Can We Detect the Color-Density Relation with Photometric Redshifts?

10.7 MB·English

by  Chuan-Chin Lai

Draft version May 3, 2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
CAN WE DETECT THE COLOR–DENSITY RELATION WITH PHOTOMETRIC REDSHIFTS?
Chuan-Chin Lai1,2, Lihwai Lin2, Hung-Yu Jian2,3, Tzi-Hong Chiueh1,3, Alex Merson4, Carlton M. Baugh5,
Sebastien Foucaud 6, Chin-Wei Chen 2, Wen-Ping Chen 7
Draft version May 3, 2016
ABSTRACT
Avarietyofmethodshavebeenproposedtodefineandtoquantifygalaxyenvironments. Whilethese
techniques work well in general with spectroscopic redshift samples, their application to photometric
6 redshift surveys remains uncertain. To investigate whether galaxy environments can be robustly
1 measured with photo-z samples, we quantify how the density measured with the nearest neighbor
0 approachisaffectedbyphoto-zuncertaintiesbyusingtheDurhammockgalaxycatalogsinwhichthe
2 3Dreal-spaceenvironmentsandthepropertiesofgalaxiesareexactlyknown. Furthermore,wepresent

y anoptimizationschemeinthechoiceofparametersusedinthe2Dprojectedmeasurementswhichyield
a the tightest correlation with respect to the 3D real-space environments. By adopting the optimized
M parameters in the density measurements, we show that the correlation between the 2D projected
optimized density and real-space density can still be revealed, and the color–density relation is also

2 visibleouttoz ∼0.8evenforaphoto-zuncertainty(σ∆z/(1+z))upto0.06. Wefindthatattheredshift
0.3<z <0.5 a deep (i∼25) photometric redshift survey with σ =0.02 yields a comparable
] performance of small-scale density measurement to a shallower ∆iz∼/(12+2z.)5 spectroscopic sample with
O
∼ 10% sampling rate. Finally, we discuss the application of the local density measurements to the
C Pan-STARRS1 Medium Deep survey, one of the largest deep optical imaging surveys. Using data
. from ∼ 5 square degrees of survey area, our results show that it is possible to measure local density
h
and to probe the color–density relation with 3σ confidence level out to z ∼ 0.8 in the PS-MDS. The
p
color–density relation, however, quickly degrades for data covering smaller areas.
-
o Subject headings: galaxies: evolution – galaxies: formation – galaxies: halos
r
t
s
a 1. INTRODUCTION tween environment and galaxy properties. Yet exactly
[ howtheenvironmentaffectstheevolutionofgalaxiesand
Recent observations have shown that various galaxy
how important it is as opposed to internal properties of
2 properties such as star formation rate, color and mor-
galaxies (e.g., stellar mass) is still unclear. Part of the
v phologyarestronglycorrelatedwithgalaxyenvironment
discrepancybetweenpreviousstudiesmaycomefromthe
8 (Hoggetal.2004;Blantonetal.2006;Hainesetal.2006;
differencesinthesampleselectionaswellasthedefinition
9 Cooper et al. 2006, 2007; Capak et al. 2007; Elbaz et al.
ofenvironment,whichmakethecomparisonsnon-trivial.
3 2007; Gallazzi et al. 2009; Cooper et al. 2012; Mostek
One of the common approaches to characterize the
1 et al. 2013; Lin et al. 2014; Darvish et al. 2014). These
0 studies indicate that galaxies located in dense environ- galaxy environment is to use the local overdensity of
. ments, such as galaxy groups and clusters, tend to be matter. For the rest of this paper, we refer the galaxy
1
environment to the observed overdensity of galaxies as
0 redder, elliptical and with lower star formation rates.
a proxy for this. A variety of methods have been used
5 Several physical processes, including ram pressure strip-
to define galaxy density field, for example, (1) the Fixed
1 ping (Gunn & Gott 1972; Quilis et al. 2000), high speed
Aperture method which counts the number of neighbor
: galaxy encounters (galaxy harassment; Moore et al.
v galaxiesinafixedvolumearoundeachgalaxy(e.g.,Gal-
1996),galaxy-galaxymergers(Mihos&Hernquist1994),
i lazzi et al. 2009; Gru¨tzbauch et al. 2011); (2) the Annu-
X and removal of warm and hot gas (strangulation; Lar-
lusmethodwhichcountsthenumberofneighborgalaxies
sonetal.1980;Baloghetal.2000;McCarthyetal.2008)
r within a circular ring around each galaxy (e.g., Wilman
a have been proposed to explain the observed relation be-
et al. 2010); and (3) the Nth nearest neighbor that de-
fines the local density by finding the distance from the
[email protected]
1GraduateInstituteofAstrophysics,NationalTaiwanUniver- individual reference galaxies to the Nth nearest galaxy
sity,Taipei,10617,Taiwan(R.O.C.) (e.g., Casertano & Hut 1985; G´omez et al. 2003; Baldry
2Institute of Astronomy & Astrophysics, Academia Sinica, et al. 2006; Haas et al. 2012). A fundamental and cru-
Taipei10617,Taiwan(R.O.C.)
cial quantity for these methods to work is the redshift
3DepartmentofPhysics,NationalTaiwanUniversity,Taipei,
information, which provides the information about the
10617,Taiwan(R.O.C.)
4Department of Physics and Astronomy, University College line-of-sight separation (in the absence of peculiar veloc-
London,GowerStreet,LondonWC1E6BT,UK ities) of two given galaxies. Observationally, there are
5Institute for Computational Cosmology, Department of two types of redshift that are used widely: spectroscopic
Physics, Durham University, South Road, Durham DH1 3LE,
redshifts and photometric redshifts (hereafter spectral-z
UK
6CenterforAstronomyandAstrophysics,ShanghaiJiaoTong and photo-z respectively). While the spectral-z samples
University,Shanghai200240,China have greater precision in the redshift measurement, they
7Graduate Institute of Astronomy, National Central Univer- suffer from incompleteness and are observationally ex-
sity,Chung-Li32054,Taiwan(R.O.C.)
2
pensive for high-redshift galaxies. vironment, and demonstrate how to optimize the choice
Todate,environmentstudiesusinglargespectroscopic ofNth nearestneighbortoimprove2Dprojecteddensity
surveys such as SDSS (York et al. 2000), DEEP2 (Davis measurement. InSection5weshowtherelationbetween
et al. 2003; Newman et al. 2013) and zCOSMOS (Lilly galaxy environment and galaxy properties in the mock
et al. 2007) have been limited to redshifts lower than galaxy catalog to verify whether or not our optimized
z ∼ 1.5 (see Tanaka et al. 2004; Haines et al. 2006; scheme is applicable. We discuss several possible factors
Cooperetal.2006,2007;Mosteketal.2013). Incontrast, that might limit our optimized scheme and apply it to
photo-z surveys provide larger sample sizes and reach to observations in Section 6. Finally we summarize our re-
higherredshifts,buttheysufferfrompoorerredshiftres- sults in Section 7. In this paper, we adopt the following
olution. Figures1–3showhowthephoto-zuncertainties cosmological parameters: H = 100h km s−1 Mpc−1,
0
distort the real galaxy environment from different view- h=0.73, Ω =0.25 and Ω =0.75.
