HerMES: detection of cosmic magnification of sub-mm galaxies using angular cross-correlation

0.3 MB·English

by  L. Wang

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed26January2011 (MNLATEXstylefilev2.2)
HerMES: detection of cosmic magnification of sub-mm
⋆
galaxies using angular cross-correlation
1
L. Wang,1† A. Cooray,2,3 D. Farrah,1 A. Amblard,2 R. Auld,4 J. Bock,3,5
1
0 D. Brisbin,6 D. Burgarella,7 P. Chanial,8 D.L. Clements,9 S. Eales,4
2
A. Franceschini,10 J. Glenn,11 Y. Gong, 2 M. Griffin,4 S. Heinis,7 E. Ibar,12
n
a R.J. Ivison,12,13 A.M.J. Mortier,9 S.J. Oliver,1 M.J. Page,14 A. Papageorgiou,4
J
C.P. Pearson,15,16 I. P´erez-Fournon,17,18 M. Pohlen,4 J.I. Rawlings,14
5
2 G. Raymond,4 G. Rodighiero,10 I.G. Roseboom,1 M. Rowan-Robinson,9

] Douglas Scott,19 P. Serra,2 N. Seymour,14 A.J. Smith,1 M. Symeonidis,14
O
K.E. Tugwell,14 M. Vaccari,10 J.D. Vieira,3 L. Vigroux20 and G. Wright12
C
h. 1Astronomy Centre, Dept. of Physics& Astronomy, Universityof Sussex, Brighton BN1 9QH, UK
p 2Dept. of Physics & Astronomy, University of California, Irvine, CA 92697, USA
- 3California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
o 4Cardiff School of Physicsand Astronomy, Cardiff University,QueensBuildings, The Parade, Cardiff CF24 3AA, UK
r 5Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
st 6Space Science Building, Cornell University,Ithaca, NY, 14853-6801, USA
a 7Laboratoire d’Astrophysique de Marseille, OAMP, Universit´e Aix-marseille, CNRS, 38 rue Fr´ed´eric Joliot-Curie, 13388 Marseille
[ cedex 13, France

8Laboratoire AIM-Paris-Saclay, CEA/DSM/Irfu - CNRS - Universit´e Paris Diderot, CE-Saclay, pt courrier 131, F-91191
1
Gif-sur-Yvette,France
v
9Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK
6
10Dipartimento di Astronomia, Universita`di Padova, vicolo Osservatorio, 3, 35122 Padova, Italy
9
11Dept. of Astrophysical and Planetary Sciences, CASA 389-UCB, Universityof Colorado, Boulder, CO 80309, USA
7
12UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
4
13Institute forAstronomy, Universityof Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
.
1 14Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK
0 15Space Science & Technology Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK
1 16Institute forSpace Imaging Science, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada
1 17Instituto de Astrof´ısica de Canarias (IAC), E-38200 La Laguna, Tenerife,Spain
: 18Departamento de Astrof´ısica, Universidad de La Laguna (ULL), E-38205 La Laguna, Tenerife,Spain
v 19Department of Physics & Astronomy, Universityof BritishColumbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
Xi 20Institut d’Astrophysique de Paris, UMR 7095, CNRS, UPMC Univ. Paris 06, 98bis boulevard Arago, F-75014 Paris, France
r
a
Accepted .Received;inoriginalform
ABSTRACT
Cosmic magnification is due to the weak gravitational lensing of sources in the dis-
tant Universe by foreground large-scale structure leading to coherent changes in the
observed number density of the background sources. Depending on the slope of the
backgroundsourcenumber counts,cosmic magnificationcausesa correlationbetween
thebackgroundandforegroundgalaxies,whichisunexpectedintheabsenceoflensing
if the two populations are spatially disjoint. Previous attempts using submillimetre
(sub-mm) sources have been hampered by small number statistics. The large num-
ber of sources detected in the Herschel Multi-tiered Extra-galactic Survey (HerMES)
Lockman-SWIRE field enables us to carry out the first robust study of the cross-
correlation between sub-mm sources and sources at lower redshifts. Using ancillary
data we compile two low-redshift samples from SDSS and SWIRE with hzi ∼ 0.2
and 0.4, respectively, and cross-correlate with two sub-mm samples based on flux
density and colour criteria,selecting galaxiespreferentially at z ∼2. We detect cross-
correlationon angular scales between ∼1 and 50 arcmin and find clear evidence that
this is primarily due to cosmic magnification. A small, but non-negligible signal from
intrinsic clustering is likely to be present due to the tails of the redshift distribution
of the sub-mm sources overlapping with those of the foreground samples.
