A new method to measure galaxy bias by combining the density and weak lensing fields

0.82 MB·

by  Arnau Pujol

Mon.Not.R.Astron.Soc.000,1–15(2012) Printed5July2016 (MNLATEXstylefilev2.2)
A new method to measure galaxy bias by combining the
density and weak lensing fields
Arnau Pujol(cid:63)1, Chihway Chang2, Enrique Gaztan˜aga1, Adam Amara2,
6 Alexandre Refregier2, David J. Bacon3, Jorge Carretero1,4, Francisco J. Castander1,
1 Martin Crocce1, Pablo Fosalba1, Marc Manera5, Vinu Vikram6,7
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2
1Institut de Ci`encies de l’Espai (ICE, IEEC/CSIC), E-08193 Bellaterra (Barcelona), Spain
ul 2Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland
J 3Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, U.K.
4Institut de F´ısica d’Altes Energies, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain
4
5University College London, Gower Street, London, WC1E 6BT, U.K

6Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA
]
O 7Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
C
.
h
p Acceptedxxxx.Receivedxxx
-
o
r
t ABSTRACT
s Wepresentanewmethodtomeasuretheredshift-dependentgalaxybiasbycombining
a
information from the galaxy density field and the weak lensing field. This method is
[
based on Amara et al. (2012), where they use the galaxy density field to construct a

3 bias-weighted convergence field κ . The main difference between Amara et al. (2012)
g
v and our new implementation is that here we present another way to measure galaxy
0
bias using tomography instead of bias parameterizations. The correlation between κ
6 g
and the true lensing field κ allows us to measure galaxy bias using different zero-lag
1
correlations, such as (cid:104)κ κ(cid:105)/(cid:104)κκ(cid:105) or (cid:104)κ κ (cid:105)/(cid:104)κ κ(cid:105). Our method measures the linear
0 g g g g
bias factor on linear scales under the assumption of no stochasticity between galaxies
0
andmatter.WeusetheMICEsimulationtomeasurethelineargalaxybiasforaflux-
.
1 limited sample (i<22.5) in tomographic redshift bins using this method. This paper
0
is the first that studies the accuracy and systematic uncertainties associated with the
6
implementation of the method, and the regime where it is consistent with the linear
1
galaxy bias defined by projected 2-point correlation functions (2PCF). We find that
:
v our method is consistent with linear bias at the percent level for scales larger than
i 30 arcmin, while nonlinearities appear at smaller scales. This measurement is a good
X
complementtoothermeasurementsofbias,sinceitdoesnotdependstronglyonσ as
r 8
a the 2PCF measurements. We apply this method to the Dark Energy Survey Science
Verification data in a follow-up paper.
Key words: gravitational lensing: weak; surveys; cosmology: large-scale structure
1 INTRODUCTION verse (galaxies and stars) and dark matter. In the ΛCDM
paradigm,structuresform intheinitial densitypeakscaus-
The formation and evolution of the large scale structures
ing dark matter to gravitationally collapse and form viri-
in the Universe is an important tool for cosmology studies.
alized objects. Galaxies are expected to follow these gravi-
But since most of the mass in the Universe is in the form
tational potentials (e.g. White & Rees 1978), and because
ofdark matter,whichcannotbedirectly observed,weneed
of this they are tracers of the dark matter density peaks.
to understand the connection between the observable uni-
Therelationbetweenthegalaxyandmassdistributionscan
be described theoretically with the galaxy bias prescription
(Kaiser1984;Fry&Gaztanaga1993;Bernardeau1996;Mo
(cid:63) E-mail:[email protected]
(cid:13)c 2012RAS
2 Arnau Pujol, Chihway Chang, Enrique Gaztan˜aga et al.
& White 1996; Sheth & Tormen 1999; Manera, Sheth & aperture mass and number counts. They also find a scale
Scoccimarro2010;Manera&Gaztan˜aga2011).Galaxybias dependence of bias on scales below 100 arcmin. The same
allowsustoconnectthedistributionofgalaxieswiththatof methodhasthenbeenappliedinmorerecentstudies(Simon
darkmatter,andagoodknowledgeofgalaxybiaswouldbe etal.2007;Julloetal.2012;Mandelbaumetal.2013;Bud-
veryimportanttoimprovetheprecisionofourcosmological dendieketal.2016).Usingasheartomographyanalysis,Si-
measurements (Eriksen & Gaztan˜aga 2015). mon(2012)combinedgalaxy-galaxylensingandgalaxyclus-
Manypapershavestudiedhaloandgalaxybiasinsimu- tering to constrain the 3D galaxy biasing parameters. Bias
lations(Cole&Kaiser1989;Kravtsov&Klypin1999;Seljak canalsobeobtainedfromthecross-correlationbetweenlens-
&Warren2004;Angulo,Baugh&Lacey2008;Faltenbacher ingfromtheCosmicMicrowaveBackgroundandthegalaxy
& White 2010; Tinker et al. 2010; Manera & Gaztan˜aga densities(Giannantonioetal.2016).Usinganothermethod,
2011;Paranjapeetal.2013;Pujol&Gaztan˜aga2014;Zent- Amaraetal.(2012)(hereafterA12)usedtheCOSMOSfield
ner, Hearin & van den Bosch 2014; Carretero et al. 2015; to measure galaxy bias by reconstructing a bias-weighted
Pujol et al. 2015), and the different ways to measure bias shearmapfromthegalaxydensityfield.Galaxybiasisesti-
(Kravtsov & Klypin 1999; Bernardeau et al. 2002; Manera matedfromthezero-lagcrosscorrelationbetweenthisbias-
& Gaztan˜aga 2011; Roth & Porciani 2011; Pollack, Smith weighted shear map from the galaxy density field and the
& Porciani 2014; Hoffmann et al. 2015; Bel, Hoffmann & shearmeasuredfromgalaxyshapes.Differentparameteriza-
Gaztan˜aga 2015). There are also several measurements of tions of bias are used to measure constant, non-linear and
bias in observations where usually the dark matter cluster- redshift-dependent bias.
