The two-and three-point correlation functions of the polarized five-year WMAP sky maps

1.1 MB·English

by  E. Gjerløw

Draftversion January23,2010
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
THE TWO- AND THREE-POINT CORRELATION FUNCTIONS OF THE POLARIZED FIVE-YEAR WMAP
SKY MAPS
E. Gjerløw1,6, H. K. Eriksen2,6,7, A. J. Banday3,8,9, K. M. Go´rski4,10,11 and P. B. Lilje5,6,7
Draft version January 23, 2010
ABSTRACT
Wepresentthetwo-andthree-pointrealspacecorrelationfunctionsofthefive-yearWMAPskymaps
0
andcomparetheobservedfunctionstosimulatedΛCDMconcordancemodelensembles. Inagreement
1
with previously published results, we find that the temperature correlation functions are consistent
0
with expectations. However, the pure polarization correlation functions are acceptable only for the
2
33GHzbandmap; the41,61,and94GHzbandcorrelationfunctions allexhibitsignificantlarge-scale

n excess structures. Further, these excess structures very closely match the correlation functions of
a the two (synchrotron and dust) foreground templates used to correct the WMAP data for galactic
J
contamination, with a cross-correlation statistically significant at the 2σ 3σ confidence level. The
3 correlation is slightly stronger with respect to the thermal dust template−than with the synchrotron
2 template.

Subject headings: cosmic backgroundradiation — cosmology: observations — methods: numerical
]
O
C 1. INTRODUCTION spectrum and cosmological parameter estimation (e.g.,
h. Observations of the cosmic microwave background G´orski 1994; Lewis & Bridle 2002; Hivon et al. 2002;
Hinshaw et al. 2003a; Verde et al. 2003; Eriksen et al.
p (CMB) have been among the most important ingredi-
2004d, and references therein), and analysis of higher-
- ents in the revolution of cosmology that has taken place
o in the last two decades, when cosmology changed from order statistics (e.g., Hinshaw et al. 1994; Kogut et al.
r 1995; Komatsu et al. 2003; Eriksen et al. 2004c, 2005
a data starved to a data rich science. The so far most
t
s influentialobservationshavebeenmadewiththeWilkin- and references therein).
a In this paper, we revisit a well-known example of the
sonMicrowave AnisotropyProbe(WMAP;Bennettetal.
[ latter category, namely, real-space N-point correlation
2003a; Hinshaw et al. 2007, 2009) satellite experiment,
functions. These functions arise naturally in studies of
3 which has measured the microwave sky at five different
non-Gaussianity, since it can be shown that any odd-
v frequencies (23, 33, 41, 61, and 94 GHz) in both tem-
ordered N-point function has an identically vanishing
0 perature and polarization. Its main results are five sky
6 mapswithresolutionsbetween13′and55′,eachcompris- expectation value, while all even-ordered N-point func-
8 ing more than three million pixels in each of the three tions haveanexpectationvalue givenby productsof the
5 Stokes parameters I, Q, and U. corresponding two-point functions. Violation of either
0. The detector technology leading to instruments like of these relations would indicate the presence of a non-
Gaussian component in the field under consideration.
1 WMAP has thus been very important in observing the
Earlier N-point correlation function analyses of
9 CMB. Almost equally important has been the progress
CMB data include studies of the COBE-DMR data
0 in computer technology and algorithms. With the
(Hinshaw et al. 1995; Kogut et al. 1996; Eriksen et al.
: enormously increased number of pixels from new high-
v 2002), the WMAP temperature data (Eriksen et al.
resolution instruments, scientists have had to develop
i 2005) and one single two-point analysis of the first-year
X cleveralgorithmsforeverystepoftherequireddataanal-
WMAP polarization data (Kogut et al. 2003). This pa-
ysis pipeline: map making (e.g., Ashdown et al. 2007;
r per is the first to consider the much more mature five-
a Hinshaw et al. 2003b, and references therein), compo-
year WMAP polarizationdata, and the first to compute
nent separation (e.g., Bennett et al. 2003b; Leach et al.
the three-point correlation function from any CMB po-
2008;Eriksenet al. 2008,andreferences therein), power
larization data. To do this, we adopt and extend the
1Electronicaddress: [email protected] N-pointalgorithms developed by Eriksen et al. (2004b).