2 L. Wang et al.
Key words: (cosmology:) large-scale structure of Universe – infrared: galaxies –
methods: statistical – submillimetre – cosmology: observations.
counts,togetherwiththelargeredshiftrange,makesub-mm
sourcesan ideal backgroundsample. Sofar therehavebeen
two attempts at measuring the weak lensing-induced cross-
1 INTRODUCTION correlation between foreground optical galaxies and back-
ground sub-mm sources, but with conflicting results. Al-
Large-scale structureatlow redshiftssystematically magni-
maini et al. (2005) measured the cross-correlation between
fies sources at higher redshifts as a result of gravitational
39 SCUBA sources and optical sources at lower redshifts
light deflection in the weak limit. On the one hand, fewer
z 0.5. They claimed evidence for a significant signal
sources will beobserved,becauselensing stretchesthesolid h i ∼
which might be caused by lensing. Conversely, Blake et al.
angleanddilutesthesurfacedensityofsources.Conversely,
(2006)didnotfindevidenceforcross-correlationduetocos-
theeffectivefluxlimitisloweredasaresultofmagnification,
mic magnification using a similar numberof sources.
whichleadstoadeepersurvey.Whetherthereisanincrease
The Herschel Multi-tiered Extra-galactic Survey (Her-
or decrease in the observed number density of sources de-
MES,Oliveretal.2010b)isthelargestprojectbeingunder-
pendsontheshapeofthebackgroundsourcenumbercounts
taken by Herschel (Pilbratt et al. 2010). In this paper, we
– an effect known as the magnification bias (Bartelmann
calculate the angular cross-correlation between foreground
& Schneider 2001; hereafter BS01). At submillimetre (sub-
galaxies selected from SDSS or the Spitzer Wide-area In-
mm) wavelengths the magnification bias is expected to be
frared Extragalactic (SWIRE; Lonsdale 2003, 2004) survey
large and positive, resulting in an increase in the observed
and background sub-mm sources detected by the Spectral
number density of sources compared to the case without
and Photometric Imaging Receiver (SPIRE; Griffin et al.
lensing (e.g. Blain & Longair 1993; Blain et al. 1996; Ne-
2010) instrument on Herschel. This paper is organised as
grello et al. 2007; Lima et al 2010).
follows: In Section 2, we give a brief introduction to mag-
Cosmic magnification also induces an apparent angu-
nification bias and the angular cross-correlation function.
lar cross-correlation between two source populations with
InSection 3,wedescribethevariousdata-setsusedas fore-
disjoint spatial distributions. It can thus be measured
groundandbackgroundsamples.Measurementsofthecross-
by cross-correlating non-overlapping foreground and back-
correlationbetweenforegroundandbackgroundsamplesare
groundsamples. Whencombinedwith numbercounts,such
presented in Section 4. Finally, discussions and conclusions
a cross-correlation can provide constraints on cosmological
aregiveninSection5.Throughoutthepaper,weuseaspa-
parameters(e.g.Ω ,σ )andgalaxybias,akeyingredientin
m 8 tially flat ΛCDM cosmology with Ω = 0.3 and H = 70
m 0
galaxy formation and evolution models (M´enard & Bartel- km s−1 Mpc−1. Magnitudes are in the ABsystem.
mann 2002). As the weak lensing-induced cross-correlation
also probes the dark matter distribution, it provides an in-
dependent cross-check of the cosmic shear measurements,
which depend on the fundamental assumption that galaxy 2 MODELLING THE CROSS-CORRELATION
ellipticities are intrinsically uncorrelated. Most previous in- FUNCTION
vestigations, using foreground galaxies selected in the opti-
In this section we briefly describe the magnification bias
calorinfrared togetherwithbackgroundquasars,havepro-
andhowitmanifestsitselfinthenumberdensityandcross-
duced controversial or inconclusive results (e.g. Seldner &
correlationbetweentwospatiallyseparatedpopulations.We
Peebles 1979; Bartelmann & Schneider 1994; Bartsch et al.
referthereadertoMoessner&Jain(1998),BS01,Cooray&
1997). The best detection to date is presented in Scranton
Sheth(2002)andreferencesthereinforacompleteintroduc-
et al. (2005), where cosmic magnification is detected at an
tion.Supposeabackgroundpopulationhasanintrinsic(i.e.
