Introducing Mexican needlets for CMB analysis: Issues for practical applications and comparison with standard needlets
S. Scodeller, O. Rudjord, F. K. Hansen, D. Marinucci, D. Geller, A. Mayeli
aa r X i v : . [ a s t r o - ph . C O ] A ug Introducing Mexican needlets for CMB analysis: Issues forpractical applications and comparison with standard needlets
S. Scodeller
Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315Oslo, Norway [email protected]
Ø. Rudjord
Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315Oslo, Norway;Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern,N-0316 Oslo
F. K. Hansen
Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315Oslo, Norway;Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern,N-0316 Oslo
D. Marinucci
Dipartimento di Matematica, Universit`a di Roma ‘Tor Vergata’, Via della RicercaScientifica 1, I-00133 Roma, Italy
D. Geller
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651
A. Mayeli
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651
ABSTRACT
Over the last few years, needlets have emerged as a useful tool for the analysisof Cosmic Microwave Background (CMB) data. Our aim in this paper is firstto introduce in the CMB literature a different form of needlets, known as Mexi-can needlets, first discussed in the mathematical literature by Geller and Mayeli 2 –(2009a,b). We then proceed with an extensive study of the properties of bothstandard and Mexican needlets; these properties depend on some parameterswhich can be tuned in order to optimize the performance for a given application.Our second aim in this paper is then to give practical advice on how to adjustthese parameters for WMAP and Planck data in order to achieve the best prop-erties for a given problem in CMB data analysis. In particular we investigatelocalization properties in real and harmonic space and propose a recipe on howto quantify the influence of galactic and point source masks on the needlet co-efficients. We also show that for certain parameter values, the Mexican needletsprovide a close approximation to the Spherical Mexican Hat Wavelets (whencetheir name), with some advantages concerning their numerical implementationand the derivation of their statistical properties.
Subject headings: (cosmology:) cosmic microwave background — cosmology:observations — methods: data analysis — methods: statistical
1. Introduction
Over the last decade, wavelet systems have grown as one of the most important toolsin the analysis of Cosmological and Astrophysical data. A lot of proposals for waveletsystems on the sphere have been advanced in the mathematical literature, see for instanceFreeden & Schreiner (1998), Antoine & Vandergheynst (2007), Dahlke et al. (2007), Holschneider & Iglewska-Nowak(2007), Rosca (2007), Wiaux et al. (2007), Starck et al. (2006) and the references therein.Some of these attempts have been explicitly motivated by Astronomy and/or Cosmology(see for instance McEwen et al. (2007) for a review). In particular in the area of CosmicMicrowave Background (CMB) data analysis, wavelets have been used for a large numberof applications (see references in the next paragraph). The interest for wavelets in this areais very easily understood; predictions from CMB theory are typically cast in the Fourierdomain, however exact Fourier analysis cannot be entertained because of the presence offoreground and masked regions. The double-localization properties of wavelet systems (inreal and harmonic domain) hence turn out to be most valuable. Moreover, addressing im-portant issues such as the possible existence of features and asymmetries in CMB maps isnearly unfeasible without ideas which are broadly related to the wavelet literature.Among spherical wavelets, particular attention has been recently devoted to so-calledneedlets, which were introduced into the Functional Analysis literature by Narcowich et al.(2006a,b); their statistical properties were first considered by Baldi et al. (2009a,b). Needletsenjoy several properties that make them worth of attention for Cosmological data analysis. 3 –In particular, they are computationally very simple, and naturally adapted to standardpackages such as HealPix (G´orski et al. 2005); they do not require any form of tangentplane approximation, but they are naturally embedded into the manifold structure of thesphere; they are compactly supported in the harmonic domain, i.e. they depend only on afinite number of multipoles which are explicitly known and can be controlled by the dataanalysts; they are quasi-exponentially localized in real space, i.e. their tails decay fasterthan any polynomial; and finally, it has been shown in Baldi et al. (2009a) that randomneedlet coefficients enjoy a very useful uncorrelation property: namely, for any fixed angu-lar distance, random needlets coefficients are asymptotically uncorrelated as the frequencyparameter grows larger and larger. As well-known, uncorrelation entails independence inthe Gaussian case: as a consequence, from the above-mentioned property it follows thatneedlet coefficients from a CMB map can be seen as nearly independent at high frequencies,making thus possible the introduction of a variety of statistical procedures for testing non-Gaussianity, estimating the angular power spectrum, testing for asymmetries, implement-ing bootstrap techniques, testing for cross-correlation among CMB and Large Scale Struc-ture data, and many others, see for instance Baldi et al. (2009a,b), Guilloux et al. (2009),Lan & Marinucci (2008b), Pietrobon et al. (2006), Marinucci et al. (2008), Fa¨y, G. et al.(2008), Rudjord et al. (2009a,b), Cabella et al. (2009). More recently, the needlet con-struction has also been extended to the case of spin/polarization data, see for instanceGeller et al. (2008, 2009), Geller & Mayeli (2009d) and Geller & Marinucci (2008).The first purpose of this paper is to introduce a new kind of needlets to the field of CMBanalysis following an approach which has been very recently advocated in mathematics byGeller & Mayeli (2009a,b,c). This approach (which we shall discuss in Section 2) can belabeled Mexican needlets. As we shall discuss below, a special case of the Mexican needletsprovides at high frequencies a close approximation to the widely used Spherical MexicanHat Wavelet (SMHW, see for instance Cayon et al. (2001); Vielva et al. (2004)), withsome advantages in terms of their numerical implementation and the investigation of theirlocalization and statistical properties. As such, the investigation of their properties in thiscase will allow to understand the stochastic properties of SMHW and compare them withstandard needlets. Mexican needlets depend on a parameter p , and we shall show how thisparameter can be tuned to improve the localization properties in the real or in the harmonicdomain.The second purpose of this paper is to provide a practical description of properties ofdifferent needlet types important for CMB analysis. Proper knowledge of the localizationproperties on the pixelized sphere as well as in multipole space is crucial for selecting andapplying the proper type of needlet to a specific problem. Although the exact mathematicalproperties of the needlets are well known, we have studied the properties which are of high 4 –importance for CMB analysis and which are too complicated to be easily deduced from themathematical results. In particular, in the presence of foregrounds and masks, it is importantto know their influence on the needlet coefficients. For the SMHW it has been shown forseveral applications (Vielva et al. 2004; Mukherjee & Wang 2004; McEwen et al. 2004;Curto et al. 2009) that an extended scale dependent mask must be used when analyzingmasked data with wavelets. Here we will study this in detail for the different needlet typeswith different parameter values.We then provide a very thorough comparison between different needlets. The previousdiscussion leads naturally to the issue about their optimal construction, i.e. how to devisenumerical recipes which will enhance their localization properties. Here, we shall comparethe numerical recipe implemented by Baldi et al. (2009a); Pietrobon et al. (2006) with analternative proposal based on Bernstein polynomials (see also Guilloux et al. (2009) for arelated numerical investigation). The latter entail weight functions with just a finite numberof bounded derivatives; to distinguish it from the previous construction we will label thisprocedure Bernstein needlets. We stress, however, that the underlying mathematical theorypresents no real novelty as compared to the results by Narcowich et al. (2006a,b). We thengo on to provided vast numerical evidence on the various forms of localization, by means ofa number of different indicators. In particular, the role of the different parameters in thedetermination of the various properties is fully exploited.As well-known, there is usually a trade-off between localization properties in the fre-quency and real domains, as a consequence of the Uncertainty Principle (“It is impossiblefor a non-zero function and its Fourier transform to be simultaneously very small”, see forinstance Narcowich & Ward (1996); Havin & Joricke (1994)); the main purpose of this pa-per is to show how Mexican and standard needlets jointly provide a flexible set of toolswhere each user can optimize this trade-off according to the needs of a specific data analysisproblem.The plan of the paper is as follows: in Section 2, we review the standard needlet con-struction, introduce Mexican needlets and compare their respective properties from a math-ematical point of view. In the following sections, we provide numerical evidence on thelocalization properties of these procedures by means of several different figures of merit;we discuss at length the interplay among the different properties and the trade-off to facewhen choosing which procedure to adopt for a given Astrophysical problem. In section 7, wesummarize the main indicators used and the properties of the different needlets measured interms of these indicators. In the Appendix, we provide some details on the numerical recipeswe adopted, some short discussion on mathematical properties and analytic fits. 5 –
2. Mexican and Standard Needlets
The construction of the standard needlet system is detailed in Narcowich et al. (2006a,b),see also Marinucci et al. (2008); we sketch here a few details to fix notation and we providein the Appendix a more detailed discussion for completeness. The introduction of Mexicanneedlets is due to Geller & Mayeli (2009a,b,c) (see also Freeden et al. (1998)); they areused here for the first time in the Astrophysical literature, so we provide below a more com-plete discussion. We point out that for p = 1, an analogous proposal was first advocated inthe Astrophysical literature by Sanz et al. (2006).The basic needlet function can be described in real space as follows: ψ jk ( x ) := p λ jk X ℓ b ℓ ( B, j ) ℓ X m = − ℓ Y ℓm ( ξ jk ) Y ℓm ( x ) . (1)Here, x refers to a position ( θ, φ ) on the sphere, Y ℓm are spherical harmonic functions, j isthe scale (frequency) of the needlet and { λ jk } is a set of cubature weights corresponding to the cubature points { ξ jk } ; for simplicity, they can be taken to be equal to the pixel areasand the pixel centres in the HEALPix (G´orski et al. 2005) grid used for CMB analysis, i.e.we shall consider λ jk = λ j = 4 π/N j , where N j is the number of pixel in the pixelizationwe are working with. The needlet function itself is contained in the function b ℓ ( B, j ) (or b ℓ for short) in harmonic space, B being one of the parameters deciding the properties ofthe needlet. The difference between the needlet systems we are going to discuss can thus betraced in the form of the weight function b ℓ .