0 Λ
points. The large-scale structures are clearly revealed in
the case without photo-z error but become less promi- 2. DATA
nent as the photo-z error increases. Despite this prob-
2.1. Simulation Data
lem, there have been some attempts to measure galaxy
environment for various studies using photo-z samples In this work, we use a theoretical mock galaxy catalog
(Capak et al. 2007; Quadri et al. 2012; Scoville et al. to understand the systematics in the local density esti-
2013; Chiang et al. 2014; Lin et al. 2016). Several works mates. The advantage of using a mock galaxy catalog
have provided viable methods that can be used to re- compared to real spectroscopic survey data is that the
cover the density fields of galaxies from photo-z samples real-space density can be directly measured and com-
(Etherington & Thomas 2015; Lin et al. 2016; Malavasi pared with the projected density. Moreover, the mock
et al. 2016). Moreover, Arnalte-Mur et al. (2009) and sample does not suffer from the incompleteness which
Schlagenhauferetal.(2012)bothdemonstratedthatthe often affects real observations. On the other hand, one
two-pointcorrelationfunctionofgalaxiescanalsobesuc- needs to be cautious when interpreting the results since
cessfully recovered from photometric samples, and they thepropertiesofgalaxiesinthesimulationmaynotbea
also discussed the influence of photo-z errors on their perfect representation of the real Universe.
measurements. The mock galaxy catalog used in this work is built
As several ongoing large sky surveys such as the based on Millennium simulation with N = 21603 in a
Panoramic Survey Telescope & Rapid Response System boxwithvolume=5003 h−3 Mpc3 fromredshiftz =127
(Pan-STARRS: Onaka et al. 2008; Kaiser et al. 2010), to the present day at z = 0 by adopting the following
DarkEnergySurvey(DES:TheDarkEnergySurveyCol- cosmological parameters: a baryon matter density Ω
b
laboration 2005; Albrecht et al. 2006), Hyper Suprime- = 0.045, a total matter density Ω = 0.25, a dark en-
0
Cam Survey (HSC: Miyazaki et al. 2012) and the up- ergydensityΩ =0.75andaHubbleconstantH =100h
Λ 0
coming Large Synoptic Survey Telescope (LSST: LSST km s−1 Mpc−1 where h = 0.73. These cosmological pa-
Science Collaboration et al. 2009) will yield large galaxy rameters match the first-year results of Wilkinson Mi-
samples with photometric redshift measurements, it is crowaveAnisotropyProbe(WMAP:Spergeletal.2003).
important to understand the potential and limitation Galaxies are put into halos using the GALFORM semi-
of the photo-z method in the studies of galaxy evolu- analytical model (Cole et al. 2000) which takes into ac-
tion, especially the environmental effects. Can we use count various galaxy formation processes including gas
photo-zsamplestomeasureenvironmentreliably? What accretion and cooling, star formation in galactic disks
arethesystematicsintheenvironmentmeasurementbe- andgalaxymergers. ThemockcatalogadoptstheLagos
tween spectral-z samples and photo-z samples? What is et al. (2012) model which takes advantage of the exten-
the optimal choice for density measurement that can re- sion to the treatment of star formation introduced into
liably recover the underlying environments? These are GALFORM in Lagos et al. (2011) to populate galax-
the questions that we aim to answer. Particularly, we ies,andisthenassembledintoalightcone(Mersonetal.
focus on the measurement of galaxy density field. We 2013). FurtherdetailedinformationisgiveninColeetal.
first study the difference between 3D real-space density (2000), Springel et al. (2005), Bower et al. (2006), Lagos
and 2D projected density measurements by using mock et al. (2011, 2012) and Merson et al. (2013).
galaxy catalogs. We adopt the Spearman’s rank correla- Weconstructedtwotypesofmockcatalogsthatmimic
tion coefficient (Spearman 1904), r , as a measure of the the observed spectral-z and photo-z catalogs. The mock
s
correlation between the 2D and the 3D real-space den- spectral-zcatalogcanbeobtainedfromprimitivesimula-
sity. The optimized parameters for the density measure- tiondatawhichstorestheintrinsicline-of-sightpositions
ment are obtained by maximizing r . We then use the of galaxies. To generate mock photo-z catalogs, we per-
s
results of optimized density measurements to show the turbthepositionofgalaxiesalongtheline-of-sightdirec-
dependence of galaxy properties on environments from tionbymakingarandomshiftwhichfollowsaGaussian-
the mock catalog. And finally, we apply our optimized distribution with standard deviation that matches the
scheme to the Pan-STARRS1 data and compare the re- photo-z error in each case, in order to simulate cases
sultswiththemeasurementsbyCooperetal.(2006)who with observed redshift uncertainties. The new redshift
use the DEEP2 spectroscopic sample in the same field. obtained can be viewed as the ”observed redshift”, and
This paper is structured as follows. In Section 2.1 and beusedtocomputethelocaldensity. Althoughthepho-
2.2 we describe the simulation and observational data tometric redshift model adopted here is oversimplified
used in our study. The environment measurements used asitdoesnottakeintoaccounttheeffectofcatastrophic
inthisstudyareintroducedinSection3. InSection4we redshiftfailures,thissimplisticmodelallowsustounder-
comparethe3Dreal-spacedensitywith2Dprojecteden- stand the effect of redshift dispersion. Later in Section
3
5.4, we consider more realistic situations in which the down to r < 24.1. More details on the data process-
p1
outlier effect is included. In this study we consider sev- ing and photo-z characteristics in the PS1 MD07 sample
eralphoto-zcaseswithuncertaintiesofσ =0.00, are given in Lin et al. (2014), and Foucaud et al. (in
∆z/(1+z)
0.02,0.04and0.06. Werestrictourenvironmentstudyto preparation).
the redshift range of 0.3<z <0.5 for most of our anal-
ysis. A central area of catalogs ∼ 16 square degrees is 3. GALAXY ENVIRONMENT MEASUREMENTS
selectedforourstudies, containing∼1,900,000galaxies
In this study, we adopt the Nth Nearest Neighbor
with i<25.8. It is worth noting that the redshift which
method to quantify galaxy environment. This method
we use for computing the 3D overdensity with the Mil-
defines the local density of each galaxy using the dis-
lenniummockspectroscopiccatalogsreferstotheintrin-
tancetotheNthnearestneighborgalaxy. Inotherwords,
sic redshift of galaxies, which does not include the effect
whether the galaxy is located in an over-dense or under-
from peculiar velocity. Therefore, the results shown for
dense environment depends on how far it is to its Nth
the spectroscopic redshift sample may be too optimistic.
nearest galaxy. In simulations, the cosmological redshift
However, since our main focus is to understand the per-
reflects the ”real” distance along the line of sight, en-
formance of density recovery in the case of photometric
abling the environment measurement to be evaluated by
errors, this mock spectroscopic catalog does provide a
using 3D Nth Nearest Neighbor method. On the other
’real’ answer of the 3D density.
