Directional Variations of the Non-Gaussianity Parameter f_NL
Øystein Rudjord, Frode K. Hansen, Xiaohong Lan, Michele Liguori, Domenico Marinucci, Sabino Matarrese
aa r X i v : . [ a s t r o - ph . C O ] M a y Draft version November 11, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
DIRECTIONAL VARIATIONS OF THE NON-GAUSSIANITY PARAMETER f NL Øystein Rudjord , Frode K. Hansen , Xiaohong Lan Michele Liguori Domenico Marinucci Sabino Matarrese Draft version November 11, 2018
ABSTRACTWe investigate local variations of the primordial non-Gaussianity parameter f NL in the WMAPdata, looking for possible influence of foreground contamination in the full-sky estimate of f NL . Wefirst improve the needlet bispectrum estimate in ( Rudjord et al. 2009) on the full-sky to f NL = 73 ± f NL estimatesclose to the galactic plane and conclude that foregrounds are unlikely to affect the estimate of f NL inthe V and W bands even for the smaller KQ85 mask. In the Q band however, we find unexpectedlyhigh values of f NL in local estimates close to the galactic mask, as well as significant discrepanciesbetween Q band estimates and V/W band estimates. We therefore conclude that the Q band is toocontaminated to be used for non-Gaussianity studies even with the larger KQ75 mask. We furthernoted that the local f NL estimates on the V+W channel are positive on all equatorial bands from thenorth to the south pole. The probability for this to happen in a universe with f NL = 0 is less thanone percent. Subject headings: cosmic microwave background — cosmology: observations — methods: statistical INTRODUCTION
Although the Cosmic Microwave Background (CMB)fluctuations are usually assumed to follow a Gaussiandistribution, the consensus is that this is only an ap-proximation valid to a certain level of precision. Thedeviations from Gaussianity seem to be small and aretherefore difficult to estimate precisely. However, accu-rate knowledge of both the size and nature of such a de-viation would be of high value in cosmology. The searchfor such anomalies in the CMB has therefore attractedmuch attention recently.Most models for inflation predict the CMB fluctua-tion to be slightly non-Gaussian. This level of non-Gaussianity is measured by the parameter f NL (see e.g.(Bartolo et al. 2004)). Alternative inflationary scenariospredict different values of f NL , thus making an accu-rate estimate of this parameter crucial for understand-ing the physics of the inflationary era. Several f NL pa-rameters have been considered in the literature, corre-sponding to different ansatz for the shape of primordialnon-Gaussianities. In this paper we focus on local non-Gaussianity, parametrized by f localNL (simply f NL in thefollowing).In (Rudjord et al. 2009), we used needlets to esti-mate a value of f NL = 84 ±
40 (local type) differ-ent from zero at the 2 σ level. This was in agree- email: [email protected] Institute of Theoretical Astrophysics, University of Oslo,P.O. Box 1029 Blindern, N-0315 Oslo, Norway Centre of Mathematics for Applications, University of Oslo,P.O. Box 1053 Blindern, N-0316 Oslo, Norway Dipartimento di Matematica, Universit`a di Roma ‘Tor Ver-gata’, Via della Ricerca Scientifica 1, I-00133 Roma, Italy Department of Applied Mathematics and TheoreticalPhysics, Centre for Mathematical Sciences, University of Cam-bridge, Wilberfoce Road, Cambridge, CB3 0WA, United King-dom Dipartimento di Fisica, G. Galilei, Universit`a di Padova andINFN, Sezione di Padova,via Marzolo 8,I-35131 Padova, Italy ment with previous estimates (Yadav & Wandelt 2008;Komatsu et al. 2009; Smith et al. 2009; Curto et al.2008, 2009; Pietrobon et al. 2008, 2009) where valuesof f NL deviating from zero with about 2 σ were also re-ported. Also in agreement with previous estimates, wefound that the value of f NL increases with smaller skycuts. This could be a result of smaller error bars due tothe increased amount of data, or it could be an indicationof foreground residuals near the galactic plane influenc-ing the value of f NL . This is our main motivation forestimating f NL on smaller parts of the sky. By havinglocal estimates of f NL we can check whether the highvalue of f NL comes from areas close to the galactic planethus indicating the influence of foreground residuals, orwhether consistent f NL values are found in different partsof the sky. The second motivation for studying the di-rectional dependence of f NL , is due to