1) Standard needlets : Let φ ( ξ ) be an infinitely differentiable (i.e., C ∞ ) functionsupported in | ξ | ≤
1, such that 0 ≤ φ ( ξ ) ≤ φ ( ξ ) = 1 if | ξ | ≤ /B , B >
1. Define b ( ξ ) = φ ( ξB ) − φ ( ξ ) ≥ ∀ ℓ > B , ∞ X j =0 b ( ℓB j ) = 1 . (2)For standard needlets we then obtain b ℓ from this function b ( ξ ) by b ℓ ( B, j ) = b ( ℓB j ). For agiven scale j , the needlet function in harmonic space is centered at a multipole ℓ ∗ ≈ B j . Thusa given scale j is mainly influenced by multipoles close to ℓ ∗ . It is immediate to verify that b ( ξ ) = 0 only if B ≤ | ξ | ≤ B . An explicit recipe to construct a function b ( ξ ) with the previousfeatures is discussed in Appendix A (compare Pietrobon et al. (2006), Marinucci et al.(2008)). The main localization property of needlets is established in Narcowich et al.(2006a), where it is shown that for any M ∈ N there exists a constant c M > ξ ∈ S : | ψ jk ( ξ ) | ≤ c M B j (1 + B j d ( ξ jk , ξ )) M uniformly in ( j, k ) , d ( ξ jk , ξ ) denotes the usual distance on the sphere. More explicitly, needlets are almostexponentially localized around any cubature point, which motivates their name.
2) Bernstein Needlets
The bound which we just provided to establish the localizationproperties of needlets depends on some constants c M which we did not write down explicitly.Such constants depend on the form of the function b ( ξ ) , and turn out to be rather large in thecase of standard needlets. In Appendix B, we give another method of construction of b ( ξ ),where such function is no longer infinitely differentiable but rather has a finite number ofbounded derivatives. The localization theory described in Narcowich et al. (2006a,b) goesthrough without any modification, as do the stochastic properties established by Baldi et al.(2009a). We do no longer have quasi-exponentially decaying tails, however, but it is pos-sible to establish a weaker result, namely the decay with a polynomial rate, depending onthe number of bounded derivatives we are allowing for b ( ξ ) . It may hence seem that thisconstruction should enjoy worse properties - but in practice this is not the case, as shown byour simulations in the sections to follow. As for standard needlets, we have b ℓ ( B, j ) = b ( ℓB j ),but note that b ( ξ ) is different for Bernstein needlets as detailed in the Appendix. As for thestandard needlets, the needlet function in harmonic space is centered at ℓ ∗ ≈ B j .
3) Mexican Needlets
The construction in Geller & Mayeli (2009a) is similar to stan-dard needlets, insofar as a combination of Legendre polynomials with a smooth function isproposed; the main difference is that for standard needlets the kernel is taken to be com-pactly supported (i.e., depending only a finite number of multipoles ℓ ), while the Mexicanneedlet construction draws information from all frequencies at any scale. More precisely, weshall consider weight functions b ℓ ( B, j ) of the form b ℓ ( B, j ) = ( ℓB j ) p e − ℓ B j , (3)for p = 1 , , , ... For instance, for p = 1 the Mexican needlet takes the form ψ jk ;1 ( x ) = p λ j X ℓ ≥ ℓ B j e − ℓ /B j ℓ + 14 π P ℓ ( d ( ξ jk , ξ )) ,and for higher p we have ψ jk ; p ( x ) := p λ jk X ℓ ≥ ( ℓ B j ) p e − ℓ /B j ℓ + 14 π P ℓ ( d ( ξ jk , ξ )) . (4)Indeed, for mathematical rigour ℓ should be replaced by the eigenvalue ℓ ( ℓ + 1) , but forCMB data analysis the difference is negligible and we shall use ℓ for notational simplicity. 7 –As mentioned before, Mexican needlets are not supported on a finite number of mul-tipoles, so the discussion of their localization properties in the harmonic domain requiressome care. Moreover, because we need to focus on an infinite number of spherical harmon-ics, from a strictly mathematical point of view exact cubature and reconstruction formulaecannot hold. Nevertheless, it must be added that the approach by Geller & Mayeli (2009a,b)enjoys some undeniable strong points, some of which we list as follows:1) Mexican needlets enjoy extremely good localization properties in the real domain;more precisely, at a fixed angular distance x their tails decay as exp( − B j x / , as j growsto infinity.2) By adjusting the parameter p , one has available a family of wavelets which can beoptimized in terms of the desired localization properties (as we shall show below, a growing p improves the localization in the harmonic domain and decreases the localization in the realdomain)3) The previously mentioned mathematical issues on the cubature points are largelynegligible from a numerical point of view4) The Monte Carlo evidence provided below proves that Mexican needlets comparefavorably with standard needlets under a variety of circumstances and for many differentindicators5) Analytic expressions can be provided for their high-frequency behavior in real space.Concerning the last point, it is important to remark the following. It can be shownthat Mexican needlets for p = 1 provide a very close approximation of the widely popularSpherical Mexican Hat Wavelets (SMHW), see Appendix C. Even in this case, though, theimplementation through needlet ideas in our view yields important benefits:a) the weight function is explicitly given, making easier the implementation and thevalidation of numerical codesb) the localization structure in harmonic domain can be analytically studied and con-trolledc) the correlation structure of random Mexican needlet coefficients is explicitly givenand can be used for statistical inferenced) the range of scales to be considered to retain the information from the data is mathe-matically determined in terms of the frequencies j, rather than by an ad hoc choice of scalesin the real domain as a function of angular distance. 8 –
3. Correlation properties of standard and Mexican needlets
In the sequel of the paper, we shall compare three properties of these needlet construc-tions, namely their localization in the real domain, the localization in the harmonic domain,and the statistical properties of needlet coefficients, primarily their correlation structure.Spherical needlet coefficients are defined as β jk = Z S T ( x ) ψ jk ( x ) dx = p λ jk X ℓ b ℓ ( B, j ) ℓ X m = − ℓ a ℓm Y ℓm ( ξ jk ) , (5)where T ( x ) is the CMB temperature field. The correlation coefficient is hence given by Corr ( β jk , β jk ′ ) = h β jk β jk ′ i q(cid:10) β jk (cid:11) (cid:10) β jk ′ (cid:11) = P ℓ ≥ b ℓ ( B, j ) ℓ +14 π C ℓ P ℓ ( d ( ξ jk , ξ jk ′ )) P ℓ ≥ b ℓ ( B, j ) ℓ +14 π C ℓ where P ℓ is the Legendre polynomial of degree ℓ and C ℓ is the power spectrum of the CMBtemperature field; the last step follows from the well-known identity (Varshalovich et al.(1988)) ℓ X m = − ℓ Y ℓm ( ξ ) Y ℓm ( η ) = 2 ℓ + 14 π P ℓ ( d ( ξ, η )) . For standard needlet coefficients, it was shown by Baldi et al. (2009a) that under generalconditions the following inequality holds | Corr ( β jk , β jk ′ ) | ≤ C ′ M (1 + B j d ( ξ jk , ξ jk ′ )) M , some C ′ M > d ( ξ jk , ξ jk ′ ) is the standard geodesic distance on the sphere. In words, for any twopoints at a finite distance on the sphere the correlation between needlet coefficients centredon this points decays to zero as the frequencies grow larger and larger. Of course, underGaussianity this simply implies that the coefficients become nearly independent at highfrequencies. For Mexican needlet coefficients, the situation is slightly more complicated, asdiscussed by Lan & Marinucci (2008a); Mayeli (2008). More precisely, let us assume thatthe CMB angular power spectrum behaves as C ℓ = h| a ℓm | i ≃ G ( ℓ ) ℓ − α , (7) 9 –where G ( ℓ ) is some smooth function, for instance the ratio of two positive polynomials.Clearly (7) provides a good approximation to CMB spectra, with spectral index α ≃ . It isthen possible to show that, for p such that α < p + 2 , there exist some constant C M > | Corr ( β jk ; p , β jk ′ ; p ) | ≤ C M (1 + B j d ( ξ jk , ξ jk ′ )) (4 p +2 − α ) , (8)where some possible