Galaxy and Mass Assembly (GAMA): Tracing galaxy environment using the marked correlation function
U. Sureshkumar, A. Durkalec, A. Pollo, M. Bilicki, J. Loveday, D. J. Farrow, B. W. Holwerda, A. M. Hopkins, J. Liske, K. A. Pimbblet, E. N. Taylor, A. H. Wright
AAstronomy & Astrophysics manuscript no. main © ESO 2021February 9, 2021
Galaxy and Mass Assembly (GAMA)
Tracing galaxy environment using the marked correlation function
U. Sureshkumar , A. Durkalec , A. Pollo , , M. Bilicki , J. Loveday , D. J. Farrow , B. W. Holwerda ,A. M. Hopkins , J. Liske , K. A. Pimbblet , E. N. Taylor , and A. H. Wright Astronomical Observatory of the Jagiellonian University, ul. Orla 171, 30-244 Kraków, Polande-mail: [email protected] National Centre for Nuclear Research, ul. Pasteura 7, 02-093 Warsaw, Poland Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32 /
46, 02-668 Warsaw, Poland Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany Department of Physics and Astronomy, 102 Natural Science Building, University of Louisville, Louisville KY 40292, USA Australian Astronomical Optics, Macquarie University, 105 Delhi Rd, North Ryde, NSW 2113, Australia Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany E. A. Milne Centre for Astrophysics, University of Hull, Cottingham Road, Kingston-upon-Hull, HU6 7RX, UK Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn 3122, Australia Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for CosmologicalLensing, 44780 Bochum, GermanyReceived 5 February 2021 / Accepted YYY
ABSTRACT
Context.
Galaxies are biased tracers of the underlying network of dark matter. The strength of this bias depends on various galaxyproperties, as well as on redshift. One of the methods used to study these dependences of the bias are measurements of galaxyclustering. Such studies are made using galaxy samples from various catalogues – frequently bearing their own biases related tosample selection methods. It is therefore crucial to understand how sample choice influences the clustering measurements, and whichgalaxy property is the least biased (the most direct) tracer of the galaxy environment.
Aims.
We investigate how di ff erent galaxy properties – luminosities in u , g, r , J , K -bands, stellar mass, star formation rate and specificstar formation rate trace the environment in the local universe. We also study the e ff ect of survey flux limits on galaxy clusteringmeasurements. Methods.
We measure the two-point correlation function (2pCF) and marked correlation functions (MCFs) using the aforementionedproperties as marks. We use nearly stellar-mass-complete galaxy sample in the redshift range 0 . < z < .
16 from the Galaxy AndMass Assembly (GAMA) survey with a flux limit of r < .
8. Further, we impose a brighter flux limit of r < . ff ects galaxy clustering analysis. We compare our results to measurements from the SloanDigital Sky Survey (SDSS) with flux limits of r < . r < . Results.
We show that the stellar mass is the best tracer of galaxy environment, the K -band luminosity being a good substitute,although such a proxy sample misses close pairs of evolved, red galaxies. We also confirm that the u -band luminosity is a good, butnot a perfect proxy of star formation rate in the context of galaxy clustering. We observe an e ff ect of the survey flux limit on clusteringstudies – samples with a higher flux limit (smaller magnitude) miss some information about close pairs of starburst galaxies. Key words. large-scale structure of Universe – galaxies: statistics – galaxies: formation – galaxies: evolution – cosmology: observa-tions
1. Introduction
Local galaxy observations reveal the large scale structure (LSS)of the Universe to be a rich network of filaments, walls, nodesand voids (de Lapparent et al. 1986; Bond et al. 1996; Alpaslanet al. 2014). According to the Λ CDM cosmological model thesestructures are built of two main elements: baryonic and darkmatter. The former exists in form of stars, gas and dust – thesecan be traced at di ff erent wavelengths using large sky surveys.Dark matter, however, although gravitationally dominant, cannotbe observed directly. Therefore, we rely on the visible baryonicmatter observations to indirectly trace the underlying dark matterdistribution. One of the methods used to connect baryonic and dark mat-ter involves measurements of the galaxy two-point correlationfunction (2pCF, Peebles 1980). This powerful statistical tool de-scribes the spatial distribution of galaxies and has been exten-sively used in the past to quantify their clustering and its var-ious dependences: on luminosity (Norberg et al. 2001; Polloet al. 2006; de la Torre et al. 2007; Meneux et al. 2009; Ze-havi et al. 2011; Marulli et al. 2013; Guo et al. 2015; Farrowet al. 2015), stellar mass (Meneux et al. 2008; Marulli et al.2013; Beutler et al. 2013; Dolley et al. 2014; Skibba et al. 2015;Durkalec et al. 2018), star formation rate (SFR, Hartley et al.2010; Mostek et al. 2013), color (Zehavi et al. 2005; Coil et al.2008; Skibba et al. 2014), and spectral type (Norberg et al. 2002;Meneux et al. 2006). The general conclusion from all these stud- Article number, page 1 of 14 a r X i v : . [ a s t r o - ph . C O ] F e b & A proofs: manuscript no. main ies is that galaxy clustering strongly depends on galaxy prop-erties. More luminous, massive, redder and early type galaxiesexhibit stronger clustering – tend to exist in denser regions ofthe universe – than their less massive, bluer and later-type coun-terparts.These observations can be explained in the framework of the Λ CDM cosmology and hierarchical model of structure forma-tion. Small density fluctuations in the early universe evolved un-der gravity to form the present LSS (Springel et al. 2005). Theinitially stronger overdensities evolved faster resulting in the for-mation of self-bound clumps of dark matter called dark matterhaloes. Such haloes provided the gravitational potential to trapthe baryonic matter and thereby form galaxies at their centers(Press & Schechter 1974; White & Rees 1978). Therefore, it isexpected that the mass of the parent halo plays a major role indefining galaxy properties like luminosity, stellar mass, color,SFR, etc. (e.g. More et al. 2009; Gu et al. 2016). Large and mas-sive haloes had potentials strong enough to form bigger, moremassive and more luminous galaxies. There is also a correlationbetween the halo formation process and the LSS surrounding it.That is, more massive haloes tend to occupy denser regions ofthe structure (Mo & White 1996). These two correlations: be-tween halo mass and galaxy properties, and between halo massand its environment prompt a correlation between galaxy prop-erties and environment.Studies of these dependences between galaxy properties andenvironment are crucial to understand structure formation in theuniverse, and over the past decade there has been remarkableprogress in the development of galaxy formation models describ-ing connections between dark matter haloes and their galaxies(see Somerville & Davé 2015, for a review). Methods used inthese models include: numerical hydrodynamic techniques (e.g.McCarthy et al. 2012; Vogelsberger et al. 2013; Kannan et al.2014), semi-analytic models (SAM, e.g. Baugh 2006; Benson2012; Linke et al. 2020) and even empirical methods in whichphysical constraints are taken entirely from observations (e.g.Mutch et al. 2013; Moster et al. 2020; Grylls et al. 2020). Weare now able to simulate the physics of galaxy formation and, tosome extent, link galaxy properties to the host halo properties.For example, the
UNIVERSEMACHINE (Behroozi et al. 2019)that reasonably parametrizes the galaxy growth-halo assemblycorrelation and the
SHARK (Lagos et al. 2018) that agrees fairlywell with observations (Bravo et al. 2020). But there is still aneed for improvement in understanding the process of galaxyformation (see Naab & Ostriker 2017, for a review). There hasnot been yet a method that would perfectly reconstruct the ob-served dependence of galaxy properties on halo mass and en-vironment and the dependences of galaxy clustering on galaxyproperties. Hence, better understanding of how di ff erent galaxyproperties trace the environment is needed to establish betterconstraints on galaxy formation and evolution models. It wouldbe preferred if this understanding came from galaxy observationsrather than simulations.There is a problem that has to be faced here: galaxy cluster-ing strength depends on the photometric passband in which thegalaxies are selected for measurement (Milliard et al. 2007; de laTorre et al. 2007; Zehavi et al. 2011; Skibba et al. 2014). In otherwords, it is common that di ff erent works report various cluster-ing strengths for galaxies selected using di ff erent methods. Forexample, in the optical range, Zehavi et al. (2005) measured cor-relation functions (CFs) in volume-limited samples of galaxiesfrom the Sloan Digital Sky Survey (SDSS) selected in bins of the r -band absolute magnitude. They observed that galaxies brighterin the r -band show stronger clustering than the fainter ones. Sim- ilar behaviour of clustering was observed for the B -band fromthe VIMOS survey (Marulli et al. 2013), g -band from PRIMUS(Skibba et al. 2014) and K -band from HiZELS (Sobral et al.2010). Measurements in the u -band, however, show an oppositetrend. Galaxies luminous in that band tend to exist in low-densityregions of the universe, whereas their u -band fainter counterpartsare preferentially found in high-density locations (Deng 2012).Additionally, measurements based on farther infra-red (IR) indi-cate stronger clustering than that measured for galaxies observedat optical wavelengths (Oliver et al. 2004; Pollo et al. 2013a,b).Moreover, Heinis et al. (2004) and Milliard et al. (2007) reportedweaker clustering of ultra-violet (UV) galaxies in the local uni-verse compared to optical and IR galaxies. All these results im-ply that galaxies selected based on di ff erent properties trace thelocal environment di ff erently. The question is: which of theseproperties is the best tracer of this environment?The main aim of this paper is to answer that question. In par-ticular, we will show how di ff erent galaxy properties trace thesmall-scale galaxy clustering. We will demonstrate which prop-erty can be a better tracer of galaxy environment. Environmenthas been defined in many di ff erent ways in the past (Muldrewet al. 2012); here we define it as the galaxy overdensity aroundthe object.We will also study the possible influence of selection meth-ods on clustering results. In particular we will show which pho-tometric passband best serves as a proxy for stellar mass in theabsence thereof, and how survey flux limitations can influenceclustering results. Frequently, in the literature, luminosity andstellar mass are considered to be one-to-one correlated – moreluminous galaxies are assumed to be more massive (Blanton& Moustakas 2009). In particular, near infra-red (NIR) lumi-nosity is known to be a good proxy of the galaxy stellar mass(Kochanek et al. 2001). Mid infra-red (MIR) fluxes, particularlythose with 3.4 µ m and 4.6 µ m wavelengths are also reliable trac-ers of galaxy stellar mass (Jarrett et al. 2013; Cluver et al. 2014).For instance, K -band selected samples are used to construct stel-lar mass limited samples with high completeness (van Dokkumet al. 2006; Taylor et al. 2009). However, it is yet unclear if sucha sample can be a perfect proxy for clustering measurements.Better understanding on this issue will give us a better idea onthe cautions to be taken while these kinds of proxy samples areused for clustering studies.In addition, galaxy surveys are inevitably flux-limited.Therefore, extra care has to be taken while working with stellarmass selected samples extracted from such surveys. They tend tomiss galaxies which are massive enough to pass the mass selec-tion, but not luminous enough to reach the flux limit of the sur-vey (Meneux et al. 2008, 2009; Marulli et al. 2013). This e ff ectmakes such samples incomplete and hence can bias the galaxyclustering measurements. In this work we will try to understandhow the flux limit of survey a ff ects the measured clustering andwhat steps are needed to account for resulting inaccuracies.In our work we will use the Galaxy And Mass Assemblyspectroscopic survey (GAMA; Driver et al. 2009). We chooseGAMA over SDSS due to its high completeness ( > . r petro < . ff er from fibre collisions that a ff ect the closegalaxy pairs in SDSS (Robotham et al. 2010). It also providesreliable measurements of absolute magnitudes in a wide wave-length range, stellar masses and SFR. The GAMA survey hasbeen used for various aspects of environmental e ff ects, in par-ticular the impact of group, cluster, local, and large-scale en-vironment on galaxy properties (Wijesinghe et al. 2012; Bur-ton et al. 2013; Brough et al. 2013; McNaught-Roberts et al. Article number, page 2 of 14. Sureshkumar et al.: Tracing galaxy environment using the marked correlation function in GAMA mark which is defined as any measurable property of the galaxy.The MCF accounts for the clustering of positions of the marks(i.e., galaxies of a given property). Hence, MCF measurementswith di ff erent galaxy properties as marks help us to study howthese di ff erent properties trace the galaxy clustering, particularlyon small scales (Sheth 2005). Marked statistics have been widelyused to show that closer pairs of galaxies are more luminous,redder and older than pairs which have larger separations (Beis-bart & Kerscher 2000; Skibba et al. 2006; Sheth et al. 2006).Additionally, Sheth et al. (2005) showed that these observa-tions are in qualitative agreement with the semi-analytic galaxyformation models. Gunawardhana et al. (2018) have recentlyused marked statistics on a set of luminosity- and stellar-mass-selected galaxy samples from GAMA with SFR, specific SFR(sSFR) and ( g − r ) rest color as marks. They observed that sSFR isa better tracer of interactions between star forming galaxies thancolor. Riggs et al. (in prep.) uses marked correlation function toexplore the clustering of galaxy groups in GAMA.In our study we compute the MCFs on stellar-mass-selectedsamples in the redshift range 0 . < z < .
