmap9.cfm MNRAS000
1, which tothe best of our knowledge has not been done before outside of therelatively local universe. As a result of our consistently high numberdensities, especially at the high mass end (see discussion in §3.3 and5), while our best-fit Ω 𝑚 is consistent within 1 𝜎 with the concor-dance value, our best-fit 𝜎 is roughly 2 𝜎 higher than the equivalentconcordance value. However, the ORELSE survey was by designtargeted around known large-scale structures, so we would expect tosee more galaxy overdensities per volume than an equivalent fieldsurvey. Though other recent studies such as Abbott et al. (2020)have found a tension in their derived cosmological parameters dueto disagreements between different mass proxies, we do not sharesimilar concerns, at least at the high-mass end, since our dynamicalmass estimates are within the error bars of the X-ray, lensing, andSZ mass measurements found in other studies. However, the issuein Abbott et al. (2020) was primarily at the low-mass end. To checkif this is potentially an issue for our results, we excluded the twolowest mass bins in our observational mass function and re-derivedthe cosmological parameters. We found no meaningful difference inour results, with Ω 𝑚 and 𝜎 being entirely within the error bars ofthe values we found in §5.Optical-wavelength mass function studies conducted for the localuniverse have recently begun to emerge. Abdullah et al. (2020) de-rived a cluster mass function using 756 Sloan Digital Sky Survey(SDSS; Albareti et al. 2017) clusters with masses estimated from the https://lambda.gsfc.nasa.gov/product/map/dr5/params/lcdm_wmap9.cfm MNRAS000 , 1–13 (2020) Hung et al.
Figure 6.
We show how the Tinker et al. (2010) halo mass function changes with Ω 𝑚 and 𝜎 at a fixed redshift of 𝑧 = .
94, which is the median redshift ofour overdensity candidate sample. Each band shows the range covered by 100 < Δ < virial theorem. Their sample had a mean redshift of 𝑧 = .
085 anda similar mass range to our work. The authors find good agreementwith the Tinker et al. (2008) model, only significantly falling shortat log ( 𝑀 / 𝑀 (cid:12) ) <
14, suggesting a possible sample incompletenesswhich we were able to avoid at the same mass threshold. However,due to their smaller density uncertainties across the mass range,Abdullah et al. (2020) were able to recover tighter cosmological con-straints of Ω 𝑚 = . + . − . and 𝜎 = . + . − . , with systematicerrors of ± .
041 and ± .
035 respectively. Though our sample is at anorder of magnitude higher redshift, our errors are only two to threetimes as large as the combined random and systematic errors foundby Abdullah et al. (2020).Cluster count studies at higher redshifts have traditionally onlybeen done with X-ray and SZ surveys, though even then mass func-tion studies have been few. Vikhlinin et al. (2009) derived a massfunction with two cluster samples. The high-redshift sample had37 clusters derived from the 400 square degree ROSAT serendip-itous survey (Burenin et al. 2007) and covered the redshift range0 . ≤ 𝑧 ≤ .
90. The low-redshift sample consisted of the 49 high-est flux clusters detected in the ROSAT All-Sky Survey and wasover 0 . ≤ 𝑧 ≤ .
25. Both samples were later observed by theChandra X-ray Observatory, providing spectral data that enabledseveral high-quality total mass estimators. Cluster masses are esti-mated using the X-ray luminosity and total mass relation. Both sam-ples approximately cover the mass range 14 < log ( 𝑀 / 𝑀 (cid:12) ) < Ω 𝑚 = . ± .
043 and 𝜎 = . ± . ± .
037 and ± . 𝜎 do not significantly change when measuredwith only the low-redshift sample and then again with the total sam-ple including the high-redshift data, which the authors argue impliesthe 𝜎 measurement is dominated by the more accurate local clusterdata.Bocquet et al. (2019) derived cosmological constraints with agalaxy cluster sample of 365 candidates over the redshift range 0 . <𝑧 < .
75 from the 2500 square degree SPT-SZ survey. Some clustersin the sample were also supplemented with optical weak gravitationallensing or X-ray measurements. Through using SZ, X-ray, and weaklensing mass proxies, the sample is estimated to cover a mass rangeof approximately 14 . < log ( 𝑀 / 𝑀 (cid:12) ) < .
3. The authors findconstraints of Ω 𝑚 = . ± . 𝜎 = . ± .
037 with theTinker et al. (2008) halo mass function.With ORELSE and VMC mapping, we have the advantage of beingsensitive to lower mass ranges than traditional X-ray and SZ surveystudies. X-ray studies, however, will soon enjoy a boon of data withthe ongoing all-sky survey by the extended Roentgen Survey withan Imaging Telescope Array (eROSITA; Merloni et al. 2012) instru-ment on the Spectrum-Roentgen-Gamma (SRG) mission, which willproduce on the order of 10,000 detections of the hot intergalacticmedium of galaxy clusters. VMC mapping itself is adaptable to anysimilar photometric and spectroscopic dataset, and thus has greatpotential when combined with future, larger optical surveys.Spectroscopic redshifts are tremendously useful for cluster studiesas they provide highly accurate information on where galaxies aredistributed along the line-of-sight, but they have been traditionallydifficult to obtain due to their large time commitment. The Subaru
MNRAS , 1–13 (2020)
RELSE Mass Function Figure 7.