logarithmic factors have been neglected, see Lan & Marinucci (2008a);Mayeli (2008) for details.We should note that while uncorrelation holds for standard needlets no matter whatthe rate of decay of the angular power spectrum, here we need α not to be ”too large”as compared to the order of the Mexican needlet we are using. The intuition behind thisresult is the following. Mexican needlets (and similarly Spherical Mexican Hat Wavelets) arenot compactly supported in the harmonic domain; in other words, whatever the frequency j, they are drawing information from the smallest multipoles, i.e. those most affected bycosmic variance. The faster the decay of the spectrum (i.e., the higher the α ), the greater theinfluence of this low frequency components on the behavior at high j . In order to compensatefor this cosmic variance effect, it is necessary to ensure that the Mexican needlet filter willgo to zero fast enough in the harmonic domain. Clearly, the higher the p , the faster ourwavelet will approach zero at low multipoles, thus compensating for Cosmic Variance effects.In practice, however, for CMB data as we mentioned before α can be taken to be equal to2, whence the correlation coefficient is seen to decay to zero even for the smallest p = 1.Indeed, our numerical results below will show that for physically realistic angular powerspectra Mexican needlets outperform standard ones in terms of uncorrelation properties,thus providing one more possible motivation for their use on CMB data.To close the introduction to Mexican and standard needlets we present plots showinga comparison between the Mexican needlets for p = 1 and SMHW (see figures 1), Mexicanneedlets for different values of the parameter p at different frequencies j (see figures 2) andthe weight function b ℓ = b ℓ ( B, j ) for Mexican and standard needlets (see the 2 top figuresin figures 3 for comparison between Mexican and standard needlets, see bottom figure infigures 3 for seeing how the Mexican weight function depend on the parameter p ).Note that for spherical Mexican needlets, the needlet function in harmonic space isno more centered at ℓ ∗ ≈ B j since one no longer has a symmetric distribution around themaximum of b ℓ , as can be seen in figures 3 . To obtain a measure of the multipole we arelooking at, we introduce therefore a weighted average defined the following way: 10 – ℓ ∗ ( j, B ) = P ℓ ℓ · b ℓ (cid:0) ℓB j (cid:1)P ℓ b ℓ (cid:0) ℓB j (cid:1) (9)which will be used for all kinds of needlets. Comparison: Mexican needlets and SMHW at low multipoles θ [deg]0123 ψ ( θ ) SMHW,R= 35.19[deg]mex,p=1,j=1SMHW,R=23.16[deg]mex,p=1,j=2SMHW,R=14.59[deg]mex,p=1,j=3
Comparison: Mexican needlets and SMHW at high multipoles θ [deg]-1000100200300400500 ψ ( θ ) SMHW,R= 0.340[deg]mex,p=1,j=11SMHW,R=0.212[deg]mex,p=1,j=12SMHW,R=0.0830[deg]mex,p=1,j=14
Fig. 1.— Comparison between Mexican needlets (grey) and SMHW (black). On the top wepresent the results at low multipoles ℓ (big angular scales), while on the bottom the resultsfor high multipoles ℓ (small angular scales). As expected the higher the multipoles the betterthe similarity. 11 – Mexican needlet j=1 θ [deg]-0.4-0.20.00.20.40.60.8 ψ ( θ ) p=1p=2p=3p=4 Mexican needlet j=6 θ [deg]-20020406080 ψ ( θ ) p=1p=2p=3p=4 Mexican needlet j=11 θ [deg]-200002000400060008000 ψ ( θ ) p=1p=2p=3p=4 Mexican needlet j=16 θ [deg]-5.0•10 ψ ( θ ) p=1p=2p=3p=4 Fig. 2.— Mexican Needlets for B = 1 . p and j . Upper left plot: j=1,upper right plot: j=6, lower left plot: j=11, lower right plot: j=16
4. Real space localization: distance of influence from a mask
For the application of the needlet transform to CMB analysis, it is crucial to knowhow well the needlet coefficients are localized on the sphere. This is particularly true in thepresence of foreground contaminants (diffuse galactic foregrounds or extragalactic sources)or in cases were parts of the CMB maps have been masked. For all practical analysis onewill always need to ask the questions ’How far away from the galactic plane are the needletcoefficients unaffected by the galaxy?’, or ’How far away from the mask are the needletcoefficients expected to behave as if the mask were not present?’.As there is an infinite number of different needlet bases, one cannot run simulationseach time one changes the basis in order to infer the localization properties for this particularkind of needlet. By comparing the influence of the mask in simulations of some needlets toexpected properties based of these needlet functions, it is our aim to obtain an understanding 12 –
Weight function for Mexican and Standard at low frequencies j l b l b l , mex, p=1 and j=1b l , std, j=1b l , mex, p=1 and j=4b l , std, j=4b l , mex, p=1 and j=6b l , std, j=6 Weight function for mexican and standard at high frequencies j l b l b l , mex, p=1 and j=11b l , std, j=11b l , mex, p=1 and j=14b l , std, j=14b l , mex, p=1 and j=15b l , std, j=15 Weight function b l for mexican needlets for different values of p l b l b l , mex, p=1 and j=14b l , mex, p=2 and j=14b l , mex, p=3 and j=14b l , mex, p=4 and j=14 Fig. 3.— Mexican and standard needlet weight functions b ℓ for B = 1 . p and j . The upper two plots show the comparison between Mexican ( p = 1) and standardneedlets for low and high frequencies whereas the lower plot compares Mexican needlets fordifferent values of p . 13 –of the effect of the shape of the needlet functions to simulated CMB maps and thereby inferformulae which can be applied to a large group of needlets without the need of running newsimulations each time.Our main goal is to find a relation for the minimum distance from a mask where theneedlet coefficients are not significantly affected by it. We perform simulations and comparethe needlet coefficients with and without the presence of the mask in order to define the sizeof the contaminated regions. We found that the starting point for the most stable way todefine the contaminated and safe regions was to construct the correlation coefficient betweenthe masked and unmasked needlet coefficients. The higher the correlation, the less theinfluence from the mask. We have obtained these correlation coefficients from an ensembleof 10000 simulations. Each simulations was created and treated according to the followingprocedure:1. Using the WMAP (Hinshaw et al. 2009) best fit power spectrum C ℓ , we generate arandom set of harmonic coefficients a ℓm and then transform to obtain the correspondingtemperature map T ( θ, φ ).2. Make a needlet transform from the map T ( θ, φ ) and obtain the needlet coefficients (atpixel k ) β nmjk , where the superscript nm stands for “no mask”;3. Save the quantity (cid:0) β nmjk (cid:1) in order to be able to be able to calculate the variance of theneedlet coefficients at the end of the iterations;4. Multiply the temperature map T ( θ, φ ) above with the mask and again make the needlettransform in order to obtain the coefficients β mjk , where the superscript m stands for“with mask”;5. Again, save the quantity (cid:0) β mjk (cid:1) ;6. Save also the square of the difference between the two maps, (cid:0) β nmjk − β mjk (cid:1) ;After running all simulations, we are interested in constructing the correlation coefficient C m,nmj ( θ ) = h β mjk β nmjk i q h ( β mjk ) ih ( β nmjk ) i = h ( β mjk ) i + h ( β nmjk ) i − h ( β mjk − β nmjk ) i q h ( β mjk ) ih ( β nmjk ) i were hi represents average over simulations and over all pixels k at the same distance θ fromthe mask. C m,nmj ( θ ) is then the correlation coefficient as a function only of the distance θ from the mask for a given scale j . 14 –The fraction of influence from the mask at a given θ is then given by 1 − C m,nmj ( θ ).We will now define the critical angle θ crit which is the distance from the border of the maskafter which the total fraction of the influence of the mask is smaller than a certain threshold τ . Thus for a threshold of τ = 0 .