several reportsof the CMB deviating from statistical isotropy in variousways (Hansen et al. 2008; Groeneboom & Eriksen 2009;Hoftuft et al. 2009; Eriksen et al. 2004; Hansen et al.2004; Vielva et al. 2004; Tegmark et al. 2003). Here wecheck whether a similar asymmetry is seen in the valueof f NL .In (Rudjord et al. 2009), the bispectrum of spheri-cal needlets (Lan & Marinucci 2008) was used to obtain f NL . The localization and uncorrelation properties ofneedlets make them a convenient tool for studying local-ized regions on the sky (see (Baldi et al. 2006)). We willhere use a similar analysis to estimate the level of non-Gaussianity on selected regions on the sky and in thisway study the spatial variations of f NL . We thereforerefer the reader to (Rudjord et al. (2009)) for a moredetailed description of the procedure, as well as for amore extensive list of references.This paper is organized as follows. In section 2 wedescribe the data used for the analysis, in section 3 wedescribe the non-Gaussian maps and in section 4 we out-line the method used for estimating f NL . We apply theprocedure to WMAP data and present the results in sec-tion 5 before we summarize and conclude in section 6. DATA
This analysis was performed using the foreground re-duced co-added
V+W frequency bands of the
WMAP five year data at Healpix resolution N side = 512. Alsothe individual Q, V and W bands were used for consis-tency tests. The Gaussian simulations were generatedusing the best fit power spectrum from the WMAP fiveyear release. We also used the beam and noise propertiessupplied by the
WMAP team. For masking out galac-tic foregrounds we used the
KQ75 as well as the smaller
KQ85 mask. NON-GAUSSIAN MAPS
The non-Gaussian maps used for the analysis havebeen generated using the algorithm developed in(Liguori et al 2003; Liguori et al. 2007) (see alsoElsner & Wandelt (2009) for recent developments). Webriefly review the general structure of the algorithm here,while addressing the reader to the above mentioned pa-pers for further details. The CMB multipoles a ℓm are re-lated to the primordial gravitational potential Φ throughthe well known formula: a ℓm = Z d k (2 π ) Φ( k ) Y ℓm (ˆ k )∆ ℓ ( k ) , (1)where ∆ ℓ ( k ) is the radiation transfer function andthe potential is written in Fourier space. However, forthe case of primordial non-Gaussianity, the primordialpotential takes a very simple expression in real space,where: Φ( x ) = Φ L ( x ) + f NL (cid:2) Φ L ( x ) − (cid:10) Φ L ( x ) (cid:11)(cid:3) . (2)In the previous expression Φ L is a Gaussian ran-dom field, characterized by a primordial power spectrum P ( k ) = Ak n − ; in the following we will refer to Φ L ( x )as the Gaussian part of the primordial potential. Theremaining non-Gaussian part of the potential is simplythe square of the Gaussian part in each point (mod-ulo a constant term, necessary to enforce the condition h Φ( x ) i = 0; however it is clear that this term affects onlythe CMB monopole)It is then convenient to work directly in real space andrecast formula (1) in the following form a ℓm = Z d x Φ( x ) Y ℓm (ˆ r )∆ ℓ ( r ) , (3)where ∆ ℓ ( r ) ≡ R dk k j ℓ ( kr )∆ ℓ ( k ) are the real spacetransfer functions, j ℓ ( kr ) is a spherical Bessel function,and r is a lookback conformal distance. This formulasuggests to structure the algorithm in the following steps1. Generate the Gaussian part Φ L of the potential ina box whose side is the present cosmic horizon.2. Square the Gaussian part point by point to get thenon-Gaussian part. http://healpix.jpl.nasa.gov
3. Expanding in spherical harmonics the Gaussianand non-Gaussian parts of the potential for differ-ent values of the radial coordinate r in the simula-tion box.4. Convolve the spherical harmonic expansions of Φ L and Φ NL with the radiation transfer function ∆ ℓ ( r )in order to obtain the Gaussian and non-Gaussianpart of the multipoles of the final NG CMB sim-ulation. For a given choice of the non-Gaussianparameter f NL a CMB map is then obtained sim-ply through the linear combination a ℓm = a Lℓm + f NL a NL ℓm (the superscripts L and NL always indi-cating Gaussian and non-Gaussian respectively).The most difficult and time consuming part in this pro-cess is actually the generation of the Gaussian part of thepotential Φ. The difficulty arise from the fact that we areworking in a box of the size of the present cosmic horizon,that in conformal time is about 15 Gpc, but at