Constraints on general primordial non-Gaussianity using wavelets for the Wilkinson Microwave anisotropy probe 7-year data
A. Curto, E. Martinez-Gonzalez, R. B. Barreiro, M. P. Hobson
aa r X i v : . [ a s t r o - ph . C O ] J un Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 14 November 2018 (MN L A TEX style file v2.2)
Constraints on general primordial non-Gaussianity usingwavelets for the Wilkinson Microwave anisotropy probe7-year data
A. Curto, ⋆ E. Mart´ınez-Gonz´alez, R. B. Barreiro, M. P. Hobson Instituto de F´ısica de Cantabria, CSIC-Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain. Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, U.K.
Accepted Received ; in original form
ABSTRACT
We present constraints on the non-linear coupling parameter f nl with the Wilkin-son Microwave Anisotropy Probe (WMAP) data. We use the method based on thespherical Mexican hat wavelet (SMHW) to measure the f nl parameter for three of themost interesting shapes of primordial non-Gaussianity: local, equilateral and orthogo-nal . Our results indicate that this parameter is compatible with a Gaussian distributionwithin the two sigma confidence level (CL) for the three shapes and the results areconsistent with the values presented by the WMAP team. We have included in ouranalysis the impact on f nl due to contamination by unresolved point sources. Thepoint sources add a positive contribution of ∆ f locnl = 2 . ± .
0, ∆ f eqnl = 37 ±
18 and∆ f ortnl = 25 ±
14. As mentioned by the WMAP team, the contribution of the pointsources to the orthogonal and equilateral form is expected to be larger than to the localone and thus it cannot be neglected in future constraints on these parameters. Takinginto account this contamination, our best estimates for f nl are − . f locnl . − f eqnl
202 and − f ortnl
34 at 95% CL. The three shapes are com-patible with zero at 95% CL (2 σ ). Our conclusion is that the WMAP 7-year dataare consistent with Gaussian primordial fluctuations within 2 σ CL. We stress howeverthe importance of taking into account the unresolved point sources in the measure-ment of f nl in future works, especially when using more precise data sets such as theforthcoming Planck data. Key words: methods: data analysis - cosmic microwave background
During the period of inflationary expansion in the very earlystages of the universe, primordial perturbations were gener-ated that are the seeds of the structures that we can observetoday (Starobinskiˇi 1979; Guth 1981; Albrecht & Steinhardt1982; Linde 1982, 1983; Mukhanov et al. 1992). These pri-mordial perturbations were linearly imprinted in the Cos-mic Microwave Background (CMB) anisotropies. Thus thestudy of the CMB anisotropies is a powerful way to under-stand the physics of the early universe. Many observationalCMB projects, for example the NASA WMAP and ESAPlanck missions, different ground based 3D observational ⋆ e-mail: [email protected] http://map.gsfc.nasa.gov/ campaigns of large scale structure and high energy accelera-tors are enabling us to understand better the properties andthe evolution of the universe. From the several observationalapproaches that are available, the search for departures fromGaussianity in the CMB anisotropies with a primordial ori-gin has become a powerful way to discriminate among dif-ferent inflationary scenarios. Inflationary models such asthe widely accepted standard, single-field, slow roll inflationpredict low levels of non-Gaussianity whereas other modelspredict levels of non-Gaussianity that may be detected us-ing the data from current experiments (Bartolo et al. 2004;Komatsu 2009; Yadav & Wandelt 2010; Komatsu 2010). Adetection of a deviation from Gaussianity with a primordialorigin would rule out many inflationary models and wouldhave far reaching implications in the physics of the earlyuniverse.The level of primordial non-Gaussianity is usually c (cid:13) A. Curto et al. parametrised by the non-linear coupling parameter f nl (Verde et al. 2000; Komatsu & Spergel 2001; Bartolo et al.2004). This parameter measures departures from zero in thevalues of the third order quantity known as the bispectrum,characterised through the shape function F ( k , k , k ).The bispectrum is related to Bardeen’s curvature per-turbations Φ( k ) through the 3-point correlation function h Φ( k )Φ( k )Φ( k ) i = (2 π ) δ ( k + k + k ) F ( k , k , k ).Depending on the physical mechanisms of the different infla-tionary models the shape function can take different forms.In this paper we measure the levels of non-Gaussianitypresent in the WMAP data corresponding to the three par-ticular shapes (local, equilateral and orthogonal) that havebeen studied by the WMAP team (Komatsu et al. 2011).The shape function F ( k , k , k ) of these types of non-Gaussianity, their CMB angular bispectra b ℓ ℓ ℓ and theinflationary scenarios that generate these non-Gaussianityare described below. • Local shape.