01, 99% of the influence of the mask is found inside thecritical angle. This area should then be masked if the remaining 1% of the total influence isaccepted for the data to be used. The thresholds we choose for both masks lie in the range[0 . , − ]. In mathematical terms, the critical angle is defined as R πθ b + θ crit dθ | − C m,nmj ( θ ) | R πθ b dθ | − C m,nmj ( θ ) | = τ, where θ b is the border of the mask. Note that θ crit is defined to be zero at the border of themask so that it directly measures the critical distance from the border of the mask .In the following two subsections we will study the result of this procedure to obtain thecritical angle applied to galactic and point source masks based on the WMAP and Planckexperiments.Since these runs were computationally heavy, we chose to do them with N side = 512 and ℓ max = 1200 and only for the Planck point source holes we needed to increase to N side = 1024and ℓ max = 2100. This has the consequence that when making the needlet transform, wherethe respective harmonic decomposition coefficients a ℓm are multiplied by b ℓ for high valuesof j , it is an “incomplete transform”, because at ℓ = 1200 respectively ℓ = 2100, b ℓ is notyet tending to zero for high values of j . The consequence is that for high ℓ ∗ ’s the values ofthe critical angle start to grow. We discard the values of the critical angle for those ℓ ∗ ’s.Additionally, to have a better estimate of ℓ ∗ we calculated it (via equation 9) using thefunctions b ℓ for ℓ max = 4500 .The calculations of the correlations C m,nmj ( θ ) were done for: 4 (point source holes) and5 (galactic cut) values of the threshold τ , three values of B = 1 .
6, 2 and 1 . B = [1 . , . , . , . , . , . , . , . , . k ∈ [1 , , ,
5] and the Mexican needlet parameter p ∈ [1 , , ℓ ∗ .We therefore make a fit to the critical angle (which is measured from where the maskends) of the form: θ crit = β/ℓ ∗ , (10) This is more relevant for Mexican needlets, since they are not compactly supported as standard ones, ascan be seen in the two top figures in figures 3.
15 –where β is a parameter to be evaluated. Note that β depends on the threshold τ and B , i.e. β = β ( B, τ ) (for Mexican needlets we found that β = β ( τ, p ) with no B dependencebut with dependence on p ). We found that for standard needlets β can be fitted to obtainthe critical angle for arbitrary thresholds τ and values of B . The fit is of the form: β ( B, τ ) = C ( B ) · (cid:18) α ( τ )( B − ξ ( τ )) + ζ ( τ ) (cid:19) , (11)where the 3 parameters α , ξ and ζ depend only on the threshold τ and the type of mask,while the parameter C depends only on B . Their functional form will be presented in therespective appendixes for the galactic and point source masks. For the mexican needlets,since they are largely independent of B , but depend on p , it is possible to make a fit of theform : β ( τ, p ) = C ( p ) · τ C ( p ) . (12)We will now discuss the details for each mask. In order not to get influenced by mask asymmetries, we use a symmetric mask extending15 ◦ on each side of the equator. Note that in section 4.3 we will show that this modelling ofa galactic cut works also for more realistic irregular galactic cuts like in the kq85-mask usedby the WMAP-team or a Planck-like galactic cut.In appendix D we show the form of the hyperbolic fits we have obtained for the criticalangles with this cut and compare the fits with actual calculated angles. Here we will onlyshow the results for a selected number of cases. For standard needlets we will use 4 differentvalues of B spanning from 1.1 to 2; for Mexican needlets we find no dependence of B andwill show results for p = 1 , ,
3. As the Bernstein needlets appear to be similar to thestandard needlets, we do not show any plots for these, but show the fits for some choices of The same holds for the Bernstein needlets (except for the additional slight dependence on the parameter k ), but since their behaviour is very close to the one of the standard needlet, we refrained to make runs formany values of B in order to be able to make reasonable fits. Instead we report the values of β in table (3). Except for the two point source masks with p = 1, where a fit of another form was necessary. Seeappendix E.
16 –parameters in the appendix. The hyperbolic fits are not valid at the lowest multipoles; theexact multipoles for which the fits are valid will be discussed in the appendix.In figure 4, we show in the upper row the critical angles for standard needlets. In theleft plot which shows the lowest multipoles, we see that for the most stringent threshold τ = 10 − where basically no influence of the mask is accepted, the lowest scales ℓ ∗ <
40 nopart of the sky can be accepted for any value of B . If a 10% influence is accepted, thenan extension of up to 20 degrees may be sufficient for lower multipoles. We can clearly seehow the localization is improved with higher values of B . The right plot shows the highermultipoles. For the most stringent threshold, an extension of 3-5 degrees is still necessaryat ℓ = 1000 whereas for a 10% influence, less than a degree is sufficient.In the lower part of the figure, we see the corresponding plots for the Mexican needlets.As expected, the higher the p , the worse the localization properties, but in all cases theMexican needlets outperform the standard needlets in localization. Even for the most strin-gent threshold, for multipoles ℓ <
20 there are still parts of the sky which may be used. At ℓ = 1000, extensions less than one degree are accepted for all thresholds. To mask point sources, the WMAP team uses circular holes of radius 0 . ◦ (WMAP-hole); this radius corresponds to 2.5 times the beam FWHM (Full Width at Half Maximum)of 14’. The Planck channels with highest resolution will have beams of FWHM 5’ and we willthus simulate Planck point sources holes with a radius of 0 . ◦ (Planck-hole) correspondingto 2.5 times 5’. To test the influence of such holes on the needlet coefficients, we have placeda hole at the north pole. Note that in section 4.3 we will show that this modelling of a singlepoint source mask works well also for more realistic point source masks, including manypoint-source holes.In appendix E we show the form of the hyperbolic fits we have obtained for the criticalangles with point source holes and compare the fits with actual calculated angles. In theappendix, we also present the multipole ranges over which the fits are applicable. Again, wewill here only show results for a range of cases.In figure 5 we show the critical angles for the WMAP-hole and in 6 for the Planck-hole.We see that for standard needlets, for ℓ ∗ < Fit of critical angle θ , galactic cut, standard needlets
10 100 1000 l *0204060 θ [ deg ] B=1.1B=1.3B=1.55B=2 τ =0.1 τ =0.01 τ =10 -4 Fit of critical angle θ , galactic cut, standard needlets, high l *
100 1000 l *0246810 θ [ deg ] B=1.1B=1.3B=1.55B=2 τ =0.1 τ =0.01 τ =10 -4 Fit of critical angle θ , galactic cut, mexican needlets
10 100 l *01020304050 θ [ deg ] p=1p=2p=3 τ =0.1 τ =0.01 τ =10 -4 Fit of critical angle θ , galactic cut, mexican needlets, high l *
100 1000 l *01234 θ [ deg ] p=1p=2p=3 τ =0.1 τ =0.01 τ =10 -4 Fig. 4.— Plots of some critical angles for the standard needlets (upper plots) and Mexicanneedlets (lower plots) for the galactic cut. The critical angle is defined to be 0 at the borderof the mask. A range of values in B , p and the threshold τ has been chosen. The plots onthe left show the smaller multipoles whereas the plots on the right are zoomed in on thelarger multipoles. ℓ ∼ − p = 1 and threshold of 10%, themask extension is still less than one degree.For the largest scales, the hole does not significantly influence the needlet coefficients;the influence is so small that the above model breaks down for small multipoles. In table1 we show the ℓ ∗ -ranges which are unaffected by the hole for different acceptance levels ofinfluence. This is important information when working with the largest scales: when themultipoles are below these limits, the point source holes may be ignored. These limits arebased on a threshold τ lim defined as τ lim = Z πθ b dθ | − C m,nmj ( θ ) | ,
18 –
Fit of critical angle θ , WMAP-hole, standard needlets l *020406080 θ [ deg ] B=1.1B=1.3B=1.55B=2 τ =0.1 τ =0.01 τ =10 -3 Fit of critical angle θ , WMAP-hole, standard needlets, high l *
100 1000 l *02468101214 θ [ deg ] B=1.1B=1.3B=1.55B=2 τ =0.1 τ =0.01 τ =10 -3 Fit of critical angle θ , WMAP-hole, mexican needlets
10 100 l *05101520253035 θ [ deg ] p=1p=2p=3 τ =0.1 τ =0.01 τ =10 -3 Fit of critical angle θ , WMAP-hole, mexican needlets, high l *
100 1000 l *01234 θ [ deg ] p=1p=2p=3 τ =0.1 τ =0.01 τ =10 -3 Fig. 5.— Plots of some critical angles for the standard needlets (upper plots) and Mexicanneedlets (lower plots) for the WMAP point source hole with radius 0 . ◦ . The critical angleis defined to be 0 at the border of the mask. A range of values in B , p and the threshold τ has been chosen. The plots on the left show the smaller multipoles whereas the plots on theright are zoomed in on the larger multipoles.which gives the total fraction of influence outside the hole. All needlet scales correspondingto ℓ ∗ ’s below the limit given in the table have a total fraction of influence smaller than thegiven τ lim . For instance if a 1% (integrated) influence from the hole is accepted, then allscales corresponding to ℓ ∗ less than the one given in the table for τ lim = 0 .