8σ significancelevel using13million galaxies and 200,000
quasars from theSloan Digital SkySurvey(SDSS)∼. unlensed) number density nu(S,z), where S is flux density
and z is redshift. As a result of lensing, the sky solid angle
The amplitude of the weak lensing-induced cross- isstretchedlocally byafactorofµ(φˆ,z)(φˆdenotesangular
correlationisdeterminedbyseveralfactors:thedarkmatter
position on the sky) and S is magnified by the same factor
powerspectrumandgrowthfunction,theshapeoftheback-
becausesurfacebrightnessispreserved.Thetwocontrasting
groundsourcenumbercountsandthebiasoftheforeground
effects modify the observed (lensed) number density in the
sources.Atsub-mmwavelengths,thepower-lawslopeofthe
following way
cumulative number count is exceptionally steep, >2.5 for
sources in the flux range 0.02 0.5 Jy at 250, 350 and n (S/µ(φˆ,z),z)
− n(S,z)= u . (1)
500 µm (e.g. Patanchon et al. 2009; Oliver et al. 2010a; l µ(φˆ,z)
Glenn et al. 2010; Clements et al. 2010). In Scranton et
al. (2005), the number count slope of the quasar sample is When the lens plane is at a much lower redshift than the
considerably flatter ( 2 for the brightest ones). In addi- source plane, the redshift-dependent magnification can be
tion, sub-mm sources∼detected in deep surveys mainly re- substituted by the magnification µ of a source at infinity.
side in the high-redshift Universe with a median redshift of Assuming the cumulative number count distribution of the
z 2 (Chapman et al. 2003, 2005; Pope et al. 2006; Aretx- background population can be described by a power-law
ag∼a et al. 2007; Amblard et al. 2010). The steep number Nu(S) S−β, we should expect a factor of
∝
Detection of cosmic magnification of sub-mm galaxies 3
Nl(S) =µβ−1 (2)
N (S)
u
changeintheobservednumbercount.Strictlyspeaking,the
number count slope β = β(S) is a function of flux density.
Inthispaper,wemakethesimplifyingassumptionthatβ is
a constant over the flux range we probe. Using the number
countsofresolvedsourcespresentedinOliveretal.(2010a),
we find that in the flux range 0.03 0.5 Jy, β = 2.53
− ±
0.16, 2.99 0.51 and 2.66 0.24 at 250, 350 and 500 µm
± ±
respectively.
Theangularcross-correlation functionbetweenpopula-
tion1atlowerredshiftsandpopulation2athigherredshifts
is defined as
w (θ)= δn (φˆ)δn (φˆ′) , (3)
cross 1 2
h i
whereδn n (φˆ)/n¯ 1 is thenumberdensity fluctuation
i i i
≡ −
andn¯ istheaveragenumberdensityoftheithsample. We
i
can decompose δn into two parts,
i
δn (φˆ)=δnc(φˆ)+δnµ(φˆ). (4)
i i i
The first term δnc is due to intrinsic clustering of galaxies
i
and is aprojection of density fluctuationsalong theline-of- Figure 1. Redshift distribution of the foreground and back-
sight, ground populations normalised so that the peak of each N(z)
is equal to unity. For the foreground sample F1, the N(z) is de-
χH
δnc(φˆ)=b dχW (χ)δ(r(χ)φˆ,a), (5) rivedfromspectroscopicredshifts.FortheforegroundsampleF2,
i iZ0 i wehaveeitherspectroscopicredshiftsorgoodqualityphotomet-
ricredshifts.The N(z) forthe two background samples,B1 and
where χ is the comoving radial distance to the horizon,
H
B2,arederived fromsub-mmcolours usingmodifiedblack-body
r(χ)isthecomovingangulardiameterdistance,W(χ)isthe
spectra.