Significant non-Gaussianity of the localform can be generated for example in multi-field inflation-ary models (Komatsu et al. 2005; Komatsu 2010), the cur-vaton model (Lyth et al. 2003), the inhomogeneous reheat-ing scenario (Dvali et al. 2004; Bartolo et al. 2004), modelsbased on hybrid inflation (Lin 2009), etc. This shape is givenby (see for example Creminelli et al. 2006; Fergusson et al.2010a; Yadav & Wandelt 2010; Komatsu et al. 2011) F ( k , k , k ) = 2 A f nl h k − ( n s − k − ( n s − ++ 1 k − ( n s − k − ( n s − + 1 k − ( n s − k − ( n s − i , (1)and its angular bispectrum is (see for exampleFergusson et al. 2010a; Yadav & Wandelt 2010; Komatsu2010) b locℓ ℓ ℓ = 2 Z ∞ x dx h α ℓ ( x ) β ℓ ( x ) β ℓ ( x ) + β ℓ ( x ) α ℓ ( x ) β ℓ ( x ) + β ℓ ( x ) β ℓ ( x ) α ℓ ( x ) i , (2)where A is the amplitude of the power spectrum P Φ ( k ) = Ak n s − , n s is the spectral index and α ℓ ( x ), β ℓ ( x )are filter functions (see for example Komatsu & Spergel2001; Komatsu et al. 2005; Fergusson et al. 2010a; Komatsu2010). • Equilateral shape.
Significant non-Gaussianity ofthe equilateral form can be generated for example bythe Dirac-Born-Infeld inflation (Silverstein & Tong 2004;Bartolo et al. 2004; Langlois et al. 2008), ghost inflation(Arkani-Hamed et al. 2004), several single-field inflation-ary models in Einstein gravity (Chen et al. 2007) etc.This shape is given by (see for example Creminelli et al.2006; Fergusson et al. 2010a; Yadav & Wandelt 2010;Komatsu et al. 2011) F ( k , k , k ) = 6 A f nl h − k − ( n s − k − ( n s − − k − ( n s − k − ( n s − − k − ( n s − k − ( n s − − k k k ) − n s ) / + n k (4 − n s ) / k − n s ) / k (4 − n s )3 + (5 perm ) oi , (3)and its angular bispectrum is (see for exampleFergusson et al. 2010a; Yadav & Wandelt 2010; Komatsu2010) b eqℓ ℓ ℓ = 6 Z ∞ dxx h − α ℓ ( x ) β ℓ ( x ) β ℓ ( x ) + (2 perm )+ β ℓ ( x ) γ ℓ ( x ) δ ℓ ( x ) + (5 perm ) − δ ℓ ( x ) δ ℓ ( x ) δ ℓ ( x ) i , (4)where γ ℓ ( x ) and δ ℓ ( x ) are filter functions (see for exampleFergusson et al. 2010a; Komatsu 2010). • Orthogonal shape.
Significant non-Gaussianity ofthe orthogonal form can be generated in general single-field models (Cheung et al. 2008; Senatore et al. 2010). Thisshape is given by (see for example Senatore et al. 2010;Yadav & Wandelt 2010; Komatsu et al. 2011) F ( k , k , k ) = 6 A f nl h − k − ( n s − k − ( n s − − k − ( n s − k − ( n s − − k − ( n s − k − ( n s − − k k k ) − n s ) / + n k (4 − n s ) / k − n s ) / k (4 − n s )3 + (5 perm ) oi , (5)and its angular bispectrum is (see for exampleYadav & Wandelt 2010; Komatsu 2010) b ortℓ ℓ ℓ = 18 Z ∞ dxx h − α ℓ ( x ) β ℓ ( x ) β ℓ ( x ) + (2 perm )+ β ℓ ( x ) γ ℓ ( x ) δ ℓ ( x ) + (5 perm ) − δ ℓ ( x ) δ ℓ ( x ) δ ℓ ( x ) i . (6)Many studies have been performed to constrain f nl , espe-cially for the local and the equilateral cases. The first con-straints on f nl were imposed using data sets with low resolu-tion or small sky coverage which led to large uncertainties in f nl . We can report analyses using the Cosmic BackgroundExplorer (COBE) data (Komatsu et al. 2002; Cay´on et al.2003), MAXIMA data (Cay´on et al. 2003; Santos et al.2003), the
Very Small Array (VSA) data (Smith et al.2004), the Archeops data (Curto et al. 2007, 2008) andthe BOOMERang data (De Troia et al. 2007; Natoli et al.2010).Once the WMAP data were available, significant im-provements were achieved in the precision of the estimationof f nl . Many studies have been developed to constrain the f nl using WMAP data and based on different estimators.We can mention the different bispectrum-based estima-tors (see for example Komatsu et al. 2003; Babich et al.2004; Fergusson & Shellard 2007; Spergel et al. 2007;Creminelli et al. 2006, 2007; Yadav & Wandelt 2008;Fergusson & Shellard 2009; Komatsu et al. 2009;Smith et al. 2009; Elsner & Wandelt 2009; Bucher et al.2010; Liguori et al. 2010; Senatore et al. 2010; Smidt et al. The improvement comes from a combination of large sky cover-age, high angular resolution and good sensitivity. This combina-tion