The ALHAMBRA survey: an empirical estimation of the cosmic variance for merger fraction studies based on close pairs

0.47 MB·

by  C. López-Sanjuan

Astronomy&Astrophysicsmanuscriptno.ms c ESO2014
(cid:13)
January21,2014
⋆
The ALHAMBRA survey : an empirical estimation of the cosmic
variance for merger fraction studies based on close pairs
C.Lo´pez-Sanjuan1,⋆⋆,A.J.Cenarro1,C.Herna´ndez-Monteagudo1,J.Varela1,A.Molino2,P.Arnalte-Mur3,
B.Ascaso2,F.J.Castander4,A.Ferna´ndez-Soto5,6,M.Huertas-Company7,8,I.Ma´rquez2,V.J.Mart´ınez6,9,
J.Masegosa2,M.Moles1,2,M.Povic´2,J.A.L.Aguerri10,11,E.Alfaro2,N.Ben´ıtez2,T.Broadhurst12,
J.Cabrera-Can˜o13,J.Cepa10,11,M.Cervin˜o2,10,D.Cristo´bal-Hornillos1,A.DelOlmo2,R.M.Gonza´lezDelgado2,
C.Husillos2,L.Infante14,J.Perea2,F.Prada2,andJ.M.Quintana2
4
1 1 CentrodeEstudiosdeF´ısicadelCosmosdeArago´n,PlazaSanJuan1,44001Teruel,Spain
0 2 InstitutodeAstrof´ısicadeAndaluc´ıa(IAA-CSIC),Glorietadelaastronom´ıas/n,18008Granada,Spain
2 3 InstituteforComputationalCosmology,DepartmentofPhysics,DurhamUniversity,SouthRoad,DurhamDH13LE,UK
4 InstitutdeCie`nciesdel’Espai(IEEC-CSIC),FacultatdeCie´ncies,CampusUAB,08193Bellaterra,Spain
n
5 InstitutodeF´ısicadeCantabria,AvenidadelosCastross/n,39005Santander,Spain
a
6 ObservatoriAstrono`mic,UniversitatdeVale`ncia,C/Catedra´ticoJose´Beltra´n2,46980Paterna,Spain
J
7 GEPI,ParisObservatory,77av.DenfertRochereau,75014Paris,France
0 8 UniversityDenisDiderot,4RueThomasMann,75205Paris,France
2 9 Departamentd’AstronomiaiAstrof´ısica,UniversitatdeVale`ncia,46100Burjassot,Spain
10 InstitutodeAstrof´ısicadeCanarias,V´ıaLa´cteas/n,LaLaguna,38200Tenerife,Spain
] 11 DepartamentodeAstrof´ısica,FacultaddeF´ısica,UniversidaddelaLaguna,38200LaLaguna,Spain
O
12 DepartmentofTheoreticalPhysics,UniversityoftheBasqueCountryUPV/EHU,Bilbao,Spain
C 13 DepartamentodeF´ısicaAto´mica,MolecularyNuclear,FacultaddeF´ısica,UniversidaddeSevilla,41012Sevilla,Spain
. 14 DepartamentodeAstronom´ıayAstrof´ısica,FacultaddeF´ısica,PontificiaUniversidadCato´licadeChile,782-0436Santiago,Chile
h
p Received12August2013–Accepted17January2014
-
o
ABSTRACT
r
t
s Aims.Ourgoalistoestimateempirically,forthefirsttime,thecosmicvariancethataffectsmergerfractionstudiesbasedonclose
a
pairs.
[
1 Mkmeths−o1d,sa.nWdmeecaosmupreutietitnhethmee4r8gesrubfr-aficetlidosnoffrothmepAhLoHtoAmMetrBicRAredssuhrivfetyc.lWoseesptauidrsywthiethdi1s0trhi−b1utkiopnco≤ftrhpe≤me5a0shu−r1edkpmcearngder∆frvac≤tio5n0s0,
v thatfollowalog-normal function, andestimatethecosmicvarianceσv astheintrinsicdispersionoftheobserved distribution.We
3 developamaximumlikelihoodestimatortomeasureareliableσvandavoidthedispersionduetotheobservationalerrors(including
9 thePoissonshotnoiseterm).
9 Results.Thecosmicvarianceσv ofthemergerfractiondepends mainlyon(i)thenumber densityof thepopulations under study,
4 bothfortheprincipal(n1)andthecompanion(n2)galaxyintheclosepair,and(ii)theprobedcosmicvolumeVc.Wefindasignificant
. dependenceonneitherthesearchradiususedtodefineclosecompanions,theredshift,northephysicalselection(luminosityorstellar
1 mass)ofthesamples.
0 Conclusions.We have estimated from observations the cosmic variance that affects the measurement of the merger fraction by
4 closepairs. Weprovide aparametrisationof thecosmicvariance withn , n ,and V ,σ n 0.54V 0.48(n /n ) 0.37.Thanks tothis
1 prescription,futuremergerfractionstudiesbasedonclosepairscouldacc1oun2tproperlcyfovrt∝hec1−osmicc−varia2nce1o−ntheirresults.
:
v
Keywords.Galaxies:fundamentalparameters–Galaxies:interactions–Galaxies:statistics
i
X
r
a 1. Introduction Survey, Davisetal. 2007), ELAIS (European Large-Area ISO
Survey,Rowan-Robinsonetal.2004),COSMOS(Cosmological
Our understanding of the formation and evolution of galaxies
Evolution Survey, Scovilleetal. 2007), MGC (Millennium
acrosscosmictimehavebeengreatlyimprovedinthelastdecade
Galaxy Catalogue, Liskeetal. 2003), VVDS (VIMOS VLT
thanks to deep photometric and spectroscopic surveys. Some
Deep Survey, LeFe`vreetal. 2005, 2013), DEEP (Deep
examples of these successful deep surveys are SDSS (Sloan
Extragalactic Evolutionary Probe, Newmanetal. 2013),
Digital Sky Survey, Abazajianetal. 2009), GOODS (Great
zCOSMOS (Lillyetal. 2009), GNS (GOODS NICMOS
Observatories Origins Deep Survey, Giavaliscoetal. 2004),
Survey, Conseliceetal. 2011), SXDS (Subaru/XMM-Newton
AEGIS (All-Wavelength Extended Groth Strip International
Deep Survey, Furusawaetal. 2008), or CANDELS (Cosmic
AssemblyNIRDeepExtragalacticLegacySurvey,Groginetal.
⋆ Based on observations collected at the German-Spanish
2011;Koekemoeretal.2011).
Astronomical Center, Calar Alto, jointly operated by the Max-
Planck-Institutfu¨rAstronomie(MPIA)atHeidelbergandtheInstituto Onefundamentaluncertaintyinanyobservationalmeasure-
deAstrof´ısicadeAndaluc´ıa(IAA-CSIC). mentderived fromgalaxy surveysis the cosmic variance (σ ),
v
⋆⋆ e-mail:[email protected] arisingfromtheunderlyinglarge-scaledensityfluctuationsand
1
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
leading to variances larger than those expected from simple
Poissonstatistics. Themostefficientwaytotacklewithcosmic
variance is split the survey in several independentareas in the
52.8
sky. This minimises the sampling problem better than increase
thevolumeinawidecontiguousfield(e.g.,Driver&Robotham )
g
2010).However,observationalconstraints(depthvsarea)leadto e 52.6
d
manyexistingsurveystohaveobservationaluncertaintiesdomi- (
c
natedbythecosmicvariance.Thus,aproperestimationofσv is De 52.4
needed to fully describe the error budgetin deep cosmological
surveys.