16 using absolutemagnitudes in the u , g, r , J , K -bands, stellar mass, SFR and sSFR(SFR per unit stellar mass) of galaxies as marks. Additionally,we explore the e ff ect of apparent flux limits on the correlationbetween small scale clustering and galaxy properties. For thispurpose, we impose various flux limits to the parent sample fromthe GAMA survey. Further we compare the MCFs using di ff er-ent marks to see how these functions are a ff ected by the changein flux limit. We also compare the measurements in our GAMAsample with those from SDSS. We select samples from SDSSas it provides a larger number of galaxies to brighter flux limitsthan GAMA.This paper is structured as follows. In Sect. 2 we de-scribe properties of the GAMA survey and our sample selectionmethod. Di ff erent clustering techniques and their definitions aredescribed in Sect. 3. Clustering measurements are presented inSect. 4. The results are discussed and compared with other worksin Sect. 5 and finally concluded in Sect. 6.Throughout the paper, a flat Λ CDM cosmological modelwith Ω M = . Ω Λ = . h = H /
100 km s − Mpc − . All galaxy prop-erties are measured using h = .
7. The distances are expressedin comoving coordinates and are in units of h − Mpc.
2. Data
Galaxy And Mass Assembly (GAMA) is a spectroscopic andmultiwavelength galaxy survey which aims to test the Λ CDMmodel of structure formation and to study the galaxy evolutionthrough the latest generation of ground-based and space-borne,wide-field survey facilities (Driver et al. 2011; Liske et al. 2015).It provides a comprehensive survey of galaxy populations bybringing together data from eight ground-based surveys and fourspace missions. GAMA covers three equatorial regions named G09, G12 and G15 and two southern regions G02 and G23. Fordetailed description on the GAMA survey, we refer the reader toDriver et al. (2009), Robotham et al. (2010), Driver et al. (2011)and Liske et al. (2015), while below we briefly describe the sur-vey details important in the context of our work.We exploit the main r -band limited data from GAMA IIequatorial regions with targets drawn primarily from SDSS DR7(Abazajian et al. 2009). For extinction-corrected r -band Pet-rosian magnitudes (Petrosian 1976) limited at r petro < . .
48% in the equatorial regions. This ex-cellent completeness of GAMA is achieved by repeated survey-ing of the same field (Robotham et al. 2010). This observationtechnique removes biases against close pairs, making GAMA anideal survey for clustering measurements.In this study we select galaxies from the GAMA II main sur-vey (
SURVEY_CLASS ≥
4) in the equatorial regions with spec-troscopic redshifts in the range 0 . < z < .
16. The redshiftsof GAMA objects were measured using the software autoz , asdescribed in Baldry et al. (2014) and are corrected for the localflow using Tonry et al. (2000) model, tapered smoothly to thecosmic microwave background rest frame for z ≥ .
03. Liskeet al. (2015) provide a detailed assessment of the quality and reli-ability of these redshifts. We are using only secure redshifts withquality flag nQ ≥
3, which assures that the redshift has > VIS_CLASS = VIS_CLASS = VIS_CLASS = tellar M asses L ambdarv
20 DMU (Tay-lor et al. 2011; Wright et al. 2016). The stellar masses are basedon the methods of Taylor et al. (2011) applied to the lambdar photometry of Wright et al. (2016). They are based on the stellarpopulation synthesis (SPS) modelling of broadband photome-try using stellar evolution models by Bruzual & Charlot (2003),assuming a Chabrier (2003) initial mass function and Calzettiet al. (2000) dust law. The absolute magnitudes are inferred fromthe SPS fits and are corrected for internal dust extinction and k -corrected to z =
0. As the fits are constrained to the restframewavelength range of 300 - 1100 nm, the J and K -band absolutemagnitudes we use are extrapolations of the fit to data. The stel-lar masses and absolute magnitudes are fluxscale correctedas described in Taylor et al. (2011) in order to account for thedi ff erence in aperture matched and Sérsic photometry. Galax-ies with physically unrealistic fluxscale values are not con-sidered for our analysis. The SFRs and sSFRs are taken fromthe DMU M ag P hysv
06 and are estimated using the energy bal-ance Spectral Energy Distribution (SED)-fitting code magphys (da Cunha et al. 2008). All the quantities are derived for the con-cordance ( Ω M , Ω Λ , h ) = (0 . , . , .
7) cosmology. For cluster-ing measurements, we use the GAMA random galaxy catalogue(R andomsv
02 DMU) by Farrow et al. (2015). In the random cata-logue, we assign stellar mass to each random galaxy by matchingon the
CATAID of the real galaxy.
Our full sample counts 34455 galaxies in the redshift range 0 . < z < .
16 with apparent flux limit r petro < . Article number, page 3 of 14 & A proofs: manuscript no. main
Table 1.
Properties of the GAMA equatorial regions used in this work.The columns represent the number of galaxies, median redshift and thearea of the regions.
GAMA region N gal z median Area[deg ]G09 7863 0.14 60G12 12652 0.13 60G15 13940 0.13 60Total 34455 0.13 180given in Table 1. The aforementioned redshift range is chosen tooptimally select volume-limited samples that include low-massgalaxies. All these galaxies have reliable spectroscopic redshiftand well-measured absolute magnitudes, stellar mass, SFR andsSFR.To study the environmental dependence of luminosity in dif-ferent bands, stellar mass, SFR and sSFR, we define a nearlystellar-mass-complete sample by applying an additional stellarmass cut of log (M (cid:63) / M (cid:12) ) min = .
3. This is referred to as Sam-ple A
1. As a representative example of the selection technique,in Fig. 1 we show the selection cut for the sample A r < . ff erent flux limits, we se-lect two stellar-mass-selected samples in the same redshift rangewith a stellar mass cut of log (M (cid:63) / M (cid:12) ) min = . ff erentflux limits: Sample B r < . B r < .
8. Thestellar mass and redshift distribution of these samples are shownin Fig. 2. The properties of all the selected samples are givenin Table 2. All the selected samples contain a su ffi cient numberof galaxies for reliable clustering measurements. For all the se-lected samples, random samples are selected from R andomsv r -bandapparent magnitude cut and stellar mass cut. Apart from the comparisons between B B
2, we also com-pare the results with the Sloan Digital Sky Survey (SDSS;York et al. 2000) to understand better how brighter flux lim-its can a ff ect our measurements. We use the LSS catalogueand the corresponding random catalogue generated from SDSSIII Baryon Oscillation Spectroscopic Survey (BOSS; Dawsonet al. 2013) Data Release 12 (DR12; Alam et al. 2015). TheSDSS BOSS DR12 catalogue encompasses massive galaxiespartitioned into two non-overlapping redshift bins named as‘LOWZ’ and ‘CMASS’ which cover galaxies in the redshiftranges z < .
43 and z > .
43, respectively. Reid et al. (2016)describe methods used in the target selection of the SDSS galaxydata sets, and give details of the MKSAMPLE code used to cre-ate the LSS catalogue and random catalogue. The total sky cov-erage of the LOWZ DR12 sample is 8337.47 deg . In this work,we make use of LOWZ galaxies in the North Galactic Cap in theredshift range 0 . < z < . r -band apparent magnitude from S pec P hoto A ll table by match-ing the angular position within 2 (cid:48)(cid:48) . Stellar masses are then as-signed by cross-matching with the table stellar M ass S tarform - ing P ort using specObjID . The stellar masses are estimated https: // data.sdss.org / sas / dr12 / boss / lss / Fig. 1.
