Using correlations between CMB lensing and large-scale structure to measure primordial non-Gaussianity
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 1–5 (2013) Printed 4 October 2018 (MN L A TEX style file v2.2)
Using correlations between CMB lensing and large-scale structureto measure primordial non-Gaussianity
Tommaso Giannantonio , ⋆ and Will J. Percival Ludwig-Maximilians-Universit¨at M¨unchen, Universit¨ats-Sternwarte M¨unchen, Scheinerstr. 1, D-81679 M¨unchen, Germany Excellence Cluster Universe, Boltzmannstr. 2, D-85748 Garching bei M¨unchen, Germany Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK
ABSTRACT
We apply a new method to measure primordial non-Gaussianity, using the cross-correlationbetween galaxy surveys and the CMB lensing signal to measure galaxy bias on very largescales, where local-type primordial non-Gaussianity predicts a k divergence. We use theCMB lensing map recently published by the Planck collaboration, and measure its externalcorrelations with a suite of six galaxy catalogues spanning a broad redshift range. We thenconsistently combine correlation functions to extend the recent analysis by Giannantonio etal. (2013), where the density-density and the density-CMB temperature correlations wereused. Due to the intrinsic noise of the Planck lensing map, which affects the largest scalesmost severely, we find that the constraints on the galaxy bias are similar to the constraintsfrom density-CMB temperature correlations. Including lensing constraints only improves theprevious statistical measurement errors marginally, and we obtain f NL = 12 ± (1 σ ) fromthe combined data set. However, the lensing measurements serve as an excellent test of sys-tematic errors: we now have three methods to measure the large-scale, scale-dependent biasfrom a galaxy survey: auto-correlation, and cross-correlation with both CMB temperature andlensing. As the publicly available Planck lensing maps have had their largest-scale modes atmultipoles l < removed, which are the most sensitive to the scale-dependent bias, we con-sider mock CMB lensing data covering all multipoles. We find that, while the effect of f NL indeed increases significantly on the largest scales, so do the contributions of both cosmicvariance and the intrinsic lensing noise, so that the improvement is small. Key words:
Cosmic microwave background; Large-scale structure of the Universe; Inflation.
The quest to measure primordial non-Gaussianity (PNG) has beena thriving field for the past decade. PNG has long been consid-ered an open window onto the physics of the early universe, af-fording the exciting possibility of ruling out the canonical slow-rollinflation model and finding evidence for new primordial physics(Byrnes & Choi 2010).PNG has been traditionally probed with the bispectrum of theCMB anisotropies, which is expected to vanish at first order in afully Gaussian universe. While possible hints of departures fromGaussianity have occasionally appeared at low significance fromanalyses based on WMAP data (Yadav & Wandelt 2008), PNGremained weakly constrained (Bennett et al. 2013; Hinshaw et al.2013), encouraging a sustained growth of theoretical models ofinflation producing non-Gaussian initial conditions (Suyama et al.2010). The constraints on PNG from the CMB bispectrum havenow improved dramatically with the first-year release of the Planck ⋆ [email protected] CMB data, from whose bispectrum it was found that local f NL =2 . ± . (68 % c.l.) (Planck Collaboration 2013c) after subtractionof non-primordial contributions; this result has put significant pres-sure onto multi-field inflation, reducing the scope of possible dis-coveries. However, some non-Gaussianity at the level of f NL ∼ isexpected even in the canonical model (Bartolo et al. 2004), mean-ing it is worthwhile to look for methods to further improve the ex-isting constraints. Furthermore, it is worth cross-checking all con-straints with independent methods.The discovery by Dalal and collaborators (Dalal et al. 2008;Matarrese & Verde 2008; Slosar et al. 