hand,observationally,thelocaldensityisoftenestimated
by using a projected method as the measured redshift is
2.2. Observation data
a combination of both the cosmological distance and the
2.2.1. Spectroscopic Observation peculiar velocity.
The DEEP2 Galaxy Survey (Davis et al. 2003; New- Herewedefinetwosetsofgalaxysamples: theprimary
man et al. 2013) was designed to study the galaxy pop- andthesecondarysample. Theprimarysamplecontains
ulation and large-scale structure at z ∼ 1. It uses galaxies brighter than a particular magnitude (mpi), and
the Keck II telescopes with the DEIMOS spectrograph isusedwhenpresentingtheresults. Thesecondarysam-
(Faberetal.2003)andcovers∼3.5squaredegreesofthe plereferstogalaxiesusedinthesearchforneighbors,and
sky with measured spectra and has targeted ∼ 60,000 is restricted to those galaxies which are brighter than a
galaxies down to a limiting magnitude of RAB < 24.1. particular magnitude (msi). The limiting magnitude of
About ∼ 60% of the galaxies are sampled over the red- the secondary sample is particularly important because
shift interval 0.2 < z < 1.4. The overall redshift success it sets the galaxy number densities in the calculation of
rate is about ∼70%. DEEP2 comprises four widely sep- density field. Figure 4 shows the median distances to
arated fields. One of the DEEP2 fields, the Extended the 3D Nth nearest neighbor with various choices of msi.
Groth Strip (EGS), is enclosed by the Pan-STARRS1 The coloured shadows show the range between the min-
Medium Deep Survey Field (MD07). In this study we imum and maximum distances to the 3D Nth nearest
match the DEEP2 spectroscopic redshift catalog to the neighbor. There are several parameters that should be
Pan-STARRS1 MD07 catalog. For galaxies that are consideredintheNth NearestNeighbormethod: (1)the
common in the two catalogs, we compare the local den- choice of the Nth Neighbor, which represents the scale
sitymeasurementscomputedusingtheDEEP2spectral-z of environment, (2) the magnitude limit of the primary
and Pan-STARRS1 photo-z respectively (see Section 6) sample (mp), (3) the magnitude limit of the secondary
i
. sample(msi),and(4)thevelocitywindow(Vcut)thatde-
fines the redshift boundaries of the neighbors considered
2.2.2. Photometric Observation inthe2Dprojectedmethod. Theseparametersshouldbe
adjusted according to different science goals and galaxy
Pan-STARRS1(hereafterPS1)isa1.8metertelescope
samples. One of the goals of this work is to provide an
equippedwithaCCDdigitalcamerawith1.4billionpix-
empirical framework which determines these parameters
els and 3-degree field of view, located on the summit of
by calculating galaxy densities with different combina-
Haleakala on Maui in the Hawaii Islands (Onaka et al.
tions of parameters in order to understand the influence
2008; Kaiser et al. 2010). The PS1 observations are ob-
of the parameters.
tained in a set of five broadband filters, which we have
designated as g ,r ,i ,z and y . There are two
p1 p1 p1 p1 p1 3.1. 2D Projected Nth Nearest Neighbor Galaxy
major components of the PS1 survey which started ob-
Environment
servations in 2010: the 3π survey and the Medium Deep
Survey (MDS) which comprises ten fields spread across In the 2D projected method, the local density of each
the sky. One of the MDS fields, namely MD07, is cho- galaxy is computed as the surface density averaged over
sen for this study because it overlaps with the EGS field the area enclosed by the Nth closest galaxy within the
whichhasthespectroscopicfromtheDEEP2survey,and velocity interval V :
cut
enables a direct comparison with the environment mea-
surements using the DEEP2 spectral-z sample (Cooper n+1
Σ = , (1)
et al. 2006). Photo-z in MD07 are computed by running n πr2
n
the EAZY code (Brammer et al. 2008) on PS1 5-band
photometryplustheu∗-banddatatakenbyEugeneMag- where n is the Nth closest galaxy for each reference
nier et al. with CFHT MEGACAM as part of the PS1 galaxyandr isthedistancefromthereferencegalaxyto
n
efforts. Comparisons against the DEEP2 spectroscopic the Nth closest galaxy on a 2D surface. There is no sim-
redshifts(Newmanetal.2013)showthatthePS1photo- ple way to determine the choice of velocity window V
cut
z reaches an uncertainty of 0.05 with outlier rate of 7% inthe2Dnearest-neighbormethod. Inprinciple,itisnot
4
meaningfultoadoptaV thatistoosmallcomparedto 2014). TheSpearman’srankcorrelationcoefficientisde-
cut
the redshift uncertainty of the data. Conversely, adopt- fined as:
ing a large velocity cut enlarges projection effect, which
leads to greater errors in the density measurement. For 6(cid:80)d2
r =1− i , (4)
instance, the separation between two galaxies that are s s(s2−1)
close in the projected plane may actually be widely sep-
arated in the third dimension, and vice versa (Muldrew where d is the difference between the ranks of the two
i
et al. 2012). Previous studies have utilized the velocity variables and s is the sample size. A coefficient r
s
intervalV ofavalueclosetothedistanceuncertainties = 0 corresponds to no correlation between two vari-
cut
intheline-of-sightdirection, astheyfoundthattheden- ables, while r = 1 (-1) corresponds to a perfect-positive
s
sity estimate does not significantly vary when changing (perfect-negative) correlation. In our analysis, we first
V aroundthisvalue. Wewillfurthertestthisapproach measure the 2D projected density Σ as well as the 3D
cut n
using the mock catalog in Section 4.1. real-space density ρ for each galaxy, and then convert
n
all the density measurements into an overdensity as de-
3.2. 3D Nth Nearest Neighbor Galaxy Environment finedinthelastsection. Afterrankingthe3Dreal-space
overdensity and 2D projected overdensity, we compute
The galaxy environment defined by the 3D Nth Near-
the difference d between the ranks of the two overden-
estNeighbormethodissimilartothatdefinedbythe2D i
sities, and then use Formula 4 to calculate Spearman’s
projected Nth Neighbor except that the projected cir-
rank correlation coefficient r .
cular area is replaced by the enclosed-spherical volume. s
Figure5isanexampleshowingthecorrelationbetween
Thevolumedensityofgalaxiesisevaluatedusing3DNth
the3Dand2DmeasurementsusingN =6, 30, 60and
Nearest Neighbor method as: 2D
90 respectively, and how the r coefficient changes with
s
different choices of N .
n+1 2D
ρ = , (2)
n (4/3)πr3
n 4.1. 3D Galaxy Environment vs. 2D Projected Galaxy
wherenistheNth closestgalaxyofeachreferencegalaxy Environment with Different Parameters
and r is the distance from the reference galaxy to the We first probe the effect of the velocity interval V
n cut
Nth closest galaxy in the three-dimensional space. We on the 2D projected density measurement. We restrict
compute the real-space density using the 3D Nth Near- the sample with mp < 25 and ms < 25 when calculat-
i i
est Neighbor method of the simulation where informa- ing2Dprojectedoverdensity, 1+δ2D, and3Dreal-space
6
tion about the three-dimensional positions of galaxies is overdensity, 1+δ3D. Figure 6 shows the scatter plots of
known. Wetreatthe3Ddensityasthe”true”densityto 6
the 2D projected overdensity versus the real-space over-
be compared with the 2D density to quantify how well density in log scale using the 6th nearest neighbor. Here
the real-space density can be recovered by the 2D pro-
we consider the following four different choices of V in
cut
jected density under various conditions. We note that
the 2D projected measurement for a galaxy sample with
practically the 3D density is rarely used even in a spec-
photo-zerror=0.04(1+z): ±0.005(1+z),±0.02(1+z),
troscopicredshiftsample,becausetheobserved’redshift’
±0.04(1+z) and ±0.06(1+z). It can be seen that the
includescontributionsfromboththeHubbleflowandthe
difference in r among the four cases of V is not sig-
s cut
peculiar velocity of galaxies, which is not possible to dif-
nificant when the size of velocity interval is close to the
ferentiate observationally.