Selection of galaxy subsample A r < .
8. The topand right histograms show the distribution of redshift and stellar massrespectively. The red lines represent the stellar mass cut and the redshiftlimit of the sample A Fig. 2.
Redshift and stellar mass distribution of galaxies in stellar-mass-selected samples B r < .
8, brown dots) and B r < .
8, greencircles). The top and right histograms show the distribution of redshiftand stellar mass respectively. from the best-fit SED obtained from the stellar population modelof Maraston et al. (2009). The fits are performed on the observed u g riz -magnitudes of BOSS galaxies with the spectroscopic red-shift determined using an adaptation of Hyper-Z code of Bol-zonella et al. (2000). The magnitudes used are extinction cor-rected model magnitudes that are scaled to the i -band cmodel magnitude. GAMA and SDSS stellar masses are derived usingdi ff erent photometry. Despite the systematic di ff erences between Article number, page 4 of 14. Sureshkumar et al.: Tracing galaxy environment using the marked correlation function in GAMA
Fig. 3.
Redshift and stellar mass distribution of stellar-mass-selected samples mentioned in Table 2. Left panel represents the C r < .
8, blackdots), C r < .
8, red circles) and C r < .
8, green squares) galaxy samples from GAMA and the right panel shows the C r < .
8, bluedots) and C r < .
8, orange squares) galaxy samples from SDSS. The top and right histograms of both the panels show the distribution ofredshift and stellar mass respectively.
Table 2.
Properties of the galaxy samples in the redshift range 0 . < z < .
16, as used in this study. The columns represent the minimum stellarmass cut, sample label, survey of origin, flux limit, number of galaxies, mean absolute magnitudes in u , g, r , J , K -bands, mean stellar mass and 16-,50- (median) and 84-percentiles of SFR and sSFR of the corresponding sample. The uncertainty with each property mean represents its standarddeviation within the sample. log (cid:16) M (cid:63) M (cid:12) (cid:17) min Sample Survey r lim N gal M mean ± σ u M mean ± σ g M mean ± σ r M mean ± σ J M mean ± σ K log (cid:16) M (cid:63) M (cid:12) (cid:17) mean ± σ SFR (16% , , (M (cid:12) yr − ) sSFR (16% , , ( × − yr − )9.3 A − . ± . − . ± . − . ± . − . ± . − . ± .
07 10 . ± .
50 (0 . , . , .
50) (1 . , . , . B − . ± . − . ± . − . ± . − . ± . − . ± .
65 10 . ± .
27 (0 . , . , .
91) (0 . , . , . B − . ± . − . ± . − . ± . − . ± . − . ± .
60 10 . ± .
28 (0 . , . , .
90) (0 . , . , . C − . ± . − . ± . − . ± . − . ± . − . ± .
54 11 . ± .
22 (0 . , . , .
05) (0 . , . , . C − . ± . − . ± . − . ± . − . ± . − . ± .
53 11 . ± .
21 (0 . , . , .
06) (0 . , . , . C − . ± . − . ± . − . ± . − . ± . − . ± .
52 11 . ± .
21 (0 . , . , .
32) (0 . , . , . C − . ± . − . ± . − . ± .
35 – − . ± .
39 11 . ± .
23 (0 . , . , .
0) (0 . , . , . C − . ± . − . ± . − . ± .
38 – − . ± .
39 11 . ± .
25 (0 . , . , .
0) (0 . , . , . SDSS and GAMA photometry, we find relatively good agree-ment between GAMA and SDSS stellar masses of overlappinggalaxies, taking into account that the common sample is rathersmall (388 objects). The median o ff set between the masses is0 .
18 dex, with a scatter on the order of 0.2 dex. Hence we usethe same stellar mass cuts in GAMA and SDSS to define samplesfor comparison.For better statistics, we fix the brightest magnitude limit inour work to be r < .
8. For the comparison between GAMAand SDSS, we define five stellar-mass-selected samples with thesame stellar mass cut of log (M (cid:63) / M (cid:12) ) > .
8, but di ff erent fluxlimits. This gives subsamples C C C r < . , . , . C C r < . , . K -band absolute magnitudes of SDSS samples men-tioned in Table 2 are derived from the same SED fits from whichSDSS stellar masses are derived. The absolute magnitudes in the u , g, r -bands of SDSS samples are taken from P hotoz table (Becket al. 2016). The angular distribution of random galaxies for eachSDSS sample are taken from the LSS random catalogue and theredshift is randomly assigned from a smoothened N(z) distribu-tion of the corresponding real galaxy sample.
3. Measurement methods
The galaxy two-point correlation function (2pCF), ξ ( r ), is a sta-tistical tool used to measure the clustering of galaxies. It is de-fined as the excess probability above random of observing a pairof galaxies at a given spatial separation r in a volume element dV (Peebles 1980), i.e., dP = n [1 + ξ ( r )] dV , (1)where n is the number density of galaxies. Article number, page 5 of 14 & A proofs: manuscript no. main
It has been observed that CF mostly follows a power-law(Groth & Peebles 1977) given by ξ ( r ) = (cid:32) rr (cid:33) − γ (2)where r and γ are respectively the correlation length and slope.In practice, due to the limitations of galaxy surveys, variousestimators of ξ ( r ) have been proposed to minimize the e ff ects re-lated to the limited number of objects and limited survey areas(e.g. Davis & Peebles 1983; Hamilton 1993). The Landy & Sza-lay (1993) estimator is the most widely used due to its capabilityto minimize the above mentioned problems, and is defined by: ξ ( r ) = (cid:104) DD ( r ) (cid:105) − (cid:104) DR ( r ) (cid:105) + (cid:104) RR ( r ) (cid:105)(cid:104) RR ( r ) (cid:105) , (3)where DD ( r ) is the observed number of galaxy-galaxy pairs withthe separation in the bin centered at r in the real galaxy sample, RR ( r ) is the expected number of such pairs from a random galaxydistribution, DR ( r ) is the number of cross pairs of galaxies be-tween the real and random sample and (cid:104)(cid:105) refers to the quantitynormalized by the total number of such pairs. The random galaxysample reflects the same sky distribution and redshift distributionof the real galaxy sample. The number of random galaxies usedfor the computation is set to be significantly greater (5-10 times)than the number of real galaxies to avoid shot noise on smallerscales.To account for the distortions in CF measurements causedby galaxy peculiar velocities, the comoving redshift space sepa-ration between the galaxies is split into two components: parallel( π ) and perpendicular ( r p ) to the line-of-sight. The CF thus takesform of a two-dimensional grid ξ ( r p , π ). Integrating ξ ( r p , π ) overthe line-of-sight ( π ) direction gives us the projected 2pCF, ω p ( r p )which can be used to recover the real space CF devoid of redshiftspace distortions (Davis & Peebles 1983). It is defined as ω p ( r p ) = (cid:90) π max ξ ( r p , π ) d π. (4)The limit of integration π max has to be reasonable enough to in-clude all the correlated pairs and reduce the noise in the estima-tor. Following Appendix B of Loveday et al. (2018), we choosethe value of π max to be 40 h − Mpc.There have been many studies in GAMA using 2pCF in thepast. The dependence of projected galaxy clustering on variousproperties was studied by Farrow et al. (2015). Loveday et al.(2018) used 2pCF to measure the pairwise velocity distribu-tion in a set of luminosity-selected samples from GAMA. Thesmall scale clustering properties of star forming galaxies wereused by Gunawardhana et al. (2018) to study the interactions be-tween galaxies. Christodoulou et al. (2012) used CF as a tool tocheck the robustness of their photometric redshift estimates. Jar-rett et al. (2017) analyzed the spatial distribution of mid-infraredWide-field Infrared Survey Explorer (WISE) sources observedin the G12 equatorial region of GAMA. Large-scale clusteringof radio galaxies in the Very Large Array Faint Images of theRadio Sky at Twenty-cm (FIRST) survey over the GAMA sur-vey area was studied by Lindsay et al. (2014). The clusteringmeasurements in GAMA were also used by Alam et al. (2020)to model the redshift space distortions. The clustering propertiesof low-redshift ( z < .