01 may be used.
As pointed out in the 2 preceding subsections the masks we used for finding the criticalangles θ crit and fits to them were idealized. In reality one is interested in more complicated 19 – Fit of critical angle θ , PLANCK-hole, standard needlets l *0510152025 θ [ deg ] B=1.2B=1.45B=1.7B=2 τ =0.1 τ =0.01 τ =10 -3 Fit of critical angle θ , PLANCK-hole, standard needlets, high l *
100 1000 l *0246810 θ [ deg ] B=1.2B=1.45B=1.7B=2 τ =0.1 τ =0.01 τ =10 -3 Fit of critical angle θ , PLANCK-hole, mexican needlets
10 100 l *051015 θ [ deg ] p=1p=2p=3 τ =0.1 τ =0.01 τ =10 -3 Fit of critical angle θ , PLANCK-hole, mexican needlets, high l *
100 1000 l *01234 θ [ deg ] p=1p=2p=3 τ =0.1 τ =0.01 τ =10 -3 Fig. 6.— Plots of some critical angles for the standard needlets (upper plots) and Mexicanneedlets (lower plots) for the Planck point source hole with radius 0 . ◦ . The critical angleis defined to be 0 at the border of the mask. A range of values in B , p and the threshold τ has been chosen. The plots on the left show the smaller multipoles whereas the plots on theright are zoomed in on the larger multipoles.types of masks. In this section we show that the results obtained are also valid with goodprecision for more general masks.To investigate this issue we implemented the following procedure: • Choose a needlet, a frequency j , a threshold τ , a value of B and obtain the correspond-ing critical angle θ crit using the fits given in the plots above or in the equations in theappendices. • Choose a realistic mask M and extend it with θ crit along the border to obtain M mod . • Run simulations to obtain the correlation C m − nm for the realistic mask and define an 20 –extended realistic mask M real by setting to zero all points of the map C m − nm wherethe value of the correlation is less than defined by the given threshold . • If the θ crit obtained from the idealistic mask is the correct angle also for a more realisticmask, the two extended masks M mod and M real should be similar. In order to measurehow well the angle works for the realistic mask, we take the difference M mod − M real .We did this for the following masks:1. WMAP point source mask2. WMAP KQ85 galactic cut3. For a Planck-like galactic cut, we used the small KP12 mask used for the WMAP 1year release.We have tested several values of τ , B , j and p for both galactic and point source masksand find excellent agreement between the extended masks; this suggests that the idealizedmodels above work very well for realistic circumstances. For the lowest multipoles, thedifference grows. Only at the lowest values of j where the mask extensions anyway are solarge that they are of little use for realistic analysis, does the model start to be significantlydifferent from the realistic mask extension. We will show three examples here at threedifferent scales and for three different masks.In figure 7 we show an example of the difference M mod − M real for WMAP-point sourcesfor standard needlets with high multipole. The grey area shows the original mask and theblue/red points show where they differ at the border of the mask extension. One can seethat the idealized WMAP-hole works well in zones with many holes. As expected, the modelworks less well when the distance between the point sources are comparable to the criticalangle θ crit , but if θ crit is used as a mask extension, this is of little significance as the fullspace between the sources will be masked anyway.In figures 8 we show selected results for the the galactic masks. Again we see that thedifference between the critical angle and the realistic angle is always smaller than the maskextension, but that the difference starts to increase for the very smallest multipoles. More precisely, for a given threshold τ , set to zero all points of the map C m − nm where C m − nm ≤ ˆ C m − nmj ( θ cr ) with ˆ C m − nmj being the correlation of the idealistic masks from the preceding two subsections.This takes into account that we define the critical angle via the total influence ( ∝ R πθ crit + θ b | − ˆ C m − nmj ( θ ) | dθ )and not only via ˆ C m − nmj ( θ crit ) ≥ − τ .
21 – -1.0 1.0
WMAP kq85 point sources, standard, B=1.6, l *=768 (-13.0, 70.0) Galactic Fig. 7.— Check of generality for WMAP point sources holes for standard needlets withB=1.6 at multipole ℓ ∗ = 768 and with threshold τ = 0 .
01. The plot shows the differencebetween M mod − M real . The light grey area shows the original mask.
5. Harmonic localization properties
In order to study the harmonic localization properties we looked at σ h defined thefollowing way: σ h = P ℓ max ℓ =0 ( ℓ − ℓ ∗ ) b ℓ P ℓ max ℓ =0 b ℓ , (13)where b ℓ is the needlet function in harmonic space. In the calculations below we have used ℓ max = 4500 (and N side = 4096). We remind that ℓ ∗ was formed as a weighted average of ℓ
22 – -1.0 1.0
WMAP kq85 galactic cut, standard, B=1.6, l *=300 -1.0 1.0 PLANCK-like galactic cut, standard, B=1.6, l *=46 Fig. 8.— Check of generality for KQ85 galactic cut (above) and “Planck” galactic cut forstandard needlets with B=1.6 the former at multipole ℓ ∗ = 300 and with threshold τ = 0 . ℓ ∗ = 46 and with threshold τ = 0 .
1. The plot shows the differencebetween M mod − M real . The light grey area shows the original mask.with b ℓ as the weight. The expression for σ h is formed as the weighted average of ( ℓ − ℓ ∗ ) thus being a measure of the width of b ℓ . Since the Mexican needlets do not have a finitesupport, we expect that their harmonic localization is worse than the one from the othertwo needlets; this is indeed shown to be the case in the following. 23 –We find that one can write σ h = Cℓ ∗ , meaning that θ crit σ h = constant . We can thus, as expected, immediately deduce that the better the pixel space localization, theworse the harmonic space localization. It also implies that when the pixel space localizationis strongly or weakly dependent on some parameter B , p or k , this will also be the case forthe harmonic space localization.We note that the localization of the Bernstein needlets in harmonic space for k > σ h of the Mexican needletsdoes not depend on B , but in the case of the Bernstein and standard needlets it does, whenincreasing B , it happens that the Mexican needlets for high values of p have better har-monic localization properties than the other needlets (according to this measure of harmoniclocalization).