2008; Desjacques et al.2009; Desjacques & Seljak 2010a,b; Desjacques et al. 2010;Giannantonio & Porciani 2010; Valageas 2010; Desjacques et al.2011a,b) that the bias of dark matter haloes and galaxies becomesstrongly scale-dependent in the presence of PNG opened up a newavenue for PNG measurement. Constraints on PNG from bias mea-surements of different galaxy samples were found to be compet-itive, and comparable with, CMB bispectrum results before thePlanck data release (Slosar et al. 2008; Xia et al. 2010a,b, 2011).The strongest robust constraints obtained with this technique were c (cid:13) T. Giannantonio & W. J. Percival recently described by Giannantonio et al. (2013), where a compila-tion of six galaxy catalogues and their external correlations with theCMB temperature anisotropies was used to measure f NL = 5 ± (68 % c.l.) for the local configuration under the most conservativeassumptions.The measurement of PNG from scale-dependent large-scalebias is complicated by observational systematics, such as stellarcontamination in galaxy samples, which acts to produce large-scalepower mimicking a PNG signal (Ross et al. 2011, 2012). If we canmodel the phase information of the systematic we can ignore theaffected modes (Leistedt et al. 2013), or we can weight the galax-ies to create an unbiased field (Ross et al. 2013). However, thesesystematics can be most easily controlled by using measurementsof cross-correlations between different galaxy samples or betweensamples and other data that trace the density field, for which weexpect uncorrelated observational systematics.In this letter we focus on a newer addition to existing large-scale structure (LSS) methods to measure PNG: the galaxy biasand thus PNG can also be measured by cross-correlating galax-ies and the matter density field, reconstructed from gravitationallensing (Jeong et al. 2009). The special case of CMB lensing(Lewis & Challinor 2006) is particularly useful, because it allowsconsistent tomographic correlations with galaxy surveys; earlyforecasts showed that this method can provide competitive PNGconstraints (Jeong et al. 2009; Takeuchi et al. 2010, 2012). CMBlensing maps have now been reconstructed, and their correlationswith galaxy surveys have been confirmed, using data from thePlanck satellite (Planck Collaboration 2013b), the South Pole Tele-scope (van Engelen et al. 2012; Bleem et al. 2012), and the Ata-cama Cosmology Telescope (Das et al. 2011; Sherwin et al. 2012).We can now for the first time apply this method to constrain PNGusing public CMB lensing data from Planck.We update the existing analysis by Giannantonio et al. (2013)as follows: in addition to the density-density correlations betweensix galaxy catalogues, and to their cross-correlations with the CMBtemperature anisotropies that we update to Planck, we also mea-sure and use their cross-correlations with the recently releasedPlanck CMB lensing map (Planck Collaboration 2013b). We testhow these correlations can improve the combined PNG constraints,and we show that they also represent an additional, partially inde-pendent cross-check on the results. In this letter we only consider the simplest local PNG model,parametrized as Φ( x , z ⋆ ) = ϕ ( x , z ⋆ ) + f NL (cid:2) ϕ ( x , z ⋆ ) − h ϕ i ( z ⋆ ) (cid:3) (1)where f NL quantifies the amount of PNG. Here Φ( x , z ⋆ ) is the po-tential at primordial times z ⋆ and ϕ an auxiliary Gaussian potential.In the presence of PNG, the bias becomes scale-dependent, and iswell described by a correction ∆ b loc ( k, f NL ) = 2 δ c b L f NL /α ( k ) , (2)where α ( k, z ) = k T ( k ) D ( z )3 Ω m H g (0) g ( z ⋆ ) , T ( k ) is the density transferfunction, D ( z ) is the linear growth function, g ( z ) ∝ (1 + z ) D ( z ) is the potential growth function, δ c = 1 . is the spherical col-lapse threshold and b L ≡ b − is the Lagrangian bias.We constrain f NL via the galaxy bias, as measured by the 2Dangular correlation functions w AB ( ϑ ) between all pairs of maps A, B , whose theoretical predictions are calculated numerically as a
Figure 1.
