Smoothing the Redshift Distributions of Random Samples for the Baryon Acoustic Oscillations : Applications to the SDSS-III BOSS DR12 and QPM Mock Samples

3.4 MB·

by  Shao-Jiang Wang

Mon.Not.R.Astron.Soc.000,1–9(2017) Printed11January2017 (MNLATEXstylefilev2.2)
Smoothing the Redshift Distributions of Random Samples
for the Baryon Acoustic Oscillations : Applications to the
SDSS-III BOSS DR12 and QPM Mock Samples
7
1 Shao-Jiang Wang,1,3,5(cid:63) Qi Guo,2,4 Rong-Gen Cai,1,3,5
0
2
1 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,

n No.55 Zhong Guan Cun East Street, Beijing 100190, P.R. China
a 2 National Astronomical Observatories, Chinese Academy of Sciences,
J 20A Datun Road, Chaoyang, Beijing 10012, P.R. China
0 3 School of Physical Sciences, University of Chinese Academy of Sciences,
1 No.19A Yuquan Road, Beijing 100049, P.R. China
4 School of Astronomy and Space Science, University of Chinese Academy of Sciences,
] No.19A Yuquan Road, Beijing 100049, P.R. China
O 5 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
C
.
h
p AcceptedXXX.ReceivedYYY;inoriginalformZZZ
-
o
r
t ABSTRACT
s
We investigate the impact of different redshift distributions of random samples on
a
[ theBaryonAcousticOscillations(BAO)measurementsofDV(z)rdfid/rd fromthetwo-
point correlation functions (2PCF) of galaxies in the Data Release 12 (DR12) of the
1
Baryon Oscillation Spectroscopic Survey (BOSS). Big surveys, such as BOSS, usually
v
assignredshiftstotherandomsamplesbyrandomlydrawingvaluesfromthemeasured
7
redshift distributions of the data, which would necessarily introduce fiducial signals
2
of fluctuations into the random samples, weakening the signals of BAO, if the cosmic
4
2 variance cannot be ignored. We propose a smooth function of redshift distribution
0 which fits the data well to populate the random galaxy samples. The resulting cos-
. mological parameters match the input parameters of the mock catalogue very well.
1
ThesignificanceofBAOsignalshasbeenimprovedby0.29σ forlow-redshift(LOWZ)
0
sample and by 0.15σ for constant-stellar-mass (CMASS) sample, though the absolute
7
1 valuesdonotchangesignificantly.Giventheprecisionofthemeasurementsofcurrent
: cosmological parameters, it would be appreciated for the future improvements on the
v
measurements of galaxy clustering.
i
X
Key words: galaxy survey, correlation function, random catalogues
r
a
1 INTRODUCTION to constrain the cosmological parameters to a high precise
level.
The Baryon Acoustic Oscillations (BAO) is an important
toolinmoderncosmology.Itistherelicimprintofthemat- ThemeasurementsofBAOhavemadeagreatprogress
terperturbationswhenthesoundwavesstoppropagatingin over the last decade. From the seminar works of the Two
thebaryon-photonfluidaftertherecombinationoftheUni- Degree Field Galaxy Redshift Survey (2dFGRS Cole et al.
verse.Itisalsoawell-understoodlinear-theoryphenomenon 2005) and the Sloan Digital Sky Survey (SDSS Eisenstein
whichusuallyservesasthestandardrulerforthelarge-scale etal.2005),followedbyenlargedmeasurementsoftheWig-
measurements of our Universe. When combined with other gleZ survey (Blake et al. 2011a) and the Six Degree Field
cosmologicalmeasurements,suchastheSNIa,gravitational Galaxy Survey (6dFGRS Beutler et al. 2011), the measure-
lensing,CosmicMicrowaveBackground(CMB)etc,itallows ments of BAO finally reached the milestone of 1 percent
to explore the expansion history of the whole Universe and precisioninSDSS-IIIBaryonOscillationSpectroscopicSur-
vey (BOSS) Data Release Eleven (DR11 Anderson et al.
2014b),whichwasconsistentwiththeresultsofrecentDR12
(Alam et al. 2015) of SDSS-III BOSS. In the final data re-
(cid:63) E-mail:[email protected] lease (Alam et al. 2016) of SDSS-III BOSS, the measure-
(cid:13)c 2017RAS
2 Shao-Jiang Wang, Qi Guo & Rong-Gen Cai
ments of BAO were carried out using a combined sample shiftdistributioninsubsection3.2.Themethodvalidationis
consisting of the CMASS, LOWZ, LOWZ2 and LOWZ3. giveninsubsection3.3usingQPMmockcatalogues(White
Anisotropy features have been found in BAO signals et al. 2014). In section 4 we apply the smooth method to
along radial and transverse directions, which arise from the DR12 data catalogues and compare with results using
two main effects as follows. One is from the late-time wiggly method. The section 5 is the conclusions.