3) sub-mm galaxies detected at 250 µ m inthe Herschel-ATLAS (Eales et al. 2010) using the redshift infor-mation from GAMA was done by van Kampen et al. (2012). vanUitert et al. (2018) used angular correlation function of GAMA galaxies as one of the probes to constrain cosmological param-eters. In our work, we examine how galaxy clustering dependson various galaxy properties like luminosities in di ff erent pass-bands, stellar mass and star formation rate using marked corre-lation function. The 2pCF characterizes the galaxy clustering. It can, and suc-cessfully has been (as described in Sect. 1), used to study clus-tering dependences on various properties of galaxies. This isdone by defining the galaxy samples based on the property ofinterest (e.g., luminosity, color or stellar mass). However, afterthe selection and further during the analysis, these properties areleft unconsidered and each galaxy is weighted equally during CFmeasurements. On the other hand, marked statistics allow us tostudy the properties of galaxy clustering by taking into accountthe physical properties (called marks ) of each galaxy in the sam-ple. These marks can be discrete or continuous values like lu-minosity, color, stellar mass, SFR, morphology, etc. (Sheth et al.2005).The marked correlation function (MCF) allows the e ffi cientstudy of the spatial distribution of galaxy properties and theircorrelation with the environment (Skibba et al. 2013). The two-point MCF is defined as: M ( r ) = + W ( r )1 + ξ ( r ) , (5)where ξ ( r ) is the galaxy 2pCF defined by Eq. (3) and W ( r ) isthe weighted CF obtained with the same estimator, but with paircounts computed by weighting each real galaxy in the pair. Thatis, W ( r ) = (cid:104) WW ( r ) (cid:105) − (cid:104) WR ( r ) (cid:105) + (cid:104) RR ( r ) (cid:105)(cid:104) RR ( r ) (cid:105) (6)We adopt multiplicative scheme for pair weighting, e.g., WW ( r ) = (cid:88) i j w i × w j , (7)where w i is the weight of the i th galaxy given by the ratio of itsmark to the mean mark across the sample.The projected two-point MCF is defined as M p ( r p ) = + W p ( r p ) / r p + ω p ( r p ) / r p (8)Essentially, MCF at a scale r tells us if galaxies in pairs separatedby r tend to have larger or smaller values of their mark than themean mark in the entire sample (Sheth & Tormen 2004). For a given property, a stronger MCF signal at a certain scale in-dicates greater probability of finding galaxy pairs for which thegiven property has a larger value for both the galaxies. Hence theproperty that is more dependent on environment is the one corre-sponding to a larger MCF. However, comparing di ff erent MCFsobtained using di ff erent properties is not straightforward (Skibbaet al. 2013). When computing the MCF in a traditional approach,the value of the physical property of a galaxy is considered as itsmark and the CF is directly weighted by the ratio of the givenmark to the mean mark of the sample. Hence, the amplitude of Article number, page 6 of 14. Sureshkumar et al.: Tracing galaxy environment using the marked correlation function in GAMA
MCF depends on the distribution of the marks and the variationsin their formulation (e.g., log or linear). This makes it impossi-ble to directly compare di ff erent MCFs measured using di ff erentproperties as marks, if these properties have di ff erent distribu-tion or formulation. Skibba et al. (2013) developed a solution tothis problem. Each galaxy is given a rank based on the relativestrength of the value of its property in the sample, i.e., a galaxywith the lowest value is given the lowest rank and another onewith a greater value is given a higher rank. This is called rank-ordering the marks. The rank of each galaxy is then used as itsmark to weight the CF. Since all the ranks have a uniform distri-bution on [1, N], the amplitudes of the MCFs thus obtained usingdi ff erent properties can be compared . However, since the weightis given by the rank rather than the property value, any informa-tion contained in the shape of the distribution of the property willbe lost. As we are interested in relative importance of di ff erentproperties for correlation measurements, all the MCFs shown inthis work are rank-ordered. Since the galaxies are clustered and the pair counts in di ff erentbins of r p can include the same galaxies, the values of ω p for dif-ferent bins are correlated. Hence the statistical errors associatedwith clustering measurements are estimated using the covariancematrix obtained from various methods of internal error estima-tion. In our work, we use the jackknife method (Norberg et al.2009) in which we divide the entire sky region into 24 subsam-ples of equal area and create 24 jackknife samples ( N jk =
24) byomitting each of the subsamples in turn.The associated covariance matrix is given by C i j = N jk − N jk N jk (cid:88) k = (cid:16) ω k p ( r i ) − ¯ ω p ( r i ) (cid:17) (cid:16) ω k p ( r j ) − ¯ ω p ( r j ) (cid:17) (9)where ω k p ( r j ) represents the measurement of ω p at r p = r j in the k th jackknife sample and ¯ ω p is the average from N jk samples.The square root of the diagonal elements of C i j gives the errorbar for the ω p at the corresponding bin.We estimate r and γ (the power-law fit parameters of ξ ( r ))from the projected function ω p ( r p ). Using the parametrizationgiven in Eq. (2), the integral in Eq. (4) can be analytically per-formed to give the power-law fit parameters as ω p ( r p ) = r p (cid:32) r p r (cid:33) − γ Γ (cid:16) (cid:17) Γ (cid:16) γ − (cid:17) Γ (cid:16) γ (cid:17) (10)where Γ ( n ) is Euler’s Gamma function (Davis & Peebles 1983).Due to the correlation between bins, we minimise the χ us-ing the full covariance matrix, defined as, χ = (cid:88) i , j (cid:16) ω modp ( r i ) − ω p ( r i ) (cid:17) C − i j (cid:16) ω modp ( r j ) − ω p ( r j ) (cid:17) (11)where C − i j is the inverse of the covariance matrix, ω p ( r i ) is themeasured value of correlation function at r p = r i and ω modp is thepower-law model value given by Eq. (10). The uncertainties inthe power-law parameters r and γ presented in this work are de-fined by the 68.3% joint confidence levels (Chapter 15.6 of Presset al. 1992). We refer the reader to Fisher et al. (1994) and Polloet al. (2005) for detailed description on the fitting procedure toestimate the power-law parameters. As the errors in W p ( r p ) and ω p ( r p ) are strongly correlated,simply summing them in quadrature gives an overestimate forthe error in M p ( r p ). A much better approximation of the uncer-tainty is obtained by randomly scrambling the marks among thegalaxies and remeasuring M . This is repeated ∼
100 times andthe standard deviation around the mean gives the uncertainty in M (Skibba et al. 2006).
4. Results
In this section, we present our results of the environmen-tal dependence of galaxy luminosity, stellar mass, SFR andsSFR. We measure two-point correlation functions (2pCF) andrank-ordered marked correlation functions (MCFs) for variousstellar-mass-selected samples described in Table 2 selected fromGAMA in the redshift range 0 . < z < .
16. Each 2pCF has beenfitted with a power-law model and the MCFs have been mea-sured using eight di ff erent properties as marks: absolute mag-nitudes in u , g, r , J , K -bands, stellar mass, SFR and sSFR. Fordetails, we refer the reader to Sect. 3, especially Eq. (4) and (8).In almost all the samples, we could reliably measure the correla-tion functions in the range 0 . < r p <
10 h − Mpc. The errors in ω p ( r p ) are obtained from 24 jackknife realizations and the errorsin M p ( r p ) are obtained by randomizing the marks as described inSect. 3.4. The best-fitting power-law parameters for 2pCFs in allthe samples are given in Table 3. Table 3.
Best-fitting power-law parameters for all subsamples used inthis work. See Table 2 for details of the samples. log (M (cid:63) / M (cid:12) ) min Sample Flux limit r (h − Mpc) γ A < . . ± .
33 1 . ± . B < . . ± .
58 1 . ± . B < . . ± .
68 1 . ± . C < . . ± .
62 1 . ± . C < . . ± .
52 1 . ± . C < . . ± .
55 1 . ± . C < . . ± .
18 1 . ± . C < . . ± .
29 1 . ± . In Fig. 4 we show the 2pCF and MCFs obtained for galaxieswith a flux limit of r < . A ω p ( r p ), which at first approximation, obeys apower-law model. The best-fit parameters are: correlation length r = . ± .
33 h − Mpc and slope γ = . ± .
02. This canbe compared with the results of Farrow et al. (2015), althoughtheir samples vary from ours. To be compared with our sam-ple A
1, the most appropriate sample of theirs is the one withmass limits 10 < log M (cid:63) / M (cid:12) h − < . . < z < .
18. The parameter values for that sample weremeasured to be r = . ± .
46 h − Mpc and γ = . ± . σ . This deviation incorrelation length is expected as their stellar mass cuts vary fromours.Right panel of Fig. 4 presents rank-ordered MCFs M p ( r p )obtained for di ff erent galaxy properties used as marks (as de-scribed in the legend). Presented MCFs strongly deviate fromunity on small scales ( r p < h − Mpc), for all luminosity, stel-lar mass, SFR and sSFR marks. This deviation then decreases,
Article number, page 7 of 14 & A proofs: manuscript no. main
Fig. 4.
Projected two-point correlation function ω p ( r p ) with a power-law fit (filled markers; left panel) and rank-ordered projected marked corre-lation functions M p ( r p ) (unfilled markers; right panel) for the Sample A ω p ( r p ) are squareroot of the diagonals of the covariance matrix obtained from 24 jackknife samples. The inset in the left panel shows the associated 1 σ , 2 σ and 3 σ error contours (solid, dashed and dotted respectively) of the power-law fit parameters. In the right panel di ff erent symbols represent measurementswith di ff erent marks (as labelled), and the error bars are obtained by random scrambling of the marks. The error bars of M p ( r p ) are too small to bevisible. but still remains, at larger scales. In general, a stronger devia-tion of MCF from unity, means stronger correlation of the corre-sponding galaxy property with the environment (as described inSect. 3.3). As shown in Fig. 4, stellar mass MCF deviation fromunity is greater than any of the luminosity MCFs. This meansthat stellar mass is the best direct indicator of galaxy environ-ment.Among the MCFs measured using luminosities in di ff erentpassbands, K -band MCF has the highest amplitude and the u -band MCF has the lowest. This means that the K -band luminos-ity traces the environment better than any other band of thoseused here. Moreover, the K -band MCF shows similar, thoughnot exactly the same, behaviour as the stellar mass MCF - the K -band luminosity and stellar mass are therefore correlated withenvironment in a similar fashion. This, in turn, confirms that K -band luminosity can be a used as a tracer (or a proxy) of stellarmass when we lack the measurements of stellar mass (Kochaneket al. 2001; Baldry et al. 2012).On the other hand SFR and sSFR MCFs behave opposite tothe stellar mass MCF – it presents low values (high deviationfrom unity) on small scales and high values (close to unity) onlarge scales. This means that there is only a small number ofclose pairs of galaxies with strong star formation activity. Thatis, the densest regions of the local universe are mainly popu-lated by old or quiescent galaxies. Similar, although weaker, be-haviour is presented by the u -band MCF. This confirms that the u -band luminosity traces galaxy SFRs (or at least serves as aproxy; Madau & Dickinson 2014). In order to study the impact of survey flux limit on the correla-tions observed in Sect. 4.1, we select two distinct samples with the same stellar mass cut log (M (cid:63) / M (cid:12) ) > .
4, but di ff erent fluxlimits: sample B r < . B r < . B B ff erent galaxy properties as marks.For the same flux limit ( r < . B A
1. This is evident from the significantdi ff erence between correlation lengths of these samples (see Ta-ble 3). By definition, A B
1. The fact that B A g, r , J , K -band MCFs of B A
1, with g, r , J , K -band MCFs falling below unity.When it comes to sample B
2, most of the MCFs show ahigher amplitude relative to B B B ff ect and Sect. 5.5 deals with a discussion onthis e ff ect. The stellar mass can still be used as a good indica-tor of galaxy environment - MCF, with the stellar mass used asmark, exhibits stronger deviation from unity than MCF with anyof the luminosity marks.However, the change in the flux limit does not a ff ect our2pCF measurements. The correlation lengths of samples B B σ . The best fit power-law parametersfor sample B r = . ± .
58 h − Mpc and
Article number, page 8 of 14. Sureshkumar et al.: Tracing galaxy environment using the marked correlation function in GAMA
Fig. 5.
