Table Of ContentMon.Not.R.Astron.Soc.000,000–000 (0000) Printed20January2011 (MNLATEXstylefilev2.2)
Wilkinson Microwave Anisotropy Probe 7-yr constraints
on f with a fast wavelet estimator
NL
B. Casaponsa,1 2⋆ R. B. Barreiro,1 A. Curto,1 E. Mart´ınez-Gonz´alez,1 P. Vielva1
1 Institutode F´ısica de Cantabria, CSIC-Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain.
1 2 Dpto. de F´ısica Moderna, Universidad de Cantabria, Avda. los Castros s/n, 39005 Santander, Spain.
1
0
2
Accepted Received;inoriginalform
n
a
J
ABSTRACT
9 A new method to constrain the local non-linear coupling parameter fNL based on a
1
fast wavelet decomposition is presented. Using a multiresolution wavelet adapted to
theHEALPixpixelization,wehavedevelopedamethodthatis∼102timesfasterthan
O] previous estimators based on isotropic wavelets and ∼ 103 faster than the KSW bis-
pectrum estimator, at the resolution of the Wilkinson Microwave Anisotropy Probe
C
(WMAP) data. The method has been applied to the WMAP 7-yr V+W combined
h. map, imposing constraints on fNL of −69 < fNL < 65 at the 95 per cent CL. This
p result has been obtained after correcting for the contribution of the residual point
- sources which has been estimated to be ∆fNL = 7±6. In addition, a Gaussianity
o
analysisofthedatahasbeencarriedoutusingthethirdordermomentsofthewavelet
r
t coefficients, finding consistency with Gaussianity. Although the constrainsts imposed
s
a on fNL are less stringent than those found with optimal estimators, we believe that
[ a very fast method, as the one proposed in this work, can be very useful, especially
bearinginmindthelargeamountofdatathatwillbeprovidedbyfutureexperiments,
2
suchas the Plancksatellite.Moreover,the localisationof waveletsallows one to carry
v
outanalysesondifferentregionsofthesky.Asanapplication,wehaveseparatelyanal-
2
ysedthetwohemispheresdefinedbythedipolarmodulationproposedbyHoftuft et al.
3
(2009). We do not find any significant asymmetry regarding the estimated value of
6
0 fNL in those hemispheres.
.
9 Key words: methods: data analysis - cosmic microwave background
0
0
1
:
v
1 INTRODUCTION while alternative models may introduce a certain level of
i
X non-Gaussianity in the CMB. A convenient parametriza-
Thecosmicmicrowavebackground(CMB)isoneofthepil-
tion valid for a large set of non-standard inflationary mod-
r lars that providesupport to theBig Bang theory.Thefluc-
a els which includes the quadratic corrections of the pri-
tuations of the CMB naturally arise in an inflationary sce-
mordial curvature perturbation is (Salopek & Bond 1990;
nario.Theunderstandingofthisveryearlystageofthehis-
Gangui et al. 1994; Verdeet al. 2000; Komatsu & Spergel
tory of the Universe is a challenging issue for the scientific
2001):
community due to the implications on large scale structure
formation and fundamental particle physics at high ener-
gies. A large number of inflationary models have been pro- φ(r)=φL(r)+fNL φ2L−<φ2L> , (1)
posedintheliterature(foranoverviewseeforinstanceLyth (cid:2) (cid:3)
2008)butthetaskoftestingsuchscenariosisnottrivial,and where φL are Gaussian linear perturbations and fNL char-
thereistheneedofnewexperimentsandpowerfulstatistical acterises the amplitude of the non-linear contribution in
tools to discriminate among them. In this sense, the statis- real space. This local form appears in non-standard multi-
tical properties of the CMB temperature anisotropies are a fieldinflationarymodels(Babich et al.2004;Komatsu et al.