non-linear evolutions of galaxy clustering. Eisenstein et al. Thefiducialcosmologyweusedinthepaperisthesame
(2007b); Padmanabhan et al. (2012) proposed to recon- asthoseusedintheSDSS-IIIBOSSDR12(e.g.Cuestaetal.
struct the linear power spectrum to improve the BAO sig- 2016): h = 0.70, Ωm = 0.29, ΩΛ = 0.71, Ωbh2 = 0.02247,
nals. The other is from both the redshift-space distortions ns = 0.97, σ8 = 0.80, T0 = 2.7255K. Note that we apply
(RSD) (Kaiser 1987) and the mismatched fiducial cosmol- our method to the pre-reconstruction data and thus only
ogy,namelytheAlcock-Paczynskieffect(Alcock&Paczyn- calculate the monopole of the clustering. The estimate of
ski 1979). The degeneracy between the Hubble parameter multi-poles requires the reconstructed data which will be
H(z) and the angular diameter distance D (z) can be bro- presented in future works.
A
ken by projecting galaxy clustering correlation along ra-
dial/transversedirections(Okumuraetal.2008;Gaztanaga
et al. 2009; Blake et al. 2011b; Chuang & Wang 2012),
ormonopole/quadrupolemoments(Padmanabhan&White
2008; Taruya et al. 2011; Chuang & Wang 2013; Xu et al.
2 DATA AND METHOD
2013) or the newly proposed wedge method (Kazin et al.
2012,2013).Moredetaileddiscussionsonthesystematicsof In this section, we will first describe the data and mock
BAO can be found in Vargas-Magan˜a et al. (2016). catalogues used in our study and then review the essential
The two point correlation function (2PCF) of galaxies formalism to extract the signals of BAO from the observed
isoneofthemostpopularmethodtoestimatetheBAOsig- galaxies positions.
nals. It measures the pairwise excesses of the data galaxies
in comparison with otherwise randomly distributed galax-
ies. The redshift distribution N(z) for the random galaxy
catalogues is supposed to be smooth, given the relatively
2.1 Data and Mocks
slow evolution of the tracer galaxies. However, if the vol-
umeissmall,thestructuresalongline-of-sightdirectionusu- We use the public data catalogues from SDSS-III BOSS
ally manifest themselves as violent fluctuations in N(z). If DR12 (Alam et al. 2015), which consist of LOWZ and
oneassumestheredshiftdistributionsofrandomsamplesto CMASSsamplesthatoccupy3.7Gpc3and10.8Gpc3,respec-
be the same as the data samples, it would bias the result- tively. The LOWZ sample has totally 361,762 galaxies cov-
ingclusteringmeasurements.Intheliteratures,manyworks ering the redshift range 0.15 < z < 0.43, of which 248,237
adoptedasmoothedfittingfunctiontodescribetheunderly- galaxiesarefromtheNorthGalacticCapand113,525galax-
ingN(z).ShaunCole(Cole2011)developedanewalgorithm ies are from the South Galactic Cap. The CMASS sam-
to generate the random catalogue from the data catalogue ple has total 777202 galaxies covering the redshift range
consistofasimpleflux-limitedsample.Itautomaticallypro- 0.43 < z < 0.70, of which 568776 galaxies are from the
videsthedistributionofanunbiasedN(z)withoutassuming North Galactic Cap and 208,426 galaxies are from South
a prior fitting formula. For large galaxy survey, the volume Galactic Cap. The total number of data galaxies we used
is usually big enough so that the cosmic variance could be for our analysis is 1,138,964.
minor.Inthiscase,redshiftsofrandomgalaxiesareusually The random galaxy catalogues corresponding to the
generated by randomly drawing from the measured galaxy DR12 data galaxy catalogues are generated by the MK-
redshifts with some weights (e.g. see Reid et al. 2016, sec- SAMPLEcodeasdescriedinReidetal.(2016)toreproduce
tion5.2).Whetherarandomcataloguegeneratedinthisway thegeometry,redshiftdistribution,andcompletenessofthe
couldinheritfluctuationsoftheclusteringsignalsandleadto survey.Intheserandomsamples,theredshiftdistributionis
aweakenedBAOsignaldependsonthevolumeofthesurvey. thesameasthatintherealdata.InSec.3wewilldiscusstwo
Ross et al. (2012) demonstrated the systematic difference differentkindofredshiftassignmentsofrandomcatalogues.