Linear theory predictions of the angular power spectra we con-sider including their theoretical uncertainties given by cosmic variance,shot noise, and intrinsic lensing noise (top panel) for two models with f NL = 0 , , and signal-to-noise estimations for a single galaxy cata-logue. We assume the specifications of the NVSS survey and the WMAP7best-fit cosmology, and apply a Healpix smoothing for N side = 64 ( ∼ arcmin) as used in the analysis. The central panel shows the absolute signal-to-noise ratio of the three spectra we measure, per multipole and cumulative(the latter starting from l = 2 or from l = 10 ). The bottom panel shows theapproximated detection power of a model with f NL = 100 , defined as thesignal-to-noise ratio of the difference between the two models. Legendre transformation of the corresponding angular power spec-tra C ABl = (2 /π ) R dk k P ( k ) W Al ( k ) W Bl ( k ) . The sources W l ( k ) describe the redshift projection over the survey visibilityfunction dN/dz ( z ) of the physical sources, which are different forgalaxy counts, ISW, and CMB lensing, calculated using a modifiedversion of C AMB (Lewis et al. 2000).The total signal-to-noise ratio to be expected for a singlegalaxy catalogue and its external correlations (GG, TG, ϕG ) isshown in Fig. 1 for one case (corresponding to NVSS specifica-tions; see below for details of the data sets we consider). Here weinclude in the theoretical uncertainties cosmic variance, shot noise,and the intrinsic lensing noise provided by Planck. We can see in c (cid:13) , 1–5 onstraining non-Gaussianity with CMB lensing and LSS the central panel that the total signal-to-noise of both TG and ϕG signals is barely affected by the modes at l < ; the constrain-ing power on f NL however, defined as the signal-to-noise ratio ofthe difference between a Gaussian and a non-Gaussian model, isreduced if the largest scales are excluded, as the scale-dependentbias is most visible precisely for these modes (bottom panel). Wecan also see that the constraining power on f NL of the new ϕG correlations should be comparable with the TG part if all modeswere available, and marginally less if using the cut data at l > only. The galaxy auto-correlation functions (ACFs) are expectedto constrain f NL more strongly because the bias enters in quadra-ture in this case. It is finally important to notice from the centralpanel that the total signal-to-noise of the ϕG correlations is actu-ally high, comparable with the ACF; but the largest contributionarise at smaller scales, thus limiting the constraining power on thescale-dependent bias. We consider the compilation of six galaxy catalogues introduced byGiannantonio et al. (2008), updated in Giannantonio et al. (2012)and used to constrain PNG using density and density-CMB temper-ature correlations in Giannantonio et al. (2013). Briefly, this con-sists of the IR galaxies of 2MASS at a median redshift z ≃ . ,the radio-galaxies of NVSS and X-ray background of HEAO (bothspanning a broad redshift range), and three photometric samplesfrom the Sloan Digital Sky Survey (SDSS), i.e. the main galaxiesat z ≃ . , the luminous red galaxies (LRGs) from the photomet-ric CMASS sample from Data Release 8 (DR8) at z ≃ . andthe DR6 photometric quasars, which also feature a broad redshiftdistribution.We replace the previously used WMAP CMB data with thenewly released Planck maps. We use the temperature S MICA mapwith the strictest provided galaxy mask, as well as the CMBlensing map reconstructed from the off-diagonal covariances be-tween different multipoles in the temperature map together withits mask. The Planck collaboration removed information for thelargest scales (modes with l < ) from this map: although thescale-dependent bias affects mostly the largest scales, as shown inFig. 1, the increased noise means that we do not expect a drasticdegradation in constraining power on f NL . We test this further withmock data below.We first measure all projected two-point angular correlationfunctions w g i g j ( ϑ ) between pairs of catalogues i, j at angular sep-arations ≤ ϑ ≤ deg using a pixel-based estimator withinthe H EALPIX scheme (G´orski et al. 2005) at N side = 64 (pixelsize ∼ arcmin): this yields 21 correlation functions. Some ofthe auto-correlation functions (ACFs) present an excess power atlarge angular separations compared with the Gaussian Λ CDM pre-dictions, especially the quasars and the NVSS galaxies; a de-tailed analysis of the systematics of these samples was presentedin Giannantonio et al. (2013), where it was shown that such sig-nals are likely due to residual systematic contaminations, as alsodemonstrated by Pullen & Hirata (2013); Leistedt et al. (2013).Following these systematics tests, it was decided to take the mostconservative approach and to keep the raw NVSS data uncorrectedfor the existing r.a. and declination-dependent systematics, to avoidthe risk of biasing the constraints on f NL . The NVSS and QSOACFs are then discarded from the cosmological analysis. We adoptthe same choice here, while keeping all the cross-correlations be-tween the different data sets. Figure 2.
The full extended data set used in this analysis. The first rowshows the new set of galaxy-CMB lensing correlation functions. The secondrow is the ISW effect (compared between WMAP and Planck), and theremaining rows are the galaxy-galaxy correlation functions. Error bars areMonte Carlos and are highly correlated. The ISW 2MASS error bars are . σ . The ACF of the raw NVSS data presents a significant excess powerwith respect to the Λ CDM expectations, which is modeled by adding to themocks the r.a. and dec density fluctuations observed in the data. The NVSSand QSO ACFs are not used for the cosmological results due to their knownsystematics.