improves the signal-to-noise ratio of f nl which for the localcase is proportional to log ( ℓ max ) (Yadav & Wandelt 2010).c (cid:13) , 000–000 onstraints on general primordial non-Gaussianity f nl given its linear dependenceand the fact that in certain ideal conditions bispectrum-based estimators may be the optimal way to measure f nl . However, given that the data are contaminated bydifferent non-Gaussian parasite signals and in most casesonly a fraction of the sky can be used, it is convenientto use additional tools that can help to understandthese effects better. We can mention the tests per-formed using the spherical Mexican hat wavelet (SMHW)(Mukherjee & Wang 2004; Curto et al. 2009a,b, 2011),a HEALPix-based wavelet (Casaponsa et al. 2011a,b),a joint analysis with the SMHW and neural networks(Casaponsa et al. 2011b), needlets (Marinucci et al. 2008;Pietrobon et al. 2009; Rudjord et al. 2009; Pietrobon et al.2010; Rudjord et al. 2010; Pietrobon et al. 2010; Pietrobon2010; Cabella et al. 2010), the Minkowski functionals(Hikage et al. 2006; Gott et al. 2007; Hikage et al. 2008;Matsubara 2010; Takeuchi et al. 2010), the N-PDF distri-bution (Vielva & Sanz 2009, 2010) or a Bayesian approach(Elsner et al. 2010a; Elsner & Wandelt 2010b). Otherworks use the 3D distribution of matter on large scales (seefor example Dalal et al. 2008; Matarrese & Verde 2008;Slosar et al. 2008; Seljak 2009; Desjacques & Seljak 2010;Xia et al. 2010; Baldauf et al. 2011; Hamaus et al. 2011) toconstrain the local f nl .In this paper we focus on the measurement of non-Gaussianity for the previous mentioned shapes using theestimator based on wavelets that has been formerly usedto constrain local f nl (Curto et al. 2009a,b, 2010, 2011).We use the technique described by Fergusson et al. (2010a)to produce non-Gaussian maps with the local, equilateraland orthogonal bispectra for WMAP resolution in realis-tic conditions of partial sky coverage and anisotropic noise.These maps are later used to evaluate the expected valuesof the wavelet third order moments α ijk for each type ofnon-Gaussianity. We finally impose constrains on f nl foreach shape using the wavelet estimator for the WMAP fore-ground reduced and raw data maps. As shown later, unre-solved point sources produce a significant bias in f nl thatshould be considered in the analyses of WMAP data and inthe forthcoming analyses of Planck data, especially for theequilateral and orthogonal shapes.This paper is organized as follows. Section 2 presentsthe non-Gaussian maps that we have used to estimate thequantities needed for this analysis. In Section 3 we presentthe method and the estimator used in this analysis to con-strain f nl . The results of the analysis using WMAP data arepresented in Section 4 and the conclusions are presented inSection 5. Non-Gaussian Monte Carlo simulations are needed in or-der to calibrate the wavelet estimator. We have simulatedour non-Gaussian maps following the algorithm describedby Fergusson et al. (2010a). The non-Gaussian a NGℓm coeffi-cients can be written in terms of the bispectrum and the Gaussian a Gℓm coefficients: a NGℓm = 16 X ℓ ,m ,ℓ ,m b ℓℓ ℓ G mm m ℓℓ ℓ × (cid:18) ℓ ℓ ℓ m m m (cid:19) a G ∗ ℓ m C ℓ a G ∗ ℓ m C ℓ . (7)Using the fact that the shape functions of the local, equilat-eral and orthogonal bispectra are separable, we are able toreduce the number of sums in Eq. 7. This can be done in astraightforward way using Eqs. 2, 4 and 6 in Eq. 7. However,as stated by Hanson et al. (2009); Fergusson et al. (2010a),there are terms that may produce spurious divergences atlow multipoles, large enough to affect the power spectrum ofthe final map. Fergusson et al. (2010a) located the divergentterms and provided equations for the local and equilateralshapes without these terms. A similar procedure can be per-formed with the orthogonal shape. In the next equations wepresent the non-Gaussian a NGℓm coefficients for each of theshapes without divergent terms. • Local bispectrum a NGℓm = Z ∞ dxx α ℓ ( x ) Z