source of information about the processes that have gener- 2009).Foracompletereviewon non-Gaussianity duetoin-
atedtheprimordialfluctuations.Inparticular,thestandard, flationarymodelsseeBartolo et al.(2004).Inadditiontoin-
slow roll, single field inflationary model predicts a nearly flationarymodels,thereareotheralternativescenariosthat
GaussiandistributionoftheCMBtemperatureanisotropies, can be constrained, such as the ekpyrotic model where a
negativevalueoffNL isexpected(Lehners2010).Moreover,
thereareotherprocessesthatcanintroducedeviationsfrom
⋆ e-mail:[email protected] Gaussianityinthethirdordermoments(asforegroundcon-
(cid:13)c 0000RAS
2 B. Casaponsa et al.
tamination, non-linear gravitational effects, topological de- equal to the number of original pixels. However, we have
fects, etc). kept the HW with its intrinsic redundancy for three main
Since the quadratic parametrization was proposed, reasons: first, to improve the computational time, second
an important effort has been made to set observational to obtain a wavelet decomposition as isotropic as the HW
constraints on local fNL with a wide variety of meth- allows 2, and third, because, as it is shown later, redun-
ods including the bispectrum (Yadav & Wandelt 2008; dancy helps to improve the sensitivity in the detection of
Smith et al. 2009; Komatsu et al. 2010), wavelet-based fNL. Similarly to the SHW, the HW is a discrete, orthog-
methods (Cayo´n et al. 2003; Mukherjee & Wang 2004; onal wavelet, adapted to a hierarchical pixelization (such
Curto et al. 2009a,b; Pietrobon et al. 2009; Rudjord et al. as HEALPix3, G´orski et al. 2005), whereas the SMHW is
2009), Minkowski functionals (Hikage et al. 2008) or the a continuous, non-orthogonal wavelet and does not have a
N-pdf(Vielva& Sanz2009,2010).Mostoftheseworksfind hierarchical decomposition structure. The HW presents an
that the data are compatible with fNL = 0, but the con- optimalspacelocalization,whilethescalelocalizationisnot
straints are not yet sufficiently tight to discriminate among as good as that of the SMHW. It is important to point out
a large set of inflationary models. The best current limit that, in the case of the SMHW and of needlets, a transfor-
is given by Komatsu et al. (2010) and is −10 < fNL < 74 mationofthedataintosphericalharmonicspaceisrequired.
at the 95 per cent confidence level. These constraints have However,thisisnotthecasefortheHWand,therefore,the
been obtained with a bispectrum estimator, which is com- computational cost is significantly reduced.
putationally very demanding. However, Curto et al. (2010) The resolution of a HEALPix map is characterised by the
haverecentlyshownthatanestimatorbasedontheSMHW N parameter, such that the number of pixels in which
side
can provide constraints on fNL as stringent as the optimal the sphere is divided corresponds to N = 12Ns2ide. Nside
estimator, the bispectrum, while reducing considerably canonlytakepowersoftwoasvalues.TheHWdecomposes
the CPU time. With the arrival of new data from high the temperature map at resolution J, where N =2J, in
side
resolution experiments such as the ESA Planck satellite1 waveletcoefficientmapsatalltheallowed HEALPixresolu-
(Tauber et al. 2010), it becomes even more important tionsdowntothelowerconsideredresolutionj .Thewavelet
0
the availability of even faster and simpler methods. With functions are defined as follows:
this aim we present the application for CMB of a wavelet ϕ (x)
adapted to the HEALPix pixelization similar to the tool Ψ0,j,k(x)=ϕj+1,k0(x)− j,k4 (2)
proposed by Shahram et al. (2007).
ϕ (x)
Ψ (x)=ϕ (x)− j,k
1.j,k j+1,k1 4
The paper is organised as follows. In Section 2 we in-
ϕ (x)
troduce the HEALPix wavelet decomposition. In Section 3 Ψ2,j,k(x)=ϕj+1,k2(x)− j,k4
the method to constrain the fNL parameter as well as the ϕ (x)
proposed Gaussianity test are described. In Section 4 we Ψ (x)=ϕ (x)− j,k
3,j,k j+1,k3 4
presenttheresultsoftheapplicationofthistechniquetothe
WMAP 7-yr data. Finally, our conclusions are summarised where ϕ(x)j,k is thescaling function
in Section 5.