52.2
The impact of the cosmic variance in a given survey
and redshift range can be estimated using two basic meth-
ods: theoretically by analysing cosmological simulations (e.g., 213.5 214.0 214.5 215.0
Somervilleetal. 2004; Trenti&Stiavelli 2008; Stringeretal. RA (deg)
2009; Mosteretal. 2011), or empirically by sampling a larger
survey (e.g., Driver&Robotham 2010). Unfortunately, previ-
Fig.1.SchematicviewoftheALHAMBRAfield’sgeometryin
ous studies estimate only the cosmic variance affecting num-
the sky plane. We show the eight sub-fields (one per LAICA
ber density measurements, and do not tackle the impact of σv chip) of the field ALHAMBRA-6. The black and red squares
in other important quantities as the merger fraction. Merger
markthe two LAICA pointingsin this particularfield. The ge-
fraction studies based on close pair statistics measure the cor-
ometryofthe othersevenfieldsis similar. [Acolourversion of
relation of two galaxy populations at small scales ( 100h−1 thisplotisavailableintheelectronicedition].
≤
kpc), so the amplitude of the cosmic variance and its depen-
dence on galaxy properties,probedvolume,etc. should be dif-
ferentthanthoseinnumberdensitystudies.Inthepresentpaper finalsurveyparametersandscientificgoals,aswellasthetech-
we take advantage of the unique design, depth, and photomet- nical propertiesof the filter set, were describedby Molesetal.
ric redshift accuracy of the ALHAMBRA1 (Advanced, Large, (2008). The survey has collected its data for the 20+3 optical-
HomogeneousArea,Medium-BandRedshiftAstronomical)sur- NIRfilters in the 3.5mtelescope atthe Calar Alto observatory,
vey (Molesetal. 2008) to estimate empirically, for the first using the wide-field camera LAICA (Large Area Imager for
time, the cosmic variance that affect close pair studies. The Calar Alto) in the optical and the OMEGA2000 camera in the
ALHAMBRA survey has observed 8 separate regions of the NIR.Thefullcharacterisation,description,andperformanceof
northern sky, comprising 48 sub-fields of 180 arcmin2 each the ALHAMBRA opticalphotometricsystem waspresentedin
∼
that can be assumed as independent for our purposes. Thus, Aparicio-Villegasetal.(2010).Asummaryoftheopticalreduc-
ALHAMBRA provides 48 measurements of the merger frac- tion can be foundin Cristo´bal-Hornilloset al. (in prep.),while
tion across the sky. The intrinsic dispersion in the distribution oftheNIRreductioninCristo´bal-Hornillosetal.(2009).
ofthesemergerfractions,thatwecharacteriseinthepresentpa- TheALHAMBRAsurveyhasobserved8well-separatedre-
per,isanobservationalestimationofthecosmicvarianceσ . gions of the northern sky. The wide-field camera LAICA has
v
The paper is organised as follows. In Sect. 2 we present fourchipswitha15 15 field-of-vieweach(0.22arcsec/pixel).
′ ′
×
the ALHAMBRA survey and its photometric redshifts, and in The separation between chips is also 15. Thus, each LAICA
′
Sect.3wereviewthemethodologytomeasureclosepairmerger pointing provides four separated areas in the sky (black or red
fractions when photometric redshifts are used. We present our squares in Fig. 1). Six ALHAMBRA regions comprise two
estimationandcharacterisationofthecosmicvarianceforclose LAICA pointings.In these cases, the pointingsdefinetwo sep-
pair studies in Sect. 4. In Sect. 5 we summarise our work and arate strips in the sky (Fig. 1). In our study we assumed the
present our conclusions. Throughoutthis paper we use a stan- four chips in each strip as independent sub-fields. The photo-
dardcosmologywith Ω = 0.3,Ω = 0.7, H = 100hkm s 1 metric calibration of the field ALHAMBRA-1 is currently on-
m Λ 0 −
Mpc 1,andh=0.7.MagnitudesaregivenintheABsystem. ongoing, and the fields ALHAMBRA-4 and ALHAMBRA-5
−
compriseonepointingeach(seeMolinoetal.2013,fordetails).
Wesummarisethepropertiesofthe7ALHAMBRAfieldsused
2. TheALHAMBRA survey
inthepresentpaperinTable1.Attheend,ALHAMBRAcom-
The ALHAMBRA survey provides a photometric data set prises 48 sub-fields of 180 arcmin2, that we assumed inde-
∼
pendent, in which we measured the merger fraction following
over 20 contiguous, equal-width ( 300Å), non-overlapping,
∼ the methodology described in Sect. 3. When we searched for
medium-band optical filters (3500Å– 9700Å) plus 3 standard
closecompanionsnearthesub-fieldboundarieswedidnotcon-
broad-band near-infrared (NIR) filters (J, H, and K ) over 8
s sidertheobservedsourcesintheadjacentfieldstokeepthemea-
different regions of the northern sky (Molesetal. 2008). The
surements independent. We prove the independence of the 48
survey has the aim of understanding the evolution of galax-
ALHAMBRAsub-fieldsinSect4.6.
ies throughout cosmic time by sampling a large enough cos-
mological fraction of the universe, for which reliable spectral
energy distributions (SEDs) and precise photometric redshifts 2.1.BayesianphotometricredshiftsinALHAMBRA
(z ’s)areneeded.ThesimulationsofBen´ıtezetal.(2009),relat-
p
WerelyontheALHAMBRAphotometricredshiftstocompute
ing the image depth and z ’s accuracyto the numberof filters,
p
themergerfraction(Sect.3).Thephotometricredshiftsusedall
have demonstrated that the filter set chosen for ALHAMBRA
overpresentpaperarefullypresentedandtestedinMolinoetal.
can achieve a photometricredshiftprecision that is three times
(2013),andwesummarisetheirprincipalcharacteristicsbelow.
better than a classical 4 5 optical broad-band filter set. The
− TheALHAMBRA z ’swereestimatedwithBPZ2.0,anew
p
1 http://alhambrasurvey.com version of BPZ (Ben´ıtez 2000). BPZ is a SED-fitting method
2
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
Table1.TheALHAMBRAsurveyfields
Field Overlapping RA DEC sub-fields/area
name survey (J2000) (J2000) (#/deg2)
ALHAMBRA-2 DEEP2 013016.0 +041540 8/0.377
ALHAMBRA-3 SDSS 091620.0 +460220 8/0.404
ALHAMBRA-4 COSMOS 100000.0 +020511 4/0.203
ALHAMBRA-5 GOODS-N 123500.0 +615700 4/0.216
ALHAMBRA-6 AEGIS 141638.0 +522450 8/0.400
ALHAMBRA-7 ELAIS-N1 161210.0 +543015 8/0.406
ALHAMBRA-8 SDSS 234550.0 +153505 8/0.375
Total 48/2.381
1.6 350
1.4 δz = 0.011 300 µ = −0.07
1.2 η = 2.1 % σ = 1.00
250
1.0 C = 0.49
200
p 0.8 N
z 150
0.6
100
0.4
50
0.2
0
−4 −2 0 2 4
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
z ∆
s
z
Fig.2. Photometric redshift (z ) versus spectroscopic redshift
p Fig.3.Distributionofthevariable∆ forthe3813galaxiesinthe
z
(z )forthe3813galaxiesintheALHAMBRAareawithi 22.5
s ALHAMBRAareawithi 22.5andameasuredspectroscopic
≤
and a measured z . The solid line marks identity. The sources ≤
s redshift. The red line is the best least-squares fit of a Gaussian
aboveandbellowthedashedlinesarecatastrophicoutliers.The
function to the data. The median, dispersion and the factor C
accuracy of the photometric redshifts (δ ) and the fraction of
z derivedfromthe fitare labelledinthe panel.[A colourversion
catastrophicoutliers(η)arelabelledinthepanel.[Acolourver-
ofthisplotisavailableintheelectronicedition].
sionofthisplotisavailableintheelectronicedition].
