Statistical imprints of CMB B-type polarization leakage in an incomplete sky survey analysis
Larissa Santos, Kai Wang, Yangrui Hu, Wenjuan Fang, Wen Zhao
SStatistical imprints of CMB B -type polarization leakage in an incomplete sky survey analysis Larissa Santos, ∗ Kai Wang, Yangrui Hu, Wenjuan Fang, and Wen Zhao † CAS Key Laboratory for Researches in Galaxies and Cosmology,Department of Astronomy, University of Science and Technology of China,Chinese Academy of Sciences, Hefei, Anhui 230026, China (Dated: August 12, 2018)One of the main goals of modern cosmology is to search for primordial gravitational waves by looking ontheir imprints in the B -type polarization in the cosmic microwave background radiation. However, this signalis contaminated by various sources, including cosmic weak lensing, foreground radiations, instrumental noises,as well as the E -to- B leakage caused by the partial sky surveys, which should be well understood to avoid themisinterpretation of the observed data. In this paper, we adopt the E / B decomposition method suggested bySmith in 2006, and study the imprints of E -to- B leakage residuals in the constructed B -type polarization maps, B (ˆ n ) , by employing various statistical tools. We find that the effects of E -to- B leakage are negligible for the B -mode power spectrum, as well as the skewness and kurtosis analyses of B -maps. However, if employingthe morphological statistical tools, including Minkowski functionals and/or Betti numbers, we find the effectof leakage can be detected at very high confidence level, which shows that in the morphological analysis, theleakage can play a significant role as a contaminant for measuring the primordial B -mode signal and must betaken into account for a correct explanation of the data. PACS numbers: 95.85.Sz, 98.70.Vc, 98.80.Cq
I. INTRODUCTION
The temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation are powerful cosmo-logical observables to understand the physics of the early universe. During the past decades, much effort has been made tocharacterize the CMB, including three satellite generations: the cosmic background explorer (COBE), the Wilkinson microwaveanisotropy probe (WMAP) and Planck. These experiments were mainly devoted to measure the CMB temperature anisotropies,precisely mapping these tiny fluctuations in the sky [1, 2]. However, according to the standard cosmological model, the fullinformation of CMB is encoded in the statistical properties of both temperature, T (ˆ γ ) , and the linear polarizations, describedby the stocks parameters, Q (ˆ γ ) and U (ˆ γ ) , which are defined with respect to a fixed coordinate system in the sky, and dependon the choice of coordinate system. It is then convenient to decompose the linear polarization into the curl-free ( E -mode) anddivergence-free ( B -mode) components since they have the advantage of being rotationally invariant [3, 4].Recent results from the Planck satellite also show precise measurements of the E -mode field [5]. In the standard cosmologicalmodel, both CMB temperature and E -mode anisotropies are mainly generated by the primordial density perturbations. Inaddition, the auto-correlation power spectra C T T(cid:96) and C EE(cid:96) and the cross-correlation power spectrum C T E(cid:96) sensitively dependon the cosmological parameters and cosmological models. Thus, the observations of WMAP and Planck satellites on thesespectra have tightly constrained most cosmological parameters and inflationary parameters. However, the B -mode polarizationencodes quite different cosmological information: In the large scales, the B -mode polarization is supported to be generated bythe primordial gravitational waves [3, 4, 6], which is the smoking-gun evidence of inflation [7]. In the small scales, the B -modeis mainly produced by the deflection of the CMB photons by cluster of galaxies during their travel between the last scatteringsurface and the observer, known as the CMB lensing [8–10]. During the past few years, many ground-based experiments weredesigned to measure the B -mode polarization signal, such as SPTPol [11], POLARBEAR [12], ACTPol [13], BICEP2 andKeck Array [14–17]. These experiments, including the Planck satellite [18], have detected the lensed B -mode signal in the highmultipole range at the high confidence level. Thus, the detection of the primordial B -mode signal is then the main goal of futureCMB experiments [19]. Among them we can cite BICEP3, AdvACT, CLASS, Simons Array, SPT-3G, C-BASS, QUIJOTE,EBEX, QUBIC, QUIET, PIPER, Spider, LSPE, et al. [20] as ground-based experiments, and LiteBIRD [21], CMBPOL [22],COrE [23], PRISM [24], PIXIE [25] as the next satellite generation of CMB experiments.In the real observations, the detection of cosmological B -mode is limited by various contaminations, including instrumentalnoises, instrumental systematical errors [26], as well as the polarized foregrounds [27, 28]. Another source of contaminationis the so-called E - B mixture, which arises from an incomplete sky analysis of the CMB polarization signals [29]. Our abilityto decompose CMB polarization signal in a partial sky coverage is crucial, since even for satellite missions the presence of ∗ [email protected] † [email protected] a r X i v : . [ a s t r o - ph . C O ] D ec non-cosmological contaminations must be masked out. So, this mixture exists in all the CMB polarization analysis, and couldbecome the dominant