the sametime a cell in this box must have a size no bigger than20 Mpc in order to resolve the last scattering surface,where most of the CMB signal is generated. It turnsout (see Liguori et al (2003); Liguori et al. (2007) fordetails and more explanations) that a convenient way toachieve this is to work directly in spherical coordinates,use a non uniform discretization of the simulation box(since no points are needed in a large region of the boxwhere photons are just free streaming) and generate themultipoles of the expansion of Φ L ( x ) through the follow-ing two step approach:1. Generate uncorrelated radial multipoles n ℓm ( r ),gaussianly distributed and characterized by the fol-lowing spectrum: (cid:10) n ℓ m ( r ) n ∗ ℓ m ( r ) (cid:11) = δ D ( r − r ) r δ ℓ ℓ δ m m ; (4)where δ D is the Dirac delta function.2. Filter the multipoles n ℓm with suitable functionsin order to produce a Gaussian random field withthe properties of the multipole expansion of theprimordial Gaussian potential Φ L . It can be shownthat the expression of the filter functions is: W ℓ ( r, r ) = 2 π Z dk k p P Φ ( k ) j ℓ ( kr ) j ℓ ( kr ) , (5)where P Φ is the primordial power spectrum, andthe filtering operation takes the formΦ L ℓm ( r ) = Z dr r n ℓm ( r ) W ℓ ( r, r ) . (6)In the last expression Φ L ℓm ( r ) are the desired quan-tities i.e. the multipoles of the expansion of theGaussian part of the primordial potential for agiven r . METHOD
The bispectrum has shown to be the most powerful toolfor estimating f NL . The bispectrum is zero for a Gaus-sian field, and any significant deviations from a zero bis-pectrum is therefore a non-Gaussian signal. Needlets area new type of spherical wavelets which were introducedby (Narcowich et al. 2006). For our analysis we usethe bispectrum of needlet coefficients (Lan & Marinucci2008) to estimate f NL . The localization properties ofneedlets make it possible to obtain the bispectrum inseveral different regions of the sky with very little addi-tional costs in terms of CPU time compared to one fullsky analysis. Although the needlet bispectrum does notyield optimal error bars on f NL (see (Smith et al. 2009)for the optimal method), the advantage is the possibil-ity of a fast and easy calculation of local estimates forconsistency checks.The needlet coefficients are denoted β Bjk , j is frequency, k the direction on the sky (we will take k as the pixelnumber in the Healpix grid) and the parameter B char-acterizes the localization in frequency domain. Indeed,needlets allow for a tight control of localization in har-monic space and uncorrelation in pixel space; these prop-erties are valuable for statistical inference and are notshared by other wavelet constructions, please refer to(Baldi et al. 2006)), (Marinucci et al. 2008) for detailsand further references (see also (Pietrobon et al. 2006)and (Fay et al. 2008)). For instance, for a given valueof B a needlet coefficient only contains information onmultipoles in the range ℓ = [ B j − , B j +1 ]. Thus, the pa-rameter B controls localization in harmonic space: smallvalues of B correspond to small ranges of frequencies j ,while the reverse is true for larger B .The needlet bispectrum with base B may be expressedas I j j j (dΩ) = pixels ∈ dΩ X k β j k β j k β j k σ j k σ j k σ j k (7)where σ Bjk are the standard deviation of the needlet co-efficients β Bjk for a Gaussian map and dΩ is the region ofthe sky for which the bispectrum is calculated.For the estimation of f NL we used a similar proce-dure as in Rudjord et al. (2009), but with a few smallchanges. We used a higher value for B , resulting in fewerneedlet scales. Intuitively one would expect this to givea poor result, since each needlet scale would cover a largeinterval in ℓ -space. However, a higher value for B alsogives better localization properties in pixel-space, thusminimizing the influence of the mask and thereby reduc-ing the error bars. Additionally, this greatly reduces thecomputational cost of the analysis.We will calculate the needlet bispectra using theneedlet coefficients calculated on the pixels k of a N side =512 map. We will use all the N side = 512 pixels k whichare inside an N side = 2 pixel as our smallest region dΩ.We thus obtain 48 bispectra I j j j ( p ) where p denotes apixel of the N side = 2 pixelization. Having obtained these48 bispectra it is straightforward to construct the bispec-trum on sky patches with different shapes