ric versus spectroscopicredshiftdistribution (Ilbertetal. 2006;
Brammeretal.2008),
based in a Bayesian inference where a maximum likelihood is z z
p s
weightedbyapriorprobability.Thelibraryof11SEDs(4ellip- δz =1.48×median |1+−z |!. (1)
ticals,1lenticular,2spirals,and4starbursts)andthepriorprob- s
abilitiesusedbyBPZ2.0inALHAMBRAaredetailedinBen´ıtez The variable η is defined as the fraction of galaxies with
(in prep.). The ALHAMBRA photometryused to compute the z z /(1 + z ) > 0.2. We illustrate the high quality of
p s s
photometric redshifts is PSF-matched aperture-corrected and t|he−ALH|AMBRA photometric redshifts in Fig. 2. We refer to
based on isophotal magnitudes. In addition, a recalibration of Molinoetal.(2013)foramoredetaileddiscussion.
the zero pointof the imageswas performedto enhancethe ac- Theoddsqualityparameter,noted ,isaproxyforthepho-
curacyofthez ’s.SourcesweredetectedinasyntheticF814W O
p tometricredshiftaccuracyofthesourcesandisalsoprovidedby
filter image, noted i in the following, defined to resemble the BPZ2.0.Theoddsisdefinedastheredshiftprobabilityenclosed
HST/F814W filter. The areas of the images affected by bright on a K(1+z) regionaroundthe main peak in the probability
stars, as well as those with lower exposure times (e.g., the ±
distributionfunction(PDF)ofthesource,wheretheconstantK
edgesoftheimages),weremaskedfollowingArnalte-Muretal. isspecificforeachphotometricsurvey.Molinoetal.(2013)find
(2013).ThetotalareacoveredbytheALHAMBRAsurveyafter thatK = 0.0125istheoptimalvaluefortheALHAMBRAsur-
maskingis2.38deg2.Finally,astatisticalstar/galaxyseparation vey.The parameter [0,1]is related with the confidenceof
is encodedin the variableStellar Flagof the ALHAMBRA thez ,makingpossiOble∈toderivehighqualitysampleswithbet-
p
catalogues, and throughout present paper we keep as galaxies ter accuracy and lower rate of catastrophic outliers. For exam-
thoseALHAMBRAsourceswithStellar Flag 0.5. ple,a 0.5selectionfori 22.5galaxiesyieldsδ =0.0094
≤ O≥ ≤ z
Thephotometricredshiftaccuracy,estimatedbycomparison and η = 1%, while δ = 0.0061 and η = 0.8% for 0.9
z
O ≥
with spectroscopic redshifts (z ’s), is δ = 0.0108 at i 22.5 (seeMolinoetal.2013,forfurtherdetails).We exploretheop-
s z
≤
withafractionofcatastrophicoutliersofη=2.1%.Thevariable timal odds selection in ALHAMBRA for close pair studies in
δ isthenormalizedmedianabsolutedeviationofthephotomet- Sect.4.3.
z
3
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
Reliablephotometricredshifterrors(σ )areneededtocom- z < 0.9).Weleftrmax 50h 1kpcasafreeparameterandesti-
putethemergerfractioninphotometricsamzpples(Sect.3).Inad- mateitsoptimalvaplue≤inSec−t.4.3.Finally,weset∆vmax = 500
ditiontothez ,wehavethez+ andz ofeachsource,definedas kms 1 followingspectroscopicstudies(e.g.,Pattonetal.2000;
p σ −σ −
theredshiftsthatenclose68%ofthePDFofthesource.Wees- Linetal. 2008).With the previousconstraints50%-70%ofthe
timatedthephotometricredshifterrorofeachindividualsource selected close pairs will finally merge (Patton&Atfield 2008;
as σ = C (z+ z ). The constant C is estimated from the Belletal.2006;Jianetal.2012).
zp × σ − −σ
distributionofthevariable Tocomputeclosepairswedefinedaprincipalandacompan-
ionsample.Theprincipalsamplecomprisesthemoreluminous
z z z z
p s p s
∆ = − = − . (2) ormassivegalaxyofthepair,andwelookedforthosegalaxies
z σ C (z+ z )
zp × σ− −σ in the companion sample that fulfil the close pair criterion for
eachgalaxyoftheprincipalsample.Ifoneprincipalgalaxyhas
The variable ∆ should be normally distributed with zero
z morethanoneclosecompanion,wetookeachpossiblepairsep-
mean and unit variance if the σ ’s from ALHAMBRA are a
zp arately(i.e.,ifthecompaniongalaxiesBandCareclosetothe
good descriptor of the z ’s accuracy (e.g., Ilbertetal. 2009;
p principalgalaxyA,westudythepairsA-BandA-Casindepen-
CarrascoKind&Brunner 2013). We find that ∆ is described
z dent).Inaddition,throughpresentpaperwedonotimposeany
well by a normal function when C = 0.49 (Fig. 3, see also
luminosityormassdifferencebetweenthegalaxiesintheclose
Molinoetal.2013).Notethat,with thedefinitionofz+ andz ,
σ −σ pairunlessnotedotherwise.
C =0.5wasexpected.ThisresultalsoimpliesthattheGaussian
Withthepreviousdefinitionsthemergerfractionis
approximation of the PDF assumed in the estimation of the
mergerfraction(Setc.3) isstatistically valid,evenif theactual N
p
PDFoftheindividualsourcescouldbemultimodaland/orasym- fm = , (3)
N
metricatfaintmagnitudes.We estimatedC fordifferenti-band 1
magnitudes and odds selections, finding that the C values are where N is the numberof sourcesin the principalsample and
1
consistent with the globalone within 0.1. Thus, we conclude N the number of close pairs. This definition applies to spec-
± p
thatσ providesa reliablephotometricredshifterrorforevery troscopic volume-limited samples, but we rely on photometric
zp
ALHAMBRAsource. redshifts to compute f in ALHAMBRA. In a previous work,
m
Lo´pez-Sanjuanetal.(2010a)developastatisticalmethodtoob-
tain reliable merger fractions from photometric redshift cata-
2.2.Sampleselection
logues as those from the ALHAMBRA survey. This method-
Throughout present paper we focus our analysis in the galax- ology has been tested with the MGC (Lo´pez-Sanjuanetal.
iesoftheALHAMBRAfirstdatarelease2.Thiscataloguecom- 2010a) and the VVDS (Lo´pez-Sanjuanetal. 2012) spectro-
prises 500k sources and is complete (5σ, 3 aperture) for scopic surveys, and successfully applied in the GOODS-
′′
i 2∼4.5 galaxies (Molinoetal. 2013). We explored differ- South (Lo´pez-Sanjuanetal. 2010a) and the COSMOS fields
ent≤apparent luminosity sub-samples from i 23 to i 20. (Lo´pez-Sanjuanetal. 2012). We recall the main points of this
That ensures excellent photometric redshifts≤and provid≤es re- methodology below and we explore how to apply it optimally
liable merger fraction measurements (Sect. 4.3), because the overtheALHAMBRAdatainSect.4.3.