d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M β ( x, ~n ) (8) • Equilateral bispectrum a NGℓm = Z ∞ dxx n − α ℓ ( x ) Z d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M β ( x, ~n ) − δ ℓ ( x ) Z d ~nY ∗ ℓm ( ~n ) M δ ( x, ~n ) M δ ( x, ~n )+6 γ ℓ ( x ) Z d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M δ ( x, ~n ) o (9) • Orthogonal bispectrum a NGℓm = Z ∞ dxx n − α ℓ ( x ) Z d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M β ( x, ~n ) − δ ℓ ( x ) Z d ~nY ∗ ℓm ( ~n ) M δ ( x, ~n ) M δ ( x, ~n )+18 γ ℓ ( x ) Z d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M δ ( x, ~n ) o (10)where α ℓ ( x ) = 2 π Z ∞ k dkg Tℓ ( k ) j ℓ ( kx ) (11) β ℓ ( x ) = 2 π Z ∞ k dkP Φ ( k ) g Tℓ ( k ) j ℓ ( kx ) (12) γ ℓ ( x ) = 2 π Z ∞ k dkP / ( k ) g Tℓ ( k ) j ℓ ( kx ) (13) δ ℓ ( x ) = 2 π Z ∞ k dkP / ( k ) g Tℓ ( k ) j ℓ ( kx ) , (14) g Tℓ ( k ) is the radiation transfer function that can be eval-uated using for example the CAMB or gTFast software, P Φ ( k ) is the linear power spectrum, j ℓ ( kx ) is the spherical http://camb.info/ http://gyudon.as.utexas.edu/ ∼ komatsu/CRL/index.htmlc (cid:13) , 000–000 A. Curto et al.
Table 1.
Quadrature in x integration used to compute a ( NG ) ℓm .We have used greater density of points near reionization and re-combination as suggested by Smith & Zaldarriaga (2006). Unitsfor x are Mpc.0 x ,
500 64 points, Gauss-Legendre quadrature9 , x ,
000 128 points, Gauss-Legendre quadrature11 , x ,
800 64 points, Gauss-Legendre quadrature13 , x ,
600 170 points, Gauss-Legendre quadrature14 , x ,
000 42 points, Gauss-Legendre quadrature16 , x ,
000 42 points, Gauss-Legendre quadrature
Bessel function (of the first kind) and the M ( x, ˆ n ) maps aredefined as M α ( x, ˆ n ) = X ℓ,m α ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ (15) M β ( x, ˆ n ) = X ℓ,m β ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ (16) M γ ( x, ˆ n ) = X ℓ,m γ ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ (17) M δ ( x, ˆ n ) = X ℓ,m δ ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ . (18)In Figure 1 we plot the power spec-trum of the non-Gaussian terms a ( aBB ) ℓm ≡ R ∞ dxx α ℓ ( x ) R d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M β ( x, ~n ), a ( dDD ) ℓm ≡ R ∞ dxx δ ℓ ( x ) R d ~nY ∗ ℓm ( ~n ) M δ ( x, ~n ) M δ ( x, ~n ) and a ( gBD ) ℓm ≡ R ∞ dxx γ ℓ ( x ) R d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M δ ( x, ~n ).The power spectrum of the Gaussian part is also plot-ted. We can see that these three terms add negligibleextra-power to the full Gaussian plus non-Gaussian map.Once the a NGℓm terms are computed as a function of thebispectrum and the a Gℓm , the a ℓm coefficients of a simulationwith a given f nl can be written as a ℓm = a Gℓm + f nl a NGℓm .In this paper we have generated a set of 300 non-Gaussian maps for the local, equilateral and orthogonal f nl .We have assumed a Λ CDM model using the parametersthat best fit the WMAP 7-year data (Komatsu et al. 2011).We have computed a power spectrum C ℓ and a transfer func-tion g Tℓ ( k ) using these parameters as inputs for the CAMBsoftware (Lewis et al. 2000) up to ℓ max = 1535. The inte-grals in Eqs. 8, 9 and 10 have been performed using a Gauss-Legendre quadrature. We have used a large density of pointsnear reionization and recombination (see Table 1 for moredetails). A large number of points has been chosen in orderto achieve convergence in the values of the Fisher matrix ofthe bispectrum. σ F ( f nl ) for the three shapes. In Figure 2 theFisher matrix σ F ( f nl ) (Komatsu & Spergel 2001) obtainedwith the three bispectra is plotted for different ℓ max values.Note that these values are comparable with the values pre-sented for example by Yadav & Wandelt (2010). Once the a ℓm of the simulations with non-Gaussianity are generated,we transform them into WMAP maps for each radiome-ter by convolving with the appropriate window functions in the spherical harmonic space and by adding a Gaussian in-strumental noise simulation in the real space (Bennett et al.2003). We use