1 if x∈P
ϕ(x) = j,k (3)
j,k (cid:26) 0 otherwise,
and P is the pixel at position k at resolution j, which
j,k
2 THE HEALPIX WAVELET
at the next higher resolution is divided into four daughter
A large set of different wavelets have been used in pixels Pj+1,k0, Pj+1,k1, Pj+1,k2, Pj+1,k3.
the astrophysics literature. In particular, different spher-
ical wavelets have been applied to CMB Gaussianity The wavelet decomposition of a temperature map can
analysis during the last decade, including the spher- be written in terms of the basis functions and a set of
ical Haar wavelet (SHW, Barreiro et al. 2000), the coefficients:
SMHW (Mart´ınez-Gonz´alez et al. 2002; Vielva et al. 2004; ∆T Nj0−1
Mukherjee & Wang 2004; Cruz et al. 2005; Curto et al. T (xi)= λj0,kϕj0,k(xi)+
2009b), elliptical SMWH McEwen et al. (2005), direc- Xk=0
tional spherical wavelets (McEwen et al. 2005, 2006, 2008) J 3 Nj−1
and needlets (Pietrobon et al. 2009; Rudjord et al. 2009; + γm,j,kΨm,j,k(xi), (4)
Cabella et al. 2010). For a review on wavelet applications jX=j0mX=0 Xk=0
to cosmology see McEwen et al. 2007.
In this work, we will present an application using the where Nj is the number of pixels at resolution j. λj,k and
γ are the approximation and detail coefficients respec-
so-called HEALPix wavelet (HW). In the previous work of m,j,k
tively.Fromapracticalpointofview,toperformthedecom-
(Shahram et al. 2007) a linear operator is applied to the
position, we start with the original resolution, i.e. j = J.
HWtoobtainwaveletcoefficientscorrespondingtovertical,
horizontal and diagonal orientations. This operation leads
to a wavelet coefficients without redundancy, obtaining a
2 It is worth mentioning that HW detail coefficients help to
number of wavelet coefficients (details plus approximation)
highlight the isotropy properties of the field as compared to the
directionalorienteddetailsoftheSHW.Thisisimportantbecause
thelocalnon-Gaussianitiesareexpected tobeisotropic.
1 http://www.rssd.esa.int/index.php?project=planck 3 http://healpix.jpl.nasa.gov/
(cid:13)c 0000RAS,MNRAS000,000–000
WMAP constraints on f 3
NL
At this resolution, the approximation coefficients λ cor- spectrum that best fits the WMAP-7yr data, and that was
J,k
respond to the pixels of the original tempature map. The used for the Gaussian simulations (the original simulations
approximation coefficientsatthenextresolution aresimply were obtained using the WMAP 5-yr power spectrum).
obtainedbydegradingthemaptotheinmediatlylowerres- Again,weconstructthemapsfortheVandWreceivers,ap-
olution (i.e., by averaging the corresponding four daughter plying thecorresponding beam and pixel transfer functions
pixels): and adding the appropriate level of noise. Finally a single
V+W combined map is constructed for each non-Gaussian
3
1 simulation.
λ = λ , (5)
j,k 4 j+1,ki
Xi=0
Ontheotherhand,thedetailcoefficientsatresolution j+1
aresimplyobtainedbysubtractingtheapproximationatres-
olution j from theapproximation at resolution j+1.Thus,
thedetail coefficients are defined as: 3.1 Cubic statistics
γ0,j,k =λj+1,k0 −4λj,k (6) In this section we define the third order moments of the
γ =λ −4λ wavelet coefficients that are used to constrain fNL. Sim-
1,j,k j+1,k1 j,k
ilar statistics have been used in other previous works
γ =λ −4λ
2,j,k j+1,k2 j,k (Curto et al. 2009b; Rudjord et al. 2009).
γ3,j,k =λj+1,k3 −4λj,k We perform the wavelet decomposition of the considered
mapstartingatresolutionN =512(J =9)anddownto
A schematic diagram of how to obtain the approximation side
N = 2 (j = 1) (when using a higher value of j we are
and detail coefficients is given in Fig. 1. side 0 0
losing efficiency whereas for j = 0 the results are not sig-
0
nificantly improved while the computational time increases
by a 30%). We obtain 8 detail maps and 1 approximation
3 METHODOLOGY
map. In addition, we also include in the analysis the origi-
nalmapandthe8intermediateapproximationmaps(which
The main purpose of this work is to constrain the parame-
terfNL definedin Eq. (1) using theWMAP-7yrdata4.For areobtainedduringthewavelet decomposition toconstruct
the detail coefficients). Although, in principle, these addi-
this analysis, we only consider the (foreground reduced) V
tional approximation maps contain redundant information,
and W channels, since they are less afected by foreground
they seem to provide additional information regarding the
contamination. A single CMB map is obtained through a
third order statistics, since a larger number of third order
noise-weightedlinearcombinationoftheVandWreceivers.