normalised radial distribution of the sources, a is the scale
factor, δ(r(χ)φˆ,a) is the dark matter density perturbations
and bi is the bias factor assumed to be scale- and time- thelensingofthebackgroundsourcesbyforegroundsources
independent.Thesecondterminequation(4)δnµ isdueto
i χH g
magnification bias, w (θ) = 3b Ω (β 1) W 2dχ
fb 1 m − Z 1 a
0
δnµ = Nl−Nu =µβ−1 1=2(β 1)κ. (6) ∞ k
i Nu − − Z 2πP(χ,k)J0(krθ)dk. (11)
0
In the last step, we have used the weak lensing limit, µ =
1+2κ. The convergence κ is a weighted projection of the The third term hδnµ1(φˆ)δnµ2(φˆ′)i is due to weak lensing by
density field along theline of sight (BS01), large-scale structure in front of both the foreground and
background sources. The last term δnµ(φˆ)δnc(φˆ′) repre-
κ (φˆ)= 3Ω χHdχg (χ)δ(rφˆ,a), (7) sentslarge-scalestructuretracedbythhe1backgro2undisources
i 2 mZ i a lensing the foreground sources which is only present if the
0
two samples have overlapping redshift distributions. The
where g(χ) is theradial weighting function defined as
last two terms are negligible. To derive the expected cross-
g(χ) r(χ) χH r(χ′−χ)W(χ′)dχ′. (8) correlations(wccandwfb)betweenourforegroundandback-
≡ Z r(χ′) ground samples in Section 4, we use the CAMB software
χ
package (Lewis, Challinor & Lasenby 2000), which is based
The angular cross-correlation between the two populations
on CMBFAST (Seljak &Zaldarriaga 1996), togenerate the
is then
non-linearmatterpowerspectrumusingthefittingformulae
w (θ) = δnc(φˆ)δnc(φˆ′) + δnc(φˆ)δnµ(φˆ′) of Smith et al. (2003).
cross h 1 2 i h 1 2 i
+ δnµ(φˆ)δnµ(φˆ′) + δnµ(φˆ)δnc(φˆ′) . (9)
h 1 2 i h 1 2 i
The first term hδnc1(φˆ)δnc2(φˆ′)i is due to the intrinsic clus- 3 DATA-SETS
tering of the two populations tracing the same large-scale
structure, For the first foreground sample, referred to as F1, we se-
χH ∞ k lect 7,761 sources with r < 19.4 from the SDSS DR7 in
wcc(θ)=b1b2Z W1W2dχZ 2πP(χ,k)J0(krθ)dk, (10) Lockman-SWIRE observed by Herschel-SPIRE. The star-
0 0 galaxy separation is done in the same way as in Stoughton
where P(χ,k) is the dark matter power spectrum and et al. (2002). The redshift distribution N(z) of the sample
J (x) = sin(x)/x is the zeroth-order Bessel function. Note F1 is derived from spectroscopic redshifts obtained in the
0
thatW vanishesifthetwopopulationshavedisjointspatial Galaxy and Mass Assembly (GAMA) survey (Baldry et al.
cc
distribution.The second term δnc(φˆ)δnµ(φˆ′) is caused by 2010). The median redshift of F1is 0.2. Thesecond fore-
h 1 2 i ∼
4 L. Wang et al.
samples, while B1 has a small overlap with the foreground,
Table 1.Summaryofforegroundandbackground samples.The
columnsarethesamplename,thenumberofsources,themedian in which case wcc is non-zero.
redshiftandtheselectioncriterion.Forthetwobackgroundsam-
ples,B1andB2,welistthenumberofsourcesintheoverlapping
regionwithF1andF2,respectively.