We then measure the six cross-correlation functions (CCFs)between the galaxy catalogues and the CMB temperatureanisotropies w Tg i ( ϑ ) , updating our analysis to the Planck first yeardata release (Planck Collaboration 2013a). The measured level ofthese correlations is consistent with the assumption that they areproduced by the integrated Sachs-Wolfe effect (ISW). This corre-sponds to the ‘fair’ sample of Giannantonio et al. (2013). We finallyadd to our data set the six CCFs between the galaxy cataloguesand the Planck CMB lensing map (Planck Collaboration 2013b): w ϕg i ( ϑ ) . These correlations allow a redshift tomography of theCMB lensing sources, effectively mapping the dark matter distri-bution in redshift bins. We thus obtain the 33 correlation functionsshown in Fig. 2. Notice that we have nulled the angular power spec-trum at l < in the galaxy-lensing spectrum for consistency withthe Planck data. We calculate the covariance matrix between all ×
13 = 429 data points using a Monte Carlo method, generat-ing 10,000 realisations based on a fiducial Gaussian Λ CDM model,including shot noise in the counts, the intrinsic lensing noise fromPlanck, and all expected correlations between the maps (see Ap-pendix of Giannantonio et al. 2008). We also include the r.a. anddeclination-dependent systematics in the mock NVSS data, so thatthe mean of the mocks used to estimate the covariance agrees withthe observed ACF (Giannantonio et al. 2013). c (cid:13) , 1–5 T. Giannantonio & W. J. Percival
Data: Planck TT, WP, and Priors f NL (68%) GG b i , κ i ± T G b i , κ i ± ϕG b i , κ i ± Mock all- l ϕG b i , κ i ± GG none ± GG + T G none ± GG + ϕG none ± GG + ϕG + T G , none ± Mock all- l , GG + ϕG + T G none ± as above, no intrinsic noise none ± Table 1.
Measurements of f NL , for different combinations of data. Wherethe data come from mock catalogues including lensing on scales l < ,only the errors are provided. We calculate the likelihood of the theoretical parameters given thefirst-year Planck temperature power spectrum (with WMAP po-larisation) to impose tight priors on most cosmological parame-ters, while our compilation of correlation functions will constrain f NL . We consider different subsets of our data, exploring the pa-rameter space with a modified version of the latest C OSMOMC code (Lewis & Bridle 2002), including the official Planck like-lihood code. As discussed in more detail in Giannantonio et al.(2013), in addition to the standard Λ CDM cosmological parame-ters, we always vary a set of ten nuisance parameters to accountfor uncertainties in our modelling of the data: one free bias pa-rameter for each i -th catalogue b i , one stellar contamination frac-tion κ i for each of the SDSS samples, and one PSF smooth-ing for the HEAO data α HEAO . As in Giannantonio et al. (2013)we assume that the Gaussian part of the bias of most samplesevolves as b i ( z ) = 1 + (cid:2) b i − (cid:3) /D ( z ) , while for the quasarswe assume b QSO1 ( z ) = b QSO0 /D . ( z ) ; for further details seeGiannantonio et al. (2013), where it was found that the results donot depend too strongly on these assumptions. We also use the stan-dard set of nuisance parameters introduced in the Planck likelihoodpackage.We summarise our results in Table 1. When using the PlanckTT data with WMAP polarisation (WP), and the GG correlationfunctions only, we find f NL = 15 ± (all results at σ ). Theaddition of the LSS-CMB temperature correlations (ISW) improvesthis to f NL = 14 ± . Note that this error is consistent with,although slightly worse than the error found in Giannantonio et al.(2013): a consequence of the different corrections assumed for theCMB data. As we are considering large-scales only, the WMAP andPlanck data provide similar signal-to-noise. If instead of the ISWwe add the CMB lensing correlations, we find f NL = 11 ± ,while the final, fully combined results (including all correlations)yields f NL = 12 ± .To better compare the constraining power of the different partsof our data set, we also test the results on f NL when using the GG,TG, and ϕG parts only. In order to make the comparison moremeaningful for these runs, we included Gaussian priors on the biasand stellar contamination parameters equal to their posteriors fromthe full run. The results presented in Table 1 show that the con-straining power on f NL of the TG part is marginally stronger thanthe ϕG at l > , while the GG part is a factor of ∼ better.This is in qualitative agreement with our signal-to-noise calcula-tions shown in Fig. 1. Figure 3.