Projected two-point correlation functions ω p ( r p ) with power-law fit (filled markers; top panels) and rank-ordered projected marked corre-lation functions M p ( r p ) (unfilled markers; bottom panels) for samples B B ff erent flux limits, as labelled. The error barsfor ω p ( r p ) in the top panels are obtained from 24 jackknife samples. In bottom panels, di ff erent symbols represent di ff erent marks considered forthe M p ( r p ) measurement (as labelled) and the error bars are obtained by random scrambling of the marks. γ = . ± .
04, whereas the same parameters for sample B r = . ± .
68 h − Mpc and γ = . ± . To further check the e ff ect of survey flux limit on clustering mea-surements, we extend our studies to even lower magnitude cuts.For that we use data from the SDSS survey. We select a totalnumber of five samples: three from GAMA ( C , C , C
3) and twofrom SDSS ( C , C (cid:63) / M (cid:12) ) > . ff erent flux limits. The de-tails are given in Table 2. Figure 7 shows the results of 2pCF andMCF (with stellar mass used as mark) measurements for each ofthese samples. Additionally, the best-fit power-law parametersfor each 2pCF are given in Table 3. The correlation lengths obtained for the GAMA samples C C C σ . Similarly, SDSS sam-ples C C σ . However, it is observed that the SDSS sample C C . σ , although they have the same flux-limit andthe stellar mass limit. This is expected as the SDSS galaxies areselected to be luminous red galaxies (Eisenstein et al. 2001).In case of MCFs, the GAMA samples ( C C C
3) arein agreement between each other within the errorbars. However,the stellar mass MCFs of samples C C ff erent from eachother on most of the scales. We also observe di ff erences betweenMCFs of the SDSS samples C C Article number, page 9 of 14 & A proofs: manuscript no. main
Fig. 6.
Projected correlation functions ω p ( r p ) for samples B B M B / M B between marked correlationfunctions obtained for these two samples (unfilled markers; right panel). Di ff erent symbols represent the ratio between di ff erent marked correlationfunctions measured with corresponding galaxy property chosen as mark. Small o ff sets along x-axis have been added for clarity. Error bars for ω p ( r p )are obtained from 24 jackknife samples. The inset in the left panel shows the associated 1 σ , 2 σ and 3 σ error contours (solid, dashed and dottedrespectively) of the power-law fit parameters. In the right panel, the errors are calculated in quadrature.
5. Discussion
The marked correlation function (MCF) is a useful tool to studythe environmental dependence of galaxy properties. Empirically,it is a ratio of terms involving the weighted and the unweightedcorrelation function (CF). For a given galaxy property, all galaxypairs in a sample carry a weight - product of property values forboth galaxies in units of its mean value. At each spatial scale, theMCF signal depends on the value of the weights at that scale -larger amplitude of rank-ordered MCF, measured using a partic-ular galaxy property, implies its stronger correlation with envi-ronment.As described in Sect. 4.1 and shown in Fig. 4, we observedi ff erent amplitudes for MCFs measured using di ff erent galaxyproperties. All galaxy properties - luminosities, stellar mass,SFR and sSFR - correlate with the environment, each in di ff erentway. For example, the stellar mass and luminosity (in g, r , J , K -bands) MCFs take values higher than unity on small scales -indicating an abundance of close galaxy pairs with these prop-erty values greater than the sample’s average. This agrees withthe well-known observation that the most massive and luminousgalaxies (in g, r , J , K -bands) are mostly found in dense regions(Norberg et al. 2002; Coil et al. 2006; Pollo et al. 2006; Meneuxet al. 2009; Bolzonella et al. 2010; Abbas & Sheth 2006; Ab-bas et al. 2010; Zehavi et al. 2011; Marulli et al. 2013; Skibbaet al. 2014; Farrow et al. 2015; Durkalec et al. 2018; Cochraneet al. 2018). This phenomenon can be explained in the frame-work of the hierarchical structure formation (Press & Schechter1974; White & Rees 1978; Mo & White 1996; Springel et al.2005). u -band luminosity and star formation rate dependenceon the environment We observe that the u -band marked correlation shows di ff er-ent behaviour in comparison to other passbands ( g, r , J , K , seeFig. 4). The u -band MCF takes values smaller than unity onscales r p < − Mpc, indicating low probability of finding pairsof two galaxies similarly bright in this band. This is in com-plete opposition to the results from other bands, where the most g, r , K -luminous galaxies were the most strongly clustered. Thisspecial behaviour of u -band MCF has also been observed byDeng (2012) and it is in agreement with the semi-analyticalgalaxy formation models (see Fig. 2 of Sheth et al. 2005).The u -band and ultra-violet (UV) light are thought to beprimarily emitted by starburst galaxies with young stellar pop-ulations (Cram et al. 1998). The UV-selected galaxies exhibitlow clustering in the local universe (Heinis et al. 2004, 2007;Milliard et al. 2007). Heinis et al. (2004) measured a correla-tion length of r = . + . − . h − Mpc for low-redshift UV galaxiesfrom the FOCA survey and Milliard et al. (2007) found it to be3 . ± . Article number, page 10 of 14. Sureshkumar et al.: Tracing galaxy environment using the marked correlation function in GAMA
Fig. 7.
Projected correlation functions (filled markers; left panel) and stellar mass marked correlation functions (unfilled markers; right panel) inGAMA and SDSS surveys. Di ff erent symbols represent the measurements in di ff erent samples as labelled. The error bars for ω p ( r p ) are obtainedfrom 24 jackknife samples and that for M p ( r p ) are obtained by random scrambling of the marks. The inset in the left panel shows the associated1 σ , 2 σ and 3 σ error contours (solid, dashed and dotted respectively) of the power-law fit parameters. Small o ff sets along x-axis have been addedfor clarity. (young) galaxies formed in less dense regions exhibit strong starformation activity. Our observations agree with the SFR MCFmeasurements by Sheth et al. (2005) and, to some extent, withrecent GAMA studies by Gunawardhana et al. (2018) - their re-sults show however weaker deviation of SFR and sSFR MCFfrom unity than ours. This can be due to the apparent absence ofthe SFR-density relationship as a consequence of selecting starforming sample of galaxies (McGee et al. 2011; Wijesinghe et al.2012). Gunawardhana et al. (2018) measurements were made forsamples of actively star forming galaxies.The similar behaviour of u -band and SFR MCFs has a prac-tical interpretation. The SFR of a galaxy can be estimated byapplying a scaling factor to the luminosity measurements sensi-tive to star formation (Condon 1992; Kennicutt 1998; Madau &Dickinson 2014). Since the u -band light is dominated by star-burst galaxies with young stellar populations, it is more closelycorrelated to SFR than to stellar mass in galaxies (Hopkins et al.2003) and is hence considered to be an indicator of SFR. Ourresult agrees that u -band can be a good, but not a perfect proxyof SFR in the context of galaxy clustering. The amplitude of rank-ordered MCFs, computed using variousmarks, can be used to find the galaxy property with the strongestenvironmental dependence. Our observations suggest that thisparameter is the stellar mass. We are therefore in agreementwith the past results that the distribution of massive galaxies isstrongly correlated with dark matter overdensities (Kau ff mannet al. 2004; Scodeggio et al. 2009; Davidzon et al. 2016). Thisdependence is expected from the hierarchical structure forma-tion theory according to which the local density contrast is con-nected to the mass of the hosting halo, which is further related to the galaxy stellar mass (Moster et al. 2010; Wechsler & Tinker2018).Our observations can be also interpreted in terms of galaxyevolution. Galaxy stellar mass and environment are often consid-ered as two separate aspects of this evolution (Peng et al. 2010).Strong correlation between stellar mass and environment tellsus that it can be misleading - these two parameters should notbe treated as independent entities (De Lucia et al. 2012). Es-pecially since stellar mass plays an important role in shapinggalaxy star formation history (Gavazzi et al. 1996; Kau ff mannet al. 2003; Heavens et al. 2004), which is yet another parameterthat is strongly correlated with the galaxy’s local environment(Kau ff mann et al. 2004; Blanton et al. 2005).It is important to note a feature of our studies that might biasour results, namely the galaxy sample selection. In this work wemeasure MCFs in stellar mass selected samples only. That is,our samples are nearly complete only in stellar mass and not inother properties. Our observation - stellar mass MCF being moreenhanced than other MCFs - could be an outcome of this bias.However, in our preliminary analysis (Sureshkumar et al. 2020),we measured the same MCFs - for u , g, r , K -band luminositiesand stellar mass marks - using samples selected based on thecorresponding property. We observed similar trends as here, rul-ing out the e ff ect of sample selection on our observation. K -band luminosity as a proxy for stellar mass In our work we also studied the behaviour of MCFs measuredusing di ff erent photometric wavebands and we observe clear dif-ferences between CFs marked with u , g, r , J , K luminosities (seeFig. 4). This means that luminosity in di ff erent passbands cor-relates di ff erently with environment. Most significantly, K -band(the reddest considered) shows a stronger environmental depen-dence than bluer bands. This means that there is higher prob- Article number, page 11 of 14 & A proofs: manuscript no. main ability to find close pairs of galaxies similarly luminous in the K -band than in other bands. In other words, galaxies luminousin K -band are strongly clustered. This observation is in agree-ment with various clustering studies (e.g. Oliver et al. 2004; dela Torre et al. 2007). In particular, Sobral et al. (2010), usingsample of H α emitters from HiZELS survey, show a strong in-crease in galaxy clustering with increasing K -band luminosity,but a weak trend in case of the B -band. This di ff erence in clus-tering strength between various bands is reflected in the varyingamplitude of MCFs in Fig. 4.Comparing the amplitudes of stellar mass and K -bandMCFs, we observe that these galaxy properties trace the envi-ronment in a similar way. This means that the K -band lumi-nosity can be used as the second-best galaxy property (amongthose considered here), after galaxy stellar mass, to trace the en-vironment. In other words, a sample that is complete in K -bandluminosity can be a good substitute of a stellar-mass-completesample.This observation is in agreement with the existing results.It has been shown that longer-wavelength luminosities (e.g., K -band) are dominated by evolved stellar populations and are leasta ff ected by dust extinction, making them directly related to thegalaxy stellar mass (Cowie et al. 1994; Gavazzi et al. 1996;Kau ff mann & Charlot 1998; Kochanek et al. 2001; Baldry et al.2012; Jarrett et al. 2013; Cluver et al. 2014). Kau ff mann & Char-lot (1998) pointed out that infrared light is a much more robusttracer of stellar mass than optical light out to z ∼ −
2. Theyobserved that galaxies of the same stellar masses have the same K -band luminosities, independent of their star formation histo-ries. Additionally, near-IR (mainly K -band) luminosity functionswere well utilized to estimate stellar mass distribution in the lo-cal universe (Cole et al. 2001; Kochanek et al. 2001; Bell et al.2003; Drory et al. 2004; Zhu et al. 2010; Meidt et al. 2014).Using a matched GAMA- WISE catalog, Cluver et al. (2014) ex-plored the usability of mid-IR wavelengths (W1 and W2) forstellar mass estimation.However, the relation between K -band luminosity and stel-lar mass is not entirely direct (van der Wel et al. 2006; Kannap-pan & Gawiser 2007). In our results, we also observe the di ff er-ences between stellar mass and K -band MCFs on small scales(see Fig. 4). Galaxy close pairs ( r p < − Mpc) show strongersignal when weighted using stellar mass than when weightedwith K -band luminosity. Given the fact that the latter is propor-tional to stellar mass, the di ff erence between both MCFs sug-gests that the correlation between stellar mass and K -band lumi-nosity does depend on environment. This conclusion is in qual-itative agreement with the predictions of semi-analytic galaxyformation models proposed by Sheth et al. (2005). Their mea-surements were made on a volume-limited sample with z = . × h − M (cid:12) which correspondsto log (M (cid:63) / M (cid:12) ) > .