combinations can be constructed. In fact, we have tested
The KQ75 mask (which covers around a 29 per cent of the
that with the inclusion of the approximation maps and the
sky) is applied subsequently.
originalmap,theresultsareimprovedbya30%.Therefore,
Inordertocalibrateourestimator,weneedbothGaus-
we have a total of 18 maps for each analysed signal. The
sian and non-Gaussian simulations. To generate the Gaus-
statistics are constructed as the third order moments of all
sian simulations,wecomputethepowerspectrumthatbest
thepossiblecombinationsofthese18maps,wherethecoef-
fits the WMAP-7yr data accordingly to the parameters es-
ficients are weighted to take into account the presence of a
timated by Komatsu et al. (2010). For this purpose we use
mask. In order to calculate these weights, one performs the
the On-linetool CAMB (Lewis et al. 2000). Wethen apply
wavelet decomposition oftheconsidered mask (thathasze-
thecorrespondingbeamandpixelfunctionstosimulatethe
ros in the masked pixels and ones in the rest). The wavelet
data at each of the considered receivers (2 for V and 4 for
coefficients of the mask at each detail and approximation
W). A Gaussian noise realisation is subsequently added to
the CMB maps with a variance per pixel given by σ0 , scale are used to construct the weight wj(i) of the coeffi-
where σ is the detector sensitivity of each of the receNivoebsrs cient at position i at resolution j. This makes sense if one
0
andN isthenumberofobservationsateachpixel.Finally bearsin mindhowthewavelet decomposition isperformed.
obs
Forinstance,toconstructtheapproximationmapatresolu-
thesix maps are combined in thesame way as thedata.
tion J−1 at a given position i, the four daughter pixels at
Regarding thenon-Gaussian simulations, we haveused
resolution J have to be averaged. If the four pixels are un-
the1000simulationsgeneratedbyElsner & Wandelt(2009)
that are publicly available5. The previous authors provide masked,thiscorrespondstoaweightof1intheoriginalmap
and also in the approximation map at position i. However,
theharmoniccoefficientsfortheGaussianandnon-Gaussian
if one of the orignal pixels is masked, this pixel would have
parts of the simulation. A non-Gaussian simulation with a
given value of fNL is then constructed as: a weight of zero, and the average would be done only over
three pixels. Thus the weight of the corresponding approx-
alm =al(mG)+fNLal(mNG) , (7) imation coefficient would be 3/4. Therefore, this weighting
schemetakesintoaccountthefactthatdifferentcoefficients
where we have normalized a(G) and a(NG)6 to the power
lm lm contain different amount of information, depending on the
consideredmask.Also,contrarytothecaseofotherwavelet
estimators,thismeansthatthemaskdoesnotneedtobeex-
4 The data are available at the LAMBDA web page:
tended but, in fact, it is reduced when increasing the scale.
http://lambda.gsfc.nasa.gov/
5 http://planck.mpa-garching.mpg.de/cmb/fnl-simulations/ This is due to the fact that a larger pixel is kept for the
6 TheamplitudeoftheaNG hasbeencorrectedbyafactorof 3 analysis, with the appropriate weight, if at least one of the
lm 5
asindicatedbytheauthors daughterpixels was unmasked.
(cid:13)c 0000RAS,MNRAS000,000–000
4 B. Casaponsa et al.
d00 d01 d10 d11
d02 d03 d12 d13
− x¯ 0i d20 d21 d30 d31 d d
xij 00 01
x00 x01 x10 x11 = d22 d23 d32 d33
dij
x02 x03 x12 x13 x¯00
− d d
x20 x21 x30 x31 = x 0i 02 03
x¯0i x¯00 x¯01 d 0i
x22 x23 x32 x33 = 3
j=X0 4xij
x¯ x¯ x¯ b b b
02 03 00= 3 x¯
i=X0 4x¯0i 00
b b b
Figure 1. Diagram of the construction of the approximation and detail coefficients. Approximation coefficients are computed as the
average ofthefourdaughter pixels.Detailcoefficients arecomputed as thesubtraction ofthat averagefromtheoriginalpixelsandare
representedbyd.