ing is assumed from a model or from simulations (Zehavi In this paper we explore and extend the method from
et al. 2011; Coupon et al. 2012; Cacciato et al. 2012; Jullo A12.Weanalyzewhetherthegalaxybiasmeasuredwithour
etal.2012;Mar´ınetal.2013;Durkalecetal.2015;DiPorto method is consistent with the linear bias obtained from the
et al. 2014; Crocce et al. 2016). In most of these studies, projected2-pointcorrelationfunctions(2PCF).Wefindthat
however, the results depend strongly on assumptions of the our method can be affected by different parameters in the
cosmological parameters. implementation such as redshift binning, the redshift range
used,angularscales,surveyareaandshotnoise.Finally,we
Gravitationallensingistheeffectoflightdeflectiondue
showhowtomeasuretheredshift-dependentgalaxybiasby
totheperturbationsinthegravitationalpotentialfrommass
using tomographic redshift binning. Although this method
distribution. It is a powerful tool to measure the mass dis-
isverysimilartotheonepresentedinA12,therearefewno-
tribution in the Universe, since the gravitational potential
table differences. First of all, in A12 they explore different
isaffectedbybothbaryonicanddarkmatter.Weaklensing
smoothing schemes for the density field, while we explore
refers to the statistical study of small distortions (around
pixelizing the maps and applying a Top Hat filter. In A12
1%) in the shapes of a large number of galaxies due to
the lensing shear is estimated for each galaxy, and the bias
this effect. Several recent, ongoing and future galaxy sur-
is measured from the predicted and measured shear of the
veysaimtoobtainlargeweaklensingdatasetsthatwillal-
galaxies, while we measure galaxy bias from the generated
lowustobetterconstraincosmology,includingtheCanada-
lensingmaps.Finally,A12fitdifferentparametricbiasesus-
France-HawaiiTelescopeLensingSurvey(CFHTLenS;Hey-
ingawiderangeofredshiftforthegalaxydensityfield,while
mans et al. 2012; Erben et al. 2013), the Hyper Suprime-
here we implement a tomographic measurement, where we
Cam (HSC; Miyazaki et al. 2006), the Dark Energy Sur-
measure bias in redshift bins by using the density field of
vey (DES; The Dark Energy Survey Collaboration 2005;
galaxies in each particular bin. We apply this method to
Flaugher 2005), the Kilo Degree Survey (KIDS; Kuijken
the DES Science Verification (SV) data in a second paper
etal.2015),thePanoramicSurveyTelescopeandRapidRe-
(Chang et al. 2016, hereafter Paper II).
sponseSystem(PanSTARRS;Kaiseretal.2010),theLarge
The paper is organized as follows. In §2 we give an
Synoptic Survey Telescope (LSST; LSST Science Collabo-
overview of the theory for our analysis. In §3 we present
ration et al. 2009), Euclid (Laureijs et al. 2011) the The
the method used to measure bias from the galaxy density
Red Cluster Sequence Lensing Survey (RCSLenS; Hilde-
andweaklensingfieldsandthenumericaleffectsassociated
brandt et al. 2016), and Wide-Field Infrared Survey Tele-
with the implementation of the method. In §4 we present
scope (WFIRST; Green et al. 2012). From the shape of the
the results of the different tests and the final measurement
galaxies one can statistically infer the lensing fields, which
ofredshift-dependentgalaxybias.Wefinallyclosein§5with
containinformationoftheprojectedmatterdistributionand
discussion and conclusions.
canbeusedtogenerate2Dand3Dmassmaps(Masseyetal.
2007; Van Waerbeke et al. 2013; Vikram et al. 2015).
Thecombinationofweaklensingandgalaxydensityin-
formation gives us a powerful handle for measuring galaxy 2 THEORY
bias. One way is by studying the cross-correlation between
2.1 Galaxy Bias
the aperture mass and number counts statistics, which are
measurements of both dark matter and galaxy densities Thedistributionofgalaxiestracesthatofdarkmatter,and
(van Waerbeke 1998; Schneider 1998). In Hoekstra et al. one of the common descriptions for this relation is galaxy
(2002) they use the Red-Sequence Cluster Survey (RCS) bias, which relates the distribution of galaxies with that of
andtheVIRMOS-DESCARTsurveytomeasuregalaxybias darkmatter.Thereareseveralwaystoquantifygalaxybias
at z (cid:39) 0.35 from the zero lag cross-correlation between (Bernardeau et al. 2002; Manera & Gaztan˜aga 2011; Roth
(cid:13)c 2012RAS,MNRAS000,1–15
A new method to measure galaxy bias by combining the density and weak lensing fields 3
& Porciani 2011; Hoffmann et al. 2015; Bel, Hoffmann & Although these relations appear to measure the same
Gaztan˜aga2015),andoneofthemostcommononesisfrom parameter b , the results can be affected by the stochastic-
1
the ratio of the 2PCFs of galaxies and dark matter: ity in the relation between δ and δ, that can come from
g
different effects, such as the stochasticity of bias and the
ξ (r)=b2(r)ξ(r), (1)
g projection effects.
where b(r) is the galaxy bias, and ξ (r) and ξ(r) are Galaxy bias from equations (6-8) depend on the
g
the scale-dependent galaxy and matter 2PCFs respectively, smoothing scale used to measure δ and δg. For small scales
which are defined as: nonlinearities in the relation between δ and δg appear, and
b isnolongerconsistentwithequation(3).Throughoutthe
1
paper we will use these equations of bias for distributions
ξ (r )=(cid:104)δ (r )δ (r )(cid:105), ξ(r )=(cid:104)δ(r )δ(r )(cid:105). (2)
g 12 g 1 g 2 12 1 2 projected in the sky. Then, the relations in this analysis
where δ =(ρ −ρ¯ )/ρ¯ is the density fluctuation of galax- depend on angular distance (for equation (3)) or smooth-
g g g g
ies (ρ is the galaxy number density), and δ =(ρ−ρ¯)/ρ¯is ing angle (for equations (6-8)). The relation between both
g
the density fluctuation of dark matter (ρ is the dark mat- scales of bias (smoothing and separation) is complex, since
terdensity).Ascanbeseenfromthisequation,galaxybias the smoothing of δ and δg on a scale Θ involves the corre-
generally depends on the scale r (defined as the distance lationsofallthescalesbelowΘ.However,inthelinearand
12
between r and r ). However, it has been shown that at local regime bias is consistent with both scales and then all
1 2
sufficientlylargescalesinthelinearbiasregime,biasiscon- the estimators can be compared.
stant (e.g. Manera & Gaztan˜aga 2011).
Bias can also be defined from the projected 2PCFs:
2.2 Weak Lensing
ω (θ)=b2(θ)ω(θ), (3)
g Weakgravitationallensing(seee.g.Bartelmann&Schneider
whereω (θ)andω(θ)refertotheprojected2PCFofgalaxies 2001; Refregier 2003) measures the small changes of galaxy
g
anddarkmatterrespectively.Thisdefinitionofbiaswillbe shapesandbrightnessesduetotheforegroundmassdistribu-
used in the analysis of this paper. In this case, the bias tionintheline-of-sightofthe(source)galaxies.Bystudying
dependence is on separation angle θ instead of distance r. this effect statistically, assuming that (lensed) galaxies are
In the local bias model approach (Fry & Gaztanaga randomlyorientedintheabsenceoflensing,onecaninferthe
1993),thedensityfieldofgalaxiesisdescribedasafunction massdistributionintheforegroundofthesesourcegalaxies.
of its local dark matter density, so that δ = F[δ]. We can As the light distortion is affected by gravity, weak lensing
g
express this relation as a Taylor series: allows us to measure the total mass distribution, including
baryonic and dark matter.