PDFs of i 23 sources are defined well by a single Gaussian We used the following procedure to define a close pair
peak(Molin≤oetal.2013).InSect.4.7wealsostudythecosmic system in our photometric catalogue (see Lo´pez-Sanjuanetal.
variance in luminosity- and stellar mass-selected samples. The 2010a,fordetails):firstwesearchforclosespatialcompanions
sABo−LubHrcaAensdMwlBuemRreAinaoclssaiottaieplosrgoavuniedd(etshdeeebsyMteBlolalPirnZmo2.ae0stsaaelns.do2f0ar1the3e,inAfocLlruHfduAerdtMhienBrRdtheAe- msoufamaxiinpmgruintmhcaiφptatplhoegsasgliaablxlaeyx,fyowrisitahlogrcievadetesnhdrifmattaxzz11ina−nth2deσufizn1r.csetTrihtnaissitnadtnyecfiσen.ze1Isf, twahsee-
p
tails). The mass-to-light ratios from Tayloretal. (2011) and a findacompaniongalaxywithredshiftz2 anduncertaintyσz2 at
Chabrier(2003)initialmassfunctionwereassumedintheesti- rp ≤ rpmax, we study both galaxies in redshift space. For con-
mationofthestellarmasses. venience, we assume below that every principal galaxy has, at
most,oneclosecompanion.Inthiscase,ourtwogalaxiescould
beaclosepairintheredshiftrange
3. Measuringofthemergerfractioninphotometric
[z ,z+]=[z 2σ ,z +2σ ] [z 2σ ,z +2σ ]. (4)
samples − 1− z1 1 z1 ∩ 2− z2 2 z2
The linear distance between two sources can be obtained from Because of variationin the range [z−,z+] of the functiondA(z),
wtahthieveierrepvrezoljoecacintteyddazsleopnaagrreatthtihoeenl,irnrepeds=ohfiφfstdigAoh(ftz,1th)∆,evanp=drintchce|izpi2rar−le(szmt1-|f/or(ra1em+leurmze1li)--, riamesdpkeoyfisnpeeatithrheaistcrzoe1nd−dsiht2iiσoftnzi1nrmtpmeiirngvha≤tlnirfoptth≤beesrakpmypaxapiaaritraatclozl1nzd+i∈t2ioσ[nzz−1i.,szWn+oe],ttashanutds-
1 2
nous/massivegalaxyinthepair)andthecompaniongalaxy,re- isfied at every redshift. After this, our two galaxies define the
spectively; φ is the angular separation, in arcsec, of the two close pair system k in the redshift interval [z−k,z+k], where the
galaxiesontheskyplane;andd (z)istheangulardiameterdis- indexkcoversalltheclosepairsystemsinthesample.
A
tance,inkpcarcsec 1,atredshiftz.Twogalaxiesaredefinedas Thenextstep is to define the numberofpairsassociated to
−
a close pair if rmin r rmax and ∆v ∆vmax. The PSF of each close pair system k. For this, and because all our sources
theALHAMBRpA gr≤ounpd≤-baspedimagesis≤. 1.4 (mediansee- havea photometricredshift,wesupposeinthefollowingthata
′′
ingof 1 ),whichcorrespondsto7.6h 1kpcinourcosmology galaxyiin whateversample is describedin redshiftspace by a
′′ −
at z =∼0.9.To ensure well de-blendedsourcesand to minimise Gaussianprobabilitydistribution,
colour contamination,we fixed rmin to 10h 1 kpc (φ > 1.8 at
p − ′′ 1 (zi zp,i)2
P (z z ,σ )= exp − . (5)
2 http://cloud.iaa.es/alhambra/ i i| p,i zp,i √2πσzp,i "− 2σ2zp,i #
4
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
With the previous distribution we are able to treat statisti- 3.2.Themergerrate
cally all the available information in redshift space and define
Thefinalgoalofmergerstudiesistheestimationofthemerger
thenumberofpairsatredshiftz insystemkas
1 rateR ,definedasthenumberofmergerspergalaxyandGyr 1.
m −
Themergerrateiscomputedfromthemergerfractionbyclose
z+
m pairsas
ν (z )=C P (z z ,σ ) P (z z ,σ )dz , (6)
k 1 k 1 1| p,1 zp,1 Z 2 2| p,2 zp,2 2
z−m C
R = m f , (12)
m m
wherez1 ∈[z−k,z+k],theintegrationlimitsare Tm
whereC isthefractionoftheobservedclosepairsthanfinally
z−m =z1(1−∆vmax/c)−∆vmax/c, (7) mergeafmteramergertimescaleTm.Themergertimescale and
z+ =z (1+∆vmax/c)+∆vmax/c, (8) themergerprobabilityC shouldbeestimatedfromsimulations
m 1 m
(e.g., Kitzbichler&White 2008; Lotzetal. 2010a,b; Linetal.
thesubindex1[2]referstotheprincipal[companion]galaxyin 2010; Jianetal. 2012; Morenoetal. 2013). On the one hand,
thesystemk,andtheconstantCk normalisesthefunctiontothe Tm depends mainly on the search radius rpmax, the stellar mass
totalnumberofpairsintheinterestrange, oftheprincipalgalaxy,andthemassratiobetweenthegalaxies
inthepair,withamilddependenceonredshiftandenvironment
2Nk = z+k P (z z ,σ )dz + z+k P (z z ,σ )dz . (9) a(Jnidaneentvairlo.n2m01e2n)t.,Ownitthheaomthiledrhdaenpden,Cdemndceepoenndbsotmhariendlyshoinftrapmnadx
p Zz−k 1 1| p,1 zp,1 1 Zz−k 2 2| p,2 zp,2 2 themassratiobetweenthegalaxiesinthepair(Jianetal.2012).
Despite of the efforts in the literature to estimate both T and
m
Note that ν = 0 if z < z or z > z+. The function ν tells C ,differentcosmologicalandgalaxyformationmodelsprovide
k 1 −k 1 k k m
us how the numberof pairs in the system k, noted Nk, are dis- differentvalueswithinafactoroftwo–three(e.g.,Hopkinsetal.
p
tributedinredshiftspace.TheintegralinEq.(6)spansthosered- 2010). To avoid model-dependent results, in the present paper
shiftsinwhichthecompaniongalaxyhas∆v ∆vmaxforagiven we focus therefore in the cosmic variance of the observational
redshiftoftheprincipalgalaxy.Thistranslates≤toz+m−z−m ∼0.005 mergerfraction fm.
inourredshiftrangeofinterest.
With thepreviousdefinitions,the mergerfractionin the in-
4. Estimationofthecosmicvarianceformerger
tervalz =[z ,z )is
r min max
fractionstudies
fm = PzmkaxRPzzmmina(xzνkz(z1,)σdz1)dz . (10) 4In.1t.hTishseeocrteiotincawlebarcekcgalrlotuhnedtheoretical backgroundand define
PiRzmin i i| p,i zp,i i the basic variables involved in the cosmic variance definition
If we integrate over the whole redshift space, zr = [0, ), and characterisation. The relative cosmic variance (σv) arises
Eq.(10)becomes ∞ fromtheunderlyinglarge-scaledensityfluctuationsandleadto
varianceslargerthanthoseexpectedfromsimplePoissonstatis-
tics.FollowingSomervilleetal.(2004)andMosteretal.(2011),
Nk
f = k p, (11) the mean N and the variance N2 N 2 in the distribution
m PN1 of galaxiehs aire given by the firhst aind−shecoind moments of the
probabilitydistributionP (V ),whichdescribestheprobability
N c
where Nk isanalogousto N inEq.(3).Inordertoestimate of counting N objectswithin a volume V . The relativecosmic
k p p c
the observationalerror of f , noted σ , we used the jackknife varianceisdefinedas
P m f
technique (Efron 1982). We computed partial standard devia-
N2 N 2 1
tions,δk,foreachsystemkbytakingthedifferencebetweenthe σ2 = h i−h i . (13)
measured f andthesamequantitywiththekthpairremovedfor v N 2 − N
m h i h i
thesample, fk,suchthatδ = f fk.Foraredshiftrangewith
N systems,tmhevarianceiskgivemn−bymσ2 =[(N 1) δ2]/N . The second term represents the correction for the Poisson shot
p f p− k k p noise.Thesecondmomentoftheobjectcountsis
P
N 2
3.1.Bordereffectsinredshiftandintheskyplane N2 = N 2+ N + h i ξ(r r )dV dV , (14)
h i h i h i V2 Z | a− b| c,a c,b
c Vc
When we search for a primary source’s companion, we de-
fine a volume in the sky plane-redshift space. If the primary whereξisthetwo-pointcorrelationfunctionofthesampleunder
source is near the boundaries of the survey, a fraction of the study(Peebles1980).CombiningthiswithEq.(13),therelative
search volume lies outside of the effective volume of the sur- cosmicvariancecanbewrittenas
vey.Lo´pez-Sanjuanetal.(2010a)findthatbordereffectsinthe
1
skyplanearerepresentative(i.e.,1σdiscrepancy)onlyatrmax & σ2 = ξ(r r )dV dV . (15)
p v V2 Z | a− b| c,a c,b
70h 1 kpc.Thus,werestrictedthesearchradiusinourstudyto c Vc
−
rpmax ≤50h−1kpc. Thus, the cosmic variance of a given sample depends on the
We avoid the incompleteness in redshift space by includ- correlationfunctionofthatpopulation.Wecanapproximatethe
ing in the samples not only the sources inside the redshift galaxycorrelationfunctioninEq.(15)bythelineartheorycor-
range[z ,z )understudy,butalsothosesourceswitheither relationfunctionfordarkmatterξ ,ξ = b2ξ ,wherebisthe
min max dm dm
z +2σ z orz 2σ <z . galaxy bias. The bias at a fixed scale depends mainly on both
p,i zp,i ≥ min p,i− zp,i max
5
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
redshift and the selection of the sample under study. With this
16
definitionofthecorrelationfunctionwefindthat
i≤ 22
14
b 0.3 ≤ z < 0.9
σ , (16) 12
v ∝ V1 α σ = 0.33
c− 10
σ = 0.25
wherethepowerlawindexαtakesintoaccounttheextravolume N 8 v
dependencefromthe integralof the correlationfunctionξ in
dm 6
Eq.(15).