The third order statistics are then definedas: 3.2 Gaussianity test and fNL constraints
We first perform a Gaussianity test in order to probe
Sjkl = Nl−1W1 (i)NXi=l−01Wjkl(σij)σǫik,jσǫli,kǫi,l, (8) wχ2heetshtiemratthoer:data is compatible with Gaussianity using the
jkl
nstat
Xi=0 χ2 = (vi−hvii)Ci−j1(vj−hvji), (10)
iX,j=1
where ǫi,j = yi,j −µj and yi,j are the wavelet coefficients
mapsat position i at resolution j. Notethat j goes from j0 wherevi isthevectorofthethirdorderstatisticscomputed
to J, k goes from j to J and l goes from k to J. µj and from the considered map (to simplify notation, hereinafter
σj aretheweightedmeanandthedispersionforthemapat we define v1 ≡ S111,v2 ≡ S112,...). hvii and Cij are the
resolution j. W (i)is theweight associated to thewavelet mean and covariance matrix of the statistics obtained with
jkl
coefficients at position i and scales j,k,l and is given by: 10000 Gaussian simulations. To perform the Gaussianity
test, the value of the χ2 is computed for the WMAP data,
and compared to thedistribution of theestimator obtained
Wjkl(i)= 3 wj(i)wk(i)wl(i). (9) from an additional set of 1000 Gaussian simulations.
p
The second analysis that hasbeen performed is thees-
Note that some of these statistics are redundant (lin-
early dependent between them), so we restrict our analysis timation of fNL from the data. As the wavelet decompo-
sition is linear, we can obtain the wavelet coefficients from
tothesetofnon-rendudantstatistics, whichgivesatotalof
theGaussian and non-Gaussian parts separately. Thus, the
n =232 quantities.
stat
The process for computing these statistics requires ∼ wavelet coefficients for a given valueof fNL are given by
N × n number of operations, where N is the number
of pixeslstatand n the number of statistics computed. This yi =yi(G)+fNLyi(NG) . (11)
stat
numberissignificantlylowerthanthatofthefullbispectrum
5 Takingintoaccountthaty(NG) arearound4ordersofmag-
that needs N2 operations. Using the KSW algorithm pre- nitudesmallerthany(G),whenwecompute<y3>theNG
sentedinKomatsu et al.(2005)thenumberofoperationsis
high-ordertermscanbeneglectedanditcanbeshownthat
3
reducedto∼rN2,whereristhenumberofsamplingpoints fNL is proportional to the wavelet estimators, as it is also
(of theorderof 100). On theotherhand,theSMHW scales thecase for other statistics (such as the bispectrum):
3
as ∼ nsN2, where ns is the number of scales involved (of
the order of 10). Thus, at WMAP resolution (Nside = 512 vi =aifNL , (12)
andN ∼3×106)wehavethatthemethodpresentedinthis
work is 102 times faster than the SMHW, 103 times faster where ai can be computed from simulations with a simple
than KSW bispectrum estimator and 107 faster than the linear regression.
general bispectrum estimator. In order to estimate the fNL parameter, we perform a
(cid:13)c 0000RAS,MNRAS000,000–000
WMAP constraints on f 5
NL
χ2minimisation.Inparticular,χ2(fNL)isdefinedasfollows
1.5
nstat
χ2 =iX,j=1(vi−hviifNL)Ci−j1(fNL)(vj−hvjifNL), (13) cs 1 996958%%%
i
where hviifNL is the mean of the statistics for a given ist 0.5 data
value of fNL obtained from the 1000 non-Gaussian simula- at
t
tions and Cij(fNL) is the corresponding covariance matrix. r s 0
For fNL << 1000 is reasonable to use the approximation rde
Cij(fNL)≃Cij, where Cij is the covariance matrix for the d o−0.5
Gaussian case. ir
h
Errorbarsontheparameterestimationatdifferentcon- t
−1
fidencelevelsare foundusingtheGaussian simulations. We
also compute the minimum variance in a semi-analytical
−1.5
manner. It is well known that the diagonal of the inverse 0 50 100 k 150 200 250
of the Fisher matrix provides an estimation of the variance
oftheparameters.InordertoestimatetheFishermatrix,we
approximatethedistributionofthestatisticsbyaGaussian. Figure2.ThecubicstatisticsvifromWMAP-7yV+Wdataare
shown. Shadow areas correspond to the 68, 95 and 99 per cent
UsingthisapproximationandtakingintoaccountEqs.(12)
confidencelevelsofthedistributionobtainedfrom1000Gaussian
and(13),thevariancefromtheFisherMatrixcanbewritten
simulations.Thestatisticshavebeenplottedfromlowertohigher
as: variance.