and sizes. Wesee from eq. 7 that the bispectrum of a larger patch is then simply I j j j (region) = X p I j j j ( p ) , (8)where the sum over p goes over the N side = 2 pixelswithin the desired region.In order to estimate f NL we perform a χ analysis forevery map and region to be investigated χ ( f NL ) = d T ( f NL ) C − d ( f NL ) (9)where the data vector is d = I obs j j j − h I j j j ( f NL ) i = I obs j j j − f NL h ˆ I j j j i . (10) h ˆ I j j j i is here the average first order non-Gaussianbispectrum obtained using non-Gaussian simulations(Liguori et al. 2007). See (Rudjord et al. 2009) for de-tails. The corresponding covariance matrix C is evalu-ated by means of Monte-Carlo simulations.Differentiating to find the value for f NL which givesthe lowest χ yields the “Generalized Least Squares” es-timate: f NL = D ˆ I j j j E T C − I j j j D ˆ I j j j E T C − D ˆ I j j j E . (11)We estimate local f NL values according to the proce-dure described in (Rudjord et al. 2009), but with oneimportant difference: We calculate and save the bispec-tra calculated on each individual N side = 2 pixel for eachsimulation. Then, when the bispectrum and thereby f NL is estimated for a larger region, the bispectra for the dif-ferent N side = 2 pixels are added up according to eq. 8and the correlation matrix is constructed from this finalbispectrum for each larger region. RESULTS
First we estimated f NL on the full sky using B = 1 . ℓ max =1500. This was the best trade-off (lowest error bars)between needlet coefficients for low values of B beingmore affected by the mask but having more frequencies j and for high values of B being less affected by the maskbut having fewer scales. Otherwise we followed the sameprocedure as in Rudjord et al. (2009). The results arepresented in table 1. freq. channel KQ75 KQ85V + W ±
31 78 ± V ±
35 55 ± W ±
37 72 ± Q − ± − ± TABLE 1 f NL estimates and σ error bars. These estimates arefound using the full CMB sky. As we see from the results, the combined V + W chan-nel gives a 2 σ deviation from Gaussianity. The individual V and W channels are consistent (within 2 σ ), but the Q channel deviates significantly from the others, suggestingpossible contamination by foregrounds.We repeat the above analysis on localized regions onthe sphere. First the bispectra were found for an ensem-ble of Gaussian simulations on each of the 48 ( N side = 2)Healpix patches on the sky. We then combined thesepatches in three different ways according to eq. 8.1. Hemispheres
The larger regions were defined tobe hemispheres. For each of the 48 directions de-fined by the ( N side = 2) pixel centers, f NL wasestimated on a hemisphere centered on this direc-tion.2. 45 ◦ discs Same as for the hemispheres, estimat-ing f NL on discs with 45 ◦ radius instead of hemi-spheres.3. Equatorial rings
These regions were defined to bethe 7 constant latitude rings in which the N side = 2healpix pixels are ordered. Each of these ringscovers a relatively small part of the sky, and theerror-bars are therefore large. However they arenot overlapping, and the f NL estimates on the dif-ferent rings can therefore be considered nearly in-dependent (except for the largest scales).The f NL estimates from the 45 ◦ discs with the smaller KQ85 mask are shown in figure 1, where each pixel rep-resents the f NL estimate on a disc centered on the pixel.The lower plot shows the same map, but f NL for each re-gion has been normalized by its standard deviation. Wesee that even for the smaller mask, there is no evidencefor particularly high values of f NL in the galactic region,but note that most of the values are positive. In fig-ure 2 we show the corresponding map of f NL estimatesover hemispheres while using the KQ75 mask. As an ad-ditional test of consistency we have produced the sameestimates based on bispectra obtained from N side = 4pixels instead of N side = 2 pixels. The result is shown inthe same figure. We can see that the two maps show thesame structures on the sky.As seen from figure 1 the distribution of f NL on the skyshows a dipole. For the f NL /σ map of the 45 ◦ discs ofthe V+W channel (KQ85), the dipole has a maximum at θ = 129 ◦ , φ = 96 ◦ with an amplitude of T f NL ℓ =1 = 61. Weinvestigated whether such a dipolar distribution of f NL was common in Gaussian simulations and found the valueof the dipole amplitude to be in good agreement withsimulated maps. The dipole is therefore to be expected.Investigation of the hemisphere results as well as for the KQ75 mask using both the combined