ofsphericallyaveragedcorrelationfunctionscalculatedwith SinceweonlyhaveoneUniversetoobserve,simulations
this redshift assignment scheme and that with“spline”ap- oftheobservedgalaxyclusteringarecrucialfordetermining
proachesarenegligible.However,theirtypicalsplinemethod the errors of the measurements. The mock catalogues were
adopted∆z=0.01toconstructtherandomcataloguecorre- generatedforthispurposetocalculatethecovariancematrix
sponding to 80∼90Mpc, which is smaller than the typical forthemeasured2PCF.TheQPMmockcatalogues(White
scale of BAO, suggesting the random catalogue generated et al. 2014) are generated using rapid, low-resolution par-
using this spline approach could also inherit the BAO sig- ticle mesh simulations, which mimic the observed data on
nals along the redshift direction. both angular selection function and redshift distribution as
In this paper, we propose to use a new smoothed red- wellasthenoiselevel.Thefiducialcosmologyisthesameas
shift distribution function for the random galaxy samples. we used in this paper. We use 1000 QPM mock catalogues
We describe the data and mock catalogues in subsection for each sample (LOWZ or CMASS). The mock catalogues
2.1,reviewthecorrelationfunctioninsubsection2.2andthe canbeusednotonlytoestimatetheerrors,butalsototest
fitting template in subsection 2.3. In section 3 we describe the reliability of certain methods. We will test our method
how we generate the random catalogues with a wiggly red- of generating random sample with the smooth redshift dis-
shiftdistributioninsubsection3.1andwithasmoothedred- tribution with the QPM mock catalogues in subsection 3.3.
(cid:13)c 2017RAS,MNRAS000,1–9
Random Samples : DR12 data and QPM mocks 3
2.2 Correlation Function and 1/(1 + k2µ2Σ2)2 for the Finger-of-God (FoG) effect
s
(Park et al. 1994). β = f/b where f ≈ Ω (z)0.55 is the
The two-point correlation function (2PCF) estimates the m
growth rate and b is the galaxy bias with respect to dark
galaxyclusteringbycountingtheexcessofdata-datagalaxy
matter. The de-wiggled power spectrum (Eisenstein et al.
pairsrelativetothoseofrandom-randomgalaxypairs.Here
2007a; Xu et al. 2013; Anderson et al. 2013, 2014a; Vargas-
we use the Landy and Szalay estimator (Landy & Szalay
Magan˜aetal.2015)modelsthedegradationtheBAOfeature
1993),
due to non-linear structure growth,
DD(r)−2DR(r)+RR(r)
ξ(r)= , (1)
RR(r) Pdewiggle(k,µ)=[Plinear(k)−Pnowiggle(k)]
(cid:34) (cid:35)
where DD(r), DR(r) and RR(r) are the normalized num- ×exp −k2µ2Σ2(cid:107)+k2(1−µ2)Σ2⊥ +P (k). (9)
ber count of data-data, data-random and random-random 2 nowiggle
galaxypairs,respectively.Thisestimatorcouldminimizethe
variancetothePoissonlevelandaccountforthesurveyge-
where both the linear theory power spectrum P (k)
linear
ometry.
(Lewis et al. 2000) and no-wiggle power spectrum
Each galaxy carries several weights to correct for pos-
P (k) (Eisenstein & Hu 1998) are well approximated
nowiggle
sible observational effects, therefore the number counts for
in details in Eisenstein & Hu (1998).
galaxy pairs should be weighted as following,
We only focus on the monopole of 2PCF with the pre-
(cid:88) reconstructedsamples.Theanisotropicfittingofhighermo-
GG(r )= w(G )w(G )Θ (d ), G=D,R; (2)
α i j rα ij ment of 2PCF with reconstructed samples will be reserved
Gi,Gj
for future works. In this study, we set the streaming scale
Θrα(dij)=(cid:26) 01,, roαth(cid:54)erwdiijse<, rα+∆r; (3) Σtrsan=sv4ehrs−e1cMompcp,otnheenrtaΣdial=co6mhp−o1nMenptcΣo(cid:107)ft=he10stha−n1dMarpdcnaonnd-
⊥
wheredij istheseparationbetweenpairwisegalaxiesGiand linearGaussiandampingscaleΣ2non−linear =(Σ2(cid:107)+Σ2⊥)/2as
G , and G stands for galaxy i,j. The specific choices of inferredfromVargas-Magan˜aetal.(2015).Thespatialsepa-
j i,j
the weights for the data galaxy w(D) and random galaxy rationsofinterestarechosenfrom4h−1Mpcto204h−1Mpc
w(R) will be presented in section 3. withbinningsizeof8h−1Mpc.Thefittingrangeisbetween
30 ∼ 180h−1Mpc also with binning size of 8h−1Mpc as in
the SDSS-III BOSS DR12 (Cuesta et al. 2016). Therefore
2.3 Fitting Template the total number of points to be fitted is 19 and the fitting
parameters to be used are B ,α,β,A ,A ,A , leaving 13
The fitting model (Xu et al. 2013; Anderson et al. 2013, 0 0 1 2
degrees of freedom. The initial values and Gaussian priors
2014a)forthemonopolemomentof2PCFisparameterized
[0±0.4,1±0.1,0.4±0.2] are adopted for [logB2,α,β].