45 in our cosmological model. Using theobserved di ff erence between K -band and stellar mass MCFs (seeFig. 5 of Sheth et al. (2005)), they concluded that the correlationbetween both these properties depends on environment.There is yet another caveat in using K -band luminosity se-lected samples as a proxy for stellar mass selection. By using thisapproximation one will miss the most evolved, red galaxies intheir sample - especially on small scales. Cochrane et al. (2018)observed that K -band derived stellar masses are underestimatedwith respect to full Spectral Energy Distribution (SED) stellarmasses. This means that some red galaxies might wrongly endup below the applied stellar mass cut, and not be selected when K -band stellar mass approximation is used. This is extremelyimportant for high-detail clustering studies. It has been shown that red galaxies tend to occupy dense environments (see e.g.,Zehavi et al. 2005; Coil et al. 2008; Zehavi et al. 2011; Palamaraet al. 2013). Samples unrepresented due to the K -band selectionmight, therefore, show weaker than actual clustering properties. Samples B B ff er-ent flux limits in the r -band. Namely, sample B r < . B r < . B B (cid:63) / M (cid:12) ) > . r < . ff er significantly between B B ff erence. In the right panel of Fig. 6, we show theratio of MCFs between samples B B
1. It is to be noted thatcorrelation studies based on the u -band luminosity are influencedat small scales by the imposed r -band flux thresholds. A di ff er-ence between these two samples is visible in the u -band MCFon the smallest scale ( r p ∼ . − Mpc) in the right panel ofFig. 6, where brighter sample shows a weaker MCF signal. Thismeans that, close pairs of galaxies which are similarly brighter in u -band drop out from the sample with the lower magnitude limit.The same behaviour is reflected in the SFR and sSFR MCFs. It isan important observation because the u -band luminosity is a verygood tracer of star formation processes. So, even though samples B B r < .
8) lose information aboutstarburst galaxies. This e ff ect has to be properly corrected for,for example by using methods discussed in Meneux et al. (2008,2009).As mentioned in Sect. 4.2, we observe a lowering of mostof the MCF amplitudes while shifting from sample A B B B
2, particularly at the larger scales. It isto be noted that, for the flux limit r < .
8, sample B A
1. Additionally, it is evidentfrom the stellar mass histograms in Fig. 2 that sample B B (cid:63) / M (cid:12) ) min = .
4. This suggests that the di ff erentbehaviour of MCFs we observe in samples A B ff ects can be observed in MCFmeasurements based on two di ff erently flux-limited SDSS sam-ples (see Fig. 7). The stellar mass MCF varies on average by ∆ M p ( r p ) = . ± .
05 on all scales. These di ff erences oc-cur even though both samples have the same stellar mass limit,which means that the brighter sample is incomplete.We do not observe this mass incompleteness e ff ect in GAMAsamples, where amplitudes of 2pCFs and MCFs do not changesignificantly with the apparent flux limit. This can be associatedwith the variation in the number of galaxies that enter the flux-limited sample even with the same stellar mass limit. The SDSSsample with the brighter flux limit r < . C
5) has ∼ r < . C ff erences between number of galaxies inGAMA samples ( C C
2, and C
3) are very small (see Table 2).So the lack of flux limit dependence of correlation functions in
Article number, page 12 of 14. Sureshkumar et al.: Tracing galaxy environment using the marked correlation function in GAMA the GAMA survey can be due to little variation in the number ofgalaxies between the GAMA samples C C
2, and C ff ect further, we would have to understandthe clustering properties of the galaxies missing in brighter flux-limited samples. Such studies were previously done by Meneuxet al. (2008, 2009); Marulli et al. (2013). However, they werebased only on measurements of projected CFs. To understanddetailed connections between these galaxies, we would like tomeasure behaviour of di ff erent MCFs. As for now, the real-dataGAMA samples for this kind of studies would consist only of300-400 galaxies (which are missing in the brighter samples).So this could only be done using catalogues built from the sim-ulations coupled with semi-analytical galaxy formation models.This kind of study is beyond the scope of the present paper.
6. Summary and conclusion
In this paper, we studied the environmental dependence ofgalaxy properties, like luminosity (in u , g, r , J , K -bands), stellarmass, SFR and sSFR in the redshift range 0 . < z < .
16 usinga spectroscopic sample of galaxies from the GAMA survey. Wechecked which of these properties is a better tracer of the envi-ronment, and showed how the results of clustering measurementscan be influenced by selecting samples using di ff erent properties.In order to achieve these aims, we measured the two point pro-jected correlation function and marked correlation functions in anearly stellar-mass-complete sample with a flux limit r < . (cid:63) / M (cid:12) ) > .
3. The marked correlationfunctions were measured using di ff erent marks: luminosities in u , g, r , J , K -bands, stellar mass, SFR and sSFR. Additionally westudied the dependence of MCF on the survey flux limit, by re-peating the same measurements in two samples with di ff erentflux limits ( r < . r < .
8) but the same stellar masscut (log (M (cid:63) / M (cid:12) ) > . (cid:63) / M (cid:12) ) > . ff erent flux limits.The summary of our main results and conclusions of ourstudy are as follows: – We observed that di ff erent galaxy properties trace the envi-ronment di ff erently in the separation scales r p < − Mpc.Based on the behaviour of marked correlation functions inFig. 4, we concluded that the close pairs of galaxies are moreluminous in g, r , J , K -bands than distant pairs. It is also moreprobable to find close pairs of massive galaxies (with massesabove sample average) than pairs including one less massivegalaxy. However, this trend is reversed if the luminosity ismeasured in the u -band. The u -band luminous galaxies tendto occupy less dense regions and the faint u -band galaxiestend to exist in more dense regions. The same is true for ac-tively star forming galaxies, which tend to occupy less denseenvironments. – From the comparisons of the amplitudes of rank-ordered lu-minosity, stellar mass, SFR and sSFR marked correlationfunctions in Fig. 4, we concluded that stellar mass is a bettertracer of galaxy environment than any of the other properties. – The significant di ff erence between stellar mass and lumi-nosity marked correlation signals in Fig. 4 shows that thecorrelation between stellar mass and luminosity depends onthe environment. We showed that a sample complete in K -band luminosity can be a good substitute for stellar-mass-complete sample. But, in such a case we tend to miss closerpairs of evolved, red galaxies. – From the similarity in behaviour of di ff erent MCFs, we con-firmed the usefulness of u -band luminosity as a good, but nota perfect proxy of SFR in the context of galaxy clustering. – From the comparative study of marked correlation functionsin di ff erent samples with the same mass selection, but dif-ferent apparent magnitude limits, we concluded that closerpairs of star forming galaxies drop out of the sample whenthe survey gets shallower in terms of limiting magnitude.Our measurements are the first of this kind in the redshiftrange 0 . < z < .
16 and with galaxies as faint as r < . Acknowledgements.