photo-z error. For example, the r of 0.434 is obtained
s
Finally, in order to contrast the most-dense environ-
in the case of V = ±0.02(1+z) (upper-right panel of
cut
mentswiththeleast-denseenvironments,weconvertthe
Figure 6), while the value of r increases to 0.473 in the
s
initial primordial density into an overdensity. The over-
case of V = ±0.06(1+z) (bottom-right panel of Fig-
cut
densityisconventionallydefinedastheinitialprimordial
ure 6). The difference in r is small (∼0.039) between
s
density divided by the median density as follows:
these two cases even though the V differs by a fac-
cut
tor of 3, which confirms the finding in previous studies
D
1+δ = i , (3) that the 2D projected measurement is not sensitive to
n DMdn Vcut when Vcut is comparable to the photo-z uncertainty
(Gallazzi et al. 2009; Muldrew et al. 2012). We further
where D is the measured density of a galaxy (i.e., D ∈
i i repeat similar exercises using samples with a larger red-
{ρ ,Σ }),andD isthemediandensitycomputedby
n n Mdn shiftuncertaintyupto0.08(1+z)andathigherredshifts
counting galaxies within a bin of ∆z = 0.04. The term
(0.6 < z < 0.8) and find that this conclusion still holds.
1+δ is the so-called overdensity, and δ can be δ3D or
n n n Therefore throughout this work, we set V to be the
δ2D, depending on D = ρ or Σ . cut
n i n n typical photo-z uncertainty of the galaxy sample.
Next we consider the effect of the secondary magni-
4. QUANTIFYING DIFFERENCES BETWEEN tude limit employed on the galaxy sample when search-
ENVIRONMENT MEASUREMENTS ingforneighbors. Abrighter(fainter)ms probesalarger
i
In this section, we compare the 2D projected den- (smaller) scale of environment for a fixed Nth nearest
sity obtained under various conditions to the 3D real- neighbor. It is therefore expected that the correlation
space density. To quantify their differences, we adopt between the 2D projected and 3D real-space environ-
the Spearman’s rank correlation coefficient, r , which is ment could depend on the choice of ms. Again we se-
s i
commonlyusedtomeasurethestrengthofarelationship lect the sample with photo-z error = 0.02(1+z) and set
between two ranked variables (Spearman 1904; Curran V = ±0.02(1+z) for the reason given above. Figure
cut
5
7 shows the scatter plots of the 2D projected overden- stronglycorrelatedwiththe3Dreal-spaceenvironments.
sity versus real-space overdensity using the 6th nearest However,forthesamplescontaminatedbyphoto-zerrors
neighbor, and both 2D and 3D environments are mea- (green, blue and cyan dots), the recovering performance
suredusingms <21, ms <23andms <25respectively. becomes gradually worse as the photo-z error increases.
i i i
Asitcanbeseen,thelargestr isobtainedwhenthems In addition, the correlation between 2D projected and
s i
in 2D measurement is equal to the ms in 3D measure- 3Dreal-spaceenvironmentsdoesnotdependonlyonthe
i
ment. Furthermore, for identical ms used in the 2D and redshift accuracy, but is also scale dependent. For ex-
i
3D measurements, the environments measured by using ample, at a given photo-z error, r decreases with N ,
s 3D
fainter ms (and hence smaller scales) have better corre- which suggests that small-scale environments are more
i
lation than those measured using a brighter ms. This is easier to recover. A possible explanation is that the 2D
i
consistent with the results from Shattow et al. (2013), projectedenvironmentscalculatedbyusingphoto-zsam-
which also shows that for samples with photo-z error, ples might include more contaminations when we probe
the 2D projected environments have a weaker correla- the larger scale of environments.
tion with 3D real-space environments on larger scales. One interesting feature in the bottom panel of Figure
Finally, Figure 8 shows the 2D versus 3D scatter 10 is that the best choices of N are in general not
2D
plots in the density measurement to understand how equal to N , but only half of N , for producing the
3D 3D
thegalaxyenvironmentisaffectedbythephoto-zerrors. largest r except for the case of error-free. However, we
s
Hereweconsidermp <25andms <25,andvaryphoto- note that the relation between the optimized N and
i i 2D
zerrorfrom0,0.02(1+z),0.04(1+z)to0.06(1+z). V N dependsonthechoicesofredshiftinterval. Detailed
cut 3D
is correspondingly set to be ±0.001(1+z), ±0.02(1+z), discussions are given in Appendix A.
±0.04(1+z) and ±0.06(1+z) respectively. It is worth As the value of density measurements also depends on
noting that even under the perfect situation where the thechoiceofthesecondarymagnitudelimit,itisinterest-
redshift error is zero, the correlation is still not per- ing to see how the correlation between 2D and 3D mea-
fect owing to the projection effect. The correlation be- surements change by varying the secondary magnitude
tween 3D real-space and 2D projected environments be- limits in both 3D and 2D environment measurements.
come gradually worse when the photo-z error increases. Figures 11 – 13 show the largest r as a function of N
s 3D
However, therestillexistssomecorrelationespeciallyfor fordifferent3Dsecondarymagnitudelimitsrespectively.
galaxies located in high density regions, while the envi- In each figure, the red, green and blue dots are for sam-
ronmentinlowerandintermediatedensitiesislessdistin- ples with secondary magnitude limits corresponding to
guishable. This is consistent with the result from Capak ms < 21, ms < 23, and ms < 25 in 2D environment
i i i
et al. (2007), who also shows that the galaxy environ- measurement respectively, at a fixed secondary magni-
ment is difficult to measure for galaxies located in low tude limit in 3D local density and a fixed photo-z error
dense regions when a redshift error is present. = 0.02(1+z). Our results show that when the 3D envi-
ronment is defined using brighter secondary magnitude
4.2. Optimizing 2D Environment Parameters
limits, there is no significant difference in the recover-
So far we have compared 3D real-space environments ingperformanceamongdifferentchoicesof2Dsecondary
with various 2D projected environments to show their magnitude limit that are fainter than the 3D secondary
correlationandwehaveadoptedr toquantifythegood- magnitude limit. On the other hand, adopting a 2D sec-
s
nessofthecorrelation. Wenowexpandthistoconstruct ondarymagnitudelimitthatisbrighterthanthe3Dsec-
an optimization scheme to determine the value of N ondary magnitude limit results in a poorer environment
2D
which gives the best correlation (largest r ) between the recovery. This means that a deeper sample is favored
s
2D and 3D environments. Figure 9 shows the r cal- when constructing the 2D density field.