as 0
The χ2 goodness-of-fit indicator is defined by
A A
ξ (r)=B2ξ (αr)+A + 1 + 2, (4)
model 0 theory 0 r r2 χ2 =(cid:88)[ξ (r )−ξ (r )]C−1[ξ (r )−ξ (r )],
model i data i ij model j data j
where the first nuisance parameter B is a normalization
0 i,j
factor, and the other three nuisance parameters A0,1,2 are (10)
introduced to account for any broad-band derivations from where the covariance matrix is given by
thetheoretical2PCFξ (r).Thedilationparameterαis
theory
defined by C = 1 N(cid:88)mock[ξ (r )−ξ¯ (r )][ξ (r )−ξ¯ (r )],
D (z)/r ij N −1 n i mock i n j mock j
α= V d . (5) mock n=1
(DV(z)/rd)fiducial (11)
Here ξ¯ (r ) is the mean value averaging over the corre-
TheBAOpeakpositionshiftstowardssmaller(larger)scale mock i
sponding2PCFfromallQPMmockcataloguesatseparation
if α>1 (α<1) with respect to the fiducial cosmology.
bin r . To have a unbiased estimation of the inverse covari-
The l-th moment of theoretical 2PCF ξ(l) (r) is the i
theory ance matrix (Hartlap et al. 2006), one has to multiply the
Fourier transformation
covariance matrix (11) by
(cid:90) k2dk
ξ(l) (r)=il j(kr)P(l) (k) (6)
theory 2π2 l nonlinear C → Nmock−1 C , (12)
ij N −N −2 ij
of the l-th multipole of non-linear power spectrum mock data
P(l) (k),
nonlinear wherethenumberofmocksisNmock =1000andthesizeof
Pn(ol)nlinear(k)= 2l2+1(cid:90) 1 dµLl(µ)Pnonlinear(k,µ), (7) dotahtaervfievcteoprairsaNmdeattear=sfo1r9.thOenfiexceadnvaallsuoemoafrαgiannadlizoebotavienrtthhee
−1 probability density function (PDF) as
The anisotropic non-linear power spectrum P (k,µ)
nonlinear
(Fisher et al. 1994) is modeled as (cid:18) χ2(α )(cid:19)(cid:30)(cid:90) (cid:18) χ2(α )(cid:19)
p(α )=exp − i dα exp − i , (13)
(1+βµ2)2 i 2 i 2
P (k,µ)= P (k,µ), (8)
nonlinear (1+k2µ2Σ2)2 dewiggle
s basedonwhichwecancalculatethemean(cid:104)α(cid:105)andthevari-
where(1+βµ2)2accountsfortheKaisereffect(Kaiser1987) ance σ2 =(cid:104)α2(cid:105)−(cid:104)α(cid:105)2.
α
(cid:13)c 2017RAS,MNRAS000,1–9
4 Shao-Jiang Wang, Qi Guo & Rong-Gen Cai
Table 1.GalaxysampleandrandomsamplewithwigglymethodandsmoothmethodbothfortheDR12dataandfortheQPMmocks.
Ng(z)istheobservedredshiftdistribution,Nr isredshiftdistributiongeneratedwithwigglemethod.Ng,i istheredshiftdistributionof
the i-th mock sample, Ns is the observed redshift distribution weighted by wsys, Nw is the observed redshift distribution weighted by
wsys andwFKP andNw,i istheredshiftdistributionofthei-thmocksampleweightedbywsys andwFKP.Herewsys isthesystematic
weightaccountingfortheobservationalsystematiceffects,andwFKP istheFeldman-Kaiser-PeacockweighttoreducethePoissonnoise.
WigglyMethod SmoothMethod
DR12Data QPMMock DR12Data QPMMock
redshift weightfor redshift weightfor redshift weightfor redshift weightfor
population pair-count population pair-count population pair-count population pair-count
galaxy
sample Ng(z) wsys·wFKP {Ng,i(z)}1i=0010 wsys·wFKP Ng(z) wsys·wFKP {Ng,i(z)}1i=0010 wsys·wFKP
rsaanmdpolme NNsr((zz)) ≈50 wFKP NNsr,i((zz)) ≈50 wFKP Nfitwti(nzg) 1 (cid:8) Nfiwtt,iin(zg) (cid:9)1i=0010 1
3 RANDOM CATALOGUES The theoretical Poisson noises can also be corrected
during pair-counting by assigning each galaxy of pairs
In this section, we describe the wiggly scheme and smooth
the Feldman-Kaiser-Peacock (FKP) weight (Feldman et al.
schemeofredshiftdistributionsusedtogeneratetherandom
1994) given by
galaxy catalogues, which are summarised in Table 1.