contamination for the detection of primordial gravitational waves [29]. In order to solve this problem,numerous practical methods have been developed to separate the E -mode and B -mode in the partial sky [30–34]. However,most of these methods suffer from one or several of the following drawbacks - they are slow in practice, they are difficult torealize in pixel space, and/or they lead to partial information loss. For instance, the method suggested by Bunn et al. (this methodis adopted by BICEP2/Keck Array collaboration [14, 17]) involves constructing an eigenbasis, and it has a high computationalcost [30, 35]. Among all these methods, the methods proposed in [32–34] are based on a common algebraic framework of theso-called χ -fields. Therefore, they are fast and can be efficiently applied to high resolution maps due to the use of a fast sphericalharmonics transformation. However, it is important to mention that the residual of the E -to- B leakage is unavoidable even ifthe proper method is applied in the data analysis [36]. In the previous work [37], the authors carefully compared these threemethods, and they found that although all of them allow a significant reduction of the level of the E -to- B leakage, the methodof Smith [32] ensures the smallest error bars in all experimental configurations and leads to the smallest leakage residuals. Forthese reasons, in the present article, we shall focus on this method due to our limited computational resources, and study theinfluence of residuals of E -to- B leakage on the CMB B -mode polarization.In most previous works [32, 37], the authors focused only on the power spectrum of the B -type polarization and that the E -to- B leakage is tiny in the Smith’s E / B decomposition method. However, the B -mode is a highly non-gaussian field due tothe different kind of components in the observed maps. Thus, in addition to the power spectrum, various non-gaussian statisticaltools are also helpful to separate the different components and/or constrain the cosmological parameters. In the previous work[38], we have applied the Minkowski functionals (MFs) to quantify the deviation from Gaussianity of B -mode maps, and studiedthe effects of instrumental noises, the CMB mask, and the influence of foreground residuals. As an extension of this work, in thepresent paper, by applying a similar analysis we will focus on characterizing the imprint of the E - B mixture residual in the lensed B -mode map. As well known, MFs characterize the morphological properties of convex, compact sets in an n -dimensionalspace [39–45]. Moreover, the E -to- B leakage has completely different morphological properties from the stochastic B -modegenerated by cosmological sources. We expect the influence of the E -to- B leakage to be significant in the MFs analysis, even ifthe amplitude of this leakage is small. As the a complementary analysis, in this paper we shall also apply other statistical tools,including the Betti numbers, skewness and kurtosis, to investigate the imprints of E -to- B leakage. Throughout this paper, inorder to focus on the effect of E -to- B leakage residuals, we shall consider the case with only cosmic variance and ignore theeffects of the other B -mode sources, including the instrumental noises and foregrounds, which have been detail studied in theprevious work [38].The paper is divided as follows: In Sec. II we briefly introduce the theory of CMB polarization. In Sec. III, we derive theequations needed to obtain the E / B decomposition in a partial sky analysis. Sec. IV is divided in 3 subsections: Sec. IV A isdevoted to the introduction of the MF statistics, Sec. IV B details the methodology used in this paper, and Sec. IV C describesthe results obtained for the MFs. In Sec. V, we use other statistics in the analysis for completeness, being the Betti numbersdescribed in Sec. V A, and both skewness and kurtosis explained in Sec. V B. Finally, in Sec. VI, we draw our conclusions. II. E - AND B -MODE POLARIZATION OF CMB The linearly polarized CMB polarization field is completely described by two Stokes parameters, Q and U that can be com-bined into a spin-(2) and spin-(-2) fields P ± (ˆ n ) = Q (ˆ n ) ± iU (ˆ n ) . For full sky, the spin fields can be expanded over spin-weightedharmonic functions basis as follows [46]: P ± (ˆ n ) = (cid:88) (cid:96)m a ± ,(cid:96)m ± Y (cid:96)m (ˆ n ) . (1)Alternatively, the polarization field can be written as the curl-free E , and divergence-free B components, which are defined interms of the coefficients a ± ,(cid:96)m in the harmonic space as E (cid:96)m ≡ −
12 [ a ,(cid:96)m + a − ,(cid:96)m ] , B (cid:96)m ≡ − i [ a ,(cid:96)m − a − ,(cid:96)m ] . (2)In the same way as the temperature field, we can define the E (ˆ n ) and B (ˆ n ) polarization sky maps in terms of sphericalharmonics, E (ˆ n ) ≡ (cid:88) (cid:96)m E (cid:96)m Y (cid:96)m (ˆ n ) , B (ˆ n ) ≡ (cid:88) (cid:96)m B (cid:96)m Y (cid:96)m (ˆ n ) . (3)The power spectra can then be written as C EE(cid:96) ≡ (cid:96) + 1 (cid:88) m (cid:104) E (cid:96)m E ∗ (cid:96)m (cid:105) , (4) C BB(cid:96) ≡ (cid:96) + 1 (cid:88) m (cid:104) B (cid:96)m B ∗ (cid:96)m (cid:105) , (5)where the brackets denote the average over all realizations. For a Gaussian field, all the statistical properties can be obtainedby analyzing the second-order power spectra. It is important to emphasize, however, that here we are dealing with a highlynon-Gaussian lensed B -map due to the contribution of CMB lensing to the final map [38]. In this case, different statistics, as,for example, the MFs, are necessary to describe the field. Moreover, we notice that the E / B decomposition in a full sky analysisis straightforward, providing a direct link to the primordial cosmological perturbations, especially the GWs imprint on the CMB B -mode polarization. Nevertheless, we know that Galactic foregrounds are present even in full sky surveys, and they should bemasked to reduce the contaminations. So, in the realistic case, we must derive the E -type and B -type maps from the incomplete Q and U observables. III. E - AND B -MODE DECOMPOSITION IN PARTIAL SKY If the polarization fields are not measured in full sky, but on a fraction only, Eqs.