∞ The gravitational potential Φ of a given density distri-
δg =(cid:15)+b0+b1δ+ b22δ2+...=(cid:88)bi(z)δi+(cid:15), (4) bution δ can be defined as:
i=0 3H2Ω
∇2Φ= 0 mδ, (9)
where bi are the coefficients of the Taylor expansion and (cid:15) 2a
represents the galaxy shot noise. The density contrasts δ
g where H and Ω are the Hubble parameter and the mat-
andδaresmoothedtoacertainscalebyawindowfunction, 0 m
ter density parameter today respectively, and a is the scale
so the relation also depends on that physical scale. It also
factorassumingaspatiallyflatUniverse.AssumingGeneral
assumesnorandomscatterbetweenδ andδ,and(cid:15)isnegli-
g Relativityandnoanisotropicstress,thelensingpotentialfor
gibleforlargesmoothingscales.Inthelinearregime,δ(cid:28)1,
a given source at position (θ,χ ) is given by the weighted
and as b = 0 because (cid:104)δ (cid:105) = (cid:104)δ(cid:105) = 0, then the equation s
0 g line-of-sight projection of Φ:
becomes:
(cid:90) χs χ(χ −χ)
δg =b1δ (5) ψ(θ,χs)=2 dχ χs Φ(θ,χ), (10)
0 s
According to Manera & Gaztan˜aga (2011), at large
where θ is the angular position on the sky, χ refers to the
scales this definition of bias is consistent with the bias ob-
comoving radius and χ is the comoving distance to the
tained from the 2PCFs: for r (cid:38)40h−1Mpc, b from equa- s
12 sources. The distortion of the source galaxy images can be
tion (1) is approximately constant and consistent with b
1 described by the convergence κ and shear γ fields that are
from equation (5). This b can then be measured from the
1 defined as:
different zero-lag correlations between δ and δ:
g 1
κ= ∇2ψ, (11)
b = (cid:104)δgδ(cid:105) (6) 2
1 (cid:104)δδ(cid:105)
1
γ =γ +iγ = (ψ −ψ )+iψ , (12)
(cid:104)δ δ (cid:105) 1 2 2 ,11 ,22 ,12
b = g g (7)
1 (cid:104)δ δ(cid:105) whereψ =∂ ∂ ψ.Focusingontheconvergencefield,com-
g ,ij i j
bining equations (9), (10) and (11) we obtain:
(cid:115)
b1 = (cid:104)δ(cid:104)δgδδg(cid:105)(cid:105) (8) κ(θ,χs)= 3H202cΩ2m (cid:90) χsdχχ(χχs−χ)δ(aθ(χ,χ)) ≡K[δ] (13)
0 s
(cid:13)c 2012RAS,MNRAS000,1–15
4 Arnau Pujol, Chihway Chang, Enrique Gaztan˜aga et al.
For simplicity, we define q(χ,χ ) as the lensing kernel When computing κ numerically, the integral is approxi-
s g
of the integral of δ at χ with the source at χ : mated by a sum over all lenses in the foreground of the
s
3H2Ω χ(χ −χ) sources:
q(χ,χ )= 0 m s (14)
s 2c2 χ a(χ) N
s κ (θ)(cid:39)(cid:88)q¯iδi(θ)∆χi, (17)
so that g g
(cid:90) χs i=1
κ(θ,χ)= q(χ,χs)δ(θ,χ)dχ. (15) where we have split the foreground galaxies into N redshift
0 bins. ∆χi refers to the redshift bin width of the ith bin
Notethatκcorrespondstoaweightedintegralofthematter incomovingcoordinates,q¯i isthemeanlensingweightthat
density fluctuations in the line-of-sight of the source galax- correspondstothatredshiftbinandδi(θ)isthegalaxyden-
g
ies. sity fluctuation in that redshift bin at position θ, where θ
now represents a pixel in the sky plane. δi(θ) is calculated
g
throughδi(θ)=(ρi(θ)−ρ¯i)/ρ¯i,whereρi(θ)isthedensity
g g g g g
3 METHOD of galaxies projected in the line-of-sight in the ith redshift
bin and position (pixel) θ, and ρ¯i is the mean density of
3.1 Simulation g
galaxies in the redshift bin, calculated from all the galax-
For the analysis we use the MICE Grand Challenge sim- ies inside the redshift bin. This measurement of ρ¯i gives a
g
ulation (Fosalba et al. 2015a,b; Crocce et al. 2015), an goodestimateofthemeandensityiftheredshiftbiniswide
N-body simulation of a ΛCDM cosmology with the fol- enough.Fornarrowbinsofredshiftwidthbelow∆z=0.03a
lowing cosmological parameters: Ω = 0.25, σ = 0.8, smoothingofρ¯i asafunctionofredshiftisneededtoobtain
m 8 g
n = 0.95, Ω = 0.044, Ω = 0.75, h = 0.7. It has a a good estimate of the mean density, as discussed in §3.3.
s b Λ
volume of (3.072h−1Gpc)3 with 40963 particles of mass Notice that δi(θ) is calculated taking into account all the
g
2.927×1010h−1M . The galaxy catalogue has been run galaxies inside the volume of the cell corresponding to each
(cid:12)
according to a Halo Occupation Distribution (HOD) and a pixel and redshift bin. This means that it corresponds to a
SubHaloAbundanceMatching(SHAM)prescriptions(Car- projectioninredshiftofthegalaxydensityfieldweightedby
retero et al. 2015). The parameters of the model have been the volume of the corresponding cell.
fitted to reproduce clustering as a function of luminosity In Figure 1 we show a schematic picture of the effects
andcolourfromtheSloanDigitalSkySurvey(Zehavietal. of equation (17). Dashed black line shows q(z,z ), defined
s
2011), as well as the luminosity function (Blanton et al. from equation (14) in redshift coordinates, while red solid
2003,2005a)andcolour-magnitudediagrams(Blantonetal. lineshowsq¯i inredshiftbinsof∆z=0.2.Weusedz =1.3
s
2005b).WeusetheMICECATv2catalogue,anextensionof for this figure. The blue shaded region represents δ (z) in a
g
the publicly available MICECATv1 catalogue1. The main random(justfortheexample)pixelintheskyusingnarrow
difference between MICECATv1 and MICECATv2 is that redshift bins (∆z = 0.05). The blue solid line represents
MICECATv2 is complete for i < 24 from z = 0.07 to δi for the redshift bins of ∆z = 0.2. Equation (17) then is
g
z = 1.4, while MICECATv1 is complete for an absolute equivalenttotheintegraloftheproductoftheblueandred
magnitude of Mr < −19. The catalogue also contains the solid lines.