2Electronicaddress: [email protected]
3Electronicaddress: [email protected] 2. METHODSANDDEFINITIONS
4Electronicaddress: [email protected] 2.1. N-point correlation functions
5Electronicaddress: [email protected]
6Institute ofTheoretical Astrophysics,UniversityofOslo,P.O. An N-point correlation function C of a stochastic
N
Bo7xC1e0n2t9reBolifndMerant,heNm-0a3t1ic5sOfosrlo,ANpporliwcaatyions, University of Oslo, field X(nˆ) is defined as
P.O.Box1053Blindern,N-0316Oslo,Norway
8Centre d’Etude Spatiale des Rayonnements 9, av du Colonel C (θ ,θ ,...,θ )= X(nˆ )X(nˆ ) X(nˆ ) , (1)
N 1 2 k 1 2 N
Roche,BP4434631028ToulouseCedex4,France ···
9Max-PlanckInstitutefu¨rAstrophysik,Karl-SchwarzschildStr. (cid:28) (cid:29)
1,D-85748,Garching,Germany where nˆ ,...,nˆ spansanN-polygondefinedbykpa-
10JPL, M/S 169/327, 4800 Oak Grove Drive, Pasadena CA ramete{rs,1 θ ,...N,}θ . Inthemostgeneralcaseinwhich
91109,USA { 1 k}
11Warsaw UniversityObservatory, AlejeUjazdowskie4,00-478 noassumptionsaremadeconcerningthestatisticalprop-
Warszawa,Poland erties of the field X, one needs k = 2N parameters to
2
P
α rameters after transformation becomes
3
3
θ
I’ 1 0 0 I
3 ′
Q = 0 cos2α sin2α Q , (2)
α U′ 0 sin2α cos2α! U!
1 −
θ CM where α is the local rotation angle between the two
2 P coordinate systems. Zaldarriaga & Seljak (1997) and
1
Kamionkowskiet al.(1997)givecompletedescriptionsof
the statistics of these fields.
This property has some important consequences when
θ
computing N-point correlation functions: since we as-
1
sume that the CMB field is isotropic and homogeneous,
α
2 the value we obtain for a given element of the correla-
tion function should not depend on the coordinate sys-
P tem. Onecannotthereforeblindly adoptQandUasthe
2 quantities to correlate,since these are in fact coordinate
dependent.
Fig.1.—Centerofmassandrotationanglesofatriangle. Each
pointiisrotatedbyanangleαifromtheglobalcoordinatesystem Forthetwo-pointfunction,thisproblemisconvention-
intothelocalsystemdefinedbythetriangle’scenter ofmass. allysolvedbydefiningalocalcoordinatesysteminwhich
thelocalmeridianpassesthroughthetwopointsofinter-
est (Kamionkowskiet al. 1997). The Stokes parameters
in this new “radial” system are denoted by Qr and Ur,
andmeasurethe polarizationeither orthogonallyorper-
uniquely describe such a polygon12, namely, the individ-
pendicularly (Qr) to the connecting line, or 45◦ rotated
ual positions of each vertex of the polygon. However,
with respect to it (Ur). The two-point function defined
in many applications one assumes that X is isotropic
in terms of Qr and Ur then becomes coordinate system
and homogeneous, and one can therefore average over
independent.
position. In such cases, the number of parameters are
We now generalize this idea to higher-order N-point
reduced by three on a two-dimensional surface, corre-
correlation functions. As with the definition of the N-
sponding to translation and rotation.
point polygon parameters, we also here have some free-
In this paper, we will consider only two- and
dom when choosing the reference points for the coordi-
three-point correlation functions defined on the two-
nate independent quantities. For instance, two natural
dimensional sphere, and we therefore need, respectively,
choices for Qr and Ur are to define them either with re-
one and three parameters to describe our polygons (ie., i i
spect to the edges or with respect to the center of mass
line and triangle). In the two-point case, the only
of the polygon,
natural parameter choice is the angular distance, θ =
nˆ
parocicnotsc(anˆs1e·nˆth2)erbeetiswseoemnethferetewdoompoitnotsc,hwoohsiele. iFnotrhseimthprleice-- nˆCM = |Piinˆii|. (3)
ity, we parameterize the triangle by the lengths of the
Inthis paper we choosethe latter definition. The result-
threeedges,(θ ,θ ,θ ),wheretheedgesareorderedsuch P
1 2 3 ing geometry is illustrated in Figure 1.
thatθ correspondstothelongestofthethreeedges,and
1 Given these new rotationally invariantquantities, X
the remaining are listed according to clockwise traversal I,Qr,Ur , defined with respect to the local center o∈f
of the triangle. { }
mass, the polarized correlation functions are simply de-
fined as
2.2. Polarized correlation functions
Our goal in this paper is to measure the N-point cor- CN(θ1,θ2,...,θk)= X1(nˆ1)X2(nˆ2) XN(nˆN) , (4)
···
relation functions of the polarized WMAP maps. To (cid:28) (cid:29)
do so, we have to generalize the methods described by Notethatinthecaseofthetwo-pointfunction,thereare
Eriksen et al. (2004b), in order to account for the fact six independent polarized correlation functions (II, IQr,
that the CMB polarizationfield is a spin-2 field. Explic- IUr, QrQr, QrUr and UrUr), while in the three-point
itly, the full CMB fluctuation field may be described in functioncase,thereare27independentcomponents(III,
terms of the three Stokes parameters, I(nˆ), Q(nˆ), and IIQr, ..., UrUrQr, UrUrUr).