(2) cannot be derived directly. So, theseparation of pure E - and B -mode from the observed Q and U maps is not trivial, due to the existence of ambiguous mode[30], which can be successfully avoided by different ways [30–34]. As mentioned above, in this paper, we adopt the methodsuggested in [32], which is based on the algebraic framework of the so-called χ -field (denoted as B -field in the present paper).For the full-sky observations, according to [47], we can define a new set of fields E and B from the polarization fields Q and U as follows: E (ˆ n ) = −
12 [¯ ð ¯ ð P + (ˆ n ) + ðð P − (ˆ n )] , (6) B (ˆ n ) = − i [¯ ð ¯ ð P + (ˆ n ) − ðð P − (ˆ n )] , (7)where ð (¯ ð ) corresponds to the spin-raising (lowering) operator for an arbitrary function f with spin s , ð f ≡ − sin s θ (cid:18) ∂∂θ + i sin θ ∂∂φ (cid:19) ( f sin − s θ ) , (8) ¯ ð f ≡ − sin − s θ (cid:18) ∂∂θ − i sin θ ∂∂φ (cid:19) ( f sin s θ ) . (9)From the definition, we know that the new map E in Eq. (6) is the standard scalar field, and B in (7) is the pseudo-scalar field inthe two-dimensional sphere. Thus, they can be expanded in the spherical harmonics as follows, E (ˆ n ) ≡ (cid:88) (cid:96)m E (cid:96)m Y (cid:96)m (ˆ n ) , B (ˆ n ) ≡ (cid:88) (cid:96)m B (cid:96)m Y (cid:96)m (ˆ n ) , (10)where the expanding coefficients are E (cid:96)m = (cid:90) E (ˆ n ) Y ∗ (cid:96)m (ˆ n ) d ˆ n, B (cid:96)m = (cid:90) B (ˆ n ) Y ∗ (cid:96)m (ˆ n ) d ˆ n. (11)These coefficients are related to the regular multipoles E (cid:96)m and B (cid:96)m by E (cid:96)m = N (cid:96), E (cid:96)m , B (cid:96)m = N (cid:96), B (cid:96)m , (12)where we have N (cid:96),s = (cid:112) ( (cid:96) + s )! / ( (cid:96) − s )! [47]. The corresponding power spectra are then obtained as C EE (cid:96) ≡ (cid:96) + 1 (cid:88) m (cid:104)E (cid:96)m E ∗ (cid:96)m (cid:105) = N (cid:96), C EE(cid:96) , (13) C BB (cid:96) ≡ (cid:96) + 1 (cid:88) m (cid:104)B (cid:96)m B ∗ (cid:96)m (cid:105) = N (cid:96), C BB(cid:96) . (14)Considering the mask window function W (ˆ n ) , the masked B -type polarization map becomes B W (ˆ n ) , and the pseudo multi-pole coefficients can be defined as follows [48], ˜ B (cid:96)m = (cid:90) d ˆ nW (ˆ n ) B (ˆ n ) Y ∗ (cid:96)m (ˆ n ) . (15)So, the pure B -type map B W (ˆ n ) and the coefficient set ˜ B (cid:96)m are mathematically equivalent by definition. In the previous work[33], we have developed to method to directly construct the pure map B W (ˆ n ) from the masked observables Q and U . Whilein the method [32], we should first construct the pure coefficients ˜ B (cid:96)m and then translate them into the map B W (ˆ n ) . However,it is important to emphasize that in both methods, it is impossible to construct the pure B -maps directly. In this paper, we shalladopt the latter method, which is briefly reviewed as follows.In this method, the concept of pure pseudo-multipoles is put forward and defined as, B pure (cid:96)m ≡ − i (cid:90) d ˆ n (cid:110) P + (ˆ n ) (cid:2) ¯ ð ¯ ð ( W (ˆ n ) Y (cid:96)m (ˆ n )) (cid:3) ∗ − P − (ˆ n ) [ ðð ( W (ˆ n ) Y (cid:96)m (ˆ n ))] ∗ (cid:111) . (16)It can be proved that this definition is equivalent to the Eq. (15) [32], which shows that in principle the pure pseudo-multipolemethod can successfully extract the pure B -type polarization signal and avoid the E - B mixing part. To calculate the expressionof Eq. (16), we use the property of spin raising and lowering operators and obtain that [37] B pure (cid:96)m = − i (cid:90) d ˆ n (cid:20) P + (cid:18) (cid:0) ¯ ð ¯ ð W (cid:1) Y ∗ (cid:96)m + 2 N (cid:96), (cid:0) ¯ ð W (cid:1) ( Y ∗ (cid:96)m ) + N (cid:96), W ( Y ∗ (cid:96)m ) (cid:19) − P − (cid:18) ( ðð W ) Y ∗ (cid:96)m − N (cid:96), ( ð W ) ( − Y ∗ (cid:96)m ) + N (cid:96), W ( − Y ∗ (cid:96)m ) (cid:19)(cid:21) . (17)This expression is used in the following calculations.Note that, in this method, the first and second derivatives of the window function W are used. If we adopt the top-hatwindow function, these derivatives diverge in the numerical computations. So, in practice, we should properly smooth thewindow function to avoid the divergence at the observed patch boundaries. Thus, an appropriate sky apodization will play animportant role in suppressing the E -to- B leakage. In the previous work [36], we carefully compare the leakage residual and theinformation loss for different smoothing methods, and found that the Gaussian smoothing method presented by [49] induces thesmallest leakage in the final B -map . The Gaussian smoothing kernel used to smooth the edges of W is defined as [49], W i = (cid:90) δ i − δc −∞ √ πσ exp (cid:18) − x σ (cid:19) dx = 12 + 12 erf (cid:18) δ i − δ c √ σ (cid:19) δ i < δ c δ i > δ c (18)where σ = θ F / √ and θ F denotes the full width at half maximum of the smoothing kernel. δ i the smallest angular distancebetween the i -th observed pixel and the boundary of the mask. δ c is an adjustable parameter referred as the apodization length.Let β denotes the jump range at δ i = δ c and δ i = 0 , which is (for details see [49]): β = 12 − erf (cid:32) δ c √ σ (cid:33) . (19) β is a small and adjustable parameter, which must be chosen by investigating the reconstruction numerical accuracy of asmoothed mask performed using the HEALPix package [36]. Here we should mention that performing a spherical harmonictransformation of the foreground mask and its later reconstruction by inverse transformation leads to an oscillation patternaround jump discontinuities (i.e. the reconstructed mask has non-zero values where they were originally zero), called the Gibbsphenomenon. The discrepancy between the original mask and the reconstructed one must be corrected by choosing a windowfunction in which multipoles higher than the truncation point are suppressed. In the previous works [32], the authors discovered the optimal smoothing function to minimize the errors of the constructed power spectra C BB(cid:96) , which isdependent of the multipole (cid:96) . However, different from them, in this paper we shall only focus on the statistical properties of the constructed B -map. So, wechoose the window function for different aim, which should minimize the E - B leakage and reduce the information loss. IV. THE IMPRINT OF E -TO- B LEAKAGE AND MINKOWSKI FUNCTIONALS ANALYSISA. Minkowski Functionals
The MFs describe the morphological properties of convex, compact sets in an n -dimensional space. They provide a powerfulstatistical tool that can also be used in partial sky maps, detecting non-gaussianities without previous knowledge of their intensityor angular dependence [39–45]. On a two-dimensional CMB field defined on the sphere, S , the morphological properties ofthe data can be characterized as a linear combination of three MFs: the area, contour length and integrated geodetic curvatureof an excursion set (the latter also known as the difference between the numbers of hot and cold spots)[40, 44]. For a pixelizedCMB sphere, an excursion set is given by the number of pixels in which the temperature exceeds the threshold ν . For a giventhreshold ν , it is convenient to define the excursion set Q ν and its boundary ∂Q ν of a smooth scalar field u as follows, Q ν = { x ∈ S | u ( x ) > νσ } , (20) ∂Q ν = { x ∈ S | u ( x ) = νσ } . (21)Thus, the area v ( ν ) , the contour length v ( ν ) and the integrated geodetic curvature v ( ν ) , can be written as [44]: v ( ν ) = (cid:90) Q ν da π , v = (cid:90) ∂Q ν dl π , v = (cid:90) ∂Q ν κdl π , (22)where da and dl are the surface element of S and the line element along ∂Q ν , respectively. The geodetic curvature is representedby k . The MFs can be numerically calculated for a given pixelized map u ( x i ) as follows [44, 50]: v ( ν ) = 1 N pix N pix (cid:88) k =1 Θ( u − ν ) , (23) v i ( ν ) = 1 N pix N pix (cid:88) k =1 I i ( ν, x k ) , ( i = 1 , . (24)In Eq. (23), the Heaviside step function is represented by Θ . In Eq. (24), we define I such that I ( ν, x k ) = δ ( u − ν )4 (cid:113) u θ + u φ , I ( ν, x k ) = δ ( u − ν )2 π u ; θ u ; φ u ; θφ − u θ u ; φφ − u φ u ; θθ u θ + u φ . (25)Expressing the covariant derivatives at a point x = ( θ, φ ) , parameterized through the azimuth angle θ and the polar angle φ ofthe unite sphere, we have [44]: u ; θ = u ,θ , u ; φ = 1sin θ u ,φ , u ; θθ = u ,θθ , (26) u ; θφ = 1sin θ u ,θφ − cos θ sin θ u ,φ , u ; φφ = 1sin θ u ,φφ + cos θ sin θ u ,θ . (27)In this paper, we shall investigate the statistical properties of the E -to- B leakage and its possible imprint in the CMB B -modepolarization field due to a partial sky analysis. We used the algorithm developed by [51] and [52] for calculating the MFs. B. Method
First, we generate two groups of Monte Carlo simulations containing 500 full-sky Q and U lensed maps each. The firstgroup of simulation corresponds to a tensor-to-scalar ratio r = 0 , and the second corresponds to r = 0 . . We used the LensPixsoftware [58] with cosmological parameters h Ω b = 0 . , h Ω c = 0 . , h = 0 . , A s = 2 . × − , n s = 0 . , τ reio = 0 . and N side = 1024 . As mentioned above, in the numerical calculations, the numerical errors caused by the highmultipoles are quite significant for two reasons: First, the errors caused by the Gibbs phenomenon is dominant by the highmultipoles [49]; Second, the BB power spectrum is quite tilde blue, and the high multipoles dominate the numerical errors inthe HEALPix-based computations [33]. Similar to the previous works [33, 36], in order to smooth the high multipoles, we applya Gaussian smoothing with the parameter full width half maximum FWHM = 30 (cid:48) . We also smooth the edges of the Planckpolarization mask UT78pol, using Eqs. (18) and (19) with δ c = 1 ◦ and β = 10 − [36] to obtain our window function, shown inFig. 