an estimator that is based on third-order statis-tics generated by the different possible combinations of thewavelet coefficient maps of the SMHW evaluated at certainangular scales. See for example Antoine & Vandergheynst(1998); Mart´ınez-Gonz´alez et al. (2002); Vielva (2007);Mart´ınez-Gonz´alez (2008) for detailed information aboutthis wavelet. This estimator is described and used to searchfor blind non-Gaussian deviations and constrain local f nl inCurto et al. (2009a,b, 2011).We consider the same set of angular scales R i selectedin Curto et al. (2011). After evaluating the wavelet coeffi-cient map w ( R i ; b ) for each angular scale R i we computethe third order moments q ijk for each possible combinationof three angular scales { i, j, k } . As mentioned in Curto et al.(2011), the expected values of the cubic statistics are linearlyproportional to f nl h q ijk i f nl = α ijk f nl , (19)where the α ijk term is linearly related to the bispectrum. Weevaluate these α ijk quantities for the local, equilateral andorthogonal bispectra by averaging the values of the estima-tors obtained with the non-Gaussian simulations describedin the previous Section. We then compute a χ statistic inorder to constrain each f nl χ ( f nl ) = X ijk,rst ( q obsijk − α ijk f nl ) C − ijk,rst ( q obsrst − α rst f nl ) , (20)where q obsijk is the value of the statistics obtained for the ac-tual data map and C is the covariance matrix among the dif-ferent statistics q ijk . The covariance matrix is estimated us-ing the q ijk statistics corresponding to 10,000 WMAP Gaus-sian simulations. A detailed study described in Curto et al.(2011) was carried out in order to compute correctly itsinverse avoiding possible degeneracies. The α ijk statisticsare estimated using the set of 300 non-Gaussian simulationstransformed into WMAP V + W maps. Although we foundanalytical expressions for the covariance matrix C ijk,rst andthe α ijk quantities (see Curto et al. 2011), those expressionsare only valid for the particular ideal case of full sky mapsand white isotropic noise. For a realistic case, the analyticalexpressions become more complicated and the best practicalapproach to compute those quantities is using simulations.Finally, this estimator is also applied to a set of Gaus-sian maps in order to obtain an empirical estimate of theuncertainties of f nl . Additionally we also compute the valueof the f nl Fisher matrix using the wavelet coefficients σ F ( f nl ) = 1 P ijk,rst α ijk C − ijk,rst α rst . (21) Note that for this estimator there is no need to subtract anylinear term due to the anisotropic noise as in the case of theKSW estimator. The reason is that the non-ideal aspects of theanalysis (as the mask, the anisotropic noise, etc.) are included inthe covariance matrix and the α ijk coefficients.c (cid:13)000
Bessel function (of the first kind) and the M ( x, ˆ n ) maps aredefined as M α ( x, ˆ n ) = X ℓ,m α ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ (15) M β ( x, ˆ n ) = X ℓ,m β ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ (16) M γ ( x, ˆ n ) = X ℓ,m γ ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ (17) M δ ( x, ˆ n ) = X ℓ,m δ ℓ ( x ) a Gℓ,m Y ℓ,m (ˆ n ) C ℓ . (18)In Figure 1 we plot the power spec-trum of the non-Gaussian terms a ( aBB ) ℓm ≡ R ∞ dxx α ℓ ( x ) R d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M β ( x, ~n ), a ( dDD ) ℓm ≡ R ∞ dxx δ ℓ ( x ) R d ~nY ∗ ℓm ( ~n ) M δ ( x, ~n ) M δ ( x, ~n ) and a ( gBD ) ℓm ≡ R ∞ dxx γ ℓ ( x ) R d ~nY ∗ ℓm ( ~n ) M β ( x, ~n ) M δ ( x, ~n ).The power spectrum of the Gaussian part is also plot-ted. We can see that these three terms add negligibleextra-power to the full Gaussian plus non-Gaussian map.Once the a NGℓm terms are computed as a function of thebispectrum and the a Gℓm , the a ℓm coefficients of a simulationwith a given f nl can be written as a ℓm = a Gℓm + f nl a NGℓm .In this paper we have generated a set of 300 non-Gaussian maps for the local, equilateral and orthogonal f nl .We have assumed a Λ CDM model using