U(nˆ). Here I is the usual (scalar) temperature fluctua- Algorithmically, we compute these functions from a
tion field, and Q and U are two parameters describing given pixelized sky map, m(nˆ), simply by averaging the
the linear polarization properties of the radiation in di- correspondingproductsoverallavailablepixelmultiplets
rection nˆ. that satisfy the geometrical constraints of the polygon
AccordingtostandardCMBconventions,QandUare under consideration,
defined with respect to the local meridian of the spheri-
calcoordinatesystemofchoice. However,Q andU form 1 Np
Cˆ (θ ,θ ,...,θ )= mX1 mX2 mXN .
aspin-2field,andthismeansthatifoneperformsarota- N 1 2 k Np p1(i) p2(i)··· pN(i)
tional coordinate system transformation, the Stokes pa- Xi
(5)
12 Inthispaper,werestrictourinteresttofieldsdefinedonthe Here i = 1,...,Np runs over the number of available
two-dimensionalsphere. pixel multiplets, and pj(i) is the j’th pixel in the i’th
3
0.5 0.07
0.06
2K) 0.0 2K)0.06 2K)
µ µ µ
wo-point function (---110...055 wo-point function (0000....00002435 wo-point function ( 00..0024
IQ t-2.0 QQ t0.01 UU t 0.00
-2.5 0.00 -0.02
0 60 120 180 0 60 120 180 0 60 120 180
Angular separation (degrees) Angular separation (degrees) Angular separation (degrees)
Fig.2.—Comparisonbetweentheanalyticallycalculatedtwo-pointfunction(black)andthetwo-pointfunctioncomputedfromthemap
withthecorrespondingCl’s(red). Theshownfunctionsare,fromlefttoright,theIQ,QQ,andUUcorrelationfunctions.
pixel multiplet. For full details on how to find the rel- functions through a standard χ2 statistic,
evant pixel multiplets corresponding to a given polygon
efficiently, see Eriksen et al. (2004b). Nbin (C(b) µ(b))2
The correlation functions are binned with a bin size χ2 = − , (6)
σ2
tunedtothepixelsizeofthemap. Explicitly,sinceweuse b=1 b
◦ X
a HEALPix resolution of N = 16, with 3 pixels,
◦ side ∼ wherethesumrunsoverallN-pointconfigurations/bins,
we adopt a bin size of 6 , for a total of 30 bins between
◦ ◦ and the mean µ(b) = C(b) and variance σ2(b) =
0 and 180 .
If all factors in an N-point multiplet are taken from (C(b) µ(b))2 are derived from the simulations.
− (cid:10) (cid:11)
the same map, the resulting function is named an auto- However,before computing the χ2 asdescribedabove,
(cid:10) (cid:11)
correlation; otherwise, it is called a cross-correlation. we Gaussianize the distribution of each bin as follows
Themainadvantageofcross-correlationsisthatthenoise (Eriksen et al. 2005):
is typically uncorrelated between maps, while the signal
(ideally)isstronglycorrelated. Therefore,thenoisecon- R = 1 s e−12t2dt, (7)
trreiabsuotni,onontoe atycpriocsasl-lcyorurseelastaiountoa-vceorrargeelasttioonzseraos. aFoprrtohbies Nsim+1 √2π Z−∞
of systematic errors (e.g., to check whether the assumed whereRistherankofthevalueunderconsideration(i.e.,
noise model is correct), but cross-correlationsfor a final the number of simulations with a lower value than the
cosmological analysis. In this paper, we consider both currently considered), and s is the corresponding Gaus-
types. sianized rank. The reason for performing this transfor-
In Figure 2, we show a comparison of three two-point mation is that the probability distribution for a given
correlation functions computed from the same signal- bin is highly non-Gaussian, and for even-ordered corre-
only simulation, using two different methods: the red lation functions strongly skewed toward high values. A
curvesshowthecorrelationfunctionscomputedfromthe directχ2 evaluationwilltherefore tend to givetoo much
power spectrum of the realization, employing the ana- weight to fluctuations that have high values compared
lytic expression given by, e.g., Smith (2006). The black to fluctuations that have low values. By first explicitly
curves show the correlation functions computed directly Gaussianizing, a symmetric response is guaranteed.
from the pixelized map using the algorithms described The final results are quoted as the fraction of simula-
above. Slight differences are expected due to different tionswithahigher χ2thantherealdata. Bysplittingthe
treatment of pixel windows, but clearly the agreement simulations into two disjoint sets, and repeating the χ2
between the two are excellent, giving us confidence that analysis for each set, we estimate that the Monte Carlo
our machinery works as expected. uncertaintyinthe resultingsignificancesis lessthan1%.