1. With this in mind, we point out the steps of our analysis, which are divided in two parts. In the first case (named as realcase in this paper), in order to mimic the realistic data analysis, we simulate the full-sky Q and U maps, and mask by adoptingthe proper mask. Then, applying the E / B decomposition method in [32, 37] we obtain the partial ‘pure’ B -type polarizationmap B (ˆ n ) . In the second case (named as ideal case in this paper), we derive the coefficients B (cid:96)m from the full-sky Q and U maps, and translate them to the corresponding B (cid:96)m . Thus, we can construct the full-sky B -maps by the standard route. Then,we mask them by applying exactly same mask in the first case. Comparing the B -maps in these two cases, we find that the mapsderived in the real case include the residual E -to- B leakage caused by the E / B decomposition method. However, the mapsderived from the ideal case are free from it [33]. So, the difference between these two kinds of B -maps reflect the imprints ofresidual E -to- B leakage, which is the main goal of this article. To realize it, we do our analysis by the following steps: • Real case : First, we obtain the E / B decomposition from partial Q and U lensed sky maps as numerically described in[37]. For each of the derived B W -map, we can define the pseudo power spectrum as ˜ C BB (cid:96) = (cid:96) +1 (cid:80) m ˜ B (cid:96)m ˜ B ∗ (cid:96)m , where ˜ B (cid:96)m = (cid:82) B W (ˆ n ) Y ∗ (cid:96)m (ˆ n ) d ˆ n . In Fig. 2, we plot the average power spectra ˜ C BB (cid:96) for the model with r = 0 and r = 0 . with solid black and blue lines, respectively. As mentioned above, these power spectra include two parts: One is the CMB B -type polarization, the other is the residual E -to- B leakage. In order to show the contribution of the residual leakage,we do the exact same analysis to the model with r = 0 and no CMB lensing (i.e. no CMB B -mode). Thus, the derived B W -maps only include the residual E -to- B leakage. From Fig. 1, we find that the residuals are quite significant aroundthe two poles and two belts at θ = 48 ◦ and θ = 132 ◦ , which are caused by the structure of the HEALPix package. Thecorresponding power spectrum is also present in Fig. 2 (blue line), which is much smaller than the spectra including CMBsignals, in particular in the low-multipole range. However, in the high-multipole range, the residuals become more andmore important, and dominate the power spectrum at (cid:96) (cid:38) , which is consistent with the previous works [32, 33, 36]. • Second, it is well known that the calculation of the MFs requires smoothing the maps to be analyzed in order to removethe contribution of multipoles dominated by noise. Even though different smoothing scales of the same CMB map have ahigh correlation, they must be taken into account in order to extract all its available statistical information. This is based onthe fact that for each smoothing scale, the information of the CMB is dominant in a different multipole range [51, 54, 56].Thus, we smooth each final B -map with N side = 1024 and FWHM = 30 (cid:48) using a Gaussian filter with a smoothing scale, θ s , such that W l = exp (cid:2) (cid:96) ( (cid:96) + 1) θ s (cid:3) with θ s = 10 (cid:48) , (cid:48) , (cid:48) , (cid:48) , (cid:48) , (cid:48) , generating 6 sets of 500 maps. • Third, we apply three different sky cuts to the final B -maps: the smoothed (for each θ s ) apodized window functionderived from the Planck UT78 polarization mask (hereafter, sky cut 1), to exclude the pixels already without any CMBinformation; the Planck mask + two bands centered at ◦ and ◦ , both with width of ◦ to avoid the E -to- B striperesiduals (hereafter, sky cut 2), and Planck mask + the two residual bands + a ◦ width cut around the poles, also to avoidthe E -to- B residuals in these regions (hereafter, sky cut 3), see the upper panel right side of Fig. 1. For each smoothingscale, we excluded every pixel of the window function with values less than 0.9 in order to remove the boundary effects.We statistically analyze the final 6 sets of 500 B -map simulations for each sky cut by means of the MFs for both groupswith r = 0 and r = 0 . . Note that the binning range of the threshold ν is set from − to with 25 equally spaced bins. • Ideal case : First, we obtain the E / B decomposition in full-sky Q and U lensed sky maps directly using the HEALPixsubroutine anafast. The B (cid:96)m coefficients must be then multiplied by N (cid:96), (see Eqs. (11) and (12)) before generating thefull-sky B -maps using the synfast subroutine of HEALPix. Then, we mask these B -maps by applying the smoothed PlanckUT78pol mask. Similar to the real case, we also calculate the corresponding power spectra ˜ C BB (cid:96) , which are presented inFig. 2 in dashed lines. Comparing with the results in the real case, we find that the spectra are same in both cases, whichvalidates the effectiveness of the E / B decomposition method. • The second and third steps are exactly the same as for the real case. • We finally compare the MFs of both cases, the one that the E -to- B leakage is present (real) and the one that it is not(ideal), in order to identify the possible signature of the leakage in the MFs. We quantify the difference between the idealand real cases by means of the χ statistics, defined as χ = (cid:88) aa (cid:48) (cid:2) ¯ v ideala − (cid:104) v reala (cid:105) (cid:3) C − aa (cid:48) (cid:2) ¯ v ideala (cid:48) − (cid:104) v reala (cid:48) (cid:105) (cid:3) , (28)where (cid:104) ¯ v reala (cid:105) is the model under test. For each smoothing factor, θ s , a and a (cid:48) denote the binning number of the thresholdvalue ν and the different kinds of MF. For the total χ T , a and a (cid:48) also denote θ s . The covariance matrix is estimated fromthe average under 