1
σ2 = . (14)
nstat
a C−1a
i ij j 0.1
Xi,j fnl=100
0.08
Inpractice,thedistributionof thestatistics donotfollow a fnl=300
0.06 fnl=−100
perfect Gaussian distribution. Therefore, this variance can
s fnl=−300
beseen as a lower limit to thetrueunderlyingvariance. nt 0.04 fnl=0
e
m
o 0.02
m
er 0
d
4 RESULTS or−0.02
d
r
In this section, we present the analysis of the WMAP-7yr i−0.04
h
t
V+W combined map. On the one hand, we analyse the
−0.06
compatibility of the data with Gaussianity using the cu-
−0.08
bicstatistics definedin Eq.(8)andtheestimator presented
in Eq. (10). On the other hand, we compute the best-fit −0.10 50 100 150 200 250
fNL parameterfromthedatabyminimizingEq.(13).Error k
bars are set using simulations. In addition, we also present
a study of the contribution of the point sources to the es- Figure 3.Meanvalues ofthe cubicstatistics vi from1000 non-
timated fNL value and of the variation of fNL estimated GaussiansimulationswithfNL=0,±100,±300.
fromtwoindependenthemispheres(definedbyHoftuft et al.
2009). theWMAPdataissomehowinthetailofthedistribution,
the χ2 value is not large enough to claim a deviation from
Gaussianity.
4.1 Gaussianity test
As explained in Section 3.1, we have considered a total of
4.2 Constraints on the f parameter
232 cubic statistics, constructed from 18 maps at 9 differ- NL
ent scales. Fig. 2 shows the value of vi for the WMAP 7-yr We have also performed an estimation of the non-linear
V+W data, after applying the KQ75 mask. The plot does parameter fNL. As already mentioned, for this analy-
not show any obvious deviation from Gaussianity. To fur- sis we have used the 1000 non-Gaussian simulations pro-
therstudytheconsistency of thedatawith Gaussianity,we vided by Elsner & Wandelt (2009). In Fig. 3, the mean of
also perform the χ2 test defined in Eq. (10). From 1000 the cubic statistics derived from simulations with fNL =
Gaussian simulations, we estimate the distribution of this 0,±100,±300 is presented.It can beseen that,as stated in
quantity, finding a mean value of χ2 =233, very close to Eq.(12),thestatisticsareproportionaltothevalueoffNL.
the number of degrees of freedom(cid:10)(23(cid:11)2). The value of the After obtaining the cubic statistics for the WMAP-7yr
dispersion is 69, larger than expected for a χ2 distribution data and minimising the χ2 given by Eq. (13), we estimate
with the considered degrees of freedom. However, this may that the best-fit value of fNL is 6. Using Gaussian simu-
be explained by the fact that the distribution of the dif- lations, we find that the contraints for the parameter are
ferent statistics are not purely Gaussian. For the WMAP −28 < fNL < 40 at the 68 per cent confidence level and
data, we find χ2data = 434 with a cumulative probability of −62 < fNL < 72 at the 95 per cent confidence level. It is
P(χ2 6 χ2 ) = 0.96. Although the result indicates that alsointerestingtopointouttheagreementbetweenthedis-
data
(cid:13)c 0000RAS,MNRAS000,000–000
6 B. Casaponsa et al.
persion computedsemi-analytically through theFisherma- 1.4
trix (Eq. 14) and that obtained from Gaussian simulations,
which are both estimated to be around 34. 1.2
Although the constraints provided by the HW are less
s 1
stringentthanthosefoundwithoptimalestimators(suchas nt
e
thebispectrumortheSMHW),theyaresimilarorevenbet- om 0.8
m
ter than those obtained by other mehtods such as needlets
r
e 0.6
(Pietrobon et al.2009;Rudjord et al.2009),theMinkowski d
r
functionals(Hikage et al.2008)ortheN-pdf(Vielva& Sanz o 0.4
d
2009).Moreover,asalreadypointedout,ourestimatorissig- ir
nificantlyfasterthanallthepreviouslymentionedmethods, th 0.2
providingaveryvaluabletool,especiallyforfuturehighres-
0
olution experiments such as Planck. It is also interesting to
point out that we finda moresymmetric constraint around −0.2
0 50 100 150 200 250
zero than those obtained, for instance, by Komatsu et al. k
(2010) or Curto et al. (2010).