lensing quantities (γ1, γ2 and κ) at the position of each Equation(17)isanapproximationof(16),thatassumes
galaxy, calculated from the dark matter field with a reso- that the small fluctuations in redshift of δ inside the bins
g
lution of Nside=8192 in healpix (corresponding to a pixel do not affect the results. The mean of q(χ,χ )δ (χ) inside
s g
sizeof∼0.43arcmin).Thelensingsignalwascomputedus- the bins can be approximated by the product of the means
ing the Born approximation. As the lensing value assigned q¯iδi(θ).Theseapproximationsholdatlargescalesandwhen
g
to a galaxy at a given 3D position is inherited from the q(χ,χ ) and δ (χ) are not correlated.
s g
corresponding pixel value of the dark matter lensing map Wefocusonthesimplestcase,wheregalaxybiasislin-
in which that galaxy sits in, the lensing quantities of the ear, local and redshift-independent. In this case, we can es-
galaxies do not have shape noise. timate b from the following zero-lag correlations of κ and
κ :
g
3.2 Bias estimation (cid:104)κ κ(cid:105)
b= g (18)
(cid:104)κκ(cid:105)−(cid:104)κNκN(cid:105)
In this section, we introduce the method used to estimate
galaxy bias from the lensing and density maps of galaxies
in the MICE simulation. It consists on the construction of (cid:104)κ κ (cid:105)−(cid:104)κNκN(cid:105)
b= g g g g , (19)
a template κg for the lensing map κ from the density dis- (cid:104)κgκ(cid:105)
tribution of the foreground galaxies assuming equation (5).
Substituting δ with δg in equation (13) gives: where κN and κNg are the sampling and shot-noise correc-
(cid:90) χs tion factors obtained by randomizing the galaxy positions
κg(θ)= q(χ,χs)δg(θ,χ)dχ (16) andre-calculatingκandκg.κisobtainedfromthemeanκ
0 ofthegalaxiesineachpixel.Thisisaffectedbythenumber
of source galaxies in the pixel, causing a noise in (cid:104)κκ(cid:105) that
1 http://cosmohub.pic.es/ depends on the angular resolution used, reaching a 10% er-
(cid:13)c 2012RAS,MNRAS000,1–15
A new method to measure galaxy bias by combining the density and weak lensing fields 5
0.6 30 0.020
0.016
40 0.4 25
1 0.012
×
z)s 0.2 20 0.008
z, ) 0.004
d q( 0.0 oC(15 0.000
n E
a D 0.004
δ(z)g 0.2 q¯i(z,zs) δgi(z) 10 0.008
0.4 q(z,zs) δg(z) 5 0.012
0.016
0.2 0.4 0.6 0.8 1.0 1.2
z 0 0.020
0 5 10 15 20 25 30
RA(o)
Figure 1. Schematic comparison of equations (16,17). Dahsed
black line shows q(z,zs), defined in comoving scales in equation
(14), for a fixed zs = 1.3, while red solid line shows q¯i from 30 g 0.020
equation(17)inredshiftbinsof∆z=0.2.Theblueshadedregion
representsδg(z)usingnarrowredshiftbins(∆z=0.05).Theblue 0.016
solidlinerepresentsδgi fortheredshiftbinsof∆z=0.2. 25 0.012
0.008
20
0.004
)
o
(
C15 0.000
E
D 0.004
ror for a pixel size of 5 arcmin. This noise is cancelled by 10
0.008
subtracting (cid:104)κNκN(cid:105). On the other hand, (cid:104)κ κ (cid:105) is affected
g g
byshotnoise,causinganerrorthatincreaseswiththeangu- 5 0.012
lar resolution up to a 20% for a pixel size of 5 arcmin. This 0.016
noise is cancelled by subtracting (cid:104)κNκN(cid:105). This correction
g g 0 0.020
assumesaPoissondistribution.Totesthowwellthiscorrec- 0 5 10 15 20 25 30
tion works for this method, we calculated (cid:104)κ κ (cid:105)−(cid:104)κNκN(cid:105) RA(o)
g g g g
using the dark matter particles instead of galaxies, and we
comparedtheresultswiththetrue(cid:104)κκ(cid:105)mapsfromthesim-
ulation. We did this with different dilutions (from 1/70 to 0.015
1/700)ofthedarkmatterparticles,andrecover(cid:104)κκ(cid:105)better
than 1% independently on the dilution, indicating that the 0.010 b=1.268 0.014
shot-noise subtraction is appropriate. ±
Since the galaxies used from the MICE simulation do
0.005
nothaveshapenoise,theestimatorsinthisanalysisarenot
affectedbyshapenoise.Thisisnotthecaseinobservations, g
where shape noise is the most important source of noise of 0.000
this method and needs to be corrected. Moreover, in obser-
vations we do not have κ either, and we need to obtain κ 0.005
from γ and equations (11-12) in order to use these estima-
tors. Notice that galaxy bias obtained from equations (18-
0.010
19) imply an average of bias as a function of redshift. This
is because κ involves a redshift integral of δ ∼ b(z)δ as 0.010 0.005 0.000 0.005 0.010 0.015
g g
specifiedinequation(16),sothefinalproductisaredshift-
averaged bias weighted by the lensing kernels that appear
in equations (18-19). Later in this analysis we use tomo- Figure 2. Comparison of κ vs κg. Top panel shows the κ field
graphic redshift bins, where we assume that bias does not from the source galaxies within 0.9 < z < 1.1 and using a Top
significantly change inside the bin, and we measure bias in Hatfilterof50arcminofradius.Middlepanelshowsκg obtained
each of the redshift bins. from equation (17), using the same smoothing scheme. Bottom
To measure the errors on b, we use the Jackknife (JK) panel shows the comparison between κg and κ for the pixels of
themaps,withthespecifiedbiasanderrorobtained.Theredline
method.Wedividetheareainto16subsamples.Weevaluate
correspondstoalinecrossingtheoriginanditsslopecorresponds
b 16 times excluding each time a different subsample. The
to b. It is consistent with the linear fit of the distribution of the
error of b is estimated from the standard deviation of these
points.
(cid:13)c 2012RAS,MNRAS000,1–15
6 Arnau Pujol, Chihway Chang, Enrique Gaztan˜aga et al.
16 measurements as: Thisisanindicationthatweareinthelinearregime,where
(cid:118) we can assume equation (5).
σ(b)(cid:39)(cid:117)(cid:117)(cid:116)NJK−1N(cid:88)JK(b −b)2, (20) Wenotethattheexpressionforthebiasfromequations
NJK i (18,19)assumesequation(5).However,κisaprojectionofδ
i=1
intheline-of-sightweightedbythelensingkernel,aswellas
where N refers to the number of JK subsamples used, b
JK i κ . Thus, the relation between κ and κ is a constant that
g g
isthebiasmeasuredbyexcludingtheithsubsampleandbis
comesfromtheredshiftdependenceofbiasweightedbythe
obtained from the average overall subsamples. We checked
redshift dependence of the lensing kernel. Hence, the bias
that the error are very similar if we use a different number
obtained in this example is a weighted mean of galaxy bias
of subsamples (between 9 and 100) instead of 16.
as a function of redshift. But we can take this dependence
Note that we can also measure bias from the following
into account to measure bias at different redshifts using to-
cross correlations:
mography as we explain in §3.4 below.