500 simulations C aa (cid:48) ≡ (cid:80) k =1 (cid:104)(cid:0) v k,reala − ¯ v reala (cid:1) (cid:16) v k,reala (cid:48) − ¯ v reala (cid:48) (cid:17)(cid:105) . FIG. 1: Upper panel: On the left, the Gaussian smoothed window function considering Planck UT78 polarization mask with parameters β = 10 − and δ c = 1 ◦ . On the right, the residuals of E -to- B leakage in µ K for r = 0 when CMB lensing is not taken into account. Lowerpanel: the same map shown in the right side of the upper panel rescaled.FIG. 2: The mean power spectra of 500 simulations considering r = 0 (black lines) and r = 0 . (red lines) for the ideal (dashed lines) andreal (solid lines) cases. The blue solid curve is the mean power spectrum of 500 simulations considering r = 0 and no CMB lensing for thereal case (power spectrum related to the E -to- B leakage). C. Results
In real CMB observations, the Galactic emission must be masked out even considering data obtained from satellite surveys.The CMB polarization, especially the B -mode signal, is the main target of future experiments since it can probe inflation.However, this primordial signal can be hidden behind the foregrounds: instrumental noise, different astrophysical foregroundsand CMB lensing (see [38] for a detailed study of the statistics of these different secondary B -mode signals). Moreover, dealingwith partial sky maps, leads to a leakage between E and B modes that could also mimic the primordial signal.First, let’s consider the case in which there is no primordial gravitational waves, r = 0 . After proceeding with the E / B decomposition, we applied to our simulations the sky cuts described in Sec. IV B for each smoothing parameter, θ s : TheUT78pol Planck mask alone, UT78pol + contamination bands, and UT78pol + contamination bands + poles. Our aim is toavoid both Galactic foreground and the E -to- B leakage (check Fig. 1). Calculating the MFs for the final B -map simulations andusing Eq. (31) to obtain the χ statistics, we get the results shown in Table I. They quantify the significance of the leakage foreach smoothing scale and sky cut. We find that, by increasing θ s the significance of the leakage becomes much smaller, whichsignificantly shows that the leakage is dominated by the high multipoles, which is consistent with the results of power spectrumshown in Fig. 2. We can see this result more clear in Fig. 4, where we plotted v , v , v (in Eq. (22)) for θ s = 10 (cid:48) , (cid:48) , (cid:48) inthe case r = 0 . Even though the significance of the leakage seen in Table I also decreases when we exclude the pixels where theleakage is more evident, the contamination bands and the poles, its imprint is still noticeable in the MFs analysis. We concludethat the contamination bands and the poles do not play a very important role in the overall leakage contribution, as we can alsosee in the lower panel of Fig. 1, where we rescaled the unlensed B -map with r = 0 to point out the leakage contribution. θ s = 10 (cid:48) θ s = 20 (cid:48) θ s = 30 (cid:48) θ s = 40 (cid:48) θ s = 50 (cid:48) θ s = 60 (cid:48) Sky cut 1 18.93 9.53 4.27 1.62 0.71 0.34Sky cut 2 17.27 8.55 3.80 1.42 0.63 0.30Sky cut 3 15.78 7.97 3.56 1.28 0.57 0.27TABLE I: χ for r = 0 and three different sky cuts: the UT78pol Planck mask alone, UT78pol + contamination bands, and UT78pol +contamination bands + poles. Now, considering both cases, without ( r = 0 ) and with primordial gravitational waves ( r = 0 . ), and applying the UT78Planck polarization mask only, we find no significant change in the χ between them, as shown in Table II. For the larger tensor-to-scalar ratio, the effect of leakage is slightly amplified for smaller θ s . We can compare the results for the three MFs in Fig. 4with θ s = 10 (cid:48) , (cid:48) , (cid:48) ( r = 0 ) with the same results for r = 0 . in Fig. 5. θ s = 10 (cid:48) θ s = 20 (cid:48) θ s = 30 (cid:48) θ s = 40 (cid:48) θ s = 50 (cid:48) θ s = 60 (cid:48) χ ( r = 0) χ ( r = 0 . χ for different smoothing parameters and different models for the difference between ideal and real case, considering onlythe UT78pol mask. This latter result is more obvious when we analyze the total χ combining all smoothing parameters in Eq. (31). We cansee in Table III that the effect of the E -to- B leakage is more evident for r = 0 . . Comparing Tables II and III, we can alsonotice by the large values of the total χ for both models that the MFs for different smoothing scales are very correlated. Itis easy to understand this since the E -to- B leakage is not a stochastic noise, and it is always relevant in the same sky regions.The total χ is then obtained considering the correlation coefficients ρ aa (cid:48) = C aa (cid:48) / √ C aa C a (cid:48) a (cid:48) instead of a direct sum of the χ values for each smoothing scale, θ s . Note that, in Fig. 3, we did not use the last binning value of the threshold ν since itapproaches zero for the first MF for every θ s in order to avoid numerical problems. It is then important to point out that eventhough the leakage seems not relevant for individual smoothing scales, it is definitely relevant when they are combined as shownin Table III. The E -to- B leakage should be carefully considered when analyzing partial sky CMB B -mode data in order to avoidmisinterpretation of the data. r = 0 r = 0 . χ T χ of MFs analysis for different models, considering only the UT78pol mask. V. OTHER STATISTICS