1
w = , (18)
FKP,i 1+n(z )P
i 0
3.1 Wiggly Method
where n(z ) is the measured number density at redshift
i
Therecentanalysis(e.g.Reidetal.2016)ofgalaxycluster- z with linear interpolation over bins ∆ = 0.005 and
i
ingutilisestheobservedredshiftdistributionofdatagalaxy P = 20000h−3Mpc3 is the observed power spectrum at
0
catalogues to generate the random galaxy catalogues. This k ≈0.15hMpc−1. The wiggly method assigns total weights
can be mimicked using the following procedure. With red- to each galaxy of pairs in different way depending on its
shiftbinsize∆z=0.001,theobserveddatagalaxiesredshift type,
distribution for each redshift bin z ,
α (cid:88)
N (z )=(cid:88)Θ (z ), GG(rα)= w(Gi)w(Gj)Θrα(dij);
g α zα Di Gi,Gj
Di (14) w(D )=w ·w ; (19)
(cid:26) 1, z (cid:54)z <z +∆z; i sys,i FKP,i
Θzα(zDi)= 0, oαtherwDisie, α w(Ri)=wFKP,i,
where pairwise galaxies GG can be DD, RR and DR type.
is a wiggly curve along redshift direction, of which the fluc-
tuations could contain both BAO signals and spurious fluc- ThewigglymethodalsoappliestoeachQPMmockcat-
tuationsthatcausedbyobservationalsystematiceffectsand alogue labeled by i from 1 to 1000, with their mock ran-
Poisson noises. dom catalogue having the same redshift distribution (17)
Theobservationalsystematiceffectscanbecorrectedby as the systematically weighted redshift distribution of the
assigning each data galaxy with a systematic weight (Ross mock galaxy catalogue,
et al. 2016) given by
N (z)∼N (z). (20)
r,i s,i
w (D )=w w (w +w −1), (15)
sys i star,i see,i cp,i noz,i TheweightsassignmentsforQPMmockgalaxy/randomcat-
where w w are the total angular systematic weights aloguesduringpair-countingarethesame(19)aswedofor
star,i see,i
which contribute little to those wiggles of redshift distribu- the DR12 data galaxy/random catalogues.
tion N (z). The weights w that correct for close pairs
g cp,i
(fibrecollisions)andtheweightsw thatcorrectforred-
noz,i
3.2 Smooth Method
shift failures have little impact on the measured clustering.
ThereforetheredshiftdistributionNs(z)correctedbythese Besides systematic effects and Poisson noises, the observed
systematic weights, reshift distribution also could contains fluctuations caused
(cid:88) by real structures. A true random catalogue should remove
N (z )= w (D )Θ (z ), (16)
s α sys i zα Di allthespuriousfluctuationsandonlyretainasmoothshape
Di ofredshiftdistribution.Thismotivateustoproposethefol-
The wiggly method thus generates random catalogues with lowing smooth method.
redshift distribution When assigning redshifts to the random catalogues, all
weights should be taken into account. The observed red-
N (z)∼N (z). (17)
r s
shift distribution of data galaxy catalogues, after weighted
Note that the total number of random galaxies is usually by both systematic weights and FKP weights,
largerthanthetotalnumberofdatagalaxies,N (z)=50×
r (cid:88)
Ns(z), for example in SDSS-III BOSS DR12 (Reid et al. Nw(zα)= wsys(Di)wFKP(Di)Θzα(zDi), (21)
2016). Di
(cid:13)c 2017RAS,MNRAS000,1–9
Random Samples : DR12 data and QPM mocks 5
givesrisetoalesswigglyredshiftdistributionN (z),which 0.9736±0.0163,0.9685±0.0169respectivelyforCMASSsam-
w
can be fitted with some smooth functions N (z). Here we ple.Thevolume-averagedistanceD (z)rfid/r are1224±46,
f V d d
adopt the Moffat function, 1227±39, 1238±42 respectively for LOWZ sample, and
a 1975±36, 1957±33, 1946±34 respectively for CMASS
N (z)= 0 +a +a z. (22)
f (cid:18)(cid:16) (cid:17)2 (cid:19)a3 4 5 sample. It shows that both the dilation parameter and the
z−a1 +1 volume-average distance is close to the true one with the
a2
smoothmethod.The1σerrorsarealsoreducedwhenadopt-
Wefindthisfittingfunctionworksbetterthantheoneusu- ing the smooth method.
ally adopted in the literature (Ross et al. 2012; Reid et al. Wefurtherapplythisproceduretothe1000mocksam-
2016).Thereforethesmoothmethodgeneratesrandomcat- ples to investigate whether our smooth method can also
alogues with redshift distribution, boost the BAO signals and reduce the error of dilation pa-
rameter in general.
N (z)∼N (z), (23)
r f Fitting results for the 1000 mocks are shown in Fig.3
and assigns weight for each galaxy of galaxy pairs by withthincurves.Averagedfitsareshownwiththickcurves.