4 MEASURING THE CROSS-CORRELATION
SIGNAL
Sample Ngal hzi Selectioncriterion
The cross-correlation between populations 1 and 2 is the
F1 7,761 ∼0.2 r<19.4
F2 13,888 ∼0.4 S3.6>100µJy fractionalexcessintheprobabilityrelativetoarandomdis-
tribution (Peebles 1980). We use a modified version of the
B1 2,477/1,886 ∼2.0 S350&35mJy Landy-Szalay estimator (Landy & Szalay 1993) to measure
B2 2,398/1,848 ∼2.5 S350/S250&0.85 theangular cross-correlation function,
D D D R D R +R R
wcross(θ)= 1 2− 1 R2−R 2 1 1 2, (12)
1 2
ground sample F2 is selected from sources detected by the
Spitzer Infrared ArrayCamera (IRAC;Fazio etal. 2004) in where D1D2, D1R2, D2R1 and R1R2 are the normalised
data1-data2,data1-random2,data2-random1andrandom1-
the SWIRE survey. Full details of the data processing and
catalogues can be found in Surace et al. (2005). We select random2paircountsinagivenseparation bin(seeBlakeet
13,888sourceswithS3.6 >100µJyintheregionoverlapping al. 2005 for a disccussion of different estimators of wcross).
with the Herschel-SPIRE observation in Lockman-SWIRE. Fortheforegroundsamples,wegeneraterandomcatalogues
by distributing sources using a uniform distribution. It is
The star-galaxy separation is performed in the same way
as in Waddington et al. (2007). About 17 percent of the more complicated to generate random catalogues for the
background samples. To take into account the noise prop-
sources in F2 have spectroscopic redshifts and 80 percent
∼ erties in the sub-mm maps and the angular resolution of
have good quality photometric redshifts with redshift un-
certainty <0.1 (Oyaizu et al. 2008; Rowan-Robinson et al. SPIRE, we make maps of randomly distributed sources
2008). The median redshift of F2 is z 0.4. which are processed by the SPIRE Photometer Simulator
To construct the background sam∼ples in the 13.2 deg2 (SPS; Sibthorpe et al. 2009) for observational programmes
exactly the same as the real data. The catalogues ex-
Lockman-SWIRE field, we use the single-band SPIRE cat-
tracted from the SPS simulations are then used as random
alogues generated by the SUSSEXtractor source extractor
catalogues. To reduce shot-noise in the data-random and
in HIPE (Smith et al. 2010). The cross-match between the
random-random pair counts, our random catalogues (after
250 and 350 µm catalogue is done by selecting the bright-
applying the bright star mask) contain roughly 10 times
est 250 µm source within 12.6 arcsec of a 350 µm source
moresourcesthantherealcatalogues. Weuse40bootstrap
(FWHM=25.2arcsecat350µm).Thefluxdensityatwhich
realisations of the foreground and background samples to
the integral source counts reach 1 source per 40 beams is
estimate the errors and covariance matrix.
18.7 and 18.4 mJy at 250 and 350 µm, respectively (Oliver
As described in Section 2, we need the bias factors
et al. 2010a). The first background sample, B1, comprises
of the foreground and background samples to calculate
sources brighterthan35 mJyat 350 µm.Intotal, thereare
the expected clustering-induced and lensing-induced cross-
2,477 / 1,886 such sources in the region that overlaps with
correlations. In the past, sub-mm sources have been shown
F1/F2.ThesecondbackgroundsampleB2includessources
withS /S &0.85.Intotal,thereare2,398/1,848such toclusterstrongly(Scottetal.2002,2006;Blainetal.2004;
350 250
Farrah et al. 2006; Blake et al. 2006; Viero et al. 2009).
sources in theoverlappingregion with F1 /F2. About 50%
More recently, the linear bias factor has been measured to
ofthesourcesinB1arefoundinB2aswell.Becausemostof
be 3.2 0.5 for sources with S & 30 mJy and 3.4 0.6
thebackgroundsourcesdonothavespectroscopicredshifts, 350
± ±
for sources with S /S & 0.85 (Cooray et al. 2010). To
we make use of the sub-mm colours and modified black- 350 250
derive the bias factors of the foreground samples, we esti-
bodytemplatestogeneratequalitativeredshiftdistributions
mate the angular auto-correlation function of F1 and F2,
whichareconsistent withtypicalmodelpredictions(e.g.Le
which can be described by a power-law w =Aθ−γ. The
Borgne et al. 2009; Valiante et al. 2009). The majority of auto
thesources with S &35 mJy lie at 1.5.z.3and peak amplitudeofwauto isrelatedtothecorrelationlengthofthe
at z 2, while mo35s0t of the sources with S /S & 0.85 spatial correlation function ξ(r) = (r/r0)−(γ+1) (e.g. Efs-
350 250
lie at∼2.z .3 (Amblard et al. 2010; Cooray et al. 2010). tathiou 1991),
Finally a bright star mask is applied to all samples −2
described above. We follow the procedures in Waddington A=frγ χ1−γ(N(z))2E(z)dz N(z)dz , (13)
0Z (cid:18)Z (cid:19)
et al. (2007) and mask a circle around all K 6 12 point
sourcesinthe2MASScatalogue within aradiusRgivenby where f = √πΓ[(γ 1)/2]/Γ(γ/2), E(z)= (H /c)[Ω (1+
0 m
−
log R(arcsec)=3.1 0.16K. This radius is more conserva- z)3 +Ω ]1/2 and we have assumed constant clustering in
Λ
−
tive compared to the star mask used in the public release comoving units. Finally, we derive the linear bias factor of
of SWIRE catalogues. In Table 1, we list the number of the foreground using the dark matter correlation function
sources, the median redshift and the selection criteria for b = [ξ(r )/ξ (r )]1/2. The linear bias factor of F1 and F2
0 dm 0
the foreground and background samples. Fig. 1 shows the derived in this way is 1.5 and 1.6 respectively.