We fit for Ω 𝑚 and 𝜎 using a Tinker et al. (2010) halo mass function at 𝑧 = .
94, with
Δ = 𝜎 uncertainties are given by the 16th and 84th percentiles of the 1D folded likelihood functions. We show the fittedhalo mass functions in Figure 8. Prime Focus Spectrograph (PFS ; Takada et al. 2014) is an opticaland NIR wavelength spectrograph expected to be ready for scientificuse in 2022. Situated on the 8.2-m Subaru Telescope, PFS is capa-ble of obtaining spectra of galaxies that were technologically out ofreach before. With a 1.3 degree diameter field-of-view, it is capableof simultaneous spectral observation of up to 2400 targets. The forth-coming 100 night PFS cosmology survey aims to sample galaxiesover a redshift range of 0 . ≤ 𝑧 ≤ . ℎ − Gpc , approximately a thousand times larger than ORELSE’sspectroscopic footprint. The ground-based Maunakea SpectroscopicExplorer is a 11.25-m telescope that will replace the 3.6-m Canada-France-Hawaii Telescope (CFHT). Construction on the telescope is https://pfs.ipmu.jp/intro.html https://mse.cfht.hawaii.edu/ anticipated to begin in 2023, with full science operations commenc-ing in August 2026. Its spectrographs can accomodate roughly 3,000spectra simultaneously. Combined with the telescope’s 1.5 square de-gree field-of-view, it will be able to obtain many more high-qualityspectroscopic redshifts from the ground with less time than was pos-sible before.Photometric redshifts are less accurate than spectroscopic red-shifts, but they generally have more uniform spatial distributions andthus enable more complete mapping of the density field of galaxieswhen combined with spectroscopic redshifts. Photometric redshiftscomplete to deeper magnitudes will be in no short supply with up-coming all-sky surveys. The ground-based Large Synoptic SurveyTelescope (LSST; Ivezić et al. 2019) is an optical survey expectedto begin operations by 2022, with the aim of uniformly observing18,000 square degrees of the sky 800 times over 10 years. Its six-band MNRAS000
Δ = 𝜎 uncertainties are given by the 16th and 84th percentiles of the 1D folded likelihood functions. We show the fittedhalo mass functions in Figure 8. Prime Focus Spectrograph (PFS ; Takada et al. 2014) is an opticaland NIR wavelength spectrograph expected to be ready for scientificuse in 2022. Situated on the 8.2-m Subaru Telescope, PFS is capa-ble of obtaining spectra of galaxies that were technologically out ofreach before. With a 1.3 degree diameter field-of-view, it is capableof simultaneous spectral observation of up to 2400 targets. The forth-coming 100 night PFS cosmology survey aims to sample galaxiesover a redshift range of 0 . ≤ 𝑧 ≤ . ℎ − Gpc , approximately a thousand times larger than ORELSE’sspectroscopic footprint. The ground-based Maunakea SpectroscopicExplorer is a 11.25-m telescope that will replace the 3.6-m Canada-France-Hawaii Telescope (CFHT). Construction on the telescope is https://pfs.ipmu.jp/intro.html https://mse.cfht.hawaii.edu/ anticipated to begin in 2023, with full science operations commenc-ing in August 2026. Its spectrographs can accomodate roughly 3,000spectra simultaneously. Combined with the telescope’s 1.5 square de-gree field-of-view, it will be able to obtain many more high-qualityspectroscopic redshifts from the ground with less time than was pos-sible before.Photometric redshifts are less accurate than spectroscopic red-shifts, but they generally have more uniform spatial distributions andthus enable more complete mapping of the density field of galaxieswhen combined with spectroscopic redshifts. Photometric redshiftscomplete to deeper magnitudes will be in no short supply with up-coming all-sky surveys. The ground-based Large Synoptic SurveyTelescope (LSST; Ivezić et al. 2019) is an optical survey expectedto begin operations by 2022, with the aim of uniformly observing18,000 square degrees of the sky 800 times over 10 years. Its six-band MNRAS000 , 1–13 (2020) Hung et al.
Figure 8.
We plot the best-fit Tinker et al. (2010) halo mass functions with theparameters found in Figure 7. The solid line follows the best-fit parameters,while the shaded region shows the maximum variation among the 1 𝜎 rangesfor Ω 𝑚 and 𝜎 . photometry will yield photometric redshifts for billions of galaxies.The European Space Agency mission Euclid is a space telescopeoperating at optical and NIR wavelengths planned to launch in 2022.It will measure the redshifts of galaxies out to 𝑧 ∼ With the extensive photometric and spectroscopic dataset from theORELSE survey and Voronoi tessellation Monte-Carlo mapping, wehave derived the first observational cluster mass function at opticaland NIR wavelengths outside of the relatively local universe.Our original methodolgy in Hung et al. (2020) recovered 51previously known structures and found 402 new overdensity can-didates over the redshift range 0 . < 𝑧 < .