As we can see, there is a gentle lift of BAO signals with
(cid:88)
GG(rα)= w(Gi)w(Gj)Θrα(dij); the smooth method compared to the wiggly method. As in
Gi,Gj Cuesta et al. (2016) we can use this 1000 mock samples to
(24)
w(D )=w ·w ; study the distribution of the dilation parameter in order to
i sys,i FKP,i
quantifythedifferenceofthesetwomethods.Wefoundthat,
w(R )=1
i
withwigglymethod,thedilationparameterαandits1σer-
during pair-counting. rorσ andthestandardderivationS are1.00134,0.03430,
α α
ThesamestrategiesalsoapplytoeachQPMmockcat- 0.03215forLOWZsampleand1.00055,0.02081,0.01980for
alogue, where its random sample is generated with redshift CMASS sample respectively. With smooth method, the di-
distribution given by lation parameter α and its 1σ error σ and the standard
α
derivationS are1.00352,0.03301,0.03040forLOWZsam-
N ∼N (z), (25) α
r,i f,i pleand1.00052,0.01949,0.01840forCMASSsamplerespec-
TheweightsassignmentsforQPMmockgalaxy/randomcat- tively. It is obvious that, with our smooth method, the 1σ
alogues for pair-counting are the same (24) as what we do error σα and the standard derivation Sα are reduced both
for the DR12 data galaxy/random catalogues. for the LOWZ and the CMASS mock samples, though the
As a comparison, we present various redshift distribu- magnitude is only of order O(0.001), similar to other sys-
tions in Fig.1 for SDSS-III BOSS DR12 data catalogues. tematic effects (Vargas-Magan˜a et al. 2016).
As we can see, the random catalogues generated by wiggly
method in line with the redshift distribution from the sys-
tematicallyweightedredshiftdistribution,whilethesmooth
4 RESULTS
methodeliminatesthewigglesthatarecausedbystructures.
Inthissection,weapplyoursmoothmethodtotheSDSS-III
BOSS DR12 data catalogues for both LOWZ and CMASS
3.3 Tests on Mock Samples samples and compare the results to those with wiggly
method. The results are presented in Fig.4 and Tab.2.
Before applying to the DR12 data catalogues, we test our
In the top panels of Fig.4, the 2PCF are shown for
smooth method against the wiggly method on QPM mock
LOWZsampleandCMASSsamplewithwigglymethodand
catalogues. We first select one particular mock which has
smooth method. Red curves are the best-fitting results and
a rather large scatter in N (z) and can thus reflects more
g
error bars are obtained from the square root of the diag-
clearly the difference caused by different random samples.
onal elements of the covariance matrix, which is obtained
Threerandomsamplesaregeneratedwiththewigglyreshift
from the QPM mock catalogues. As we can see, there is
distribution N (z), the smooth redshift distribution N (z)
s f
a gentle lift of BAO signals with our smooth method with
and an underlying true redshift distribution respectively.
respect to the wiggly method. This exactly meets our ex-
Here the true redshift distribution is assumed to be the av-
pectation for the smooth method, where a more randomly
eragedvalueofsystematicallyweightedredshiftdistribution
populatedcataloguealongredshiftdirectionnecessarilywipe
over 1000 QPM mock galaxy samples,
outresidualfluctuationsofclusteringsignalsinheritedfrom
1 N(cid:88)mock the measured redshift distribution. Therefore a more pro-
N (z)= N (z). (26)
s N s,i nounced BAO signal is obtained with smooth method.
mock
i=1 In the bottom panels of Fig.4, we present the like-
Theweightassignmentduringpair-countingfortherandom lihood surfaces for a grid of fixed dilation parameter α
cataloguegeneratedby(26)isassameasthewigglymethod fromisotropicallyfittingthe 2PCFfor LOWZ sample (left)
(19). As shown in the bottom panels of Fig.2, the measure- and CMASS sample (right) with wiggly method (red) and
ments of the 2PCF at BAO scale are closer to the true one smooth method (blue) using fitting templates with (solid)
with our smooth method than the results with the wiggly andwithout(dashed)BAOfeature.Eachlikelihoodsurface
method.Thedilationparametersalongwiththeir1σ errors hasbeensubtractedwiththeirminimumχ2 values.Thedif-
given by wiggly method, smooth method and true redshift ference in χ2 between the template with and without BAO
distribution are 0.9906±0.0371, 0.9930±0.0314, 1.0021± feature reflects the significance of detection of BAO. The
0.0339respectivelyforLOWZsample,and0.9830±0.0178, χ2 minimum is slightly narrowed down with our smooth
(cid:13)c 2017RAS,MNRAS000,1–9
6 Shao-Jiang Wang, Qi Guo & Rong-Gen Cai
DR12 Data LOWZ DR12 Data CMASS
22550000 5000
22000000 4000
11550000 3000
) )
z z
( (
N N
11000000 2000
550000 1000
00 0
00..1155 00..2200 00..2255 00..3300 00..3355 00..4400 00..4455 0.45 0.50 0.55 0.60 0.65 0.70
z z
Figure 1.RedshiftdistributionsoftheDR12datacatalogsforLOWZsample(left)andCMASSsample(right).Differentsamplesare
presented with different colors as indicated in the each panel. See text and Tab.1 for the definition of each sample. It shows that the
smoothmethodeliminatesthosewigglespresentedinNw(z).Allredshiftdistributionshavebeenscaledproperlyforclearness.