The mock ϕG correlations, for the case of uncut CMB lensingdata. We set the mock data to be equal to the fiducial Λ CDM model, andthe covariance matrix has been re-calculated using the full angular powerspectrum at all l . As previously mentioned, the publicly available CMB lensing mapreconstructed by the Planck collaboration has had modes l < re-moved. As it is known that in the presence of PNG scale-dependentbias is strongest on the largest scales, here we address the questionof how much better would our constraints be if we could use thefull uncut CMB lensing data. For this purpose, we replace the mea-sured w ϕg i ( ϑ ) data points with mock data that we set equal to ourfiducial Λ CDM model. We also generate a new covariance matrix,where the input fiducial model does include all multipoles in theLSS-CMB lensing correlations. We show the modified data set inFig. 3, where we can see that both the signal and the error bars inthe CMB lensing correlations have significantly increased.This can be readily understood by remembering that, in thesimplified case of cosmic variance-dominated errors, the vari-ance is proportional to the angular power spectrum, which steeplyincreases at the smallest multipoles in the CMB lensing case.In addition to this, the intrinsic lensing noise of Planck is alsolarge compared with the signal on these scales (see Fig. 1 inPlanck Collaboration 2013b). Thus, when projecting to real space,the inclusion of the modes at < l < will bring a large con-tribution for both signal and noise, as shown in Fig. 1. We haverun the full likelihood analysis on this modified data set, and findmarginally improved results, with error on f NL , ± . This is againin agreement with the signal-to-noise projection of Fig. 1. We fi-nally test how much would the results improve if we had an idealexperiment without any intrinsic lensing noise: in this case we findan error ± using all of the data. We have applied a new method to improve the large-scale struc-ture constraints on primordial non-Gaussianity, using the cross-correlations of galaxy catalogues with CMB lensing maps. Newmaps from the Planck satellite were used to measure the PNG pa-rameter f NL , finding similar errors to those from ISW based biasmeasurements. Consequently, the final combined measurements of f NL are only marginally improved by including density-CMB lens-ing correlations in addition to density and density-CMB temper-ature correlations. We have investigated the penalising effects ofcosmic variance, intrinsic lensing noise, and cuts imposed on thePlanck CMB lensing maps, finding consistency between resultsand expectations. Combining all of our measurements, we find f NL = 12 ± ( σ ).The addition of the CMB lensing correlations provides an im-portant consistency check for f NL measurements, as it is expectedto be affected by different systematics than ISW, galaxy-galaxycorrelation, and bispectrum based measurements. The method pre- c (cid:13) , 1–5 onstraining non-Gaussianity with CMB lensing and LSS sented in this letter serves also as a preliminary exercise for theDark Energy Survey (DES; ),to which we will apply a similar analysis in the near future. Be-yond PNG, the consistent combination of internal and external cor-relation functions of the LSS represents a powerful way to extractthe most cosmological information, and to reconstruct the evolutionof the Universe at the perturbative level. Based on our analysis, theaddition of CMB lensing is expected to provide more powerful cos-mological measurements on smaller scales than those used here tocontain the PNG signal. Thus, future analyses of Dark Energy andof neutrino masses will be particularly interesting (Pearson & Zahn2013). As clustering, the ISW, and gravitational lensing are sensi-tive to different combinations of the gravitational potentials andtheir derivatives, their combination could also provide a powerfultool to constrain the history of gravity and structure formation. ACKNOWLEDGEMENT
We thank Aur´elien Benoˆıt-Levy and Pablo Fosalba for useful dis-cussions on the Planck CMB lensing data. We also thank EiichiroKomatsu and Bj¨orn S¨orgel for useful comments. TG acknowledgesthe Rechenzentrum Garching of the Max Planck Society for com-putational resources. WJP acknowledges support from the UK Sci-ence & Technology Facilities Council (STFC) through the consoli-dated grant ST/K0090X/1, and from the European Research Coun-cil through the “Starting Independent Research” grant 202686,MDEPUGS.