s
culated by fitting 2D projected and 3D real-space envi-
ronment with various choice of N2D and N3D. The red, 5. THE CORRELATION BETWEEN
green, blue, cyan and magenta dots are for the cases ENVIRONMENT AND GALAXY PROPERTY
where the 2D projected environments are calculated us-
InSection4weoptimizedthechoiceofN forthe2D
ing N = 6, 30, 60, 90 and N respectively. We also 2D
2D 3D projected density measurement to yield the best corre-
mark the four cases of Figure 5, r = 0.425, 0.455, 0.503
s lation with the 3D real-space environments by using the
and0.549respectively,todemonstratehowr varieswith
s r metric. In this section, we use these optimized results
different choices of N . In the following analysis, we s
2D to study how the color–density relation in the simula-
only show the optimized choice of N corresponding to
2D tionchangeswhenvaryingthephoto-zuncertaintiesand
thecasewhichyieldsthelargestr ,asafunctionofN .
s 3D outlier rates. Although the density–color and/or halo
Figure 10 shows the largest r (upper-panel) and cor-
s mass–colorrelationsseeninthesimulationsmaynotfully
responding choice of N (bottom-panel) that yields the
2D represent the observed Universe, this provides us with a
bestcorrelationbetween2Dprojectedand3Dreal-space
guideline to understand how reliably we can study the
environments for different choices of N , as a function
2D dependence of galaxy properties on environment using
of N from mock galaxy catalogs. The red, green, blue
3D the photo-z samples.
and cyan dots are for samples with different photo-z er-
rors: 0.00,0.02(1+z),0.04(1+z),0.06(1+z)respectively.
5.1. Environment vs. Galaxy Color
Asexpected,thedensitiesarerelativelyeasiertorecover
for samples with lower photo-z errors than those with To explore the relation between galaxy color and en-
higherphoto-zerrors. Ther obtainedforthecasewith- vironment, wecomparetheapparentmagnitudeiversus
s
out photo-z error (red dots) are greater than 0.9, mean- g −i colors of galaxies located in the 20% most dense
ing that the optimized 2D projected environments are and the 20% least dense galaxy environments respec-
6
tively. We note that although conventionally the color– ally smaller with increasing photo-z error. Considering
magnitude relation is defined in the rest frame quantity thecasewithoutphoto-zerrorandgalaxiesm >20,the
i
when studying the color–density relation, here we only difference in red fraction between the 20% most dense
look at the observed quantity since the redshift range is and20%leastdenseenvironmentsrangesfrom0.5to0.6,
very small and our main purpose is to see if the density but decreases to 0.3 – 0.4 in the case of photo-z error =
dependence of color distributions can still be revealed in 0.06(1+z). Nevertheless,itisstillencouragingthateven
the photometric redshift sample, rather than quantify- in the worst case (photo-z error = 0.06(1+z)), the de-
ing the ’color–density relation’ itself. Galaxies are first pendence of the red fraction on galaxy environment can
classifiedtoberedorblueaccordingtotheirlocationsin still be seen.
theobservedColor–MagnitudeDiagram(CMD).Weuse Figure 15 presents similar information to Figure 14,
g−i=1.5and1.75asdividinglinestoseparateblueand but now for the cases with various secondary magnitude
red galaxies at 0.3 < z < 0.5 and 0.6 < z < 0.8, respec- limits. Hereweconsiderthecaseswithms <25,ms <23
i i
tively. Nextwebinthegalaxiesaccordingtotheiri-band and ms < 21. All the three cases are considered using
i
apparentmagnitudeandthencomputethepercentageof mp <25. Toremoveadditionaluncertaintiesduetopro-
i
red galaxies defined as: jection effects, the density ranking is based on the 3D
real-space environments, ρ . Our results show that the
6
N environmentmeasuredwithfaintersecondarymagnitude
f = red, (5)
red N limits yield a better correlation with galaxy properties.
bin
This result is somewhat expected because the scales of
where N is the total number of galaxies and N is the environment defined by different ms are different for
bin red i
the number of red galaxies in each bin. a given neighbor N, being greater for brighter ms. If the
i
WedeterminethechoiceofN thatyieldsthelargest color–density relation is scale dependent, it can lead to
2D
r for photo-z samples with different photo-z uncertain- thedependenceontheadoptedms. Thedegradedcolor–
s i
ties using the methodology described in Section 4.2. In density relation with brighter sample in Figure 15 could
the case where we study the color–density relation for be due to environmental effects on color being weaker
the environment scale corresponding to the 6th nearest with increasing environment scale. Furthermore, fainter
neighbor in the 3D space, i.e., ρ , it is found that the sampleisspatiallydenserandthereforecontainsmorein-
6
optimized N = 6, 6, 6, 12 is for the cases with photo- formationaboutenvironmentthanasparsesampledoes.
2D
z error = 0.00, 0.02(1+z), 0.04(1+z) and 0.06(1+z) Includingmoregalaxiesinthedensityestimatethusalso
respectively (see the bottom panel of Figure 10). We help to characterize the environments. We will inves-
note that in the case of photo-z error = 0.06(1+z), al- tigate the correspondence between galaxy environment
though N = 12 is the best choice for optimization, we and dark matter halo mass in the next section.
2D
still adopt N = 6 to show its CMD for convenience
2D 5.2. Environments vs. Dark Matter Halo Mass
asthere isalmostnosignificant differencebetween using
N = 6 or N = 12 for the optimization. Observationally it is found that galaxies located in
2D 2D
The upper-panels of Figure 14 show the CMD for the massive halos, such as groups and clusters, are in gen-
20%mostdenseenvironments(red-contour)andthe20% eral formed earlier and hence are more evolved, com-
least dense environments (blue-contour) for galaxy sam- pared to galaxies in the field (Capak et al. 2007). Semi-
ples with different photo-z errors. Here we use the 2D analytic galaxy formation models have also successfully
projectedmeasurementsΣ andsetV =±0.001(1+z), reproduced the observed trend (Lemson & Kauffmann
6 cut
±0.02(1 + z), ±0.04(1 + z) and ±0.06(1 + z) for the 1999;Bensonetal.2000;Haasetal.2012;Muldrewetal.
case with photo-z error = 0.00, 0.02(1+z), 0.04(1+z) 2012). Figure 16 shows the red fraction as a function of
and 0.06(1+z) respectively. All cases are considered for dark matter halo mass in our mock catalog based on the
mp <25andms <25. Herethecontoursconnectpoints model of Lagos et al. (2012). As can be seen, the red
i i
with equal pixel-density in the CMD. The lower-panels fraction increases rapidly toward massive halos, in good
of Figure 14 show the red fraction, f , as a function agreement with observation.