In Tab.2, we briefly summarize the fitting results of
Table2.ResultsfortheDR12datacatalogues.Intheupperpart
ofthetable,thebestfitresultsofdilationparameterαareshown 2PCF for LOWZ sample and CMASS sample. In the upper
along with their 1σ errors σα, which can be used to constrain part of the Tab.2, the best fit results of dilation parameter
the volume-average distance to the effective redshifts of LOWZ α are shown along with their 1σ errors σ , which can be
α
samplez=0.32andCMASSsamplez=0.57.Inthelowerpart used to constrain the volume-average distance to the effec-
of the table, the mean value of dilation parameter α are shown tiveredshiftsofLOWZsamplez=0.32andCMASSsample
along with their 1σ errors σα = (cid:112)(cid:104)α2(cid:105)−(cid:104)α(cid:105)2. The detection z =0.57. Although the shift of dilation parameter is minor
significanceiscalculatedfromthesquarerootofdifferenceofχ2
withsmoothmethodandwithwigglymethod,bothLOWZ
fromthede-wiggledandno-wiggledtemplateattheminimumof
and CMASS samples have exhibited an uniform reduction
χ2 surface. The upshot of these results is that there is a gentle
ontheerror ofdilationparameterwithoursmoothmethod
butuniformimprovementontheerrorofdilationparameterand
over the wiggly method. In the lower part of the Tab.2, we
thedetectionsignificancefortheDR12datacatalogues.
present the fitting results from probability density function
2PCFξ(r) α±σα DV(z)rrdfdid χ2/dof (vPalDueF)ooffdfiolratLioOnWpZarsaammeptleeraαndaCreMsAhoSwSnsaamlopnlge.wTihthemtheeainr
DR12LOWZ 1σerrorsσ =(cid:112)(cid:104)α2(cid:105)−(cid:104)α(cid:105)2 andthedetectionsignificance.
α
wigglymethod 1.0175±0.0343 1257±42 6.5/13 ThesignificanceofdetectionofBAOhasbeenimprovedby
0.29σ for LOWZ sample and 0.15σ for CMASS sample. Al-
smoothmethod 1.0153±0.0322 1254±40 8.3/13
thoughtheboostedsignalsandreducederrorarenecessarily
DR12CMASS small, it would be appreciated for the future improvements
on the measurements of galaxy clustering given the high
wigglymethod 1.0148±0.0178 2039±36 10.1/13
precise of the measurements of cosmological parameters.
smoothmethod 1.0077±0.0174 2025±35 10.5/13
(cid:113)
PDFp(α) (cid:104)α(cid:105) σα ∆χ2min
DR12LOWZ 5 CONCLUSIONS
wigglymethod 1.0199 0.0585 2.80σ
The measurements of galaxy clustering on BAO scale have
smoothmethod 1.0158 0.0465 3.09σ reachedunprecedentedprecisionsincetheDR11ofBOSSof
SDSS-III. Further improvements require more careful un-
DR12CMASS
derstanding of the potential errors. The 2PCF of galaxy
wigglymethod 1.0154 0.0173 5.25σ clustering measures the number excesses of the data-data
smoothmethod 1.0087 0.0168 5.40σ galaxy pairs with respect to the random-random galaxy
pairs. Therefore a random sample with proper redshift dis-
tributions is necessary to recover the BAO signals. In this
method compared to the wiggly method, which indicates paper, we propose to use a smooth function which fits the
a minor improvement on the constraint of dilation parame- observed galaxy redshift distribution to generate the ran-
ter. The upshot of the bottom panels of Fig.4 is that both dom galaxy catalogue. It has the advantage to remove the
the significance of detection of BAO and the constraint on BAOsignalsalongtheredshiftdirectioncomparedwiththe
the dilation parameter have been improved gently with our wigglymethodthatusuallyadoptedintheliteratures,which
smooth method over the wiggly method. assignes each random galaxy a redshift by randomly draw-
(cid:13)c 2017RAS,MNRAS000,1–9
Random Samples : DR12 data and QPM mocks 7
QPM Mock LOWZ QPM Mock CMASS
22550000 6000
5000
22000000
4000
11550000
) )
z z 3000
( (
N N
11000000
2000
550000
1000
00 0
00..1155 00..2200 00..2255 00..3300 00..3355 00..4400 0.45 0.50 0.55 0.60 0.65 0.70
z z
QPM Mock LOWZ QPM Mock CMASS
150 150
100 100
2 ] 2 ]
) )
h h
c/ 50 c/ 50
p p
M M
( (
) [ 0 ) [ 0
r r
( (
ξ ξ
2 2
r −50 r −50
−100 −100
0 50 100 150 200 0 50 100 150 200
comoving r [ Mpc/h ] comoving r [ Mpc/h ]
Figure 2. Top panels: redshift distribution for one particular QPM mock catalogue (the i-th one) for both LOWZ (left column) and
CMASS (right column) samples. Different colors are for different sample definitions as denoted in the right corner of each panel. The
fluctuation of Nw,i is rather big to illustrate the effect. Bottom panels: 2PCFs calculated with random samples generated with wiggly
method, smooth method and the true redshift distribution are shown with red, blue and black curves, respectively. Results with the
smoothmethodareclosertothetrueonecomparedtothewigglymethod.