∼
N(z) for each sample. The N(z) of the background is our The measured angular cross-correlations between the
biggest source of uncertainty.If it is agood approximation, various foreground and background samples are shown in
thenB2isalmostcompletelyseparatedfromtheforeground Fig. 2. A set of logarithmically spaced angular separation
Detection of cosmic magnification of sub-mm galaxies 5
bins are used, ranging from 1 to 50 arcmins. The green numberdensitythroughtheauto-correlationfunctionofthe
dashedlineistheexpectedlen∼sing-inducedcross-correlation background sub-mmsources, w (θ)= δn(φˆ)δn(φˆ′) . Us-
auto
h i
w (θ), the red dashed line is the expected clustering- ingequation(4),wecandecomposew (θ)intothreecom-
fb auto
induced cross-correlation w (θ) and the blue dashed line ponents, δnc(φˆ)δnc(φˆ′) , δnc(φˆ)δnµ(φˆ′) + δnc(φˆ′)δnµ(φˆ)
cc
is the sum of the two. In the left column of Fig. 2, the ex- and δnµh(φˆ)δnµ(φˆ′) , wihhich represent tiheh galaxy-galaxyi,
h i
pected clustering-induced cross-correlation w is non-zero galaxy-lensing and lensing-lensing correlation functions re-
cc
because the tail of the background N(z) overlaps slightly spectively. The lensing-lensing term is given by (Moessner
with that of the foreground N(z). Although w is much & Jain 1998)
cc
smaller thanw ,weshouldbearin mindthatw could be
fb cc χH
underestimated if a higher than expected fraction of SMGs wlensing−lensing(θ) = [3Ω (β 1)]2 (g /a)2dχ
auto m − Z 2
reside at low redshifts z .1. In the right column of Fig. 2, 0
∞
the predicted w vanishes, as B2 does not overlap with F1 k
cc P(χ,k)J (krθ)dk. (15)
orF2.Toassessthesignificanceofthelensing-inducedcross- Z0 2π 0
correlationsignal,giventhecovariancematrixobtainedfrom At zero lag, w(0)lensing−lensing = (δnµ)2 is the variance of
bootstrap realisations, we derivethe Bayes factor auto h i
the number density fluctuatuation due to lensing and thus
K = P(D|Mlensing), (14) thermsfluctuationisδnµ =(w(0)aleuntsoing−lensing)1/2 whichis
P(DM ) at a few percent level.
null
|
where P(DM ) is the probability of the data given
lensing
|
the lensing model and P(DM ) is the probability of the
null
|
data assuming there is no cross-correlation. We find that
6 DISCUSSIONS AND CONCLUSIONS
K = 6.3 for the cross-correlation between F1 and B2 and
K = 132.6 between F2 and B2. On Jeffreys’ scale (Jeffreys The unusually steep number count in the bright sub-
1961), K >3 means that there is substantial evidence that mm regime leads to an enhanced cross-correlation signal
Mlensing is more strongly supported by the data than the that is due to weak gravitational lensing. In this paper,
null hypothesis and K > 100 means that there is decisive we have measured the angular cross-correlations between
evidence that Mlensing is the favoured model compared to sub-mm sources detected by Herschel-SPIRE in Lockman-
the null. Note that there is almost a factor of two increase SWIRE and foreground sources selected in the optical or
inthesourcedensityintheforegroundsampleF2compared near-infrared. We have also derived theoretical expecta-
to F1; increasing the number of tracers of the foreground tions of the weak lensing-induced cross-correlation w and
fb
structureincreases the strength of thelensing signal. the clustering-induced cross-correlation w which are in