In order to study further the robustness of our estima-
Figure 5. Mean values of the vi statistics obtained from 1000
tor,wehaveperformedsomeadditionaltests.Inparticular, Gaussiansimulationswithandwithoutpointsources.Diamonds
we have estimated the mean value and dispersion of the represent CMB plus noise simulations, while crosses correspond
best-fit value of fNL from simulations with different values to simulations including also the point sources. To improve the
of fNL. The left panel of Fig. 4 shows the histograms of visualization, the statistics have been normalisedto unit disper-
theestimatedvaluesoffNL obtainedfromsimulationswith sion.
fNL=-40,0,20,60.Tocarryoutthesetests,wehaveused500
of the1000 non-Gaussian simulations to estimate themean
that correspond to the ones with a lower value of k in the
valueofthethirdorderstatisticshviifNL andtheremaining figure. Taking into account this result our final constraint
500 simulations to obtain estimates of fNL and construct
thehistograms.ThemeanvaluesanddispersionsoffNL are on fNL for the WMAP-7yr data is −69<fNL <65 at the
95 per cent confidencelevel.
giveninthecorrespondingpanels.Inparticular,weseethat
the method is unbiased, since the mean value of the esti-
mated fNL is veryclose to thetrueunderlyingvaluefor all
4.4 Local study of f
the considered cases. In addition, we also plot in the right NL
panel of Fig. 4 how the dispersion of the estimator varies Finally, we have analysed the data considering two in-
as a function of fNL. The standard procedure to estimate dependent hemispheres. In particular, we have consid-
this dispersion is to use Gaussian simulations but, as seen ered the hemispheres associated to the dipolar modu-
in the plot, this gives a minimum in the estimated value of lation proposed by Hoftuft et al. (2009) where the pre-
σ(fNL). However, for small values of fNL, such as the ones ferred direction is pointing towards the Galactic coordi-
foundinthispaper,thevariation issmallandthereforeone nates (l,b)=(224◦,−22◦). We have estimated the best-fit
can safely use the value of the dispersion estimated for the value and constraints on fNL for the WMAP 7-yr data
Gaussian case. in both hemispheres, following the same procedure as for
Finally, wehaverepeated thesame analysisusingaset the full-sky. After correcting the point source contribu-
of300non-GaussiansimulationsgeneratedbyLiguori et al. tion, the constraint found for the northern hemisphere is
(2007), findingvery similar constraints on fNL. −73 < fNL < 119 while for the southern hemisphere we
have −137<fNL <62, both at the 95 per cent confidence
level.Therefore,asitwasthecaseforthefull-sky,bothhemi-
4.3 Point source contribution
spheres are consistent with Gaussianity (i.e, fNL = 0). We
The background of unresolved point sources may introduce have also tested that the results from the hemispheres are
a bias in the estimation of fNL. In order to correct this consistentbetweenthem.Inparticular,wehaveobtainedthe
bias, we have studied the contribution to fNL given by a mean difference and dispersion between the fNL estimates
pointsourcebackgroundthatisaddedtotheGaussiansim- ateach hemisphereforGaussian simulations, findingvalues
ulations. For that purpose, we have produced point source of h∆fNLi=-4 and σ(∆fNL)=71. For the WMAP data, we
simulations following theprocedureof Curto et al. (2009a). have∆fNL =67,whichisperfectlyconsistent withtheval-
Inparticular,pointsourcesmapsaresimulatedaccordingto ues expected from Gaussian simulations. Therefore, we do
the density distribution given by de Zotti et al. (2005) in a not find any assymmetry for the considered hemispheres.