U.S. and A.D. are supported by the Polish National ScienceCentre grant UMO-2015 / / D / ST9 / / MNS / / / M / ST9 / / / E / ST9 / / / G / ST9 / / WK / / . Based on observa-tions made with ESO Telescopes at the La Silla Paranal Observatory under pro-gramme ID 179.A-2004. During this research, we made use of Tool for OP-erations on Catalogues And Tables (TOPCAT; Taylor 2005) and NASA’s Astro-physics Data System Bibliographic Services. This research was supported in partby PLGrid Infrastructure and OAUJ cluster computing facility. References
Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS,182, 543Abbas, U., de la Torre, S., Le Fèvre, O., et al. 2010, MNRAS, 406, 1306Abbas, U. & Sheth, R. K. 2006, MNRAS, 372, 1749Alam, S., Albareti, F. D., Allende Prieto, C., et al. 2015, ApJS, 219, 12Alam, S., Peacock, J. A., Farrow, D. J., Loveday, J., & Hopkins, A. M. 2020,arXiv e-prints, arXiv:2006.05383Alpaslan, M., Driver, S., Robotham, A. S. G., et al. 2015, MNRAS, 451, 3249Alpaslan, M., Robotham, A. S. G., Driver, S., et al. 2014, MNRAS, 438, 177Baldry, I. K., Alpaslan, M., Bauer, A. E., et al. 2014, MNRAS, 441, 2440Baldry, I. K., Driver, S. P., Loveday, J., et al. 2012, MNRAS, 421, 621Baldry, I. K., Robotham, A. S. G., Hill, D. T., et al. 2010, MNRAS, 404, 86Barsanti, S., Owers, M. S., Brough, S., et al. 2018, ApJ, 857, 71Baugh, C. M. 2006, Reports on Progress in Physics, 69, 3101Beck, R., Dobos, L., Budavári, T., Szalay, A. S., & Csabai, I. 2016, MNRAS,460, 1371Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488,3143Beisbart, C. & Kerscher, M. 2000, ApJ, 545, 6Bell, E. F., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003, ApJS, 149, 289Benson, A. J. 2012, New A, 17, 175Beutler, F., Blake, C., Colless, M., et al. 2013, MNRAS, 429, 3604Blanton, M. R., Eisenstein, D., Hogg, D. W., Schlegel, D. J., & Brinkmann, J.2005, ApJ, 629, 143Blanton, M. R. & Moustakas, J. 2009, ARA&A, 47, 159Bolzonella, M., Kovaˇc, K., Pozzetti, L., et al. 2010, A&A, 524, A76Bolzonella, M., Miralles, J. M., & Pelló, R. 2000, A&A, 363, 476Bond, J. R., Kofman, L., & Pogosyan, D. 1996, Nature, 380, 603Bravo, M., Lagos, C. d. P., Robotham, A. S. G., Bellstedt, S., & Obreschkow, D.2020, MNRAS, 497, 3026
Article number, page 13 of 14 & A proofs: manuscript no. main
Brough, S., Croom, S., Sharp, R., et al. 2013, MNRAS, 435, 2903Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000Burton, C. S., Jarvis, M. J., Smith, D. J. B., et al. 2013, MNRAS, 433, 771Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682Chabrier, G. 2003, ApJ, 586, L133Christodoulou, L., Eminian, C., Loveday, J., et al. 2012, MNRAS, 425, 1527Cluver, M. E., Jarrett, T. H., Hopkins, A. M., et al. 2014, ApJ, 782, 90Cochrane, R. K., Best, P. N., Sobral, D., et al. 2018, MNRAS, 475, 3730Coil, A. L., Newman, J. A., Cooper, M. C., et al. 2006, ApJ, 644, 671Coil, A. L., Newman, J. A., Croton, D., et al. 2008, ApJ, 672, 153Cole, S., Norberg, P., Baugh, C. M., et al. 2001, MNRAS, 326, 255Condon, J. J. 1992, ARA&A, 30, 575Cowie, L. L., Gardner, J. P., Hu, E. M., et al. 1994, ApJ, 434, 114Cram, L., Hopkins, A., Mobasher, B., & Rowan-Robinson, M. 1998, ApJ, 507,155da Cunha, E., Charlot, S., & Elbaz, D. 2008, MNRAS, 388, 1595Davidzon, I., Cucciati, O., Bolzonella, M., et al. 2016, A&A, 586, A23Davies, L. J. M., Lagos, C. d. P., Katsianis, A., et al. 2019a, MNRAS, 483, 1881Davies, L. J. M., Robotham, A. S. G., Driver, S. P., et al. 2016, MNRAS, 455,4013Davies, L. J. M., Robotham, A. S. G., Lagos, C. d. P., et al. 2019b, MNRAS,483, 5444Davis, M. & Peebles, P. J. E. 1983, ApJ, 267, 465Dawson, K. S., Schlegel, D. J., Ahn, C. P., et al. 2013, AJ, 145, 10de la Torre, S., Le Fèvre, O., Arnouts, S., et al. 2007, A&A, 475, 443de Lapparent, V., Geller, M. J., & Huchra, J. P. 1986, ApJ, 302, L1De Lucia, G., Weinmann, S., Poggianti, B. M., Aragón-Salamanca, A., & Zarit-sky, D. 2012, MNRAS, 423, 1277Deng, X.-F. 2012, AJ, 143, 15Dolley, T., Brown, M. J. I., Weiner, B. J., et al. 2014, ApJ, 797, 125Driver, S. P., Hill, D. T., Kelvin, L. S., et al. 2011, MNRAS, 413, 971Driver, S. P., Norberg, P., Baldry, I. K., et al. 2009, Astronomy and Geophysics,50, 5.12Drory, N., Bender, R., Feulner, G., et al. 2004, ApJ, 608, 742Durkalec, A., Le Fèvre, O., Pollo, A., et al. 2018, A&A, 612, A42Eales, S., Dunne, L., Clements, D., et al. 2010, Publications of the AstronomicalSociety of the Pacific, 122, 499Eisenstein, D. J., Annis, J., Gunn, J. E., et al. 2001, AJ, 122, 2267Farrow, D. J., Cole, S., Norberg, P., et al. 2015, MNRAS, 454, 2120Fisher, K. B., Davis, M., Strauss, M. A., Yahil, A., & Huchra, J. 1994, MNRAS,266, 50Gavazzi, G., Pierini, D., & Boselli, A. 1996, A&A, 312, 397Gómez, P. L., Nichol, R. C., Miller, C. J., et al. 2003, ApJ, 584, 210Grootes, M. W., Tu ff s, R. J., Popescu, C. C., et al. 2017, AJ, 153, 111Groth, E. J. & Peebles, P. J. E. 1977, ApJ, 217, 385Grylls, P. J., Shankar, F., Leja, J., et al. 2020, MNRAS, 491, 634Gu, M., Conroy, C., & Behroozi, P. 2016, ApJ, 833, 2Gunawardhana, M. L. P., Norberg, P., Zehavi, I., et al. 2018, MNRAS, 479, 1433Guo, H., Zheng, Z., Zehavi, I., et al. 2015, MNRAS, 453, 4368Hamilton, A. J. S. 1993, ApJ, 417, 19Hartley, W. G., Almaini, O., Cirasuolo, M., et al. 2010, MNRAS, 407, 1212Heavens, A., Panter, B., Jimenez, R., & Dunlop, J. 2004, Nature, 428, 625Heinis, S., Milliard, B., Arnouts, S., et al. 2007, ApJS, 173, 503Heinis, S., Treyer, M., Arnouts, S., et al. 2004, A&A, 424, L9Hopkins, A. M., Miller, C. J., Nichol, R. C., et al. 2003, ApJ, 599, 971Jarrett, T. H., Cluver, M. E., Magoulas, C., et al. 2017, ApJ, 836, 182Jarrett, T. H., Masci, F., Tsai, C. W., et al. 2013, AJ, 145, 6Kannan, R., Stinson, G. S., Macciò, A. V., et al. 2014, MNRAS, 437, 3529Kannappan, S. J. & Gawiser, E. 2007, ApJ, 657, L5Kau ff mann, G. & Charlot, S. 1998, MNRAS, 297, L23Kau ff mann, G., Heckman, T. M., White, S. D. M., et al. 2003, MNRAS, 341, 54Kau ff mann, G., White, S. D. M., Heckman, T. M., et al. 2004, MNRAS, 353,713Kennicutt, Robert C., J. 1998, ARA&A, 36, 189Kochanek, C. S., Pahre, M. A., Falco, E. E., et al. 2001, ApJ, 560, 566Lagos, C. d. P., Tobar, R. J., Robotham, A. S. G., et al. 2018, MNRAS, 481, 3573Landy, S. D. & Szalay, A. S. 1993, ApJ, 412, 64Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv e-prints, arXiv:1110.3193Lewis, I., Balogh, M., De Propris, R., et al. 2002, MNRAS, 334, 673Lindsay, S. N., Jarvis, M. J., Santos, M. G., et al. 2014, MNRAS, 440, 1527Linke, L., Simon, P., Schneider, P., et al. 2020, A&A, 640, A59Liske, J., Baldry, I. K., Driver, S. P., et al. 2015, MNRAS, 452, 2087Loveday, J., Christodoulou, L., Norberg, P., et al. 2018, MNRAS, 474, 3435LSST Science Collaboration, Abell, P. A., Allison, J., et al. 2009, arXiv e-prints,arXiv:0912.0201Madau, P. & Dickinson, M. 2014, ARA&A, 52, 415Maraston, C., Strömbäck, G., Thomas, D., Wake, D. A., & Nichol, R. C. 2009,MNRAS, 394, L107Marulli, F., Bolzonella, M., Branchini, E., et al. 2013, A&A, 557, A17McCarthy, I. G., Font, A. S., Crain, R. A., et al. 2012, MNRAS, 420, 2245 McGee, S. L., Balogh, M. L., Wilman, D. J., et al. 2011, MNRAS, 413, 996McNaught-Roberts, T., Norberg, P., Baugh, C., et al. 2014, MNRAS, 445, 2125Meidt, S. E., Schinnerer, E., van de Ven, G., et al. 2014, ApJ, 788, 144Meneux, B., Guzzo, L., de la Torre, S., et al. 2009, A&A, 505, 463Meneux, B., Guzzo, L., Garilli, B., et al. 2008, A&A, 478, 299Meneux, B., Le Fèvre, O., Guzzo, L., et al. 2006, A&A, 452, 387Milliard, B., Heinis, S., Blaizot, J., et al. 2007, ApJS, 173, 494Mo, H. J. & White, S. D. M. 1996, MNRAS, 282, 347More, S., van den Bosch, F. C., Cacciato, M., et al. 2009, MNRAS, 392, 801Mostek, N., Coil, A. L., Cooper, M., et al. 2013, ApJ, 767, 89Moster, B. P., Naab, T., & White, S. D. M. 2020, MNRAS[ arXiv:1910.09552 ]Moster, B. P., Somerville, R. S., Maulbetsch, C., et al. 2010, ApJ, 710, 903Muldrew, S. I., Croton, D. J., Skibba, R. A., et al. 2012, MNRAS, 419, 2670Mutch, S. J., Croton, D. J., & Poole, G. B. 2013, MNRAS, 435, 2445Naab, T. & Ostriker, J. P. 2017, ARA&A, 55, 59Norberg, P., Baugh, C. M., Gaztañaga, E., & Croton, D. J. 2009, MNRAS, 396,19Norberg, P., Baugh, C. M., Hawkins, E., et al. 2002, MNRAS, 332, 827Norberg, P., Baugh, C. M., Hawkins, E., et al. 2001, MNRAS, 328, 64Oliver, S., Waddington, I., Gonzalez-Solares, E., et al. 2004, ApJS, 154, 30Palamara, D. P., Brown, M. J. I., Jannuzi, B. T., et al. 2013, ApJ, 764, 31Peebles, P. J. E. 1980, The large-scale structure of the universePeng, Y.