REFERENCES
Bartolo N., Komatsu E., Matarrese S., Riotto A., 2004, Phys. Rep.,402, 103Bennett C. L., et al., 2013, Astrophys. J. Supp., 208, 20Bleem L. E., et al., 2012, Astrophys. J. Lett., 753, L9Byrnes C. T., Choi K.-Y., 2010, Advances in Astronomy, 2010Dalal N., Dor´e O., Huterer D., Shirokov A., 2008, Phys. Rev. D,77, 123514Das S., et al., 2011, Physical Review Letters, 107, 021301Desjacques V., Crocce M., Scoccimarro R., Sheth R. K., 2010,Phys. Rev. D, 82, 103529Desjacques V., Jeong D., Schmidt F., 2011a, Phys. Rev. D, 84,061301Desjacques V., Jeong D., Schmidt F., 2011b, Phys. Rev. D, 84,063512Desjacques V., Seljak U., 2010a, Classical and Quantum Gravity,27, 124011Desjacques V., Seljak U., 2010b, Advances in Astronomy, 2010Desjacques V., Seljak U., Iliev I. T., 2009, Mon. Not. R. As-tron. Soc., 396, 85Giannantonio T., Crittenden R., Nichol R., Ross A. J., 2012,Mon. Not. R. Astron. Soc., 426, 2581Giannantonio T., Porciani C., 2010, Phys. Rev. D, 81, 063530Giannantonio T., Ross A. J., Percival W. J., Crittenden R., BacherD., Kilbinger M., Nichol R., Weller J., 2013, Phys. Rev. D, inpress, arXiv:1303.1349Giannantonio T., Scranton R., Crittenden R. G., Nichol R. C.,Boughn S. P., Myers A. D., Richards G. T., 2008, Phys. Rev.D, 77, 123520 G´orski K. M., Hivon E., Banday A. J., Wandelt B. D., HansenF. K., Reinecke M., Bartelmann M., 2005, Astrophys. J., 622,759Hinshaw G., et al., 2013, Astrophys. J. Supp., 208, 19Jeong D., Komatsu E., Jain B., 2009, Phys. Rev. D, 80, 123527Leistedt B., Peiris H. V., Mortlock D. J., Benoit-L´evy A., PontzenA., 2013, Mon. Not. R. Astron. Soc., 435, 1857Lewis A., Bridle S., 2002, Phys. Rev. D, 66, 103511Lewis A., Challinor A., 2006, Phys. Rep., 429, 1Lewis A., Challinor A., Lasenby A., 2000, Astrophys. J., 538, 473Matarrese S., Verde L., 2008, Astrophys. J. Lett., 677, L77Pearson R., Zahn O., 2013, ArXiv e-prints, arxiv:1311.0905Planck Collaboration, 2013a, ArXiv e-prints, arxiv:1303.5062Planck Collaboration, 2013b, ArXiv e-prints, arxiv:1303.5077Planck Collaboration, 2013c, ArXiv e-prints, arxiv:1303.5084Pullen A. R., Hirata C. M., 2013, Publ. Astron. Soc. Pacif., 125,705Ross A. J., et al., 2011, Mon. Not. R. Astron. Soc., 417, 1350Ross A. J., et al., 2012, Mon. Not. R. Astron. Soc., 424, 564Ross A. J., et al., 2013, Mon. Not. R. Astron. Soc., 428, 1116Sherwin B. D., et al., 2012, Phys. Rev. D, 86, 083006Slosar A., Hirata C., Seljak U., Ho S., Padmanabhan N., 2008,J. Cosmol. Astropart. Phys., 8, 31Suyama T., Takahashi T., Yamaguchi M., Yokoyama S., 2010,J. Cosmol. Astropart. Phys., 12, 30Takeuchi Y., Ichiki K., Matsubara T., 2010, Phys. Rev. D, 82,023517Takeuchi Y., Ichiki K., Matsubara T., 2012, Phys. Rev. D, 85,043518Valageas P., 2010, Astron. Astrophys., 514, A46van Engelen A., et al., 2012, Astrophys. J., 756, 142Xia J.-Q., Baccigalupi C., Matarrese S., Verde L., Viel M., 2011,J. Cosmol. Astropart. Phys., 8, 33Xia J.-Q., Bonaldi A., Baccigalupi C., De Zotti G., Matarrese S.,Verde L., Viel M., 2010a, J. Cosmol. Astropart. Phys., 8, 13Xia J.-Q., Viel M., Baccigalupi C., De Zotti G., Matarrese S.,Verde L., 2010b, Astrophys. J. Lett., 717, L17Yadav A. P. S., Wandelt B. D., 2008, Physical Review Letters,100, 181301 c (cid:13)000