(cid:104)γ γ (cid:105)
b= i,g i (21)
(cid:104)γ γ (cid:105)−(cid:104)γNγN(cid:105)
i i i i
3.4 Redshift dependence
(cid:104)γ γ (cid:105)−(cid:104)γNγN(cid:105)
b= i,g i,g i,g i,g ,i=1,2 (22)
(cid:104)γNγN(cid:105) Thismethodinvolvesanintegral(orasuminpractice)along
i,g i
theredshiftdirection,andbecauseofthisthebiasobtained
As this is not the focus of the paper, and we can obtain κ
is a weighted average of the redshift dependent bias. How-
fromthesimulation,wemeasurebfromequations(18,19)in
ever, we can estimate galaxy bias in a given redshift bin
thisstudy.However,inobservationswemeasuretheshapeof
if we restrict the calculation to the foreground galaxies in
the galaxies, that is directly related to γ . Because of this,
i that redshift bin, assuming that bias does not change sig-
applying this method to data requires a conversion from
nificantly in the bin. If this is the case, we can measure the
κ to γ or from γ to κ. These conversions imply other
g i,g i redshift-dependent bias using tomographic redshift bins.
systematics due to the finite area and the irregularities of
Since κ is obtained from the contribution of all the
themask.Theconversionfromκ toγ canbeaffectedby g
g i,g galaxies in front of the sources, if we restrict the redshift
the shot noise in κ , but this noise is less dominant than
g range for the calculation of κ we need to renormalize the
shape noise, that can affect the conversion from γ to κ. g
i result by taking into account the contribution from the un-
WeaddressthisissueinPaperII,whereweuseconversions
used redshift range.
basedonKaiser&Squires(1993)(hereafterKSmethod)to
We define as κ(cid:48) the construction of a partial κ using
applythismethodtoDESSVdata.Anotheraspecttotake g g
only the galaxies projected in a given redshift bin, so:
intoaccountfordataanalysisisthatsinceshapenoiseisthe
main source of noise in the measurement, we like to avoid (cid:90)
κ(cid:48)(θ)= dχq(χ,χ )p(χ)δ (θ,χ), (23)
the terms that involve variance of lensing quantities (cid:104)κκ(cid:105) g s g
and(cid:104)γ γ (cid:105),sincethesetermsarethemostaffectedbyshape
i i
wherep(χ)istheradialselectionfunction,equalto1inside
noise.
the bin χ < χ < χ and 0 outside. To simplify the
min max
notation, when the limits are not specified in the integral,
3.3 Implementation the integral will go through the whole range between 0 and
∞. We assume that all the sources at located at χ here
In Figure 2, we illustrate our procedure. We used a ∼ 900 s
andinthefollowing,andbecauseofthiswewillnotinclude
square degree area from the MICE simulation correspond-
theargumentχ inκ(cid:48)(θ)andotherfunctions.Notethat,as
ing to 0◦ < RA < 30◦ and 0◦ < DEC < 30◦. The top s g
p(χ)=0forallχoutsidethebin,onlytherangeχ <χ<
panel shows the convergence map κ of source galaxies lo- min
χ contributes to the integral in equation (23), and p(χ)
cated at z (cid:39) 1. The middle panel shows the constructed max
implies a projection inside the bin. In order to simplify the
convergence template, κ , derived via equation (17). Both
g expression,ifq(χ,χ )isnotcorrelatedwithp(χ)δ (χ)inside
mapshavebeengeneratedbypixelizingthedistributionsin s g
the bin (which is the case, since δ (χ) decorrelates quickly
pixels of 7 arcmin of side. Then, the pixelated maps have g
in the redshift direction and hence the correlation is only
been smoothed using a circular top hat filter of 50 arcmin
important for very narrow bins), q(χ,χ ) can be described
ofradiusfromthispixelatedmap.Themapobtainedcorre- s
outside the integral as:
spondstotheangularscaleof50arcmin,andtheirstatistics
do not depend on the scale of the previous pixelization (if (cid:90)
κ(cid:48)(θ)(cid:39)q¯(cid:48)∆χ dχp(cid:48)(χ)δ (θ,χ)=q¯(cid:48)∆χδ¯(cid:48), (24)
the pixels are much smaller than the smoothing scale). We g g g
canseethatκ isabiasedversionofκatlargescales.Inthe
g
with
bottom panel we show the scatter plot of κ versus κ . The
g
biasbshownintheplotisestimatedviaequation(18),and (cid:90) χmax q(χ,χ )
q¯(cid:48) = dχ s . (25)
the error corresponds to the JK errors from equation (20). ∆χ
Inred,weshowalinecrossingtheoriginandwiththeslope χmin
correspondingtothisestimatedbias.Wehavecheckedthat ∆χ = χ −χ , and now p(cid:48)(χ) is the same selection
max min
the b value derived from the zero-lag statistics is in agree- function as p(χ) but normalized to 1, so p(cid:48)(χ)=p(χ)/∆χ.
ment with a linear fit to the scatter plot at the 0.1% level. Withthisdefinition,wemeasurethegalaxybiasinthis
(cid:13)c 2012RAS,MNRAS000,1–15
A new method to measure galaxy bias by combining the density and weak lensing fields 7
redshift bin, that we call b(cid:48), from the following expressions: redshiftbin,weusethefollowingexpressionsfortheangular
correlation functions:
1 (cid:104)κ(cid:48)κ(cid:105)
b(cid:48) = g (26) (cid:90) (cid:90)
1 f1(cid:104)κκ(cid:105)−(cid:104)κNκN(cid:105) ωA(cid:48)B(θ)=q¯(cid:48)∆χ dχA dχBq(χB)p(cid:48)(χA)p(χB)ξAB(r)
(36)
b(cid:48)2 = f12(cid:104)κ(cid:48)gκ(cid:48)g(cid:105)(cid:104)−κ(cid:48)g(cid:104)κκ(cid:105)(cid:48)gNκ(cid:48)gN(cid:105), (27) ωA(cid:48)B(cid:48)(θ)=q¯(cid:48)2∆χ2(cid:90) dχA(cid:90) dχBp(cid:48)(χA)p(cid:48)(χB)ξAB(r),
(37)
where κ(cid:48)gN is obtained by randomizing the positions of the where A(cid:48) and B(cid:48) refer to the cases where the fields A and
galaxiesintheredshiftbininordertocorrectforshot-noise,
B arerestrictedtotheredshiftbin,and∆χ=χ −χ
max min
and f1 and f2 correspond to the following ratios: defines the redshift bin width of A(cid:48) and B(cid:48).