the parametersthat best fit the WMAP 7-year data (Komatsu et al. 2011).We have computed a power spectrum C ℓ and a transfer func-tion g Tℓ ( k ) using these parameters as inputs for the CAMBsoftware (Lewis et al. 2000) up to ℓ max = 1535. The inte-grals in Eqs. 8, 9 and 10 have been performed using a Gauss-Legendre quadrature. We have used a large density of pointsnear reionization and recombination (see Table 1 for moredetails). A large number of points has been chosen in orderto achieve convergence in the values of the Fisher matrix ofthe bispectrum. σ F ( f nl ) for the three shapes. In Figure 2 theFisher matrix σ F ( f nl ) (Komatsu & Spergel 2001) obtainedwith the three bispectra is plotted for different ℓ max values.Note that these values are comparable with the values pre-sented for example by Yadav & Wandelt (2010). Once the a ℓm of the simulations with non-Gaussianity are generated,we transform them into WMAP maps for each radiome-ter by convolving with the appropriate window functions in the spherical harmonic space and by adding a Gaussian in-strumental noise simulation in the real space (Bennett et al.2003). We use an estimator that is based on third-order statis-tics generated by the different possible combinations of thewavelet coefficient maps of the SMHW evaluated at certainangular scales. See for example Antoine & Vandergheynst(1998); Mart´ınez-Gonz´alez et al. (2002); Vielva (2007);Mart´ınez-Gonz´alez (2008) for detailed information aboutthis wavelet. This estimator is described and used to searchfor blind non-Gaussian deviations and constrain local f nl inCurto et al. (2009a,b, 2011).We consider the same set of angular scales R i selectedin Curto et al. (2011). After evaluating the wavelet coeffi-cient map w ( R i ; b ) for each angular scale R i we computethe third order moments q ijk for each possible combinationof three angular scales { i, j, k } . As mentioned in Curto et al.(2011), the expected values of the cubic statistics are linearlyproportional to f nl h q ijk i f nl = α ijk f nl , (19)where the α ijk term is linearly related to the bispectrum. Weevaluate these α ijk quantities for the local, equilateral andorthogonal bispectra by averaging the values of the estima-tors obtained with the non-Gaussian simulations describedin the previous Section. We then compute a χ statistic inorder to constrain each f nl χ ( f nl ) = X ijk,rst ( q obsijk − α ijk f nl ) C − ijk,rst ( q obsrst − α rst f nl ) , (20)where q obsijk is the value of the statistics obtained for the ac-tual data map and C is the covariance matrix among the dif-ferent statistics q ijk . The covariance matrix is estimated us-ing the q ijk statistics corresponding to 10,000 WMAP Gaus-sian simulations. A detailed study described in Curto et al.(2011) was carried out in order to compute correctly itsinverse avoiding possible degeneracies. The α ijk statisticsare estimated using the set of 300 non-Gaussian simulationstransformed into WMAP V + W maps. Although we foundanalytical expressions for the covariance matrix C ijk,rst andthe α ijk quantities (see Curto et al. 2011), those expressionsare only valid for the particular ideal case of full sky mapsand white isotropic noise. For a realistic case, the analyticalexpressions become more complicated and the best practicalapproach to compute those quantities is using simulations.Finally, this estimator is also applied to a set of Gaus-sian maps in order to obtain an empirical estimate of theuncertainties of f nl . Additionally we also compute the valueof the f nl Fisher matrix using the wavelet coefficients σ F ( f nl ) = 1 P ijk,rst α ijk C − ijk,rst α rst . (21) Note that for this estimator there is no need to subtract anylinear term due to the anisotropic noise as in the case of theKSW estimator. The reason is that the non-ideal aspects of theanalysis (as the mask, the anisotropic noise, etc.) are included inthe covariance matrix and the α ijk coefficients.c (cid:13)000 , 000–000 onstraints on general primordial non-Gaussianity Figure 1.