ingavaluefromtheobservedredshiftdistribution,andcan ACKNOWLEDGEMENTS
thusboostthemeasuredBAOsignals.Usingthemockdata,
We want to thank Antonio J. Cuesta and Benjamin Alan
we demonstrate that the smooth method is capable of im-
WeaverofSDSS-IIIBOSSDR12forthehelponBAOfitting.
provingBAOsignalsandreducingtheerrorscomparedwith
WealsowanttothankYutingWang,Gong-BoZhaoandBin
thewigglymethod.WhenapplyingtotheSDSSDR12data,
Huforhelpfuldiscussions.SJWwanttothankShuangpeng
wefindwiththesmoothmethod,wecanimprovethesignifi-
Sun,ShihongLiao,JunyiJia,ZhenJiang,ChunxiangWang
canceofBAOsignalsby0.29σand0.15σfortheLOWZand
for the help on IDL coding and Wei-Ming Dai, Li-Wei Ji
CMASS samples, respectively. The rather small improve-
for the help on Fortran coding during this work. SJW also
ments is expected, because both LOWZ and CMASS sam-
wanttothankthehospitalityofMisaoSasakiwhenthework
ples cover rather large volume and the sample size is large,
wasfinishedduringthevisitattheCenterforGravitational
which means that the observed redshift distribution suffers
PhysicsofYukawaInstituteforTheoreticalPhysicsofKyoto
littlefromcosmicvariance.Forstudieswhichcontainsmaller
University. We acknowledge the use of the cluster PANGU
samples, for example, the QSO sample, a smooth method
and NOVA of NAOC. Qi Guo is supported by NSFC grant
couldbemorehelpfulinimprovingtheresults.Moreover,as
(No. 11573033, 11622325), the Strategic Priority Research
discussedinRossetal.(2012)althoughtheeffectofredshift
Program “The Emergence of Cosmological Structure” of
distribution on the monopole of the clustering could be ig-
the Chinese Academy of Sciences (No.XDB09000000) and
nored,itseffectislargeformulti-poles.Wewillexplorethis
the “Recruitment Program of Global Youth Experts” of
effect with post-reconstruction data in future works.
China, the NAOC grant (Y434011V01). Rong-Gen Cai is
supported in part by the Strategic Priority Research Pro-
(cid:13)c 2017RAS,MNRAS000,1–9
8 Shao-Jiang Wang, Qi Guo & Rong-Gen Cai
Averaging fitting of QPM LOWZ Averaging fitting of QPM CMASS
115500 150
110000
100
2 ] 2 ]
h) 5500 h)
c/ c/ 50
p p
M 00 M
( (
) [ ) [ 0
r −−5500 r
( (
ξ ξ
2 2
r −−110000 r −50
−−115500 −100
00 5500 110000 115500 220000 0 50 100 150 200
comoving r [ Mpc/h ] comoving r [ Mpc/h ]
Figure 3. The fitting results of 2PCF of all the 1000 LOWZ (left) and CMASS (right) mock samples with wiggly method (red) and
smoothmethod(blue).Eachthincurverepresentsforonemockresults.Theaveraged2PCFsarepresentedwiththickcurves.The2PCF
atBAOscaleisslightlyliftedwiththesmoothmethodoverthewigglymethodbothforLOWZandCMASSmocksamples.
DR12 Data LOWZ DR12 Data CMASS
115500 150
110000
2 ] 2 ] 100
) )
h h
c/ 5500 c/
p p
M M 50
( (
) [ 00 ) [
r r
( (
ξ ξ
2 2 0
r −−5500 r
−−110000 −50
00 5500 110000 115500 220000 0 50 100 150 200
comoving r [ Mpc/h ] comoving r [ Mpc/h ]
DR12 Data LOWZ DR12 Data CMASS
14
40
12
10 30
χ2 8 χ2
∆ ∆ 20
6
4
10
2
0 0
0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.7 0.8 0.9 1.0 1.1 1.2 1.3
dilation parameter α dilation parameter α
Figure 4.The2PCF(top)andχ2 surface(bottom)forDR12datacataloguesareshownforLOWZsample(left)andCMASSsample
(right) with wiggly method (red) and smooth method (blue). In the top panels, there is a gentle lift of BAO signals with the smooth
methodoverthewigglymethod.Inthebottompanels,itshowsaminorimprovementofdetectionsignificancefortheBAOsignals.
(cid:13)c 2017RAS,MNRAS000,1–9
Random Samples : DR12 data and QPM mocks 9
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No.QYZDJ-SSW-SYS006 and by the National Natural Sci- Park C., Vogeley M. S., Geller M. J., Huchra J. P., 1994,
ence Foundation of China under Grants No.11375247 and Astrophys. J., 431, 569
No.11435006. Reid B., et al., 2016, Mon. Not. Roy. Astron. Soc., 455,
1553
Ross A. J., et al., 2012, Mon. Not. Roy. Astron. Soc., 424,
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Ross A. J., et al., 2016, Mon. Not. Roy. Astron. Soc.
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