6. Correlation properties
Adding on to the previous discussion, we will also calculate directly the correlationbetween needlet-coefficients at a distance θ on the sphere. Using the definitions of theneedlet coefficients, it can easily be shown that for an isotropic field, the correlation function C j ( θ ), defined as the correlation between β jk and β jk ′ when the angular distance between k and k ′ is θ , can be written as C j ( θ ) = h β jk β jk ′ ih β jk i = P ℓ b ℓ ( B, j ) C ℓ ℓ +14 π P ℓ (cos θ ) P ℓ b ℓ ( B, j ) C ℓ ℓ +14 π (14)where P ℓ (cos θ ) is a Legendre polynomial, and C ℓ is the power spectrum of the harmoniccoefficients. We have checked that this expression holds using simulations. In figure 9 wesee a plot of the correlation function for some selected values of θ for standard and Mexicanneedlets. Looking at similar plots for Bernstein needlets with k ∈ { , , } we found thatthere are only negligible differences in correlation properties between them and the standardneedlets. 24 – Correlation for θ = 0.5 o l *-0.50.00.51.0 C o rr e l a t i on Correlation for θ = 4 o l *-0.50.00.51.0 C o rr e l a t i on Correlation for θ = 8 o l *-0.50.00.51.0 C o rr e l a t i on Correlation for θ = 30 o l *-0.4-0.20.00.20.4 C o rr e l a t i on stdmex, p=1mex, p=2mex, p=3 Fig. 9.— Correlation function for different values of θ . The solid line is standard needlets,the others are Mexican needlets with p = 1 (short dashes), p = 2 (dash dot) and p = 3(dashdot dot dot).Upper left: θ = 0 . ◦ , upper right: θ = 4 ◦ , lower left: θ = 8 ◦ and lower right: θ = 30 ◦ .
7. Summary
We have performed the following set of tests of localization properties of needlets whenused with CMB data: • The distance of influence from a galactic cut on needlet coefficients was found fromsimulations and a range from conservative (allowing no influence) to less conservative’distances of safety’ (expressed by the critical angle θ crit ) from the mask. We expressthe resulting distances (critical angles θ crit ) as functional fits of the form β/ℓ ∗ and haveobtained analytical fits for β for standard and Mexican needlets (see expressions 11and 12). Results for some values are shown in figure 4 (standard and Mexican) andlisted in tables 2 (Mexican) and 3 (Bernstein). 25 – • In a similar manner, the ’distance of safety’ from a point source hole (one WMAP-like, one Planck-like) was found; analytical fits for the resulting critical angles (seeagain expressions 10, 11 and 12) were obtained. Some values are shown in figures 5(WMAP-hole) and 6 (Planck-hole) and listed in table 5 (Mexican) and 6 (Bernstein). • We have then also shown that our idealized masks and the results obtained from therecan be used and are valid for more general and realistic masks. In particular, theresults are found valid both for the WMAP KQ85 mask and for a smaller Planck-likegalactic mask. The same holds for point source holes: The results obtained for onehole is shown to be correct also for point source masks with many holes. • We have also studied the correlation distance of needlet coefficients on the sphere. Weuse an analytical formula which we have found to agree well with simulations (equation14) to find the degree of correlation as a function of distance between coefficients.Results for some distances are shown in figure 9. • We have defined a measure σ h (see equation 13) of localization in harmonic space.This characterizes the number of multipoles contained in one single needlet scale j andis found to be proportional to ℓ ∗ . As a result (remembering that θ crit is proportionalto 1 /ℓ ∗ and that θ crit depends heavily on real space localization), we find that, asexpected, θ crit × σ pix is roughly a constant such that when the localization in real spaceis improved, the localization in harmonic space worsens and vice versa.For a certain type of needlet, one can thus obtain the critical angle by (1) calculating ℓ ∗ for the given scale j , using equation 9, (2a) reading the critical angle off figures 4, 5 and6 OR (2b) use the analytic fits presented in appendix D and E.Our results for the different types of needlets can be summarized as follows: • Standard needlets:
The standard needlets are much better localized in harmonicspace than the Mexican needlets (but similar to Bernstein needlets). In fact, thecontribution to a certain scale j is coming from a limited number of multipoles with noinfluence from multipoles outside this range. However, the penalty for high localizationin harmonic space is that the real space localization is lower than for other needlets.The parameter B controls the localization properties: The higher the B , the higheris the real space localization and the larger are the multipole ranges included in eachscale j . • Mexican needlets:
The Mexican needlets are much better localized in real spacethan any of the other types. In harmonic space however, a large (in principle infinite) 26 –multipole range contributes to each scale j . The Mexican needlets depend on theparameter p , larger p have worse real space localization properties but better harmonicspace localization. For p = 3 −
4, the real space localization properties of the Mexicanneedlets approach those of standard needlets for high values of B . For p = 1 and high j , the Mexican needlets are almost identical to the Spherical Mexican Hat Wavelets. • Bernstein needlets:
Bernstein needlets are in between the Mexican and standardneedlets. Their real and harmonic space localization is similar to the standard needlets,sometimes it is slightly better, sometimes it is slightly worse. The Bernstein needletsdepend both on the parameter B and (weakly) on a parameter k . As for the standardneedlets, a higher value of B increases the number of multipoles included in each scale j and improves the real space localization. The parameter k may slightly improveor worsen the localization properties depending on the exact measure used. As anexample we find that for the critical angle for a galactic cut, for k ≥ k , whereas the angle for the morestringent thresholds decreases for increasing k .
8. Acknowledgements
We would like to thank S.K. Naess for useful discussions. We acknowledge the useof the HEALPix (G´orski et al. 2005) package. FKH is thankful for an OYI grant fromthe Research Council of Norway. This research was partially supported by the ASI/INAFagreement I/072/09/0 for the Planck LFI activity of Phase E2. We acknowledge the use ofthe NOTUR super computing facilities. We acknowledge the use of the Legacy Archive forMicrowave Background Data Analysis (LAMBDA). Support for LAMBDA is provided bythe NASA Office of Space Science.
A. A numerical recipe for standard needlets
We recall here briefly the recipe for the standard needlet construction which is advo-cated in Baldi et al. (2009a); Pietrobon et al. (2006); Marinucci et al. (2008)STEP 1: Construct the function f ( t ) = ( exp( − − t ) , − ≤ t ≤ , otherwise