cc
good agreement with our measurements. We find clear ev-
idence for a lensing-induced cross-correlation between sub-
5 THE EFFECT OF WEAK LENSING ON THE mm sources at high redshifts and galaxies at low redshifts.
NUMBER COUNT OF SUB-MM SOURCES The redshift distribution of the sub-mm sources is the
biggestsourceofuncertaintyinouranalysisbecausemostof
The effect of lensing on the number count of the sub-mm thesourcesdonothavespectroscopicredshifts.Inprinciple,
sources is expressed in equation (2), under the assumption the clustering-induced cross-correlation w could contam-
cc
that the lens plane is at a much lower redshift than the inate the lensing-induced cross-correlation w if a higher
fb
sourceplane.Thepower-lawslopeoftheintrinsic/unlensed thanexpectedfraction ofsub-mmsourcesresideinthelow-
numbercountN (S)isnotaffectedbecausethelensingmag- redshiftUniverse.Astheamplitudeofw ismainlysensitive
u fb
nificationµisindependentofthefluxdensity.However,the to the mean redshift of the background population rather
overall normalisation of the number count can be modified than the exact shape of the N(z) (M´enard & Bartelmann
byafactor ofµβ−1,whereµ=1+δµ=1+2κin theweak 2002), we have carried out a simple calculation of the ex-
lensing limit. Weak lensing by large-scale structure causes pectedw andw amplitudebyvaryingthemean redshift
fb cc
δµ to follow a Gaussian function with mean magnification z (from 0.3 to 4.0) and the width σ (from 0.2 to 2.5),
z
h i
δµ =0 and its dispersion σ dependenton theredshift of assuming the N(z) of the sub-mm sources can be approx-
µ
h i
the sub-mm population (BS01). Therefore, when averaged imated by a Gaussian function. In all cases, to reproduce
over a statistically representative area, the effect of weak the measured cross-correlation signal, w is at most com-
cc
lensing on thenumbercount should benegligible. parable to w when z 3.5,σ 1.5, z 2.5,σ 1.0
fb z z
h i∼ ∼ h i∼ ∼
The effect of weak lensing on the local numberdensity or z 1.5,σ 0.5. Sothedetection of theweak lensing-
z
h i∼ ∼
ofthesub-mmsourcesalongacertaindirection canbeesti- inducedcross-correlationshouldberobust.Itshouldbepos-
matedfromthemeasuredcross-correlationbetweenthefore- sible to acurately determine N(z) in the future when the
ground and the background populations. In the right panel infrared spectral energy distributions are well understood
of Fig. 2 where the measured signal is expected to be due and/or more spectroscopic redshifts are acquired for sub-
to lensing only, we can see that the probability of finding a mm sources.
background sub-mm source close to a foreground galaxy is Limitationsinourmodellingofthecross-correlationin-
increased by a few percent above random on angular scales clude:usingascale-andtime-independentbiasfactorforthe
between 1 and 50 arcmin. Therefore, thelensing induced galaxy-dark matter power spectrum; assuming a linearised
∼
change in the number density along a certain direction is magnification; and adopting a constant power-law number
expected to beat the level of a few percent. count slope independent of flux. While for this first study
We can also estimate the effect of lensing on the local a simple model is adequate given the large error bars, an
6 L. Wang et al.
Figure 2. The angular cross-correlations between foreground and background populations. The errorbars are the rms scatter derived
from 40 bootstrap realisations of the real data. In each panel, the red dashed line is the predicted cross-correlation due to mutual
clusteringwcc(θ).Thegreendashedlineisthepredictedcross-correlationduetolensingwfb(θ).Thebluedashedlinesshowthesumof
wcc(θ)andwfb(θ).Theblackhorizontal linedenotes thezerolevel.Intherightcolumn,theexpected wcc(θ)=0.
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