(cid:104)κ(cid:48)κ(cid:105) Equations(28-29)canbepredictedtheoreticallybyas-
f1 = (cid:104)κκ(cid:105) (28) suming a cosmology. However, most of the cosmology de-
pendence of the expression is canceled out due to the ra-
and tios from f and f , so the final factor is weakly depen-
1 2
(cid:104)κ(cid:48)κ(cid:48)(cid:105) dent on cosmology. In our case we assume the cosmology
f2 = (cid:104)κ(cid:48)κ(cid:105), (29) of the MICE simulation. In Figure 3 we show f1 (top) and
f (bottom)fordifferentcosmologiesandtheories,usingan
2
where κ(cid:48) is defined as the contribution to κ of the dark angular scale of 50 arcmin, normalized by the values corre-
matter field projected in the redshift bin used, so: sponding to the MICE cosmology. Orange solid lines repre-
sent the MICE cosmology, predicted from Eisenstein & Hu
(cid:90)
κ(cid:48)(θ)=q¯(cid:48)∆χ dχp(cid:48)(χ)δ(θ,χ)=q¯(cid:48)∆χδ¯(cid:48). (30) (1998) non-linear theory obtained using Halofit (Smith
et al. 2003). The dashed green lines show the same but ob-
Forourpurposeweareinterestedintheanalyticexpres- tained from linear theory. We can see that the differences
sionsof(cid:104)κ(cid:48)κ(cid:105),(cid:104)κ(cid:48)κ(cid:48)(cid:105)and(cid:104)κκ(cid:105)tobeabletousef andf to between using linear and non-linear theory are small com-
1 2
measuregalaxybiasintomographicredshiftbins.According paredwiththefinalerrorsthatweobtainfromourmethod.
to the definitions, from equations (15,30) we can derive: Weusetheoldversionof Halofitforthisprediction,which
produces larger differences between the MICE and the the-
(cid:104)κ(cid:48)κ(θ)(cid:105)=q¯(cid:48)∆χ(cid:48)(cid:90) p(cid:48)(χ )dχ (cid:90) χsdχ q(χ )ξ(r ) (31) oretical linear Power Spectrum (Fosalba et al. 2015a). On
1 1 2 2 12
0 theotherhand,toobtainthenon-linearpredictionwecom-
putedthenon-linearitiesinanintermediateredshiftandex-
(cid:90) (cid:90) trapolatedtotheotherredshiftsusinglineargrowth,which
(cid:104)κ(cid:48)κ(cid:48)(θ)(cid:105)=(q¯(cid:48)∆χ(cid:48))2 p(cid:48)(χ )dχ dχ p(cid:48)(χ )ξ(r ) (32)
1 1 2 2 12 causes a larger disagreement between the linear and non-
linearpredictions.Becauseofallthis,thedifferencebetween
(cid:90) χs (cid:90) χs theorangeanddashedgreenlinesmaybeinterpretedasthe
(cid:104)κκ(θ)(cid:105)= q(χ1)dχ1 q(χ2)dχ2ξ(r12), (33) upper bound of the disagreement between linear and non-
0 0 linear theory.
with r2 =χ2+χ2+2χ χ cosθ, ξ(r ) is the 2PCF and θ Finally, in black dotted lines we show the predictions
12 1 2 1 2 12
is the angular separation between the two fields. forthesamecosmologybutwithΩm =0.3.Wecanseethat
For the general case, the zero-lag correlation of two the differences between both cosmologies are smaller than
fields A and B at an angular scale Θ (corresponding to a theerrorsofourbiasestimation,evenwiththefactthatthe
radius R in the given redshift bin) is given by: differences in Ωm are very large and that Ωm is the most
sensitive parameter of these predictions. Hence, we can say
4 (cid:90) R (cid:90) R (cid:90) π thatthecosmologydependenceofthismethodisveryweak.
(cid:104)κ κ (Θ)(cid:105)= dr r dr r dηω (θ),
A B πR4 1 1 2 2 AB Equations (28-29) describe the contribution of these
0 0 0
(34) zero-lag correlations of κ and κ(cid:48) in a given redshift bin for
where θ2 = r12+r22−2r1r2cosη, κA and κB can be κ, κ(cid:48), thedarkmatterfield.Asthedarkmatterfieldhasabiasof
κg orκ(cid:48)g,η istheangularseparationbetweenthevectorsr1 1 by definition, using the galaxies instead of the dark mat-
andr2andω(θ)isaprojectedtwo-pointangularcorrelation ter field to compute κ(cid:48)g instead of κ(cid:48) in equations (28-29)
function of the two fields A and B defined as: would give b(cid:48) f instead of f , where b(cid:48) is the galaxy
1,2 1,2 1,2 1,2
(cid:90) (cid:90) bias in the redshift bin used (assuming that galaxy bias is
ωAB(θ)= dχA dχBq(χA)q(χB)p(χA)p(χB)ξAB(r), constant inside the redshift bin). Then, to estimate galaxy
(35) biasinthesebins,weneedtoobtainthebiasfromequations
where p(χ ) are the corresponding selection functions of (26-27).
A,B
the fields A and B, and ξ (r) is the 3D two-point cross-
AB
correlation function, that in this case corresponds to the
3.5 Numerical effects and parameters
dark matter ξ(r).
In order to be consistent with equations (24,30), when Therearedifferentparametersthatcanaffectourimplemen-
A (and also B) refer to the dark matter field limited in a tation presented in §3.2. We have studied in which regime
(cid:13)c 2012RAS,MNRAS000,1–15
8 Arnau Pujol, Chihway Chang, Enrique Gaztan˜aga et al.
selecting galaxies by colour or luminosity, in order to mea-
1.10 sure colour and luminosity dependent bias, that would give
information about galaxy formation and evolution.
1.05
)
5
2
0.1.00 Redshift bin width
=
m0.95 Forthechoiceof∆zweneedtotakeintoaccounttwoeffects.
Ω Ononeside,theuseofwideredshiftbinswouldmeanlosing
f(10.90 Ωm=0.25 non-linear ilninfeo-romf-astigiohnt,fsrionmcetwheepsrmojaelcltstchaelegaflluacxtiuesatinionthseosfamδgeibninthtoe