In order to further investigate the imprint of residual E -to- B leakage in the constructed B -maps, in this section, we shallconsider the other statistical tools (Betti numbers, skewness, kurtosis), which are the complementary of analysis based on MFsand power spectrum. FIG. 3: The correlation coefficient values, ρ aa (cid:48) = C aa (cid:48) / √ C aa C a (cid:48) a (cid:48) , for r = 0 (left panel), and for r = 0 . (right panel) are represented bythe colors. The axis correspond to a and a (cid:48) : the binning number of the threshold value, the different kinds of MFs and the smoothing scale.Both panels correspond to the calculations when only the UT78 Planck polarization mask is considered (sky cut 1).FIG. 4: The difference between the mean values of real and ideal case for the MFs considering r = 0 over 500 simulations. From top tobottom: θ s = 10 (cid:48) , (cid:48) , (cid:48) , respectively. From left to right: the first, second and third MF, respectively. A. Betti numbers
The morphological properties of the excursion sets can be also quantified in terms of topological quantities called the Bettinumbers. They provide an intuitive understanding of the topology of isosurfaces. For a two-dimensional manifold, such as theCMB field, there are two non-zero Betti numbers. The excursion set consists of many connected regions (number of hot spots) β , and independent tunnels (number of cold spots) β . For each threshold ν , we can mathematically express β and β as line0 FIG. 5: The difference between the mean values of real and ideal case for the MFs considering r = 0 . over 500 simulations. From top tobottom: θ s = 10 (cid:48) , (cid:48) , (cid:48)(cid:48)
The morphological properties of the excursion sets can be also quantified in terms of topological quantities called the Bettinumbers. They provide an intuitive understanding of the topology of isosurfaces. For a two-dimensional manifold, such as theCMB field, there are two non-zero Betti numbers. The excursion set consists of many connected regions (number of hot spots) β , and independent tunnels (number of cold spots) β . For each threshold ν , we can mathematically express β and β as line0 FIG. 5: The difference between the mean values of real and ideal case for the MFs considering r = 0 . over 500 simulations. From top tobottom: θ s = 10 (cid:48) , (cid:48) , (cid:48)(cid:48) , respectively. From left to right: the first, second and third MF, respectively. integrals [53, 55]: β = 12 π (cid:90) C + kdl, β = 12 π (cid:90) C − kdl, (29)where C + and C − denote the contours that enclose hot and cold spots, respectively. k is the total curvature of iso-temperaturecontours for each threshold ν . Thus, the genus, g , is given by a linear combination of β and β as g ( ν ) = β ( ν ) − β ( ν ) . Thenumerical method for computing the Betti numbers is outlined in [53, 55].Following the same steps outlined in Sec. IV B and using the exact same 500 B -mode simulated maps, we calculated β and β for both real and ideal cases considering both r = 0 and r = 0 . . The imprint of the leakage in the Betti number results wereobtained by considering ∆ β i = β ideali − β reali , (30)where i = 0 , represent the Betti numbers or alternatively, the number of hot and cold spots.To quantify the difference between the ideal and real cases, we again use the χ statistics χ = (cid:88) aa (cid:48) (cid:2) ¯ β ideala − (cid:104) β reala (cid:105) (cid:3) C − aa (cid:48) (cid:2) ¯ β ideala (cid:48) − (cid:104) β reala (cid:48) (cid:105) (cid:3) . (31)For each smoothing factor θ s , a and a (cid:48) denote the binning number of the threshold value ν and the two Betti numbers. We noticeby Fig. 7 (for r = 0 ) and 8 (for r = 0 . ) that our results are qualitatively consistent with the ones obtained for the MFs: Thesignificance of the leakage becomes smaller as θ s increases. Moreover, we found that the quantitative results of the χ statisticsconsidering each individual smoothing scale for the Betti numbers are also in agreement with the results found for the MFs(compare Tables IV and V with I and Tables II), supporting the method and providing a consistency check for our calculations.Finally, as for the total χ , we see from Table VI that the imprint of the E -to- B leakage becomes less evident when comparedwith the results for the MFs (Table III). It is important to emphasize, however, that there is more information encoded in the MFsanalysis, especially considering all the correlation coefficients when the smoothing scales are combined (compare Figs. 3 and6). The calculation of the Betti numbers is strongly related to the third MF v [53, 55].1 θ s = 10 (cid:48) θ s = 20 (cid:48) θ s = 30 (cid:48) θ s = 40 (cid:48) θ s = 50 (cid:48) θ s = 60 (cid:48) Sky cut 1 24.14 11.47 4.64 1.84 0.74 0.35Sky cut 2 22.31 10.45 4.17 1.53 0.64 0.30Sky cut 3 19.17 9.11 3.93 1.46 0.60 0.28TABLE IV: χ for r = 0 and three different sky cuts, considering the Betti numbers: the UT78pol Planck mask alone, UT78pol + contamina-tion bands, and UT78pol + contamination bands + poles. θ s = 10 (cid:48) θ s = 20 (cid:48) θ s = 30 (cid:48) θ s = 40 (cid:48) θ s = 50 (cid:48) θ s = 60 (cid:48) χ ( r = 0) χ ( r = 0 . χ for different smoothing parameters and different models for the difference between ideal and real case Betti numbers,considering only the UT78pol mask. r = 0 r = 0 . χ T χ of Betti numbers analysis for different models, considering only the UT78pol mask. B. Skewness and kurtosis