From left to right, the power spectrum of the non-Gaussian a ( aBB ) ℓm , a ( dDD ) ℓm and a ( gBD ) ℓm coefficients (lower line) comparedwith the Gaussian part of the power spectrum (upper line). Figure 2.
From left to right, the Fisher matrix σ F ( f nl ) versus ℓ max for the local, equilateral and orthogonal bispectra b ℓ ℓ ℓ definedin Eqs. 2, 4 and 6. Table 2.
Constraints on the local, equilateral and orthogonal f nl for the clean and raw maps and their uncertainties obtained withthe Fisher matrix σ F ( f nl ) and with simulations (RMS).CASE raw f nl clean f nl σ F ( f nl ) MEAN RMSlocal 25.0 32.5 22.5 0.0 23.00equilateral 28.0 -53.0 145.0 1.0 156.0orthogonal -119.0 -155.0 106.0 0.0 112.00 f nl using WMAP data We use the combined WMAP 7-year V and W band maps atthe HEALPix (G´orski et al. 2005) resolution of N side = 512.We consider both raw and foreground reduced data mapsas Komatsu et al. (2011). The maximum multipole chosenin this analysis is 3 N side although the noise contaminationstarts to be significant at ℓ ∼ f nl and provide the value of theFisher and the simulated σ ( f nl ). During all the analysis weuse the WMAP KQ75 mask (Gold et al. 2011). In Table 2we summarize our results. We find that for the three cases,the parameters are compatible with zero at 95% CL. Wewould like to note the different effect that the foregroundsproduce on different shapes: whereas it is negative for thelocal shape, it is positive for the equilateral and orthogonalshapes. For all the cases, σ F ( f nl ) is lower than the valueobtained with simulations ( ∼
95% depending on the shape).We think that this small discrepancy is due to the limitednumber of simulations. We have checked that our estimatoris unbiased. We have estimated the f nl values of 100 non-Gaussian simulations with an input f nl = 100 and usedthe remaining 200 non-Gaussian simulations to estimate the α ijk . The results are f locnl = 99 . ± . f eqnl = 98 ±
150 and f ortnl = 97 ± f nl taking into account the expected errors in the mean forthe available number of realizations. Our best estimates forthe clean maps are: • Local : f nl = 32 . ± . • Equilateral: f nl = − ±
145 (68% CL) • Orthogonal: f nl = − ±
106 (68% CL)The values match well the results presented byKomatsu et al. (2011) within one sigma error-bars.The differences can be explained by the different sensitivityof the bispectrum and wavelet estimators to the possiblenon-cosmological residuals present in the data.
We have also estimated the contribution of undetected pointsources using the source number counts dN/dS derived fromde Zotti et al. (2005). We have used point source simula-tions based on this dN/dS. We have chosen a maximumflux for the bright sources such that the power spectrumfor the Q band is compatible with the value provided bythe WMAP team, A ps = 0 . ± . µK sr in an-tenna units (Larson et al. 2011). We have estimated the Using a set of 300 non-Gaussian simulations generated followingthe procedure by Liguori et al. (2003, 2007) our best estimateis f nl = 37 ±
21 (68% CL). These maps were generated in adifferent way: the non-Gaussianity is introduced in the primordialcurvature perturbation Φ( x ) = Φ L ( x ) + f nl (cid:0) Φ L ( x ) − h Φ L ( x ) i (cid:1) and then extrapolated to the CMB. This process add extra non-Gaussianity at higher moments whereas the procedure used in thispaper and in Fergusson et al. (2010a) just adds non-Gaussianityto the third order moments (bispectrum).c (cid:13) , 000–000 A. Curto et al. best-fitting f nl value for two sets of 1,000 maps. The first setconsists of 1,000 Gaussian CMB + noise maps and the sec-ond consists of the same Gaussian CMB + noise maps plusthe point source maps. For each map with point sourceswe estimate its best-fitting f nl parameter and compare withthe value obtained for the same map without point sources.The difference ∆ f nl provides an estimate of the impact on f nl due to the unresolved point sources. The point sourcesadd a contribution of ∆ f nl = 2 . ±
3, ∆ f nl = 37 ±
18 and∆ f nl = 25 ±
14 for the local, equilateral and orthogonalforms respectively.To check further these results, we have used an alterna-tive method to estimate the point source contamination to f nl given by the expression∆ f nl = P ijk,rst h q ijk i ps C − ijk,rst α rst P ijk,rst α ijk C − ijk,rst α rst , (22)where h q ijk i ps is the expected value of the third order mo-ments due to the point sources. The results are ∆ f nl = 2 . f nl = 38 and ∆ f nl = 24 which agree with the values pre-viously obtained with simulations. Taking into account thepoint source contribution, our best estimates of f nl are: • Local: f nl = 30 . ± . • Equilateral: f nl = − ±