V+W channel andthe individual Q , V and W frequency channels yieldedsimilar results.In order to check whether the 48 estimates of f NL wereinternally consistent, we combined the estimates to forma data vector d = [ f NL , f NL , ..., f NL ] for a χ test. Sim-ulations were then used to find a covariance matrix con-taining the covariances between the 48 estimates. Then a χ test was performed on Gaussian simulations as well ason the WMAP data. The χ values of the data were com-pared with those of the simulations. The results, bothfor the hemisphere estimates as well as for the 45 ◦ diskestimates, was that the χ of the WMAP data was fully
Fig. 1.—
The upper figure shows f NL estimates on 45 ◦ discscentered on the given pixels, while the lower figure shows the sameestimates divided by their standard deviation. The estimates weremade on the V+W channel using the KQ85 galactic mask. Fig. 2.—
Both the figures shows f NL estimates on hemispheres,but for the upper one the estimates are performed while usingresolution N side = 2, while for the lower one we have used N side =4. We can see that the two maps show the same structures. Theseestimates were made on the V+W channel using the KQ75 galacticmask. consistent with Gaussian simulations (within 2 σ ). Thelocal f NL estimates are therefore internally consistent.As a further test we obtained estimates of f NL on equa-torial rings. The motivation behind this approach was touncover possible foreground contamination outside the KQ75 mask. The f NL estimates for the different ringsare presented in table 2. We see that the estimates of f NL seem higher around equator, but the errors-bars arealso larger (due to the galactic mask in the equatorialregion). Since none of the rings show particularly highvalues (compared to the error bars) we do not have ev-idence to claim that foreground residuals have an influ-ence of the estimates of f NL . The internal consistency ofthe various ring estimates have been tested as describedabove and shown to be in agreement with simulations(within 2 σ ).We also estimated f NL on the equatorial rings of the V+ W map using the smaller
KQ85 mask. These resultsare presented in table 2. As expected the error-bars aresomewhat smaller then the results with the
KQ75 mask,especially around the equatorial region. The estimatesare consistent with the ones from the
KQ75 mask.We then followed through with similar investigationsof the individual Q , V and W frequency channels usingthe KQ75 mask (table 2). The V and W channels areconsistent (within 2 σ ), while only the Q channel showsa 3 σ deviation in the ring around equator exactly whereforeground residuals would be expected. We thus sus-pect a possible influence of foreground contamination inthis band. This was further checked by testing the con-sistency of the estimated f NL between the bands using a χ approach. Whereas the (V-W) and (V-Q) differenceswere found to be consistent with simulations (within 2 σ ),the (W-Q) difference was found to be larger than in 99%of the simulations.Also, a similar test was performed on the differences f NL ( V ) − f NL ( V W ) and f NL ( W ) − f NL ( V W ). We seein table 1 and on some rings in table 2 that the VWestimates seem driven by the W estimate. This is notseen in simulated maps and it was found that the smalldifference f NL ( W ) − f NL ( V W ) found for the WMAPdata is found in only 5% of the simualted maps whereasthe difference f NL ( V ) − f NL ( V W ) for WMAP was foundto be consistent (well within 2 σ ) with simulations. V+W V+W Q V W ring
KQ75 KQ85 KQ75 KQ75 KQ75 ±
95 93 ±
95 47 ±
118 66 ±
106 94 ± ±
68 6 ± − ±
83 1 ±
76 28 ±
793 80 ±
80 43 ± − ±
100 19 ±
90 13 ±
934 283 ±
183 122 ±
113 700 ±
226 462 ±
205 253 ± ±
82 128 ± − ±
103 37 ±
92 122 ±
966 39 ±
66 53 ± − ±
81 35 ±
74 15 ±
787 158 ±
93 156 ±
93 174 ±
114 138 ±
104 201 ± TABLE 2The f NL estimates and σ error-bars for equatorial ringsof the individual Q , V , and W frequency channels as wellas the co-added V+W for the masks
KQ75 and
KQ85 . We see that all of the rings for the
V+W channel (aswell as the individual V and W channels) give a estimate f NL ≥
0. These rings are nearly independent (exceptfor the largest scales) and for a sky with f NL = 0 onewould expect that each of these have a 50% probability ofbeing positive. For 7 rings one would then expect only a ≈ .