4
The bias of a particular population is usually measured
from the analysis of the correlationfunction and is well estab- 2
lished that the bias increases with luminosity and stellar mass 0
(seeZehavietal.2011;Couponetal.2012;Marullietal.2013; 0.02 0.04 0.06 0.08 0.10
Arnalte-Muretal.2013,andreferencestherein).Theestimation
f
m
ofthebiasisalaborioustask,sowedecidedtousetheredshift
andthenumberdensitynofthepopulationunderstudyinstead
ofthebiastocharacterisethecosmicvariance.Thenumberden- 16
sityisanobservationalquantitythatdecreaseswiththeincrease 14 i≤ 21
of the luminosity and the mass selection, so a b n β rela- 0.3 ≤ z < 0.9
∝ − 12
tion is expected. This inverse dependence is indeed suggested σ = 0.62
10
byNuzaetal.(2013)results. σ = 0.44
Insummary,weexpect N 8 v
6
b zγ
σv ∝ V1 α ∝ nβV1 α. (17) 4
c− c−
2
Thisequationshowsthatthenumberdensityofgalaxies,thered-
0
shift, and the cosmic volume can be assumed as independent 0.02 0.04 0.06 0.08 0.10
variablesinthecosmicvarianceparametrisation.Equation(17)
f
and the deduction above apply to the cosmic variance in the m
number of galaxies. We are interested on the cosmic variance
ofthemergerfractionbyclosepairsinstead,soadependenceon Fig.4. Distribution of the merger fraction f for i 22
m
V ,redshift,andthenumberdensityofthetwopopulationsunder (top panel) and i 21 (bottom panel) galaxies i≤n the
c
study,notedn forprincipalgalaxiesandn forthecompanion 48 ALHAMBRA sub≤-fields, measured from close pairs with
1 2
galaxies,isexpected.We usedthereforethisfourvariables(n , 10h 1kpc r 30h 1kpcat0.3 z<0.9.Ineachpanel,the
1 − p −
≤ ≤ ≤
n , z, and V ) to characterise the cosmic variance in close pair redsolidlineisthebestleast-squaresfitofalog-normalfunction
2 c
studies(Sect.4.4). tothedata.Thestarandtheredbarmarkthemedianandthe68%
The power-law indices in Eq. (17) could be different for confidenceintervalofthefit,respectively.Theblackbarmarks
luminosity- and mass-selected samples, as well as for flux- theconfidenceintervalfromthemaximumlikelihoodanalysisof
limited samples. In the presentpaper we use flux-limited sam- the data and is ourmeasurementof the cosmic varianceσ . [A
v
ples selected in the i band to characterise the cosmic variance. colourversionofthisplotisavailableintheelectronicedition].
Thischoicehasseveralbenefits,sincewehaveawellcontrolled
selectionfunction,abetterunderstandingofthephotometricred-
shifts and their errors, and we have access to larger samplesat reliablythecosmicvarianceσv.Asrepresentativeexamples,we
lowerredshiftthatin theluminosityandthestellar masscases. show in Fig. 4 the distributions of the merger fraction fm in
That improves the statistics and increases the useful redshift the 48 ALHAMBRA sub-fields for i 22 and i 21 galax-
≤ ≤
range. At the end, future studies will be interested on the cos- ies. The merger fraction was measured from close pairs with
mic variancein physicallyselected samples(i.e.,luminosityor 10h−1kpc rp 30h−1kpc.Unlessnotedotherwise,inthefol-
≤ ≤
stellar mass). Thus, in Sect. 4.7 we compare the results from lowing the principal and the companion samples comprise the
theflux-limitedi bandsampleswiththeactualcosmicvariance same galaxies. We find that the observed distributions are not
measuredinphys−icallyselectedsamples. Gaussian,butfollowalog-normaldistributioninstead,
[C3]Finally,wesetthedefinitionofthenumberdensityn.In
1 (ln f µ)2
thepresentpaperthenumberdensityofagivenpopulationisthe P (f µ,σ)= exp m− , (18)
cosmicaveragenumberdensityofthatpopulation.Forexample, LN m| √2πσfm "− 2σ2 #
if we are studying the merger fraction in a volume dominated
whereµandσarethemedianandthedispersionofaGaussian
by a cluster, we shouldnot use the numberdensity in thatvol-
functioninlog-space f =ln f .Thisis,
ume,butthenumberdensityderivedfromageneralluminosity m′ m
or mass function work instead. Thanks to the 48 sub-fields in 1 (f µ)2
ALHAMBRAwehavedirectaccesstotheaveragenumberden- PG(fm′ |µ,σ)= √2πσexp"− m′2σ−2 #. (19)
sitiesofthepopulationsunderstudy(Sect.4.4.1).
The 68% confidence interval of the log-normal distribution is
[eµe σ,eµeσ].Thisfunctionaldistributionwasexpectedfortwo
4.2.Distributionofthemergerfractionandσ estimation −
v reasons.First,themergerfractioncannotbenegative,implying
In this section we explore which statistical distribution repro- anasymmetricdistribution(Cameron2011).Second,thedistri-
ducesbettertheobservedmergerfractionsandhowtomeasure butionofoverdensestructuresintheuniverseislog-normal(e.g.,
6
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
0.6
0.14
0.12 0.5
0.10
m 0.08 v 0.4
f σ
0.06
0.3
0.04
0.02 0.2
0.0 0.2 0.4 0.6 0.8 30 35 40 45 50
O rmax [h−1 kpc]
sel p
Fig.5. Merger fraction fm as a function of the odds selection Fig.6. Cosmic variance σ as a function of rmax for i
fori 22.5galaxiesat0.3 z < 0.9.Thefilledtriangles, v p ≤
sel 22.5,21.5, and 21 galaxies at 0.3 z < 0.9 (circles, stars,
Ocircles, an≤d squaresare for rmax≤= 30,40,and 50h 1 kpc close ≤
p − andtriangles,respectively).Thehorizontallinesmarktheerror-
pairs, respectively.The open triangles are the observed merger
weighted average of the cosmic variance in each case, and the
fractionsforrmax = 30h 1 kpctoillustratetheselectioncorrec-
p − colouredareastheir68%confidenceintervals.[Acolourversion
tionfromEq.(21).Inseveralcasestheerrorbarsaresmallerthan
ofthisplotisavailableintheelectronicedition].