-j., Lilly, S. J., Kovaˇc, K., et al. 2010, ApJ, 721, 193Petrosian, V. 1976, ApJ, 210, L53Pollo, A., Guzzo, L., Le Fèvre, O., et al. 2006, A&A, 451, 409Pollo, A., Meneux, B., Guzzo, L., et al. 2005, A&A, 439, 887Pollo, A., Takeuchi, T. T., Solarz, A., et al. 2013a, Earth, Planets, and Space, 65,1109Pollo, A., Takeuchi, T. T., Suzuki, T. L., & Oyabu, S. 2013b, Earth, Planets, andSpace, 65, 273Press, W. H. & Schechter, P. 1974, ApJ, 187, 425Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numer-ical recipes in FORTRAN. The art of scientific computingReid, B., Ho, S., Padmanabhan, N., et al. 2016, MNRAS, 455, 1553Robotham, A., Driver, S. P., Norberg, P., et al. 2010, PASA, 27, 76Schaefer, A. L., Croom, S. M., Allen, J. T., et al. 2017, MNRAS, 464, 121Schaefer, A. L., Croom, S. M., Scott, N., et al. 2019, MNRAS, 483, 2851Scodeggio, M., Vergani, D., Cucciati, O., et al. 2009, A&A, 501, 21Sheth, R. K. 2005, MNRAS, 364, 796Sheth, R. K., Connolly, A. J., & Skibba, R. 2005, ArXiv e-prints, astroSheth, R. K., Jimenez, R., Panter, B., & Heavens, A. F. 2006, ApJ, 650, L25Sheth, R. K. & Tormen, G. 2004, MNRAS, 350, 1385Skibba, R., Sheth, R. K., Connolly, A. J., & Scranton, R. 2006, MNRAS, 369,68Skibba, R. A., Coil, A. L., Mendez, A. J., et al. 2015, ApJ, 807, 152Skibba, R. A., Sheth, R. K., Croton, D. J., et al. 2013, MNRAS, 429, 458Skibba, R. A., Smith, M. S. M., Coil, A. L., et al. 2014, ApJ, 784, 128Sobral, D., Best, P. N., Geach, J. E., et al. 2010, MNRAS, 404, 1551Somerville, R. S. & Davé, R. 2015, ARA&A, 53, 51Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629Stoyan, D. & Stoyan, D. H. 1994, Fractals, Random Shapes and Point Fields:Methods of Geometrical Statistics (Wiley)Sureshkumar, U., Durkalec, A., Pollo, A., Bilicki, M., & GAMA Collabora-tion. 2020, in XXXIX Polish Astronomical Society Meeting, ed. K. Małek,M. Poli´nska, A. Majczyna, G. Stachowski, R. Poleski, Ł. Wyrzykowski, &A. ó˙za´nska, Vol. 10, 346–348Taylor, E. N., Franx, M., van Dokkum, P. G., et al. 2009, ApJ, 694, 1171Taylor, E. N., Hopkins, A. M., Baldry, I. K., et al. 2011, MNRAS, 418, 1587Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Se-ries, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed.P. Shopbell, M. Britton, & R. EbertTonry, J. L., Blakeslee, J. P., Ajhar, E. A., & Dressler, A. 2000, ApJ, 530, 625van der Wel, A., Franx, M., Wuyts, S., et al. 2006, ApJ, 652, 97van Dokkum, P. G., Quadri, R., Marchesini, D., et al. 2006, ApJ, 638, L59van Kampen, E., Smith, D. J. B., Maddox, S., et al. 2012, MNRAS, 426, 3455van Uitert, E., Joachimi, B., Joudaki, S., et al. 2018, MNRAS, 476, 4662Vázquez-Mata, J. A., Loveday, J., Riggs, S. D., et al. 2020, MNRAS, 499, 631Vogelsberger, M., Genel, S., Sijacki, D., et al. 2013, MNRAS, 436, 3031Wang, L., Norberg, P., Brough, S., et al. 2018, A&A, 618, A1Wechsler, R. H. & Tinker, J. L. 2018, ARA&A, 56, 435White, S. D. M. & Rees, M. J. 1978, MNRAS, 183, 341Wijesinghe, D. B., Hopkins, A. M., Brough, S., et al. 2012, MNRAS, 423, 3679Wright, A. H., Robotham, A. S. G., Bourne, N., et al. 2016, MNRAS, 460, 765York, D. G., Adelman, J., Anderson, Jr., J. E., et al. 2000, AJ, 120, 1579Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2005, ApJ, 630, 1Zhu, Y.-N., Wu, H., Li, H.-N., & Cao, C. 2010, Research in Astronomy andAstrophysics, 10, 329]Moster, B. P., Somerville, R. S., Maulbetsch, C., et al. 2010, ApJ, 710, 903Muldrew, S. I., Croton, D. J., Skibba, R. A., et al. 2012, MNRAS, 419, 2670Mutch, S. J., Croton, D. J., & Poole, G. B. 2013, MNRAS, 435, 2445Naab, T. & Ostriker, J. P. 2017, ARA&A, 55, 59Norberg, P., Baugh, C. M., Gaztañaga, E., & Croton, D. J. 2009, MNRAS, 396,19Norberg, P., Baugh, C. M., Hawkins, E., et al. 2002, MNRAS, 332, 827Norberg, P., Baugh, C. M., Hawkins, E., et al. 2001, MNRAS, 328, 64Oliver, S., Waddington, I., Gonzalez-Solares, E., et al. 2004, ApJS, 154, 30Palamara, D. P., Brown, M. J. I., Jannuzi, B. T., et al. 2013, ApJ, 764, 31Peebles, P. J. E. 1980, The large-scale structure of the universePeng, Y.-j., Lilly, S. J., Kovaˇc, K., et al. 2010, ApJ, 721, 193Petrosian, V. 1976, ApJ, 210, L53Pollo, A., Guzzo, L., Le Fèvre, O., et al. 2006, A&A, 451, 409Pollo, A., Meneux, B., Guzzo, L., et al. 2005, A&A, 439, 887Pollo, A., Takeuchi, T. T., Solarz, A., et al. 2013a, Earth, Planets, and Space, 65,1109Pollo, A., Takeuchi, T. T., Suzuki, T. L., & Oyabu, S. 2013b, Earth, Planets, andSpace, 65, 273Press, W. H. & Schechter, P. 1974, ApJ, 187, 425Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numer-ical recipes in FORTRAN. The art of scientific computingReid, B., Ho, S., Padmanabhan, N., et al. 2016, MNRAS, 455, 1553Robotham, A., Driver, S. P., Norberg, P., et al. 2010, PASA, 27, 76Schaefer, A. L., Croom, S. M., Allen, J. T., et al. 2017, MNRAS, 464, 121Schaefer, A. L., Croom, S. M., Scott, N., et al. 2019, MNRAS, 483, 2851Scodeggio, M., Vergani, D., Cucciati, O., et al. 2009, A&A, 501, 21Sheth, R. K. 2005, MNRAS, 364, 796Sheth, R. K., Connolly, A. J., & Skibba, R. 2005, ArXiv e-prints, astroSheth, R. K., Jimenez, R., Panter, B., & Heavens, A. F. 2006, ApJ, 650, L25Sheth, R. K. & Tormen, G. 2004, MNRAS, 350, 1385Skibba, R., Sheth, R. K., Connolly, A. J., & Scranton, R. 2006, MNRAS, 369,68Skibba, R. A., Coil, A. L., Mendez, A. J., et al. 2015, ApJ, 807, 152Skibba, R. A., Sheth, R. K., Croton, D. J., et al. 2013, MNRAS, 429, 458Skibba, R. A., Smith, M. S. M., Coil, A. L., et al. 2014, ApJ, 784, 128Sobral, D., Best, P. N., Geach, J. E., et al. 2010, MNRAS, 404, 1551Somerville, R. S. & Davé, R. 2015, ARA&A, 53, 51Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629Stoyan, D. & Stoyan, D. H. 1994, Fractals, Random Shapes and Point Fields:Methods of Geometrical Statistics (Wiley)Sureshkumar, U., Durkalec, A., Pollo, A., Bilicki, M., & GAMA Collabora-tion. 2020, in XXXIX Polish Astronomical Society Meeting, ed. K. Małek,M. Poli´nska, A. Majczyna, G. Stachowski, R. Poleski, Ł. Wyrzykowski, &A. ó˙za´nska, Vol. 10, 346–348Taylor, E. N., Franx, M., van Dokkum, P. G., et al. 2009, ApJ, 694, 1171Taylor, E. N., Hopkins, A. M., Baldry, I. K., et al. 2011, MNRAS, 418, 1587Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Se-ries, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed.P. Shopbell, M. Britton, & R. EbertTonry, J. L., Blakeslee, J. P., Ajhar, E. A., & Dressler, A. 2000, ApJ, 530, 625van der Wel, A., Franx, M., Wuyts, S., et al. 2006, ApJ, 652, 97van Dokkum, P. G., Quadri, R., Marchesini, D., et al. 2006, ApJ, 638, L59van Kampen, E., Smith, D. J. B., Maddox, S., et al. 2012, MNRAS, 426, 3455van Uitert, E., Joachimi, B., Joudaki, S., et al. 2018, MNRAS, 476, 4662Vázquez-Mata, J. A., Loveday, J., Riggs, S. D., et al. 2020, MNRAS, 499, 631Vogelsberger, M., Genel, S., Sijacki, D., et al. 2013, MNRAS, 436, 3031Wang, L., Norberg, P., Brough, S., et al. 2018, A&A, 618, A1Wechsler, R. H. & Tinker, J. L. 2018, ARA&A, 56, 435White, S. D. M. & Rees, M. J. 1978, MNRAS, 183, 341Wijesinghe, D. B., Hopkins, A. M., Brough, S., et al. 2012, MNRAS, 423, 3679Wright, A. H., Robotham, A. S. G., Bourne, N., et al. 2016, MNRAS, 460, 765York, D. G., Adelman, J., Anderson, Jr., J. E., et al. 2000, AJ, 120, 1579Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2005, ApJ, 630, 1Zhu, Y.-N., Wu, H., Li, H.-N., & Cao, C. 2010, Research in Astronomy andAstrophysics, 10, 329