range of intensities between Imin = 1mJy and Imax = 1Jy. These results are in agreement with the analysis based on
Thesemapsarethenconvolvedwiththecorrespondingbeam needletsmadebyPietrobon et al.(2010)andRudjord et al.
and pixel functions and added to the simulations contain- (2010) for the WMAP-5yr data, where several divisions of
ing Gaussian CMB plus noise. The estimated value of fNL the CMB map are studied without finding a significant
whenpointsources arepresentisthencompared totheone asymmetry. In a recent work, Vielva & Sanz (2010) have
obtained from simulations containing only CMB and noise, found an asymmetry in the same hemispheres studied in
findingadifferenceof∆fNL =7±6.Fig.5showstheeffect this paper on the estimation of the fNL using the N-pdf.
thatpointsourceshaveonthevistatistics.Asonewouldex- The disagreement may be caused by the differences on the
pect,theymainlyaffectthestatisticsinvolvingsmallscales, methods. While Pietrobon et al. (2010) and Rudjord et al.
(cid:13)c 0000RAS,MNRAS000,000–000
WMAP constraints on f 7
NL
50
0.06 0.06
σ(f ) = 34 σ(f ) = 35
0.05 µ(fnl) = 1 0.05 µ(fnl) = 21 48
0.04 nl 0.04 nl 46
0.03 0.03
44
0.02 0.02
0.01 0.01 42
)
0 −100 0 100 0 −100 0 100 fNL40
fnl=0 fnl=20 (
σ
0.06 0.06 38
σ(f ) = 36 σ(f ) = 36
0.05 µ(fnl) = −39 0.05 µ(fnl) = 61 36
0.04 nl 0.04 nl
34
0.03 0.03
0.02 0.02 32
0.01 0.01 30
0 −100 0 100 0 −100 0 100 −300 −200 −100 f0 100 200 300
fnl=−40 fnl=60 NL
Figure 4.Theleftpartof thefigureshows the histogramsof theestimated fNL fromsimulationswithvalues offNL =−40,0,20,60.
The mean value and dispersion of fNL for each considered case is indicated in the corresponding panel. The right panel shows the
behaviourofσ(fNL)whenestimatedfromsimulationswithdifferentvaluesoffNL.
(2010) haveworked with thesameresolution as wedid(6.9 ACKNOWLEDGMENTS
arcmin),Vielva & Sanz(2010)focused onscales around 2◦.
We thank R. Ferna´ndez-Cobos for useful discussion on the
Also, the non-Gaussian model used by the former works is
HEALPixwaveletproperties.WealsothankM.Liguoriand
the same as the one used in this paper, whereas the model
F. Elsner for help with the non-Gaussian simulations. The
of the latter stands on the Sachs-Wolfe regime.
authors thank L. Cabellos for computational support. We
acknowledgepartialfinancialsupportfromtheSpanishMin-
isteriodeCienciaeInnovaci´onprojectAYA2007-68059-C03-
02. B. Casaponsa thanks the Spanish Ministerio de Cien-
5 CONCLUSIONS cia e Innovaci´on for a pre-doctoral fellowship. P. Vielva ac-
knowledges financial support from the Ram´on y Cajal pro-
WehavepresentedanewmethodologytoanalysetheGaus- gramme. Theauthors acknowledge thecomputer resources,
sianity of the CMB and to constrain the fNL parameter technical expertise and assistance provided by the Span-
using the so-called HEALPix wavelet. To our knowledge, ishSupercomputingNetwork(RES)nodeatUniversidadde
thedevelopedfNL estimator isthefastest method thathas Cantabria. We acknowledge the use of Legacy Archive for
beenproposed uptodate.Inparticular, forWMAPresolu- MicrowaveBackgroundDataAnalysis(LAMBDA).Support
tion(Nside=512),itis∼102 timesfasterthantheSMHW, foritisprovidedbytheNASAOfficeofSpaceScience.The
∼103 timesfasterthantheKSWbispectrumand107 times HEALPix package was used throughout the data analysis
faster than thegeneral bispectrum estimator. Moreover, al- (G´orski et al. 2005).
though the constraints imposed by our method are not as
stringent as those of the optimal estimators (based on the
bispectrumorontheSMHW),theyareverysimilaroreven
betterthanthoseproposed byalternativemethods,such as
needlets, Minkowski functionals or the N-pdf.
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