27 –It is immediate to check that the function f ( t ) is C ∞ (i.e. smooth) and compactly supportedin the interval ( − , φ ( u ) = R u − f ( t ) dt R − f ( t ) dt . The function φ ( u ) is again C ∞ ; it is moreover non-decreasing and normalized so that φ ( −
1) =0 , φ (1) = 1;STEP 3: Construct the function ϕ ( t ) = if ≤ t ≤ B φ (1 − BB − ( t − B )) if B < t ≤ if t > φ ( t ) isconstant on (0 , B − ) and monotonically decreasing to zero in the interval ( B − , − BB − t − B ) = (cid:26) f or t = B − f or t = 1and ϕ ( 1 B ) = φ (1) = 1 ,ϕ (1) = φ ( −
1) = 0;STEP 4: Construct b ( ξ ) = ϕ ( ξB ) − ϕ ( ξ )and for b ( ξ ) we take the positive root. In view of all the above four steps, we see that b ( ξ ) ∈ C ∞ with this construction method. B. A numerical recipe for Bernstein Needlets
The steps for constructing a compactly supported function b ( ξ ) having α boundedderivatives (i.e. b ( ξ ) ∈ C α , α < k + 1 /
2) can be described as follows:STEP 1: Define t = ξ − /BB − /B
28 –where
B > . STEP 2: Define polynomials (of Bernstein) B ( n ) i ( t ) = (cid:18) ni (cid:19) t i (1 − t ) n − i . Example: for n = 1 B (1)0 ( t ) = (1 − t ) , B (1)1 ( t ) = t ,for n = 2 B (2)0 ( t ) = (1 − t ) , B (2)1 ( t ) = 2 t (1 − t ) , B (2)2 ( t ) = t ,for n = 3 B (3)0 ( t ) = (1 − t ) , B (3)1 ( t ) = 3 t (1 − t ) , B (3)2 ( t ) = 3 t (1 − t ) , B (3)3 ( t ) = t . STEP 3: Define polynomials p k +1 ( t ) = k X i =0 c i B (2 k +1) i ( t ) ,where c i = 1 , for i = 1 , ..., k, and c i = 0 otherwise.Example: for k = 1 p ( t ) = X i =0 B (3) i ( t ) = (1 − t ) + 3 t (1 − t ) , p ( t ) = X i =0 B (5) i ( t ) = (1 − t ) + 5 t (1 − t ) + 10 t (1 − t ) .Note that p (0) = p (0) = 1 ,and p ′ (1) = p ′ (0) = 0 , p ′ ( t ) = 30 t (1 − t ) , Therefore the derivatives of p have value zero up to order 2 at points 0 and 1. In general p ( r )2 k +1 (1) = p ( r )2 k +1 (0) = 0 for r = 1 , ..., k . 29 –STEP 4: Define the function ϕ ( ξ ) := ξ ∈ [0 , B ] p k +1 ( t ) = p k +1 ( ξ − /BB − /B ) if ξ ∈ [ B , B ]0 if ξ > B . STEP 5: Finally, we have b ( ξ ) = ( q ϕ ( ξB ) − ϕ ( ξ ) , B ≤ ξ ≤ B . (B1)In view of the examples given in this construction, we can also easily see that theconstant C M is in the order of M M ( B − /B ) M , which is rather small compared with theone could obtain from the recipe of standard needlets. C. A comparison between SMHW and Mexican Needlets for p = 1As mentioned earlier, it is suggested from results in (Geller & Mayeli (2009a)) thatMexican needlets in the special case where p = 1 provide asymptotically a very good ap-proximation to the widely popular Spherical Mexican Hat Wavelets (SMHW), which havebeen used in many papers on CMB analysis. More precisely, the discretized form of theSMHW can be written asΨ jk ( θ ; B − j ) = 1(2 π ) √ B − j (1 + B − j + B − j ) [1 + ( y ] [2 − y t ] e − y / B − j , where the coordinates y = 2 tan θ follows from the stereographic projection on the tangentplane in each point of the sphere; here we take θ = θ jk ( x ) := d ( x, ξ jk ). Now write ψ jk ; p ( θ jk ( x )) = ψ jk ; p ( θ ) ;by following the arguments in Geller & Mayeli (2009a) and developing their bounds further,it can be argued that (cid:12)(cid:12) Ψ jk ( θ ; B − j ) − K jk ψ jk ;1 ( θ ) (cid:12)(cid:12) = B − j O (cid:0) min (cid:8) θ B j , (cid:9)(cid:1) , (C1)for some suitable normalization constant K jk > . Equation (C1) suggests that the numericalresults in this paper in the special case where p = 1 can be used as a guidance for theasymptotic theory of random SMHW coefficients at high frequencies j . 30 – D. The hyperbolic fits for the galactic cut
In this section we will list the hyperbolic fits to the critical angle for the galactic cut.We use fits of the form β ( B, τ ) = C ( B ) · (cid:18) α ( τ )( B − ξ ( τ )) + ζ ( τ ) (cid:19) , (D1)where the 4 parameters of the fit C , α , ξ and ζ will be listed in this section. For the Mexicanwe use fits of the form: β ( τ, p ) = C ( p ) · τ C ( p ) . (D2)For standard needlets we obtained fits of β for the values B = [1.1,1.2,1.3,1.4,1.45,1.5,1.55,1.6,1.7,1.8,1.9,2] as well as the thresholds τ ∈ [0 . , . , − , − , − ]. We thenfitted these values of β to the functional form in eq.11 and obtained the following fits for α , ξ and ζ , • C ( B ) = − . B + 4 . • ξ ( τ ) = 0 .
89 independent of threshold; • α ( τ ) = 0 . · τ − . ; • ζ ( τ ) = 0 . · τ − . .We have tested several values of B and τ which were not used for obtaining the abovefits, and also in these cases the critical angles were found to agree very well with the criticalangles obtained with the above fits.In order to show how well the fits work, in figure 10 we show some typical curves forthe critical angle as a function of ℓ ∗ for different thresholds. We show both hyperbolic fits(of the form θ = β/ℓ ∗ ) as well as the actually calculated points for the critical angles.For the Mexican needlets we obtain the following values of the constants C , for eq.12: • C (1) = 2 . C (2) = 2 . C (3) = 2 . • C (1) = − . C (2) = − .
14 and C (3) = − . B and τ are listed in table 3.In tables 2, 3 and 4 we show the lower limit of ℓ ∗ for which the hyperbolic fit for thecritical angle is applicable. This lower limit is due to the fact that at lower ℓ ∗ ’s the galacticcut has influence over the whole sphere. In order to be more precise, we consider the maskto have influence over the whole sphere if the critical angle is bigger than 70 ◦ . One wouldexpect that this lower limit is higher for the more conservative thresholds, which is indeedthe case. Fit of critical angle θ , galactic cut
10 100 1000 l *0204060 θ [ deg ] STANDARDB=1.1B=1.2B=1.3B=1.55B=2MEXICANp=2, τ =0.1p=3, τ =0.01 τ =0.1 τ =0.01 τ =10 -4 Fig. 10.— Plot of some critical angles of the standard and Mexican needlets for the galacticcut for different values of B , p and thresholds. The lines are the hyperbolic fits and pointsare measured values. The maximal angle possible with our definition of the critical angle being zero at the border of the maskis 75 ◦ .
32 –
E. Fits for point source holes
The functional form of the parameters entering equation 11 to fit β ( B, τ ) are: • Both holes: – C ( B ) = − . B + 4 . • WMAP-hole – ξ ( τ ) = 0 .
89 independent of threshold; – α ( τ ) = 10 f ( τ ) with f ( τ ) = − . √ | log ( τ ) | exp (cid:16)
12 ( log ( τ )+2 . . (cid:17) ; – ζ ( τ ) = 0 . · τ − . . • PLANCK-hole – ξ ( τ ) = 0 .
57 independent of threshold; – α ( τ ) = 10 f ( τ ) with f ( τ ) = − . √ | log ( τ ) | exp (cid:16)
12 ( log ( τ )+4 . . (cid:17) ; – ζ ( τ ) = 0 . · τ − . .which we obtained by fitting to the same set of values for B and τ ∈ [10 − , · − , − , − ]as for the galactic cut.For the Mexican needlets we obtain the following fit: • WMAP-hole – p = 1: β ( τ ) = 4 . · f ( τ ) with f ( τ ) ≡ − . · (log ( τ )) − . · log ( τ ) − . – p = 2: β ( τ ) = 1 . · τ − . ; – p = 3: β ( τ ) = 2 . · τ − . . • PLANCK-hole – p = 1: β ( τ ) = 4 . · f ( τ ) with f ( τ ) ≡ − . · (log ( τ )) − . · log ( τ ) − .