red
of i-band apparent magnitude for local densities corre- We now proceed to show the correlation between host
spond to the 20% most dense (red), 60% – 80% densest halo mass and overdensity in order to understand why
(orange), 40% – 60% densest (yellow), 20% – 40% dens- the galaxy environment calculated using a fainter sec-
est (green) and 20% least dense (blue). The error bars ondary magnitude limit has a better correlation with
show the 1-σ Poisson uncertainty in each bin. galaxy color. We adopt a method similar to the one
It is clear that in the simulation, galaxy colors are used in Muldrew et al. (2012) which is to plot the rela-
strongly correlated with environment: being redder in tionship between host halo mass and overdensity. The
denserenvironments. Inthecasewherethereisnophoto- upper-panels of Figure 17 show the correlations between
zerror,theredandbluecontoursoccupydistinctregions host halo mass and the 3D overdensity for the galaxy
intheCMD.Thistrendisingoodagreementwithobser- samples with different secondary magnitude limits. In
vational results (Balogh et al. 2004; Cooper et al. 2006, general we find that even in the case of zero redshift er-
2007;Cassataetal.2007). Asthephoto-zerrorincreases ror, the 3D environment measures are a poor tracer of
(from left to right), red and blue contours begin to over- massforindividualobjectsasrevealedbythelargescat-
lap. To further quantify the influence of the photo-z un- ters between 3D density field and halo mass. Similar to
certaintyontheCMD,weplottheredfractionasafunc- what is found by Muldrew et al. (2012), for low N a
3D
tion of i-band apparent magnitude for galaxies located galaxy found at high 3D density is actually more likely
in5differentdensitypercentiles(bottompanelofFigure to be in a low-mass halo than in a high-mass one. Fur-
14). The difference in the red fraction becomes gradu- thermore, our results show that for a fixed N , the low
3D
7
density region probed using a brighter ms has a wider compared to that of the spectral-z samples. To do so,
i
spread in halo mass. In contrast, the low density region werandomlychoosepartoftheentirespectral-zsamples
measuredusingafainterms isdominatedbygalaxieslo- from the mock catalogs to simulate the spectral-z sam-
i
cated in small halos (<1012 M ) where the red fraction ples with different percentage of completeness. We then
(cid:12)
drops significantly with decreasing halo mass (see Fig- apply our optimized scheme as introduced in Section 4.2
ure 16). This is because the scale of the density defined to both incomplete spectral-z samples and photo-z sam-
by a fainter magnitude limit for a fixed N3D is typically ples to compare their rs.
smaller and less contaminated by the 2-halo term, and Figures 19 – 21 show rs as a function of N3D for
therefore is a better tracer of the halo mass, except for spectral-z samples with different completeness of 10%,
very massive halos (>1014 M ). 20%, 40%, 60%, 80% and 100% (denoted by different
(cid:12)
This point is further illustrated in the lower panels colors). For comparison, we also overplot the results of
of Figure 17 where we show that smaller N yields a photo-z sample with different photo-z uncertainties pre-
3D
stronger correlation between overdensity and host halo sented in different line styles in the top-left panel. To
mass than using larger N3D. Therefore the tighter re- fairly compare rs on the same physical distance scales
lationship between the galaxy colors and densities com- among various cases, rs is calculated using the N3D that
puted using a fainter secondary magnitude limit seen in correspondstothesameNth nearestneighborinthecase
Figure 15 can be attributed to the fact that the environ- of 100% complete spectroscopic sample. A zoomed-in
ment defined using a fainter sample traces more closely version on small scales are shown in the top-right panel.
with host halo masses. The difference among the three figures (19 – 21) is the
Figure18shows1+δ2D versushalomassforthegalax- secondarymagnitudelimitapplied. Forexample,inFig-
6
ies with photo-z errors varying from 0.0 to 0.06(1+z). ure 19 we consider the case where galaxy environments
The overdensity 1+δ62D increases with host halo mass are calculated by using msi < 25.0. It shows that the
but with a large scatter, as seen in Muldrew et al. galaxy environments calculated by using an incomplete-
(2012)whichisbasedondifferentsimulations. Neverthe- nessspectral-zsamplewiththisdeepmagnitudeselection
less, the correlation becomes progressively weaker when are more reliable than those galaxy environments calcu-
photo-z uncertainty increases. In the case of photo-z = lated by using a complete photo-z sample.
0.06(1+z),thecorrelationisalmostflat,suggestingthat However, in general it is difficult to obtain spectro-
the 2D projected density is no longer a good tracer of scopic redshifts for a large sample of very faint galaxies.
halo mass when photo-z errors are non-negligible. For example, the DEEP2 survey (Newman et al. 2013)
is limited to R < 24.1 and the zCOSMOS bright sam-
ple (Lilly et al. 2007) is limited to i = 22.5. Next we
5.3. Comparison with Spectroscopic Observation
varythesecondarymagnitudelimitsofthespectroscopic
In previous sections, we present how the photo-z un- sample to see how the trend changes. Here we consider
certainty can have an impact on the measurement of lo- the two optimized results, ms < 24.1 and ms < 22.5,
i i
cal density and discuss how well the 2D projected den- to roughly mimic the results of DEEP2 sky survey and
sitytracesthereal-spacedensitywhenadoptingdifferent zCOSMOS-bright survey, respectively. Strictly speaking
choices of the size of velocity (redshift) window, magni- the DEEP2 is limited in the R-band instead of i-band,
tude limit, and the Nth nearest neighbor. Furthermore, however, here we simply adopt the i-band in order to
wehavealsostudiedhowthecolor–densityrelationisaf- reveal the trend more clearly. Our results show that
fectedbythepresenceofphoto-zerrorsandasafunction in the case of ms < 24.1, the performance of density
i
of the magnitude limit that is applied to the secondary recovery with photo-z error as low as ∼ 0.02(1+z) is
sample. In this section, we further investigate the dif- always worse than that of the spectral-z samples. How-
ference in the local density measurements between the ever,whenthemagnitudelimitofthespectral-zsamples
photo-z and spectral-z sample using mock galaxies, by decreases to ms < 22.5, the performance becomes com-
i
taking into account more realistic situations including parabletothespectral-zsamplewith10%completeness.
the incompleteness of the spectroscopic sample. Inotherwords,asthespectral-zsamplegetsbrighter,the
AswediscussinSection4,thephoto-zuncertaintyhas r coefficients between the spectral-z and photo-z sam-
s
a strong impact on the correlation between the 2D and plesbecomecloser. However,thedifferencebetweentheir
3D densities. Ideally, the density measurement based on r coefficientsgraduallybecomeslargerwhenweprobea
s
spectral-z sample is more reliable. However, the spec- larger scale of environment, as described in Section 4.2.
troscopic observations of galaxies are time-consuming That is, a deeper photo-z sample can yield similar per-
and hence are normally limited to a small sample size, formance as good as an incomplete, shallower spectral-z
brighter galaxies, and a lower redshift range. Moreover sample,butthisisonlyrestrictedtosmall-scaleenviron-
they often suffer from incompleteness due to the limited ments.
observing time as well as the fiber and/or slit collisions.