146 (68% CL) • Orthogonal: f nl = − ±
107 (68% CL)Fig. 3 contains the histograms of the best-fitting f nl valuesfor each shape corresponding to 1,000 CMB + noise Gaus-sian simulations and the values of the data after the pointsource correction. Note that the point sources add a signifi-cant contribution to the equilateral and orthogonal shapes.We agree with Komatsu et al. (2011) that the WMAP seven-year data are consistent with Gaussian primordial fluctua-tions for the three considered shapes. Planck will be ableto address this issue with more detail due to its increasedsensitivity and power to clean the signal. We have imposed constraints on primordial non-Gaussianitywith the WMAP 7-year data using the wavelet based esti-mator. In this analysis we have considered the combinedV+W maps and the KQ75 mask. In particular, we have fo-cused in three shapes with particular interest for the physicsof inflation in the early universe: the local, equilateral andorthogonal bispectra.We have simulated the non-Gaussian maps for eachof the considered shapes and estimated with these simu-lations the required quantities for our estimator. Our re-sults are compatible with the values obtained by the WMAPteam and our uncertainties are very similar to the er-ror bars obtained with the optimal bispectrum estimator(Komatsu et al. 2011).In addition we have estimated the contribution of thepoint sources. In the particular case of the local f nl , the con-tribution is ∆ f locnl = 2 . ± .
0. This is similar to the valuesobtained by the WMAP team (Komatsu et al. 2011) andits contribution to the parameter is not significant. How-ever we have detected a non-negligible contribution to theequilateral and orthogonal shapes due to the unresolvedpoint sources. In particular, we have found ∆ f eqnl = 37 ±
18 and ∆ f ortnl = 25 ±
14 (68%CL). These large values werealready predicted by Komatsu et al. (2011) although theydid not provide actual figures. This contribution should betaken carefully into account in future constraints on f nl withWMAP and Planck data. Considering the point sources, ourbest estimates of f nl are f locnl = 30 . ± . f eqnl = − ± f ortnl = − ± f nl param-eters are compatible with zero within the 2 σ CL and ourresults are in agreement with Komatsu et al. (2011).The wavelet estimator has been tested and carefullychecked with the available WMAP data in this and severalprevious works (Curto et al. 2009a,b, 2011). It is now readyand being upgraded to analyse the forthcoming Planck data.In future works we will also use the wavelet estimator jointlywith neural networks to constrain these shapes along thelines of Casaponsa et al. (2011b) where this procedure hasbeen already applied for the local f nl using WMAP data.This later process helps to speed up the calculations sinceit is not necessary to estimate the covariance matrix of thecubic statistics and it avoids all the possible complicationsin the computation of the inverse covariance matrix. ACKNOWLEDGMENTS
The authors are thankful to Eiichiro Komatsu and MicheleLiguori for their useful comments that have helped in theproduction of this paper. The authors thank J. Gonz´alez-Nuevo for providing the dN/dS counts and for his usefulcomments on unresolved point sources. The authors alsothank Biuse Casaponsa, Airam Marcos-Caballero, SabinoMatarrese and Patricio Vielva for useful comments on differ-ent computational and theoretical issues on the primordialnon-Gaussianity. The authors acknowledge partial financialsupport from the Spanish Ministerio de Ciencia e Innovaci´onproject AYA2010-21766-C03-01, the CSIC-Royal Societyjoint project with reference 2008GB0012 and the ConsoliderIngenio-2010 Programme project CSD2010-00064. A. C.thanks the Universidad de Cantabria for a post-doctoral fel-lowship. The authors acknowledge the computer resources,technical expertise and assistance provided by the Span-ish Supercomputing Network (RES) node at Universidad deCantabria. We acknowledge the use of Legacy Archive forMicrowave Background Data Analysis (LAMBDA). Supportfor it is provided by the NASA Office of Space Science. TheHEALPix package was used throughout the data analysis(G´orski et al. 2005).
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