Forthis reason,wequote casesinconsistentwith simula-
2.3. Comparison with simulations tions at more than 99% by instead providing the actual
number of simulations with higher χ2. This allows us
In this paper we perform a standard frequentist anal-
to distinguish betweena casewith 1%significance and
ysis, in the sense that we compare the results obtained
∼
one with 0.1% significance, but still recognize the im-
fromtherealdatawithanensembleofsimulationsbased
∼
portance of the Monte Carlo uncertainties. Further, we
on some model. Specifically, our null-hypothesis here is
conservatively never claim to obtain results with higher
thattheuniverseisisotropicandhomogeneous,andfilled
significance than 99% in any case, despite the fact that
with Gaussian fluctuations drawn from a ΛCDM power
manyresultsverylikelyarefarmoreanomalous,aswould
spectrum. Our ensemble consists of N =100000sim-
sim
be clear by using more simulations.
ulations.
For each realization in the ensemble, we compute the
N-pointcorrelationfunctionsinpreciselythesameman- 3. DATAANDSIMULATIONS
ner as for the real data. Finally, we compare the corre- In the following, we analyze the five-year WMAP
lationfunctionderivedfromthe datawiththe simulated sky maps, including both temperature and polarization.
4
TABLE 1
Statisticalsignificances(two-point functions)
Correlation
Frequency II IQ IU QI QQ QU UI UQ UU
KQ85
KaxKa — — — — 0.40(0.32) 0.45(0.38) — ... 0.32(0.25)
QxQ 0.08 0.85 0.65 ... 0.01(348∗) 0.01(0.01) ... ... 254∗ (128∗)
QxV 0.07 0.26 0.61 0.85 25∗ (18∗) 0.06(0.04) 0.69 0.01(338∗) 0.04(0.02)
QxKa — 0.81 0.97 — 0.01(307∗) 0.02(0.01) — 0.03(0.02) 154∗ (48∗)
QxW 0.08 0.84 0.05 0.86 3∗ (5∗) 0∗ (0∗) 0.69 0∗ (0∗) 253∗ (222∗)
VxV 0.07 0.27 0.60 ... 73∗ (49∗) 0.02(0.01) ... ... 0.01(393∗)
VxKa — 0.82 0.97 — 0.01(357∗) 21∗ (15∗) — 18∗ (10∗) 0.02(0.02)
VxW 0.07 0.84 0.05 0.27 1∗ (2∗) 42∗ (14∗) 0.57 37∗ (8∗) 106∗ (88∗)
WxKa — 0.78 0.98 — — — — — —
WxW 0.07 0.83 0.05 ... 3∗ (0∗) 71∗ (34∗) ... ... 2∗ (0∗)
W1xW1 0.09 0.70 0.11 ... 116∗ (123∗) 0.01(0.01) ... ... 5∗ (6∗)
W2xW2 0.07 0.32 0.22 ... 95∗ (83∗) 0.03(0.03) ... ... 478∗ (476∗)
W3xW3 0.07 0.91 0.82 ... 0.78(0.78) 0.94(0.94) ... ... 0.60(0.60)
W4xW4 0.10 0.53 0.03 ... 0∗ (0∗) 0.08(0.08) ... ... 1∗ (3∗)
KQ75
QxQ 0.07 0.96 0.61 ... ... ... ... ... ...
QxV 0.06 0.30 0.67 0.97 ... ... 0.62 ... ...
QxKa — 0.87 0.95 — ... ... — ... ...
QxW 0.06 0.91 0.15 0.97 ... ... 0.64 ... ...
VxV 0.06 0.31 0.65 ... ... ... ... ... ...
VxKa — 0.88 0.96 — ... ... — ... ...
VxW 0.06 0.92 0.16 0.32 ... ... 0.62 ... ...
WxKa — 0.85 0.96 — — — — — —
WxW 0.06 0.92 0.15 ... ... ... ... ... ...
W1xW1 0.08 0.68 0.16 ... ... ... ... ... ...
W2xW2 0.06 0.33 0.35 ... ... ... ... ... ...
W3xW3 0.06 0.93 0.94 ... ... ... ... ... ...
W4xW4 0.10 0.57 0.10 ... ... ... ... ... ...
Note. — Statistical significances for the two-point functions as computed from the WMAP data. For the W differencing assemblies,
the first-year maps were used. For the band data, the five-year co-added maps were used. The Monte Carlo uncertainty is 1%. The
correlationsusingboththeKQ85andtheKQ75masksareshown. AsthepurepolarizationcorrelationsareinsensitivetowhichImaskis
beingused,theseareonlydisplayedundertheKQ85header. Anentrywith’...’ signifiesthatsomeotherentryinthetablecontains the
sameinformationasthatentry, whileanentrywith’—’indicates thattheinformationinthatentrywasnotcomputed. Thelistedratios
indicate the fractionofsimulations withahigher χ2 than the observed data. Theentries markedwithan’*’ signifiesthat the fractionis
lowerthantheMonteCarlouncertainty, butstilllargerthanzero;consequently, thenumberofsimulations(outof100,000)withahigher
χ2 thantheobserveddataareshowninsteadfortheseentries. Theentriesinparenthesisarethesignificancesofthenoise-onlyhypothesis.