37 and mass range10 . < log ( 𝑀 / 𝑀 (cid:12) ) < .
8. However, we had for the most part setaside the issue of separating blended structures in favor of the mostgeneral case of finding any overdensity in the data. As the cluster massfunction reports the number density as a function of mass, we neededin this paper to take caution with what candidates in our sample weresingle structures or not. In total, we had five independent parame-ters that affected the numbers and masses of candidates we obtained https://sci.esa.int/web/euclid/ from the same dataset. We also limited our sample to the mass range13 . < log ( 𝑀 / 𝑀 (cid:12) ) < .
8, which is where we had purity and com-pleteness estimates from our tests with mock catalogs. We had 144unique sets of values for the five overdensity candidate parameters,where the median total number of overdensity candidates was 241and the median redshift was 𝑧 = .
94. We derived the cluster massfunction through treating our overdensity candidates sample with aMonte-Carlo scheme and applied purity and completeness correc-tions as functions of redshift, mass, and spectroscopic fraction.We compared our observational mass function to the Tinker et al.(2010) halo mass function, set to 𝑧 = .
94 to match the medianredshift of our sample, and using
Δ =
200 and WMAP9 cosmol-ogy. We find cosmological constraints of Ω 𝑚 = . + . − . and 𝜎 = . + . − . . While our Ω 𝑚 value agrees with the concordancevalue within 1 𝜎 , our 𝜎 value is high by approximately 2 𝜎 . Thisdiscrepancy is a consequence of our inflated observed number densi-ties, brought about because ORELSE was designed to be targeted atknown large-scale structures. In an attempt to mitigate this, we fittedfor Ω 𝑚 and 𝜎 again after removing all previously known structuresfrom our sample, which gave us constraints of Ω 𝑚,𝑛𝑘 = . + . − . and 𝜎 ,𝑛𝑘 = . + . − . , dropping the discrepancy in 𝜎 to roughly1.5 𝜎 .The Ω 𝑚 and 𝜎 constraints we present here are meant to be takenas a proof of concept that pure optical/NIR cluster abundance can bea viable cosmological probe at moderately high redshifts. Though ithas limitations when applied to data obtained through a biased surveystrategy, our methology has strong potential when combined with theseveral large optical surveys on the horizon, which will yield manymore photometric and spectroscopic redshifts than what was possibleto obtain before. Along with advancements in X-ray surveys whichwill offer complementary results for investigating cluster evolution,we can expect cluster-based constraints to grow into an even morepowerful cosmological probe in the near future. ACKNOWLEDGEMENTS
We would like to thank Steven Murray and Chris Power for their promptguidance to our questions with using HMFcalc. We would also like to thankAlexey Vikhlinin for his useful comments on our mass function analysis andcomparisons. We also thank the anonymous referee for their valuable sug-gestions. Some of the material presented in this paper is supported by theNational Science Foundation under Grant Nos. 1411943 and 1908422. Thiswork was additionally supported by the France-Berkeley Fund, a joint ven-ture between UC Berkeley, UC Davis, and le Centre National de la RechercheScientifique de France promoting lasting institutional and intellectual coop-eration between France and the United States. This study is based, in part,on data collected at the Subaru Telescope and obtained from the SMOKA,which is operated by the Astronomy Data Center, National Astronomical Ob-servatory of Japan. This work is based, in part, on observations made with theSpitzer Space Telescope, which is operated by the Jet Propulsion Laboratory,California Institute of Technology under a contract with NASA. UKIRT issupported by NASA and operated under an agreement among the Univer-sity of Hawaii, the University of Arizona, and Lockheed Martin AdvancedTechnology Center; operations are enabled through the cooperation of theEast Asian Observatory. When the data reported here were acquired, UKIRTwas operated by the Joint Astronomy Centre on behalf of the Science andTechnology Facilities Council of the U.K. This study is also based, in part,on observations obtained with WIRCam, a joint project of CFHT, Taiwan,Korea, Canada, France, and the Canada-France-Hawaii Telescope which isoperated by the National Research Council (NRC) of Canada, the InstitutNational des Sciences de l’Univers of the Centre National de la RechercheMNRAS , 1–13 (2020)
RELSE Mass Function Scientifique of France, and the University of Hawai’i. Some portion of thespectrographic data presented herein was based on observations obtained withthe European Southern Observatory Very Large Telescope, Paranal, Chile,under Large Programs 070.A-9007 and 177.A-0837. The remainder of thespectrographic data presented herein were obtained at the W.M. Keck Ob-servatory, which is operated as a scientific partnership among the CaliforniaInstitute of Technology, the University of California, and the National Aero-nautics and Space Administration. The Observatory was made possible bythe generous financial support of the W.M. Keck Foundation. We thank theindigenous Hawaiian community for allowing us to be guests on their sacredmountain, a privilege, without which, this work would not have been possible.We are most fortunate to be able to conduct observations from this site.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable request to thecorresponding author.
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