2; 33 – – p = 2: β ( τ ) = 1 . · τ − . ; – p = 3: β ( τ ) = 1 . · τ − . .As for the galactic cut, we did not make a fit of the parameter β for the Bernstein needlets,as they are very similar to the standard ones. Instead, to give at least a glimpse, we reportthe values for some values of B in table 6 only for the WMAP-hole. In table 5 we show somevalues of β for the Mexican needlets (which can also be obtained via the formula).In figure 11 we show some examples of the fits and actually calculated points. We showsome of the worst as well as some of the better fits. In tables 5, 6 and 7 we show the lowerlimits on multipoles for which the models work. Fit of critical angle θ , WMAP and PLANCK hole
100 1000 l *051015 θ [ deg ] STANDARD WMAPB=1.2, τ =0.1B=1.2, τ =0.03B=1.4, τ =10 -3 B=1.55, τ =0.1MEXICAN WMAPp=2, τ =0.1STANDARD PLANCKB=1.2, τ =0.03B=1.6, τ =0.03MEXICAN PLANCKp=1, τ =10 -3 p=2, τ =0.1 Fig. 11.— Plot of some critical angles of the standard and Mexican needlets for WMAP-holeand Planck-hole for different values of B , p and thresholds. The lines are the hyperbolic fit,the points are measured values. 34 – REFERENCES
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37 –Needlet τ lim = 10 − τ lim = 10 − τ lim = 10 − B=1.1 W:55, P: - W:19, P: - W:7, P: -B=1.2 W:46, P:240 W:16, P:56 W:9, P:39B=1.3 W:40, P:252 W:18, P:52 W:6, P:18B=1.4 W:42, P:225 W:15, P:59 W:8, P:21B=1.45 W:43, P:189 W:14, P:62 W:7, P:20STD B=1.5 W:40, P:204 W:18, P:60 W:8, P:18B=1.55 W:35, P: - W:15, P: - W:6, P: -B=1.6 W:46, P:187 W:18, P:46 W:7, P:18B=1.7 W:44, P:218 W:15, P:44 W:9, P:15B=1.8 W:38, P:219 W:21, P:68 W:6, P:21B=1.9 W:53, P:191 W:15, P:53 W:8, P:15B=2 W:36, P:145 W:18, P:72 W:9, P:18MEX B=1.1 W:29, P: - W:12, P: - W:6, P: -p=1 B=1.6 W:40, P:165 W:16, P:40 W:6, P:16B=2 W:48, P:189 W:12, P:48 W:6, P:24MEX B=1.1 W:30, P: - W:11, P: - W:9, P: -p=2 B=1.6 W:35, P:142 W:14, P:55 W:5, P:22B=2 W:33, P: - W:16, P: - W:8, P: -MEX B=1.1 W:30, P: - W:13, P: - W:10, P: -p=3 B=1.6 W:42, P:172 W:16, P:42 W:6, P:16B=2 W:40, P: - W:20, P: - W:10, P: -BERN B=1.1 W:55/60*, P: - W:19, P: - W:7, P: - ∀ k B=1.6 W:46, P: - W:17, P: - W:7, P: -B=2 W:36, P: - W:18, P: - W:9, P: -Table 1: Holes: Multipole ranges unaffected by the holes for different acceptance levels ofinfluence, where ’W’ means WMAP-hole and ’P’ Planck-hole. 55/60* means that the rangeis 55 for all k except k = 5. Hyphen means that no simulations were run for those values ofthe parameters. p β , τ = 10 − β , τ = 10 − β , τ = 10 − β , τ = 10 − ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
72 2.8, ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
73 3.4, ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥ β for Mexican needlets for different thresholds. Below thelower limit on ℓ ∗ the mask has influence over the whole sphere. NB: β is given in radians. 38 –Needlet β , τ = 10 − β , τ = 10 − β , τ = 10 − β , τ = 10 − Bern B=1.1 19.0 ℓ ∗ ≥
28 41.5 ℓ ∗ ≥
37 122 ℓ ∗ ≥
73 339 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 31.9 ℓ ∗ ≥
29 81.3 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
19 26.3 ℓ ∗ ≥
36 65.3 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
18 40.8 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
18 16.1 ℓ ∗ ≥
18 32.0 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 19.4 ℓ ∗ ≥
18 26.9 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
18 15.6 ℓ ∗ ≥
18 23.5 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 20.8 ℓ ∗ ≥
18 27.3 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
18 16.1 ℓ ∗ ≥
18 23.4 ℓ ∗ ≥ ℓ ∗ ≥
38 43.9 ℓ ∗ ≥
41 110 ℓ ∗ ≥
97 222 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 27.9 ℓ ∗ ≥
29 55.4 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
10 22.9 ℓ ∗ ≥
19 45.1 ℓ ∗ ≥ β for Bernstein needlets for different thresholds. The lowerlimit on ℓ ∗ is due to the fact that below those ℓ ∗ ’s the mask has influence over the wholesphere. We list also the values of β of the standard needlets at the same B ’s in order toshow the small difference. NB: β is given in radians.Needlet τ = 10 − τ = 10 − τ = 10 − τ = 10 − τ = 10 − B=1.1 38 41 97 189 306B=1.2 16 22 47 115 200B=1.3 5 19 31 89 150B=1.4 6 16 22 59 115B=1.45 4 15 21 43 90B=1.5 6 12 27 60 91B=1.55 4 15 23 55 85B=1.6 5 12 29 46 118B=1.7 6 16 27 45 76B=1.8 4 12 21 38 68B=1.9 5 15 15 52 100B=2 3 10 19 36 -Table 4: Galactic cut: Lower limit on the ℓ ∗ for standard needlets. This limit on ℓ ∗ is due tothe fact that below those ℓ ∗ ’s the mask has influence over the whole sphere. Hyphen meansthat there were not enough points which did not feel the influence over the whole sphere inorder to obtain a trustable fit. 39 – p β , τ = 10 − β , τ = 3 · − β , τ = 10 − β , τ = 10 − ℓ ∗ ≥
13 3.11, ℓ ∗ ≥
13 3.69, ℓ ∗ ≥
13 4.57, ℓ ∗ ≥ ℓ ∗ ≥
15 3.59, ℓ ∗ ≥
16 4.24, ℓ ∗ ≥
15 6.50, ℓ ∗ ≥ ℓ ∗ ≥
16 4.20, ℓ ∗ ≥
16 5.17, ℓ ∗ ≥
16 7.19, ℓ ∗ ≥ ℓ ∗ ≥
16 2.98, ℓ ∗ ≥
25 3.66, ℓ ∗ ≥
25 4.53, ℓ ∗ ≥ ℓ ∗ ≥
22 3.43, ℓ ∗ ≥
22 4.15, ℓ ∗ ≥
22 6.32, ℓ ∗ ≥ ℓ ∗ ≥
16 3.93, ℓ ∗ ≥
16 4.81, ℓ ∗ ≥
26 6.99, ℓ ∗ ≥ β for Mexican needlets for the different thresholds and the 2 holes.The lower limit on ℓ ∗ corresponds to the limit from where our model works. NB: β is givenin radians.Needlet β , τ = 1 · − β , τ = 3 · − β , τ = 1 · − β , τ = 1 · − Bern B=1.1 15.1 ℓ ∗ ≥
22 23.5 ℓ ∗ ≥
24 29.9 ℓ ∗ ≥
24 46.5 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 6.64 ℓ ∗ ≥
12 13.2 ℓ ∗ ≥ ℓ ∗ ≥
10 3.76 ℓ ∗ ≥
10 4.30 ℓ ∗ ≥
19 10.0 ℓ ∗ ≥ ℓ ∗ ≥
22 24.7 ℓ ∗ ≥
24 31.5 ℓ ∗ ≥
26 50.5 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 6.84 ℓ ∗ ≥
12 13.4 ℓ ∗ ≥ ℓ ∗ ≥
10 3.87 ℓ ∗ ≥
10 4.62 ℓ ∗ ≥
19 10.3 ℓ ∗ ≥ ℓ ∗ ≥
22 26.8 ℓ ∗ ≥
26 34.2 ℓ ∗ ≥
31 68.0 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 7.16 ℓ ∗ ≥
12 15.3 ℓ ∗ ≥ ℓ ∗ ≥
10 4.04 ℓ ∗ ≥
10 4.92 ℓ ∗ ≥
18 11.7 ℓ ∗ ≥ ℓ ∗ ≥
24 27.4 ℓ ∗ ≥
31 35.3 ℓ ∗ ≥
31 72.4 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 7.49 ℓ ∗ ≥
12 16.7 ℓ ∗ ≥ ℓ ∗ ≥
10 4.08 ℓ ∗ ≥
10 5.10 ℓ ∗ ≥
18 12.2 ℓ ∗ ≥ ℓ ∗ ≥
22 23.8 ℓ ∗ ≥
24 30.4 ℓ ∗ ≥
24 50.7 ℓ ∗ ≥ ℓ ∗ ≥ ℓ ∗ ≥
12 6.80 ℓ ∗ ≥
12 13.9 ℓ ∗ ≥ ℓ ∗ ≥
10 3.76 ℓ ∗ ≥
10 4.49 ℓ ∗ ≥
19 10.2 ℓ ∗ ≥ β for standard and Bernstein needlets for different thresh-olds. The lower limit on ℓ ∗ corresponds to the limit from where our model works. We listalso the values of β of the standard needlets at the same B ’s in order to show the smalldifference. NB: β is given in radians. 40 –Needlet β , τ = 1 · − β , τ = 3 · − β , τ = 1 · − β , τ = 1 · − B=1.1 W:22, P: - W:24, P: - W:24, P: - W:67, P: -B=1.2 W:16, P:33 W:16, P:33 W:19, P:56 W:39, P:140B=1.3 W:9, P:24 W:9, P:41 W:15, P:53 W:31, P:194B=1.4 W:8, P:22 W:11, P:30 W:11, P:42 W:30, P:161B=1.45 W:7, P:21 W:10, P:30 W:15, P:43 W:30, P:131B=1.5 W:6, P:18 W:8, P:27 W:12, P:41 W:27, P:91B=1.55 W:7, P: - W:10, P: - W:10, P: - W:23, P: -B=1.6 W:7, P:18 W:12, P:29 W:12, P:46 W:29, P:118B=1.7 W:6, P:16 W:10, P:27 W:10, P:45 W:27, P:129B=1.8 W:7, P:21 W:7, P:38 W:12, P:38 W:21, P:122B=1.9 W:8, P:28 W:8, P:28 W:15, P:53 W:28, P:101B=2 W:10, P:19 W:10, P:37 W:19, P:37 W:37, P:144Table 7: Holes: Lower limit on the ℓ ∗ for standard needlets, where ’W’ stands for WMAP-hole and ’P’ for the Planck-hole. This limit on ℓ ∗∗