thepoints.Thedotted,dashed,andsolidlinesmarktheaverage
f at 0.3 0.6 for rmax = 30,40,and 50h 1 kpc close
m ≤ Osel ≤ p − Table 2.Cosmic varianceσ as a functionofthe searchradius
pairs,respectively.[Acolourversionofthisplotisavailablein v
rmaxfor =0.3galaxiesat0.3 z<0.9
theelectronicedition]. p O≥Osel ≤
Coles&Jones 1991; delaTorreetal. 2010; Kovacˇetal. 2010) rmax σ σ σ
p v v v
andthe mergerfractionincreaseswith density (Linetal. 2010; (h 1kpc) (i 22.5) (i 21.5) (i 21.0)
−
≤ ≤ ≤
deRaveletal. 2011; Kampczyketal. 2013). We checked that
30 0.181 0.030 0.235 0.053 0.447 0.091
the merger fraction follows a log-normaldistribution in all the ± ± ±
35 0.184 0.027 0.246 0.045 0.433 0.079
samplesexploredinthepresentpaper. 40 0.199±0.026 0.284±0.041 0.460±0.073
The variable σ encodes the relevant information about the 45 0.195±0.024 0.289±0.040 0.447±0.067
dispersioninthemergerfractiondistribution,includingthedis- 50 0.190±0.023 0.284±0.038 0.451±0.066
± ± ±
persion due to the cosmic variance. The study of the median
Average 0.190 0.011 0.272 0.019 0.448 0.033
value of the merger fraction in ALHAMBRA, estimated as eµ, ± ± ±
and its dependence on z, stellar mass, or colour, is beyond the
scope of the present paper and we will address this issue in a
futurework.
Abestleast-squaresfitwithalog-normalfunctiontothedis- Sect.4.4.3.Thatprovidesacompletedescriptionofthecosmic
tributions in Fig. 4 shows that σ increases with the apparent variance for merger fraction studies. We stress that our defini-
brightness, from σ = 0.33 for i 22 galaxies to σ = 0.62 tionofσvdiffersfromtheclassicaldefinitionoftherelativecos-
for i 21 galaxies. However, the≤origin of the observed σ is mic variancepresentedin Sect. 4.1, which is equivalentto eσv.
twofo≤ld:(i)theintrinsicdispersionduetothecosmicvarianceσv However, σv encodes the relevant information needed to esti-
(i.e.,thefield-to-fieldvariationinthemergerfractionbecauseof mate the intrinsic dispersionin the measurementof the merger
theclusteringofthegalaxies),and(ii)thedispersionduetothe fractionduetotheclusteringofgalaxies.
observationalerrorsσ (i.e.,theuncertaintyinthemeasurement
o
ofthemergerfractioninagivenfield,includingthePoissonshot
4.3.Optimalestimationofσ intheALHAMBRAsurvey
noiseterm).Thus,the dispersionσreportedin Fig. 4is an up- v
per limit for the actual cosmic variance σ . We deal with this In the previous section we have defined the methodology to
v
limitation applying a maximum likelihood estimator (MLE) to computethe cosmic variance fromthe observeddistributionof
the observed distributions. In Appendix A we develop a MLE themergerfraction.However,asshownbyLo´pez-Sanjuanetal.
that estimates the more probablevaluesof µ and σ , assuming (2010a), to avoid projection effects we need a galaxy sample
v
that the mergerfractionfollows a Gaussian distribution in log- witheithersmallphotometricredshifterrorsoralargefractionof
space (Eq. [19]) that is affected by known observationalerrors spectroscopicredshifts.Inthepresentstudywedidnotuseinfor-
σ .We provethattheMLEprovidesanunbiasedestimationof mationfromspectroscopicredshifts,soweshouldcheckthatthe
o
µ and σ , as well as reliable uncertaintiesof these parameters. photometricredshiftsinALHAMBRAaregoodenoughforour
v
Applying the MLE to the distributions in Fig. 4, we find than purposes.Anaturalwaytoselectexcellentz ’sinALHAMBRA
p
σ islowerthanσ,asanticipated,andthatthecosmicvariance is by a selection in the odds parameter. On the one hand, this
v
increaseswiththeapparentbrightnessfromσ =0.25 0.04for selectionincreasesthe accuracyofthe photometricredshiftsof
v
±
i 22galaxiestoσ =0.44 0.08fori 21galaxies. the sample and minimises the fraction of catastrophic outliers
v
≤ ± ≤
We constraint the dependence of σ on the number den- (Molinoetal.2013), improvingthe mergerfractionestimation.
v
sity of the populations under study in Sects. 4.4.1 and 4.4.4, Ontheotherhand,oursamplebecomesincompleteandcouldbe
ontheprobedcosmicvolumeinSect.4.4.2, andonredshifton biasedtowarda populationofeitherbrightgalaxiesorgalaxies
7
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
withmarkedfeaturesintheSED(i.e.,emissionlinegalaxiesor
old populationswith a strong4000Åbreak).In this section we 0.5
studyhowthemergerfractioninALHAMBRAdependsonthe
selection and derive the optimal one to estimate the cosmic 0.4
O
variance.
Following the methodology from spectroscopic surveys v 0.3
σ
(e.g.,Linetal.2004;deRaveletal.2009;Lo´pez-Sanjuanetal.
2011, 2013), if we have a population with a total number of 0.2
galaxiesN inagivenvolumeandweobservearandomfraction
tot 0.1
f ofthesegalaxies,themergerfractionofthetotalpopulation
obs
is
f = f f 1, (20) 0.0 0.2 0.4 0.6 0.8
m m,obs× o−bs O
sel
where f is the merger fraction of the observed sample. In
m,obs
ALHAMBRA we applied a selection in the parameter , so
Fig.7. Cosmic variance σ as a function of the odds selection
O v
Eq.(20)becomes
for i 22.5 galaxies at 0.3 z < 0.9. Triangles, circles,
sel
fm = fm( sel) Ntot , (21) Oaspnedcstiqvuealyre.≤sTahreedfoortterpmd,axda=sh3e0d,,4a0n,d≤ansdol5id0hli−n1ekspmcacrlkostheepaaviresr,argee-
≥O × N( )
≥Osel σv at 0.1 ≤ Osel ≤ 0.5 for rpmax = 30,40, and 50h−1 kpc close
where N( ) is the number of galaxies with odds higher pairs,respectively.[Acolourversionofthisplotisavailablein
sel
than (i≥.e.O,galaxieswith ),N isthetotalnumberof theelectronicedition].
sel sel tot
O O≥O
galaxies(i.e.,galaxieswith 0),and f ( )isthemerger
m sel
O≥ ≥O
factionofthosegalaxieswith .Because f mustbein-
O ≥ Osel m Lo´pez-Sanjuanetal. 2011). This is, R f (rmax)/T (rmax).
dependentofthe selection,thestudyof fmasafunctionof sel m ∝ m p m p
O O Forthesamereason,thecosmicvarianceofthemergerratecan
providesthe cluesaboutthe optimaloddsselection formerger
notdependonrmax.Inotherwords,the68%confidenceinterval
fractionstudiesin ALHAMBRA. We show fm asa functionof p
selforgalaxieswithi 22.5at0.3 z<0.9inFig.5.Wefind of the merger rate, [Rme−σv,Rmeσv], should be independent of
Othat ≤ ≤ the search radius. Expanding the previous confidence interval
wefindthat
themergerfractionisroughlyconstantfor0.2 0.6.
• This is the expected result if the mergerfractio≤nOisserle≤liable [Rme−σv,Rmeσv] (22)
∝
andmeasuredinanonbiasedsample.Inthisparticularcase, [fm(rpmax)Tm−1(rpmax)e−σv, fm(rpmax)Tm−1(rpmax)eσv]=
nthuemObseelr=of0g.a2l(a0x.i6e)sswamithplieco2m2.p5r;ises98%(66%)ofthetotal [fm(rpmax)e−σv, fm(rpmax)eσv]Tm−1(rpmax).