These data are available from LAMBDA13, including all best-fit five-year WMAP ΛCDM power spectrum, in-
ancillarydata,suchasbeamtransferfunctions,noiseco- cluding multipole moments up to ℓ = 1024. This
max
variance matrices and foreground templates. The total realization is then convolved with the instrumental
data set spans five frequencies (23, 33, 41, 61, and 94 beam of each WMAP differencing assembly and the
GHz),correspondingtotheK,Ka,Q,V,andW bands. N = 512 HEALPix pixel window, and projected
pix
Our main objects of interest are the frequency co- onto a HEALPix grid. For temperature, we then add
added and foreground-reducedsky maps, which are pro- a Gaussian noise realization with pixel-dependent vari-
videdintheformofHEALPix14pixelizedskymaps. The ance, σ2 = σ2/N (p), where σ2 is the noise variance
p 0 obs 0
temperature sky maps are given at a HEALPix resolu- per observation and N is the number of observations
obs
tionofNside =512,whilethepolarizationmapsaregiven in that given pixel. The temperature component is then
at Nside =16. To bring the data into a common format, degraded to Nside =16.
we therefore degrade the temperature component to the For the polarization component, we first degrade the
same resolution as the polarization maps, simply by av- high-resolutionmap to N =16, and then add a noise
side
eraging over sub-pixels in a low-resolutionpixel. term. This is because the polarized noise description
For temperature we consider the Q, V, and W bands, for WMAP is given by a full noise covariance matrix,
and for polarization also the Ka band. This difference Cp,p′ at Nside = 16, and not as a simple Nobs count at
mirrors the official cosmological WMAP analysis, which high resolution. The correlated noise realization is gen-
alsousestheKabandforpolarization,butconsidersthe erated by drawing a vector of standard normal variates,
same band to be too foreground contaminated in tem- η N(0,1),andmultiplyingthisvectorbytheCholesky
perature. fa∼ctorofthecovariancematrix,n=Lη,whereC=LLt.
Our simulations are generated as follows: we first Fortemperature,weadopttwodifferentmasks,namely
draw a random Gaussian CMB realization from the the five-yearWMAP KQ85 and KQ75 masks. These ex-
clude 18% and 28% of the sky, respectively. For the
13 http://lambda.gsfc.nasa.gov polarizationcomponent, there is only one relevant mask
14 http://healpix.jpl.nasa.gov
5
Fig.3.— Two-point functions computed with the KQ85 mask, as functions of angular separation on the sky. Only the Ka, Q and
V auto-correlations, as well as the cross-correlations between the W intensity map and the Ka polarization map, are shown. The red
line shows the two-point function computed from the WMAP five-year co-added maps. The light, medium and dark gray shaded areas
correspondto1σ,2σ,and3σ,respectively. Thedashedbluelineisthemaximalpointofthecorrelationhistograms,whiletheblacklineis
the theoretically predicted (no noise) two-point function computed fromthe best-fit five-year WMAPΛCDM power spectrum. Each row
correspondstoaspecificcombinationofmodes,whilethefrequencybandsinvolvedaremarkedoneachplot.
6
inthefive-yearWMAPdatarelease,whichexcludes27% However,evenmoreinterestingis theoverallverypro-
of the sky. nounced large-scale excesses seen in the QQ and UU
Two different polarizedforegroundtemplates are used functions: both the Q- and V-band functions lie mostly
in the following. The first is simply the difference be- within the 2σ-3σ confidence regions, and the overall
tweentheK andKabands,smoothedtoaneffectiveres- agreement with the simulations appears quite poor.
◦
olution of 10 to reduce noise, which traces synchrotron Again, this is strongly reflected in Table 1: all QQ
radiation. Thesecondtemplateisthesamestarlighttem- correlationsareanomalousatmorethan99%confidence,
plate as used for foreground correction in the five-year andallUUcorrelationsatmorethan98%confidence. In
WMAP data release, which traces thermal dust. the case of the W band, which happens to be the most
−
anomalous of any band, we have good reasons to expect
4. RESULTS
such behavior. This band has a significantly higher 1/f
Inthissectionwepresentthetwo-andthree-pointcor- kneefrequencythananyotherband(Jarosik et al.2003),
relationfunctions computed fromthe polarizedfive-year with the W4 differencing assembly having the highest.
WMAP data. Due to the large number of available and As a result, the WMAP team has chosen not to use this
unique correlation functions one can form within this band for cosmological analysis in polarization. But the
data set, we plot only a few selected cases, and instead anomalousbehaviorofthe QandV bandsisapriorinot
show the full set of results only in the form of tabulated expected; theseareusedforcosmologicalanalysisby the
significances. WMAP team and should be clean.