/
f10.85 Ωm=0.25 linear measure δg. We have seen that this produces a deviation in
Ω =0.3 thevalueofgalaxybiasthatislargerthan5%for∆z>0.2,
m
0.80 and it can be larger than 10% for ∆z > 0.3. We explore
0.0 0.2 0.4 0.6 0.8 1.0
thisinFigure6andin§4.1.Wetakethiseffectintoaccount
z whenweestimatebiasintomographicbinsattheendofthe
paper. When we have photo-z errors, the redshift binning
1.004 effectisnotasimportantasfortheidealcase.Ifthephoto-
zerrorsdominate,thedilutionofthesmallscalefluctuations
) comefromthephoto-zerrors,andtheredshiftbinningdoes
51.002
2 not affect much. We address the effects of photo-z errors in
0. Paper II.
=1.000 On the other hand, the use of narrow redshift bins re-
m quiresasmoothingoftheestimationofρ¯ (z).Ifwecalculate
Ω g
ρ¯ for each redshift bin alone, for narrow bins ρ¯ (z) is af-
(0.998 g g
f2 fectedbythestructurefluctuationineachparticularredshift
/
2 bin, and this causes a smoothing in the final estimation of
f
0.996 δ .Thishappensbecause,whenaredshiftbinisdominated
g
by an overdensity fluctuation, ρ¯ (z) is overestimated and
g
0.0 0.2 0.4 0.6 0.8 1.0 hence δ is underestimated. On the other hand, when the
g
redshift bin is dominated by an underdensity, ρ¯ (z) is un-
z g
derestimatedandhenceδ isoverestimated.Thefinalδ (z)
g g
Figure3.f1(top)andf2(bottom)fordifferenttheorycosmolo- is then smoothed, since all the values tend to be closer to
gies,normalizedbythevaluesfromtheMICEcosmology.Dotted zero due to the calculation of ρ¯ (z). Some smoothing of ρ¯
g g
linesareobtainedforacosmologywithΩm=0.3,whiletheother in redshift is needed to avoid this effect when using narrow
twolinesrepresentΩm=0.25.Theorangesolidlinehasbeenob- bins. This is relevant for ∆z<0.03.
tainedusingnon-lineartheorywithHalofit(Smithetal.2003),
We use redshift bins of ∆z = 0.2 for the foreground
whilethedashedgreenlinehasbeenobtainedfromlineartheory
usingHalofit. galaxies. In this analysis we use the true redshift from the
simulation, but in data this method would be also affected
by photo-z errors. When photo-z errors are present, using
our method is valid, or consistent with the linear bias from narrow redshift bins is not worth, since the uncertainty in
equation (3), and what are the dependences when it is not redshift from photo-z errors dominate. We choose this red-
valid. With this, we can either calibrate our results or re- shiftbinwidthforourestimationofbiasinordertotesthow
stricttothe regimeswhereourbiasmeasurementiscarried well we can recover galaxy bias using the redshift binning
out. Here we describe the main numerical effects and our that is used in Paper II.
choice of parameters for our final implementation in §4.2
and Figure 9.
Angular scale
To generate the maps we pixelize the sky using a si-
Catalogue selection
nusoidal projection (which consists on redefining RA as
We used an area of 0o <RA,DEC <30o. This is the same (RA−15)cos(DEC) in order to obtain a symmetric map
area we used for the fiducial bias measurements from equa- with pixels of equal area) with an angular resolution of 50
tion(3),sothatourcomparisonofbothbiasisnotaffected arcmin,sothattheareaofthepixelsis(50 arcmin)2.Then
bydifferencesinareaorsamplevariance.Thisareaissimilar galaxiesareprojectedindifferentredshiftbinsaccordingto
toDESY1data,sothisstudycanbeseenasanestimation their true redshift.
of the theoretical limitations of this method on DES Y1. Thebiasestimatedfromthismethodisnotnecessarily
We apply a magnitude cut for the foreground galaxies consistent with the bias from equation (3) at small scales.
of i<22.5, to be able to compare it in Paper II with mea- These two methods are only expected to agree at large
surements in the DES SV data (Crocce et al. 2016). How- scales, in the linear bias regime. Moreover, this method
ever, other selections can be done for this method, such us requires a projection in the line-of-sight, so that different
(cid:13)c 2012RAS,MNRAS000,1–15
A new method to measure galaxy bias by combining the density and weak lensing fields 9
scales (weighted differently according to the lensing kernel)
aremixedforthesameangularscale.However,wehaveseen 2.4
thatbiasisconstantforangularscaleslargerthanΘ(cid:38)30ar-
cmin,meaningthatlinearscalesaredominantinthisregime. 2.2 b= ωg(θ)/ω(θ)
In Figure 4 we show the agreement of galaxy bias between q
b= δ δ / δ δ
equations (3) and (6-8) when we use a pixel scale of 50 ar- 2.0 g g g
cmin, as a visual example of this. › fi › fi
1.8 b= δgδg / δδ
b q
Smoothing 1.6 b= δg›δ / fiδδ› fi
We do not apply any smoothing in the pixelized maps to 1.4 › fi › fi
estimate galaxy bias in this paper. Exceptionally, for the
maps in Figure 2 we use pixels of 7 arcmin and we apply 1.2
a Top Hat filter of 50 arcmin to smooth the maps. We do
1.0
thisonlyinthisfigureinordertohaveabettervisibilityof 0.0 0.2 0.4 0.6 0.8 1.0 1.2
the structures of the maps and the shape of the area used. z
Fortherestoftheanalysisofthepaper,weusepixelsof50
arcmin and no smoothing kernel afterwards.
Figure 4.Comparisonofdifferentdefinitionsofbias.Solidcyan
lineshowsthebiasasdefinedinequation(3).Thedashedblack,
Edge effects dash-dottedgreenanddottedredlinesshowbiasaccordingtothe
differentdefinitionsfromequations(6-8).
We use a limited area and we project the sky to obtain the
maps.Whenwepixelizethemapwithadefinitepixelscale,
duetotheprojectionandtheshapeoftheareaused,partof
thepixelsintheedgesarepartiallyaffectedbytheseedges.
We exclude these pixels from the analysis.
When a smoothing kernel is applied to the pixelized
In Figure 4 we compare different estimators of
map, the pixels that are close to the edges are also affected
galaxy bias from the MICE Simulation, using an area of
bythem.Weexcludethepixelswhosedistancetotheedges 0o < RA, DEC < 30o. The solid cyan line represents
is smaller than the smoothing radius.
thebiasdefinitionfromequation(3).Wemeasureω(θ)and
ω (θ) as a function of the angular scale, and to obtain the
g
Source redshift and redshift range bias we fit the ratio as constant between 6 and 60 arcmin.
The angular correlation function involves different comov-
We estimate the κ field at z (cid:39)1.3 by calculating the mean ing scales for different redshifts, and then fixing the same
κ of the source galaxies with 1.2 < z < 1.4 in each pixel. angular scales for the galaxy bias implies a mix of physical
The redshift range used ensures we have enough density of scales.However,forlargeenoughscales,biasisconstantand
galaxies to correctly calculate κ. is not affected by this. We have checked that bias does not
Theoretically one should take into account the redshift change significantly at these scales, and using these scales
distribution of the source galaxies so that each galaxy con- to measure galaxy bias give consistent results with using
tributestoκg withitspositionχs.However,approximating largerscales.Thegalaxybiasobtainedfromequations(6-8)
these galaxies to a plane in their mean position at z (cid:39) 1.3 are shown in dashed black line, dotted red line and dash-
causes less than a 1% effect. dotted green line as specified in the legend. This has been
We use single redshift bins of ∆z = 0.2 for the fore- calculatedineachredshiftbinbypixelatingδ andδ inpix-
g
ground galaxies in the range of 0.2 < z < 1.2 to estimate els of area (50 arcmin)2 using redshift bins of ∆z = 0.2.