≤
the mergerfractionisoverestimatedfor sel 0.1.Evenif Note that the dependence on rmax is encoded in the median
• O ≤ p
onlyasmallfractionofgalaxieswithpoorconstrainsintheir mergerfraction andin the mergertime scale. Thus,the cosmic
zp’sareincludedinthesample,theprojectioneffectsbecome variance σv of the merger fraction should not depend on the
important; searchradius. We checkedthis predictionby studyingthe cos-
themergerfractionisoverestimatedfor sel 0.7.Thisbe- micvarianceasafunctionofthesearchradiusfori 22.5,21.5,
• haviourathighodds(i.e.,insampleswitOhhig≥hqualitypho- and21galaxieswith = 0.3at0.3 z <≤0.9.We find
sel
tometric redshifts) suggests that the retained galaxies are a that σ is consistent wOit≥h aOconstant value i≤rrespective of rmax
v p
biasedsub-sampleofthegeneralpopulationunderstudy. inthethreepopulationsprobed,asdesired(Table2andFig.6).
This supports σ as a good descriptor of the cosmic variance
v
In theanalysisabovewe onlyaccountedforclose compan-
andourmethodologytomeasureit.Inthepreviousanalysiswe
ions of i 22.5 galaxies with 10h 1 kpc r 30h 1 kpc,
− p − haveomittedthemergerprobabilityC ,whichmainlydepends
but we can≤use other values of rpmax or searc≤hing≤over different onrmax andenvironment(Sect.3.2).Tmhemergerfractioncorre-
samples.Ontheonehand,werepeatedthestudyforrmax = 40 p
p lateswithenvironment,sothemergerprobabilitycouldmodify
and50h−1kpc,findingthesamebehaviourthanforrpmax =30h−1 the factor eσv in Eq. (22). Because a constant σv with rpmax is
kpc(Fig.5).Theonlydifferencesarethatthemergerfractionin- observed, the impact of C in the f to R translation should
m m m
creaseswiththesearchradiusandthattheOsel =0.2pointstarts besimilarin therangeofrpmax explored.Detailedcosmological
to deviate from the expected value (the search area increases simulationsareneededtoclarifythisissue.
withrpmax andmoreaccuratezp’sareneededtoavoidprojection Finally, we studied the dependence of σv on the odds se-
effects).Ontheotherhand,weexploredawiderangeofi band lection for i 22.5 galaxies at 0.3 z < 0.9. Following the
magnitude selections, from i 23 to 20, in the three pr−evious sameargumen≤tsthanbefore,thecosm≤icvarianceshouldnotde-
rmax cases. We find again the≤same behaviour. That reinforces pendontheoddsselection.Wefindthat(i)σ isconsistentwith
p v
our argumentsabove and suggests 0.3 0.6 as accept- a constant value as a function of rmax for any , reinforcing
≤ Osel ≤ p Osel
ableoddslimitstoselectsamplesformergerfractionstudiesin ourresultsabove,and(ii)σ isindependentoftheoddsselec-
v
ALHAMBRA. tionat0.1 0.5(Fig.7). Asforthemergerfraction,we
sel
≤ O ≤
Themergerfractionincreaseswiththesearchradius(Fig.5). checked that different populations follow the same behaviour.
However,themergerrateR (Sect.3.2)isaphysicalpropertyof We set therefore = 0.3 as the optimaloddsselection
m sel
anypopulationanditcannotdependonrmax.Thus,theincrease tomeasurethecoOsm≥icOvarianceinALHAMBRA.Thisselection
p
inthemergerfractionwiththesearchradiusiscompensatedwith providesexcellentphotometricredshiftsandensuresrepresenta-
theincreaseinthemergertimescale(e.g.,deRaveletal.2009; tivesamples.
8
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
0.9 Table3.Cosmicvarianceσv asafunctionoftheprincipalsam-
ple’snumberdensityn
0.8 σv =0.45×n−10.54 1
0.7
0.6 Principal n σ
1 v
σv 0.5 sample (10−3Mpc−3)
0.4 i 23.0 6.88 0.16 0.158 0.019
≤ ± ±
i 22.5 4.79 0.14 0.190 0.023
0.3 ≤ ± ±
i 22.0 3.30 0.11 0.245 0.030
0.2 i≤21.5 2.12±0.07 0.284±0.038
≤ ± ±
i 21.0 1.28 0.05 0.451 0.066
0.1 ≤ ± ±
1 2 3 4 5 6 7 i 20.5 0.73 0.03 0.587 0.100
≤ ± ±
n [10−3 Mpc−3] i≤20.0 0.35±0.01 0.695±0.154
1
Fig.8.Cosmicvarianceσ asa functionofthenumberdensity
v
n1 oftheprincipalpopulationunderstudy.Increasingthenum- being the number density in the sub-field j and Vcj the cosmic
ber density, the principal sample comprises i 20, 20.5, 21, volume probed by it at z. In the measurement of the number
≤ r
21.5, 22, 22.5, and 23 galaxies, respectively. The probed cos- densityallthegalaxiesweretakingintoaccount,i.e.,anyodds
mic volume is the same in all the cases, Vc 1.4 105 Mpc3 selectionwasapplied( 0).Westressthatourmeasurednum-
(0.3 z < 0.9). The dashed line is the err∼or-wei×ghted least- berdensitiesareunaffeOcte≥dbycosmicvariance,andtheycanbe
squar≤es fit of a power-law to the data, σv ∝ n−10.54. [A colour used thereforeto characteriseσv. We reportour measurements
versionofthisplotisavailableintheelectronicedition]. inTable3.
We find that the cosmic variance increases as the number
density decreases (Fig. 8), as expected by Eq. (17). The error-
weightedleast-squaresfitofapower-lawtothedatais
In summary, in the following we estimate the cosmic
vAaLriHanAcMeBσRvAfrsoumb-fitheledsmweirtghe1r0fhra1ctkipocns mreasu5r0ehd1inkptchcelo4s8e σ (n )=(0.45 0.04) n1 −0.54±0.06. (24)
pairs(theσv uncertaintyislower−forlarg≤erpse≤archr−adii)andin v 1 ± × 10−3Mpc−3!
samples with = 0.3. That ensures reliable results in
sel In this section and in the following ones we used i band
O ≥ O
representative(i.e.,nonbiased)samples. −
selectedsamplestocharacteriseσ .Weshowthattheresultsob-
v
tainedwiththesei bandsamplescanbeappliedtoluminosity-
−
andstellarmass-selectedsamplesinSect.4.7.