We havealsogeneratedanensembleofnoise-onlysim-
4.1. Two-point correlations
ulations, and computed the two-point functions from
In Figure 3, we show a selection of two-point corre- these. Thecorrespondingχ2fractionsarelistedinparen-
lation functions derived from the WMAP data. Specif- theses in Table 1 for the pure polarization modes. Here
ically, all available Ka, Q and V auto-correlations are weseethat,generallyspeaking,thenoise-onlyhypothesis
shown, in addition to the cross-correlationsbetween the performsalmost,butnotquite,aswellasthesignal-plus-
KaandW bands. Table1showsthe χ2 significancesfor noise hypothesis. This is simply due to the fact that the
all possible two-point functions from the available data. WMAPpolarizationdata arestronglydominated by the
(Notethatonlyuniquecombinationsareactuallyshown; correlatednoise,anditisverydifficultfromacorrelation
empty table cells indicate that the same combinations functionpointofviewtodistinguishbetweenasmallsig-
may be found elsewhere in the table. For example, IQ nal component and a large-scale noise fluctuation.
and QI are identical for auto-correlations, but different
4.1.1. Foreground cross-correlations
for cross-correlations.)
Starting with the pure intensity correlations shown in The anomalous behavior seen in the Q- and V-band
the top row of Figure 3, we recognize the by now well- two-point functions clearly needs an explanation. And
knownbehavioroftheCMBtemperaturetwo-pointfunc- typically, when such unexpected behavior is observed,
tion, observedboth by COBE (e.g., Bennett et al. 1994) one of the the first issues to consider is residual fore-
and WMAP (e.g., Spergel et al. 2003): at small angular grounds.
separations, it is slightly low compared to the ΛCDM Tocheckwhetherthis maybe arelevantissue,wefirst
◦
model, and at separations larger than 60 , very close to simply compute the cross-correlation functions between
zero. This peculiar behavior has attracted the interest each of the three frequency bands (Ka, Q, and V) and
of both experimentalists and theorists, and some have thetwoavailableforegroundtemplates(synchrotronand
suggested that this could be the signature of a closed dust). This is done both for the real WMAP data and
topological space. However, from the view of a simple thesimulatedensembles,andtheagreementbetweenthe
χ2 test, this function remains consistent with the sim- (pureCMB+noise)simulationsandtheWMAPdatais
plest ΛCDM model at the 5%–10% level. again quoted in terms of a χ2 fraction.
Next,lookingatthetemperature-polarization(I-Q/U) The results from these calculations are as follows: for
cross-correlationswe see that the overallbehavior of the the Ka band, we find that the χ2 significances are 0.51
Q-, V, and Ka W band functions are quite different. and 0.24 for synchrotron and dust, respectively, indicat-
×
This indicates that the WMAP data still are too noise ingnosignificantforegrounddetectioninthiscase. How-
dominatedtoextracthigh-sensitivitycosmologicalinfor- ever,fortheQbandthe correspondingnumbersare0.05
mation from individual bands. This is also reflected in and 0.005, corresponding to correlations statistically
the second and third columns of Table 1, where there is signifi∼cantat2σ and 3σ,respectively. FortheV band,
a significant scatter between the different results. How- thenumbersare0.05∼and0.02,significantat2σ ormore.
ever, we do see that all results are in good agreement To study these foreground correlations further, we
withthe expectations,indicating thatthe noisemodelis compute the two-point functions from the foreground
satisfactory. templates directly, and fit these to the observed corre-
The three bottom rows of Figure 3 show the pure po- lation functions with a single free amplitude for each
larization correlation functions, which are the main tar- template, A and A , by minimizing
s d
get in this paper. And here we see several interesting
features. First, we recognize a sharp feature in the UU χ2 = (C (b) C (b) A C (b) A C (b))Σ−1(b,b′)
◦ fg obs − sim − s s − d d sim
correlation function at 141 . This is the angular sep- bb′
∼ X
aration between the A- and B sides of the differential ′ ′ ′ ′
(C (b) C (b) A C (b) A C (b)).
WMAP detectors, and it was also seen in the pure noise × obs − sim − s s − d d
temperature two-point correlation function in the first- Here, Csim(b) is the correlation function mean and
year WMAP data (Eriksen et al. 2005). Σ (b,b′) = (C(b) µ(b))(C(b′) µ(b′)) is the co-
sim
− −
(cid:10) (cid:11)
7
Fig. 4.— Template two-point functions fitted to the WMAP data two-point functions. The red line shows the WMAP data, with the
grayareabeingthe1σconfidenceintervals. Thebluelineshowsthethemeancorrelationfunctionofthesimulations,whilethegreenline
showsthemeanvalueplustheforegroundcontribution.