the bias in each of these bins. This produces a galaxy bias Theagreementbetweenthesolidcyanandthedashedblack
estimation of 5 points in the whole redshift range available linesconfirmsthatlinearbiasfromequation(3)isconsistent
(for this method) in the simulation. withlocalbiasmeasurefromequation(6)atthesescales.On
the other hand, the differences in the different expressions
of equations (6-8) implies a stochasticity between δ and δ
g
4 RESULTS that affects our estimations of bias. We see that the same
effect appears when using equation (40) to estimate galaxy
4.1 Testing
bias, and this can be explained by the projection effect due
In this study we test our method against a fiducial galaxy totheredshiftbinning,asdiscussedbelowinFigures5and
bias.Forthis,wemeasuretheangular2PCFsofmatterand 6. We take into account this effect to estimate tomographic
galaxies ω(θ) and ω (θ) in the simulation for different red- bias in §4.2.
g
shift bins, using the same area and galaxies that we use for The idea of the following analysis is to test how the
our method. We also estimate bias from the definitions in calculations of this method deviate from the expected es-
equations (6-8) in the same simulation to study the consis- timation of linear bias using different angular scales and
tency between the different bias definitions. binning. For these testing purposes, we construct here the
(cid:13)c 2012RAS,MNRAS000,1–15
10 Arnau Pujol, Chihway Chang, Enrique Gaztan˜aga et al.
1.15 1.10
ˆb( ω (θ)/ω(θ)) ˆb( δ δ / δδ )
g g
q
1.10 b˜( ωg(θ)/ω(θ)) b˜(›δgδfig /› δgfiδ ) 1.05
q
› fi › fi
1.05 1.00
ˆb m
b
1.00 0.95
0.95 0.90 b
m,1
b
m,2
0.90 0.85
0 20 40 60 80 100 120 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Θ(arcmin) ∆z
Figure5.Biasfromthezero-lagcrosscorrelationsofκandκˆgas Figure6.bm,1andbm,2,definedinequations(42,43),asafunc-
afunctionofangularscale,whereκˆg isanestimationofκg nor- tionoftheredshiftbinwidthused,∆z,forthetwoestimators.
malizedbytheredshiftdependentbiasfromdifferentestimators
as in equations (38-40). Dashed red line showsˆb((cid:112)ωg(θ)/ω(θ)).
Thesolidredlineshowsthesame,butobtainingκˆg fromthebias tion (3) (at the 1% level), while ˜b is indirectly measuring
from equation (6) to obtainˆb((cid:104)δgδ(cid:105)/(cid:104)δδ(cid:105)). The dashed blue line equation(7),whichisslightlyhigherthanbiasfromequation
shows˜b((cid:112)ωg(θ)/ω(θ)).Thesolidbluelineshowsthesame˜b,but (3).Ifweuseequation(6)forthecalculationofκˆg toobtain
normalizingκˆg fromequation(7)toobtain˜b((cid:104)δgδg(cid:105)/(cid:104)δgδ(cid:105)). ˆb((cid:104)δgδ(cid:105)/(cid:104)δδ(cid:105)) (shown in the solid red line) and equation (7)
forthecalculationofκˆ toobtain˜b((cid:104)δ δ (cid:105)/(cid:104)δ δ(cid:105))(shownin
g g g g
thesolidblueline),thenbothestimationsareconsistent,as
bias-corrected κ map, κˆ , defined as:
g g expected. As in Figure 4, the difference between both esti-
κˆg(b,θ)=(cid:88)N qiδgib(iθ)∆χi, (38) cmaantobres sceoemninags farnomindˆb(ic(cid:112)atωiogn(θ)(/aωnd(θ)a)manedasu˜b(r(cid:112)emωegn(tθ))/oωf(tθh)e)
i=1 stochasticity in the relation between δ and δ, giving a fac-
g
where bi corresponds to the linear bias measured in N bins tor of 5%.
thatcanbeobtainedfromequations(3)or(6-8).Inanalogy Inordertogodeeperintheanalysisoftheseeffectsand
with equations (18,19), we can calculate the corresponding seewhetherthesedifferencesbetweenbothestimatorscome
normalized bias between the κˆg and κ fields: fromtheintrinsicrelationbetweenδg andδ orfromnumer-
icalsystematics,weconstructedthefollowingtemplateκ :
(cid:104)κˆ (b)κ(cid:105) m
ˆb(b)= g (39)
(cid:104)κκ(cid:105)−(cid:104)κNκN(cid:105) κ (θ)=(cid:88)N qiδi(θ)∆χi, (41)
m
(cid:104)κˆ (b)κˆ (b)(cid:105)−(cid:104)κNκN(cid:105) i=1
˜b(b)= g g g g . (40)
(cid:104)κˆ (b)κ(cid:105) whichcorrespondstothesameexactcalculationthanequa-
g
tion (17) for κ , but using dark matter particles instead of
g
Note that ˆb and ˜b depend on the bias b used to obtain κˆg. galaxies. This field κm is expected to reproduce κ from the
Under this definition, ˆb = 1 and ˜b = 1 suggest that this Born approximation consistently except for the numerical
method is consistent with measuring linear bias b. differences between the method and how the original κ is
Figure 5 shows how the estimators ˆb and ˜b change as obtained, which basically come from the redshift binning
a function of the angular scale, defined by the pixel scale, and projection discussed below equation (17). In order to
for different estimators of bias used to obtain κˆg. For the avoidnoiseintheκmap,weuseκT,definedasthetruemap
dashed red and blue lines we used b(z) from equation (3) directly obtained from the high resolution map of the sim-
to obtain κˆg for our estimation of ˆb((cid:112)ωg(θ)/ω(θ)) and ulation (see Gaztanaga & Bernardeau 1998; Fosalba et al.
˜b((cid:112)ω (θ)/ω(θ)) respectively. We can see that the measure- 2008,2015b),andwecalculatethebiasofthesetwoestima-
g
ments are constant for Θ > 30 arcmin, meaning that we tors of κ as:
are in the linear regime in these scales. However, there is
(cid:104)κ κ (cid:105)
a 5% difference between the two estimators at large scales bm,1 = (cid:104)κmκT(cid:105) (42)
T T
(at small scales nonlinearities appear and the difference is
larger). This can be interpreted from Figure (4), where we
(cid:104)κ κ (cid:105)−(cid:104)κNκN(cid:105)
see that the estimators from equations (6) (represented as b = m m m m , (43)
m,2 (cid:104)κ κ (cid:105)
a dashed black line) and (7) (represented as a dash-dotted m T
greenline)areslightlydifferent.Infact,ˆbisindirectlymea- thatshouldgiveb =1iftherearenonumericalsystem-
m,1,2
suringequation(6),whichisconsistentwithbiasfromequa- atics.
(cid:13)c 2012RAS,MNRAS000,1–15