4.4.Characterisationofσ
v
At this stage we have set both the methodology to compute a
4.4.2. Dependenceonthecosmologicalvolume
robust cosmic variance from the observed merger fraction dis-
tribution(Sect.4.2)andtheoptimalsearchradiusandoddsse- In this section we explore the dependence of the cosmic vari-
lectiontoestimateσv inALHAMBRA(Sect.4.3).Nowwecan ance on the cosmic volume probed by the survey. We defined
characterisethecosmicvarianceasafunctionofthepopulations σ asσ =σ /σ (n ).Thiserasedthedependenceonthenum-
∗v ∗v v v 1
understudy(Sects.4.4.1and4.4.4),theprobedcosmicvolume berdensityofthepopulationandonlyvolumeeffectsweremea-
(Sect.4.4.2),andtheredshift(Sect.4.4.3). sured. We explored smaller cosmic volumes than in the previ-
oussectionbystudying(i)differentredshiftrangesoverthefull
ALHAMBRA area (avoiding redshiftrangessmaller than 0.1),
4.4.1. Dependenceonthenumberdensityoftheprincipal
and (ii) smaller areas, centred in the ALHAMBRA sub-fields,
sample
at 0.3 z < 0.9.All the cases, summarisedin Table 4, are for
≤
i 23galaxies.Attheend,weexploredanorderofmagnitude
In this section we explore how the cosmic variance depends
in≤volume,fromV 0.1 105 Mpc3 toV 1.4 105 Mpc3.
on the number density n of the principal population under c c
1 ∼ × ∼ ×
The power-law function that better describes the observations
study. For that, we took the same population as principal and
(Fig.9)is
companion sample. We study the dependence on the compan-
ion sample in Sect. 4.4.4. To avoid any dependence of σ on
either the probed cosmic volume and z, and to minimisevthe σ (V )=(1.05 0.05) Vc −0.48±0.05. (25)
observational errors, in this section we focus in the redshift ∗v c ± × 105Mpc3!
range 0.3 z < 0.9. This range probes a cosmic volume of
V 1.4 1≤05Mpc3ineachALHAMBRAsub-field.Toexplore Wetestedtherobustnessofourresultbyfittingthetwosets
c ∼ × ofdata(variationinredshiftandarea)separately.Wefindσ
differentnumberdensities,wemeasuredthecosmicvariancefor V 0.43 0.08 for the redshift data, while σ V 0.48 0.05 for∗vth∝e
differenti bandselectedsamples,fromi 20toi 23in0.5 c− ± ∗v ∝ c− ±
− ≤ ≤ areadata.
magnitude steps. We estimated the average number density n
1
in the redshiftrangez as the mediannumberdensityin the 48
r
ALHAMBRAsub-fields,with 4.4.3. Dependenceonredshift
zmaxPj(z z ,σ )dz Theredshiftisanexpectedparameterintheparametrisationthe
n1j(zr)= PiRzmin iVji(z| )p,i zp,i i (23) cfeorsemnticrevdasrhiaifntcsea.rHeocownesviestre,nFtigw.it9hsthhoowsesftrhoamt tthheerwesidueltsreadtsdhiiff-t
c r
9
C.Lo´pez-Sanjuanetal.:TheALHAMBRAsurvey.Anempiricalestimationofthecosmicvarianceformergerfractionstudies
Table4.Cosmicvarianceσ asafunctionoftheprobedcosmicvolumeV
v c
Redshift Effectivearea V n σ σ
c 1 v ∗v
range (deg2) (104Mpc3) (10 3Mpc 3) σ /σ (n )
− − v v 1
[0.30,0.69) 2.38 6.98 0.06 9.21 0.25 0.169 0.025 1.24 0.15
± ± ± ±
[0.69,0.90) 2.38 6.87 0.06 4.69 0.16 0.273 0.040 1.39 0.18
± ± ± ±
[0.30,0.60) 2.38 4.68 0.04 10.32 0.32 0.205 0.030 1.60 0.19
± ± ± ±
[0.60,0.77) 2.38 4.68 0.04 5.82 0.18 0.274 0.042 1.57 0.19
± ± ± ±
[0.77,0.90) 2.38 4.49 0.04 4.17 0.19 0.323 0.051 1.54 0.21
± ± ± ±
[0.30,0.55) 2.38 3.60 0.03 11.23 0.38 0.230 0.032 1.88 0.22
± ± ± ±
[0.55,0.70) 2.38 3.68 0.03 6.14 0.21 0.252 0.041 1.48 0.20
± ± ± ±
[0.70,0.82) 2.38 3.74 0.03 5.26 0.21 0.311 0.056 1.68 0.23
± ± ± ±
[0.45,0.60) 2.38 2.91 0.02 7.64 0.31 0.268 0.043 1.78 0.24
± ± ± ±
[0.30,0.45) 2.38 1.76 0.01 14.04 0.50 0.276 0.051 2.55 0.30
± ± ± ±
[0.30,0.90) 2.38 13.85 0.11 6.88 0.16 0.158 0.019 0.99 0.12
± ± ± ±
[0.30,0.90) 1.92 11.15 0.11 6.91 0.17 0.158 0.020 0.99 0.13
± ± ± ±
[0.30,0.90) 1.59 9.26 0.10 7.00 0.17 0.150 0.020 0.95 0.13
± ± ± ±
[0.30,0.90) 1.19 6.95 0.07 6.79 0.18 0.179 0.024 1.11 0.15
± ± ± ±
[0.30,0.90) 0.79 4.61 0.05 7.06 0.20 0.259 0.033 1.64 0.21
± ± ± ±
[0.30,0.90) 0.59 3.44 0.04 6.85 0.22 0.264 0.036 1.65 0.22
± ± ± ±
[0.30,0.90) 0.48 2.77 0.03 6.74 0.21 0.325 0.045 2.01 0.28
± ± ± ±
[0.30,0.90) 0.39 2.29 0.03 6.73 0.21 0.354 0.050 2.19 0.31
± ± ± ±
[0.30,0.90) 0.34 1.97 0.04 6.72 0.24 0.340 0.050 2.10 0.31
± ± ± ±
[0.30,0.90) 0.30 1.74 0.03 6.82 0.26 0.391 0.055 2.44 0.34
± ± ± ±
[0.30,0.90) 0.24 1.40 0.02 6.99 0.26 0.411 0.059 2.60 0.37
± ± ± ±
3.0 1.3
σv∗ =1.05×Vc−0.48 1.2
2.5
1.1
2.0
∗σv ∗∗v 1.0
σ
1.5 0.9
1.0 0.8
0.7 σv∗∗ =1.02±0.07
0.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Vc [105 Mpc3] z
Fig.9. Normalised cosmic variance σ as a function of the Fig.10. Normalised cosmic variance σ as a function of red-
∗v ∗v∗
probed cosmic volume V for galaxies with i 23. The circle shift for galaxies with i 23 (circles). The dashed line marks
c
≤ ≤
corresponds to same data as in Fig. 8. The stars probe differ- the error-weighted average of σ , σ = 1.02 0.07, and the
∗v∗ ∗v∗ ±
ent redshift intervals, while triangles probe sky areas smaller colouredareashowsits68%confidenceinterval.[Acolourver-
than the fiducial ALHAMBRA sub-field. The dashed line is sionofthisplotisavailableintheelectronicedition].
the error-weighted least-squares fit of a power-law to the data,
σ V 0.48. [A colour version of this plot is available in the
∗v ∝ c−
electronicedition].
4.4.4. Dependenceonthenumberdensityofthecompanion
sample
range0.3 z <0.9.Asaconsequence,theredshiftdependence
≤
of the cosmic varianceshould be smaller than the typicalerror As we show in Sect. 3, two different populations are involved
inourmeasurements.Wetestedthishypothesisbymeasuringσ inthemeasurementofthemergerfraction:theprincipalsample
v
in different, non-overlapping,redshift bins. We summarise our andthesampleofcompanionsaroundprincipalgalaxies.Inthe
measurements, performed for i 23 galaxies, in Table 5. We previoussectionstheprincipalandthecompanionsamplewere
≤
definedσ = σ /σ (n ,V ) to isolate the redshiftdependence thesame,andhereweexplorehowthenumberdensityn ofthe
∗v∗ v v 1 c 2
ofthecosmicvariance.Wefindthatσ iscompatiblewithunity, companionsampleimpactsthecosmicvariance.Weseti 20.5
∗v∗ ≤
σ =1.02 0.07,andthatnoredshiftdependenceremainsafter galaxies at 0.3 z < 0.9 as principals, and varied the i band
∗v∗ ± ≤ −
accounting for the variation in n and V (Fig. 10). This con- selectionofthecompaniongalaxiesfromi 20.5to i 23in
1 c
≤ ≤
firms our initial hypothesis and we assume therefore γ = 0 in 0.5 steps. As in Sect. 4.4.2, the variable σ = σ /σ (n ) was
∗v v v 1
thefollowing. used.
10