78% probability that all rings give f NL ≥ V+ W channel (with the
KQ75 mask) is f NL = 11, we in-vestigated the probability of this occurring in Gaussiansimulations. We found that of 4000 Gaussian simula-tions, only 0 .
35% have f NL ≥
11 in all the rings. CONCLUSIONS
In this paper we have used the bispectrum of needletsto obtain local estimates of f NL on the WMAP five yeardata. We performed the analysis on the combined V+W channel, as well as the individual Q , V and W channels,using multipoles up to ℓ = 1500 and the KQ75 galacticcut. For the combined
V+W channel we also applied the
KQ85 mask.We first made a full sky analysis, resulting in a best fitvalue of f NL = 73 ±
31 for the combined
V+W channelusing the
KQ75 mask. The individual V and W chan-nels give consistent (within 2 σ ) results, but the f NL esti-mate of the Q channel deviates significantly, suggestingcontamination of foregrounds.The estimates were then made on selected regions ofthe sky and showed how the needlet bispectrum approachis powerful for finding estimates of f NL in many differentregions, roughly at the cost of one single full sky estimate.We divided the sky into smaller regions according to fourdifferent patterns: hemispheres, 45 ◦ disks and equatorialrings. In each of these schemes f NL was estimated inevery region. The results were compared to simulationsusing a χ test, and all local f NL estimates were foundto be internally consistent (within 2 σ ) for the V and W channels.The local estimates of f NL showed a dipolar distribu-tion of f NL on the sphere, with a maximum at θ = 129 ◦ , φ = 96 ◦ . For comparison, the hemispherical power asym-metry reported in Hansen et al. (2008) was found witha maximum in θ = 107 ◦ , φ = 226 ◦ , and is therefore notexpected to be connected to the findings in this paper.Also, such a dipolar distribution was found to be com-mon in simulated Gaussian maps. For the equatorialrings we found a positive value for f NL in every ring.We compared the lower estimate of the V+W channel( f NL ≥
11 for all rings) with simulations and found thatonly 0 .
35% of Gaussian simulations have this feature.For the rings we find no significant evidence of fore-ground contamination outside the galactic
KQ75 mask.The results also seem to be fairly consistent (within 2 σ )between the individual V and W channels whereas the Q band show signs of possible foreground contaminationin the equatorial band where f NL is larger than zero atthe 3 σ level. This is confirmed by the fact that the dif-ferences in local f NL values between the Q and W bandsare larger than in 99% of the simulations.We conclude that our study shows no significantanisotropy in the estimates of f NL in the CMB sky. Noabnormal values for f NL are found close to the equatorexcept for the Q band where we suspect foregrounds toinfluence the estimate of f NL .FKH is grateful for an OYI grant from the ResearchCouncil of Norway. This research has been partiallysupported by ASI contract I/016/07/0 ”COFIS” andASI contract Planck LFI Activity of Phase E2. Weacknowledge the use of the NOTUR supercomputingfacilities. We acknowledge the use of the HEALPix(G´orski et al. 2005) package and the Legacy Archivefor Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Officeof Space Science..FKH is grateful for an OYI grant from the ResearchCouncil of Norway. This research has been partiallysupported by ASI contract I/016/07/0 ”COFIS” andASI contract Planck LFI Activity of Phase E2. Weacknowledge the use of the NOTUR supercomputingfacilities. We acknowledge the use of the HEALPix(G´orski et al. 2005) package and the Legacy Archivefor Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Officeof Space Science.