5.4. Effect of Outliers
Incontrast, thephoto-zcanberelativelyeasilyobtained
down to fainter galaxies and out to higher redshifts with So far the studies on the effect of the photo-z uncer-
amuchlargersizeofsample,butwiththedrawbackthat tainty on the density measurement are carried out by
the redshift resolution is substantially poorer compared perturbingtheredshiftsofmockgalaxieswithaGaussian
to spectral-z. Nevertheless, both spectral-z and photo-z function. However, this method does not totally mimic
samples have been used on environment studies (Cooper the realistic case because the photo-z errors may not ex-
etal.2006;Elbazetal.2007;Cassataetal.2007;Quadri actly follow the Gaussian distribution. For example, in
et al. 2012). It is thus interesting to investigate to what the cases where there are not enough numbers and/or
extend the local density from photo-z samples can be wavelength coverage of bandpasses, the feature of the
8
Lyman break (∼ 912 angstrom) can be misidentified as spectroscopic redshifts and compare these environments
Balmerbreak(∼4000angstrom)andviceversa, leading with those calculated by using photometric redshifts
to a catastrophic failure in the photo-z estimation, the from Pan-STARRS1 for the same galaxies. For compar-
so-called ‘redshift outliers’ (Brough et al. 2013). Next ison, we also utilize the mock galaxy catalogs described
we study how the outlier rate influences the correlation inSection2.1andperturbtheirredshiftstosimulatethe
between the 2D and 3D local density measurements. photo-z conditions of Pan-STARRS1 where the typical
Tosimulategalaxysampleswithoutliers,werandomly error is ∼0.06(1+z) and the outlier rate is ∼6%. Fig-
choosepartoftheentiresimulationsamplesaccordingto ure 24 shows the scatter plot for galaxies in the EGS
the desired outlier rate, and assign them a new redshift field (left-panel) and in the simulations (right-panel). In
randomlybetween0–2.0. Figure22showstheresultsus- the left panel, The 2D and 3D environments are eval-
ingsampleswithphoto-z=0.02(1+z)andfourdifferent uated by using samples from Pan-STARRS1 (photo-z)
percentages of outliers: 5%, 10%, 15% and 20%. Similar and DEEP2 (spectral-z) respectively. As it can be seen,
to Figures 19 – 21, we also mark the results for different whilewecomparethe2Dprojectedand3Denvironments
photo-z uncertainties in the case of 0% outliers with dif- intherealisticcase, theslopeofscatterplotissimilarto
ferentlinestylesforcomparison. Fromthisfigure,wecan simulationresultbutr issmallerthantheresultsofsim-
s
see that r also strongly depends on the outlier fraction, ulation. While this might be explained by the intrinsic
s
becoming worse as the outlier fraction increases. The ef- difference in the galaxy spatial distribution between the
fect is similar to the degradation of photo-z uncertainty real Universe and simulations, it is also noticed that the
and the completeness of the sample. For example, for galaxiesintheMD07/EGSfieldspananarrowerrangein
the sample with photo-z error = 0.02(1+z) and outlier the 3D overdensity because of the smaller field size such
rate = 10% (green dots), we find its optimized result is thattheextremeenvironmentsarenotwell-sampled. As
similar to the result of outlier-free sample with photo-z a result, the simulated sample includes very dense en-
error = 0.04(1+z). vironments which are more discernible and easier to be
Figure23showstheCMDforthesampleswithphoto- recovered compared to intermediate environments. Nev-
z error = 0.02(1+z) and four different percentages of ertheless, there still exists a weak correlation in compar-
outliers. The environments are measured by using Σ , ison with real data even though their r coefficients are
6 s
mp < 25.0 and ms < 25.0. As it can be seen, in smaller than those of the simulated data.
i i
the case with 5% outlier, it is comparable to a non- To know whether the color–density relation can still
outlier case with photo-z error between ∼ 0.02(1 + z) berevealedintherealisticphoto-zsample, inthe Figure
and ∼ 0.04(1 + z) and the color–density relation can 25 we plot the CMD (upper panels) and the red frac-
still be revealed. However, as the outlier rate goes up tions versus i magnitude (lower panels) for galaxies
p1
to 10%, the density measurements for under-dense en- located in the 20% most dense (red contour) and 20%
vironments (for example, the least 20% (blue) and 20% leastdense(bluecontour)galaxyenvironments. Thered
– 40% (green) dense environments) are no longer distin- fraction is defined as the ratio of the number of galax-
guishable, resulting in a weaker color–density relation. ies with g −i color redder than 1.5 to that of the
p1 p1
This is in contrast to the situations with pure photo-z full sample. For each galaxy in the right-panel of PS1,
errors for which the lowest density curves remain dis- we compute Σ with ms < 24, using the photometric
6 i
tinguishable. The outliers have larger effects in lower redshifts derived in Lin et al. (2014) with the redshift
density environments because the change in the density range of 0.3 < z < 0.5. The left panel of Figure 25,
measurements are proportionally larger in those regions which is for comparison, shows the DEEP2 result us-
when some fraction of galaxies are scatted inside or out ing galaxy densities computed by Cooper et al. (2005).
the relevant redshift window. Theirdensitymeasurementshavebeencorrectedforsev-
eral effects such as the survey edges, redshift precision,
6. COLOR–DENSITY RELATION FOR
redshift-spacedistortionandtargetselectionasdescribed
PAN-STARRS1 DATA
in Cooper et al. (2005). Therefore, their measurements
Themainpurposeofthisworkistounderstandthesys- can be regarded as the ’true’ answer in this comparison.
tematics in the 2D density measurement and its limita- The middle panel shows the result calculated with PS1
tion,withtheultimategoalofitsapplicationtotheongo- photometric redshifts only for galaxies located in the re-
ingandfuturelargephotometricsurveys. Sofarwehave gion overlapping with the EGS field which is ∼0.5 deg2
exploredvariousaspectsofthedensitymeasurementsby (∼ 1,500 galaxies), while the right panel shows the re-
usingmockgalaxycatalogsforwhichthereal-spaceden- sult based on the entire PS1/MD07 field of ∼ 5 deg2
sityisknown. Wehaveconsideredseveralfactorssuchas (∼25,000galaxies). Tominimizetheimpactoftheedge
photo-z uncertainty, magnitude limit, completeness and effects, we also exclude galaxies near the survey bound-
outlier rate that make simulation data as similar to re- aries when showing the color–density relation. Among
alistic samples as possible. However these factors are all the three samples, the color–density relation is only
still not sufficient to imitate the realistic samples. An significantlydetected(>3σ)inthe∼5deg2 PS1photo-
alternative is to compare the results of the overlapping z sample. This is because although the photo-z uncer-
samplesdirectlybetweenphoto-zandspectral-zsurveys. taintyingeneralcontaminatesthedensitymeasurement,
We adopt this approach by using the PS1 MD07 photo- which leads to some systematics in the color–density re-
metric redshift catalog (Lin et al. 2014) as it covers the lation, the random errors can be largely improved given
well-known EGS field, which has the spectroscopic red- the large volumes probed by a photometric survey. In
shifts from DEEP2, allowing for a direct comparison of other words, the reduced errors due to the larger sample
the density measurement. are sensitive enough to allow for the detection of a ’de-
Wefirstcomputetheenvironmentsusinggalaxieswith
9
graded’ relation between red fraction and environment. shallower i∼22.5 spectroscopic redshift samples with ∼
Furthermore, we also extend our study from low red- 10%completeness. Inaddition,theeffectofcatastrophic
shift range (0.3 < z < 0.5) to higher redshift range failuresinthephoto-zmeasurementsonthedensitymea-
(0.6 < z < 0.8). In this redshift bin, g −i = 2.0 surement is similar to that of the photo-z errors.