pared to the observed functions. First, the red curve
TABLE 2 showsthefunctionsderivedfromtheactualWMAPdata,
Amplitude confidence intervals
with gray bands indicating the 1σ uncertainties derived
from simulations. The blue curve shows the mean cor-
Synchrotron Dust
Band As σsig Ad σsig relationfunction of the simulations, Csim(b), and finally,
thegreencurveshowsthesame,butwiththeforeground
Ka 0.0038±0.0013 2.9 0.0115±0.0057 2.0
Q 0.0052±0.0010 5.2 0.0256±0.0042 6.1 contribution added in, Cfg(b) = Csim(b) + AsCs(b) +
V 0.0063±0.0011 5.7 0.0197±0.0047 4.2 AdCd(b). Clearly, the latter matches the real data far
better than the pure CMB-plus-noise hypothesis, with
Note. — Mean values and confidence intervals for
thebest-fitamplitudesoftheK−Kaanddusttemplate the best match seen in the UU correlation function.
two-point functions relative to the WMAP data two-
pointfunctions. 4.2. Three-point correlations
variancematrix,bothquantitiesobtainedfromthe noise We now turn to the three-point correlation functions
simulations. The indices b,b′ run over all possible pure derived from the five-year polarized WMAP data. How-
polarization two-point bins (i.e., both angular bins and ever, we note that the three-point function is primarily
QQ, QU, and UU correlations). used in the literature as a test of non-Gaussianity. In
The resulting best-fit amplitudes from this calculation the present case, this will not be case, since we have
forKa-,Q-andV bandsareshowninTable2. Here,we already seen that the data appear to be foregroundcon-
againseethattheKa-bandamplitudesaregenerallysig- taminated, and even the two-point correlation function
nificantly lowerthanthose of the Q and V bands. (Note fails a standard χ2 test. The following three-point anal-
that the uncertainties quoted in this table only include ysiswillthereforeessentiallyonlybe aconsistencycheck
statisticalerrors,notsystematicerrors. Thesignificances of the above results, and a demonstration of the general
should therefore not be consideredas true detection lev- procedure, to be further developed in the future, when
els, but are only suggestive.) cleaner data become available.
In Figure 4, the fitted correlation functions are com- We consider only cross-correlations of pure polariza-
8
TABLE 3
Three-point function results
Correlation
Frequency QQQ QQU QUQ QUU UQQ UQU UUQ UUU
Equilateralfunctions
KaxQxV 0.04(0.03) 0.31(0.26) 18∗ (4∗) 0.04(0.03) 0.16(0.13) 0.01(416∗) 0.07(0.05) 0.21(0.17)
KaxVxQ 0.45(0.40) 0.10(0.08) 0.07(0.05) 0.01(446∗) 359∗ (241∗) 0.35(0.31) 0.01(0.01) 0.11(0.09)
Collapsedfunctions
KaxQxV 0.49(0.45) 0.01(0.01) 0.23(0.18) 0.03(0.02) 0.63(0.56) 0.01(384∗) 0.21(0.18) 0.43(0.38)
KaxVxQ 323∗ (240∗) 0.01(0.01) 2∗ (1∗) 0.05(0.03) 0∗ (1∗) 0.11(0.09) 0.05(0.03) 17∗ (11∗)
QxKaxV 0.01(0.01) 0.28(0.25) 0.01(329∗) 0.17(0.14) 0.01(298∗) 231∗ (179∗) 0.07(0.05) 0.15(0.12)
QxVxKa 0.05(0.04) 0.09(0.07) 0.01(0.01) 0.10(0.08) 0.04(0.03) 0.07(0.06) 0.12(0.09) 0.12(0.09)
VxKaxQ 0.21(0.17) 0.04(0.03) 180∗ (5∗) 0∗ (2∗) 3∗ (2∗) 127∗ (79∗) 80(45∗) 0.01(456∗)
VxQxKa 0.01(0.01) 0.03(0.02) 0.58(0.51) 439∗ (301∗) 0.01(0.01) 0.38(0.33) 0.18(0.13) 0.05(0.04)
Note. — Statistical significances of the polarization-only three-point functions computed from the WMAP five-year co-added Ka, Q
andV maps. TheMonteCarlouncertainty is1%. SeeTable1forfeatureexplanation.
tion modes for the cosmologically interesting frequency tion of a signature consistent with residual foregrounds
channels. Thisleavesuswithonlyeightindependentcor- in both the Q- and V-band data, significant at 2σ-3σ.
relationfunction modes (QQQ,QUQ, UQQ, UUU etc.), This should not come as a complete surprise, though,
and only one frequency combination (KaxQxV). Fur- as a similar result was found in the three-year
ther, we consider only two general types of three-point WMAP data through a direct template fit approach by
configurations,namely, equilateral and pseudo-collapsed Eriksen et al. (2007). They used a Gibbs sampling im-
triangles. The latter is defined such that two legs of the plementationtoestimatethejointCMBpowerspectrum
triangle are of equal length, while the third leg spans an and foreground template posterior, and found non-zero
angular distance of less than 12◦. template amplitudes at similar significance levels when
TheresultsfromthesecalculationsaretabulatedinTa- marginalizing over the CMB spectrum. Given the re-
ble 3, and a few arbitrarily chosen modes of the equilat- sults found in the present paper, this still appears to be
eralthree-point functions are plotted in Figure 5. (Plot- animportantissueforthe five-yearWMAPdatarelease.
ting all functions would take too much space, without Resolving this question is of major importance for the
adding any particular new insights.) CMB community, since many cosmological parameters
Once again, we see that the χ2’s obtained from the depend critically on the large-scale WMAP polarization
real data are high, but generally not quite as striking as data, most notably the optical depth of reionization, τ,
in the two-point case. This is at least partly due to the and the spectral index of scalar perturbations, ns.
fact that the Ka band is involved in all configurations, The question of primordial non-Gaussianity in CMB
aswellasthefactthatthethree-pointfunctionisalways polarization data can not be meaningfully addressed as
“noisier” than the two-point function. We also see that long as this issue is unresolved. The three-point correla-
the noise-only hypothesis always has higher χ2’s than tionfunctionresultsshowninthispaperthereforemostly
thesignal-plus-noisehypothesis,againfollowingthetwo- serveasademonstrationoftheapproach,andasacross-
point behavior. checkonthe two-pointresults. The three-pointfunction
It is difficult to interpret these results much further, willobviouslybecomeamoreimportantandindependent
given that there are clear problems already at the two- quantity in the future, when higher-fidelity polarization
point level. We therefore leave a full interpretation of data become available.
the three-point function to a future publication, when
cleaner data, either from WMAP or Planck, have been
made available. We acknowledge use of the HEALPix15 software
(G´orski et al. 2005) and analysis package for deriving
5. CONCLUSIONS the results in this paper, and use of the Legacy Archive
for Microwave Background Data Analysis (LAMBDA).
We compute for the first time the polarized two-
H.K.E. and P.B.L. acknowledge financial support from
and three-point correlation functions from the five-year
the ResearchCouncilofNorway. The computations pre-
WMAP data, with the initial motivation of constraining
sented in this paper were carriedout on Titan, a cluster
possiblenon-GaussiansignalsonlargescalesintheCMB
ownedandmaintainedbytheUniversityofOsloandNO-
polarizationsector. However,our main result is a detec-
APPENDIX
PRECOMPUTING CONFIGURATION TABLES
In order to compute the two-and-three point functions, we have utilized the method introduced by Eriksen et al.
(2004b).
15 http://www.eso.org/science/healpix/
9
Fig.5.— Equilateral three-point functions, computed with the KQ85 mask, as functions of angular separation on the sky. Only a
representativeselectionisshown. Theredlineisthethree-pointfunctioncomputed fromtheWMAPfive-year co-addedmaps. Thelight,
mediumanddarkgrayshadedareascorrespondto1σ,2σ,and3σ,respectively. Thedashedbluelineisthemaximalpointofthecorrelation
histograms. Eachrowcorrespondstoaspecificcombinationofmodes,whilethefrequencybands involvedaremarkedoneachplot.
10
The basic idea of this method is that given the resolution and the mask of the map, and the desired number of
distance bins, the configurations that correspond to an angle θ is uniquely determined. One may then do the work of
findingtheseconfigurationsonlyonce,savingtheminlargetables. Eachtablethencorrespondstoacertainbink,and
the first row in each table contains all the unmasked pixels of the map. Each column contains all pixels at a distance
givenby kdθ from the pixel in the first elementof the column. By moving througha column in such a table, one then
traces out a circle of radius kdθ. Thus, by saving the configurations in tables, one pays the initialization costs only
once.
An additional, powerful feature of this approach becomes apparent when computing correlation functions of higher
orderthan2. Ifonethinksofthesetablesascompasses,tracingoutcirclesofagivenlengthforeachpixel,itshouldbe
clear that, just as one can construct, e.g., triangles using compasses, one can construct triangles on the sphere using
these tables. If the edges of the desired triangle is of lengths θ, α and β, one begins with the θ table, selects a pixel
from the upper row, and selects a second pixel from the column below this pixel. One then has the baseline of the
triangle. To construct a triangle with a compass when the baseline is given, one would adjust the compass so that
it spans a length equal to the length of the second edge of the triangle, and draw a circle with this radius with one
end of the baseline as the center of the circle. Then, one would again adjust the compass so that it spans the desired
third length of the triangle, and draw a circle around the other end of the baseline. Then, one would check whether
the two circles drawnintercept at any point(s). These points would then be the third vertex of the triangle. Carrying
this analogytoourtables,ourtwoalreadychosenpixels arethe endsofthe baselinearoundwhichtodrawourcircles.
We ’adjust our compass’ by finding pixel 1 in the first row of table α, we ’draw a circle around it’ by looping over all
pixels in the column below it, and do the same for pixel 2, expect that we now look in table β . By checking whether
there are any common pixels in these ’circles’, we find the triangle(s) whose edges have the desired lengths.
Since any N-point polygon can be reduced to triangles, higher-order correlation functions can also be computed
using this method.
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