p1 p1
is used to separate blue and red galaxies. Similarly we (vi) Using Durham mock galaxy catalogs in the
first show the color–density relation at redshift range redshift range of 0.3 < z < 0.5, we show that the
0.6 < z < 0.8 using our simulated dataset (Figure 26) density–dependent red fraction can still be revealed in
and PS1 samples (Figure 27, ∼ 45,000 galaxies). As photometric redshift samples with photo-z uncertainty
shown in the right panel of Figure 27, the difference up to 0.06(1 + z). Similarly with photo-z sample
in the red fraction between two extremely environments from PS1, we show that the color–density relation is
is still detectable at ∼ 2−3σ level. The Kolmogorov- also present in the sample whose photo-z uncertainty
Smirnovtest(Andradeetal.2001)onthecolordistribu- is ∼ 0.06(1 + z) and the outlier rate is ∼ 6%, but
tions for galaxies with 21<i <23 located in the most the significance strongly depends on the sample size.
p1
20% and least 20% density percentiles returns a value of Based on the results of PS1 in the two redshift bins
p << 0.1%, rejecting the null hypothesis that the color (0.3 < z < 0.5 and 0.6 < z < 0.8), we recommend
distributions of galaxies are drawn from the same popu- that the survey size should at least exceed ∼ 5 deg2
lation. in order to yield > 3σ results. Larger fields will be
required in order to reduce the Poisson errors if going
7. CONCLUSIONS to higher-redshifts as the color–density relation is less
prominentandthenumberdensityofgalaxiesisreduced.
In this work, we have studied how the 2D projected
environment correlates with the 3D real-space environ-
We thank the anonymous referee for very construc-
ment. UsingtheDurhammockgalaxycatalogs,weinves-
tive comments, which greatly improve this manuscript.
tigatevariousparametersinmeasuringthe2Dprojected
This work is supported by the Ministry of Science and
environment and find the best parameters which maxi-
Technology of Taiwan under the grant NSC101-2112-
mizetheSpearman’srankcorrelationcoefficient,defined
as rs = 1− s6(s(cid:80)2−d12i), which is a quantity for quantifying MM-O0S01T-100121--2M11Y22-,M-M00O1S-0T0110-3M-21Y132.-MH-.0-0Y1.-0J3ia1n-MaYck3noawnld-
the correlation between 3D real-space and 2D projected edges the support of NSC101-2811-M-002-075. W.-P.
overdensities. When applying the Nth nearest neighbor Chen acknowledges the support of NSC102-2119-M-008-
method to the PS1 photo-z sample, we show that color– 001. We thank Michael C. Cooper for providing the
densityrelationcanstillberevealeddespiteofthesizable DEEP2 environment measurements for our work. We
photo-z errors inherent in the data. Our main conclu- alsothankthePS1BuildersandPS1operationsstafffor
sions are as follows. construction and operation of the PS1 system and ac-
(i) The correlation between the 2D projected and 3D cess to the data products provided. The Pan-STARRS1
real-space overdensity is sensitive to the photo-z uncer- Surveys (PS1) have been made possible through con-
tainty. Smaller N3D is recommended for photo-z sam- tributions of the Institute for Astronomy, the Univer-
ples to achieve a better correlation (larger rs) between sity of Hawaii, the Pan-STARRS Project Office, the
2D projected and 3D real-space environments. Max-Planck Society and its participating institutes, the
(ii)Asthescaleof3Dreal-spaceenvironmentincreases, Max Planck Institute for Astronomy, Heidelberg and
the rs derived by using spectral-z and photo-z samples the Max Planck Institute for Extraterrestrial Physics,
show the opposite trend: the correlation becomes grad- Garching, The Johns Hopkins University, Durham Uni-
ually stronger for the spectral-samples but worse for the versity, the University of Edinburgh, Queen’s Univer-
photo-z samples. sity Belfast, the Harvard-Smithsonian Center for Astro-
(iii) The 2D projected environment measurements are physics, the Las Cumbres Observatory Global Telescope
less sensitive to the redshift interval (Vcut). The redshift Network Incorporated, the National Central University
intervalcomparabletothephoto-zuncertaintyyields2D of Taiwan, the Space Telescope Science Institute, the
projected overdensity that reasonably traces the real- National Aeronautics and Space Administration under
space density. Grant No. NNX08AR22G issued through the Planetary
(iv) The magnitude limit should also be considered Science Division of the NASA Science Mission Direc-
whencomputinglocaldensitiesofgalaxies. The2Denvi- torate,theNationalScienceFoundationunderGrantNo.
ronments measured with fainter magnitude limits yield AST-1238877, the University of Maryland, and Eotvos
better correlation with the 3D real-space environments Lorand University (ELTE).
derived from the same limiting magnitude sample for a
fixedN . Inaddition,thecolor–densityrelationismore APPENDIX
3D
prominent if the density is measured using fainter mag-
A. N VS. N IN ENVIRONMENT
nitude limits for a fixed N . This is because the over- 3D 2D
2D MEASUREMENTS
densitycomputedwithfaintermagnitudesprovessmaller
scales with the same N , and traces the hosting halo AsmentionedinSection4.2,theoptimizedN isonly
2D 2D
mass of galaxies better. half the value of N . The ratio of the two quantities in
3D
(v) Considering the case calculated by using galaxy fact depends on the size of the redshift interval when
samplesat0.3<z <0.5withmp <25.0andms <25.0, computing the 2D density. Figure 28 shows the opti-
i i
the recovering performance of small-scale environments mized scheme for the case with different size of V . We
cut
for photometric redshift samples with redshift uncer- considerthesamplewithphoto-zerror=0.04(1+z)and
tainty of 0.02(1+z) is roughly comparable to that for calculate the environment with different sizes of V =
cut
10
±0.005(1+z),±0.02(1+z),±0.04(1+z)and±0.06(1+z), effect of N is on r . In Figure 29 we plot the differ-
2D s
corresponding to red, green, blue and cyan dots respec- ence between the two r , one is evaluated by using best
s
tively. Asitcanbeseen,forthecaseswithsizeofV = choice of N , and another is evaluated by N = N ,
cut 2D 2D 3D
±0.02(1+z), ±0.04(1+z) and ±0.06(1+z), their opti- normalized by the former, as a function of N . As can
3D
mized results are quite similar although the best choices beseen, theirmaximumdifferenceisonly∼9%, 3%and
of N are different. The ratio of the optimized N to 3% in the case with V = ±0.02(1+z), ±0.04(1+z)
2D 2D cut
N , roughly, is 1:10, 1:5, 1:2.5 and 1:1.7 for the cases and ±0.06(1+z) respectively. This suggests that if the
3D
withV =±0.005(1+z),±0.02(1+z),±0.04(1+z)and size of V comparable to the photo-z uncertainty, the
cut cut
±0.06(1+z), respectively. The N in the case of V 2D local density measured with N ∼ N could be as
2D cut 2D 3D
=±0.06(1+z)istripledcomparedtoN inthecaseof good as that derived with the optimized N .
2D 2D
V =±0.02(1+z). Nextweinvestigatehowstrongthe
cut
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