In this subsection, for the cross-check, we apply the one-point statistics, skewness and kurtosis to search for the imprints of E -to- B leakage. We calculated the skewness, the lack of symmetry in a distribution, and the kurtosis, the degree to which thedistribution is peaked, for the 500 B -mode simulations considering both ideal and real cases and the 6 smoothing parametersmentioned previously ( θ s = 10 (cid:48) , (cid:48) , (cid:48) , (cid:48) , (cid:48) , (cid:48) ).The results in Tables VII and VIII show the mean values and standard deviation for both statistics considering r = 0 and r = 0 . , which show that no evidence of the leakage was found neither in the skewness nor in the kurtosis statistics (we seeno deviations in the results comparing the ideal and real cases). These analyzes indicate that, different from the MFs and Bettinumbers, the skewness and kurtosis statistics are not sensitive enough to probe the imprints of residual E -to- B leakage in theconstructed B -maps. The skewness histograms of these simulations for the ideal and real cases considering both r = 0 and r = 0 . are shown in Figs. 9 and 10, respectively. The histograms were also plotted for the kurtosis statistics, as can be seen inFigs. 11 and 12, for r = 0 and r = 0 . , respectively. These figures are consistent with the results listed in Tables VII and VIII,i.e. the differences between ideal case and real case are not significant. θ s = 10 (cid:48) θ s = 20 (cid:48) θ s = 30 (cid:48) θ s = 40 (cid:48) θ s = 50 (cid:48) θ s = 60 (cid:48) r = 0 − . ± . − . ± . − . ± . . ± . . ± . . ± . − . ± . − . ± . − . ± . . ± . . ± . . ± . r = 0 . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . TABLE VII: The skewness values (mean value and the standard deviation) for the simulated B -mode polarization maps. For each case, theupper one shows the results derived from the ideal case, while the lower one shows those derived for the real case. In both cases we use 500lensed B -maps simulations and the UT78pol mask only. θ s = 10 (cid:48) θ s = 20 (cid:48) θ s = 30 (cid:48) θ s = 40 (cid:48) θ s = 50 (cid:48) θ s = 60 (cid:48) r = 0 0 . ± .
018 0 . ± .
018 0 . ± .
019 0 . ± .
019 0 . ± .
019 0 . ± . . ± .
018 0 . ± .
018 0 . ± .
019 0 . ± .
019 0 . ± .
019 0 . ± . r = 0 . . ± .
017 0 . ± .
017 0 . ± .
017 0 . ± .
017 0 . ± .
017 0 . ± . . ± .
017 0 . ± .
017 0 . ± .
017 0 . ± .
017 0 . ± .
017 0 . ± . TABLE VIII: The kurtosis values (mean value and the standard deviation) for the simulated B -mode polarization maps. For each case, theupper one shows the results derived from the ideal case, while the lower one shows those derived for the real case. In both cases we use 500lensed B -maps simulations and the UT78pol mask only. FIG. 6: The correlation coefficient values, ρ aa (cid:48) = C aa (cid:48) / √ C aa C a (cid:48) a (cid:48) , for r = 0 (left panel), and for r = 0 . (right panel) are represented bythe colors. The axis correspond to a and a (cid:48) : the binning number of the threshold value, the two Betti numbers and the smoothing scale. Bothpanels correspond to the calculations when only the UT78 Planck polarization mask is considered (sky cut 1).FIG. 7: The difference between the mean values of real and ideal case for the Betti numbers considering r = 0 over 500 simulations. Fromtop to bottom: θ s = 10 (cid:48) , (cid:48) , (cid:48) , respectively. From left to right: the first and the second Betti numbers, respectively. VI. CONCLUSIONS
The CMB primordial B -type polarization is the main target of future CMB observations since it provides the unique op-portunity to directly probe the evolution of the universe in the inflationary stage. To avoid misunderstanding of the data, onemust take into account the different sources of the observed B -mode signal, as for example, CMB weak lensing, astrophysicalforegrounds and instrumental noises. It is well known that the unavoidable partial sky CMB analysis lead to a leakage from E to B mode, acting as an extra noise for the primordial signal. In this paper, we analyzed the effect of the E -to- B leakage in themeasurement of the CMB B -mode. In order to clearly show the effect of leakage residuals, and exclude the effects of the otherfactors, throughout this paper, we do not consider the instrumental noises (depending on the specific experiment, and the survey3 FIG. 8: The difference between the mean values of real and ideal case for the Betti numbers considering r = 0 . over 500 simulations. Fromtop to bottom: θ s = 10 (cid:48) , (cid:48) , (cid:48) , respectively. From left to right: the first and the second, respectively. scheme) and the possible foreground radiations.A number of practical methods have been developed for the E / B decomposition in the incomplete sky surveys. In this paper,we adopted the method suggested by Smith et al., which ensures the small error bars in all experimental configurations and leadsto the smallest leakage residuals. In addition, this method is based on the algebraic framework of χ -field, which avoids the highcomputational cost and can be easily applied to the high resolution CMB maps. However, even if this separation method isused, the residual of the E -to- B leakage is left in the constructed B -maps. In the present article, we employed the MFs, Bettinumbers, skewness and kurtosis statistics to study the morphological imprint of the leakage in the B -mode polarization maps.We compared the ideal case, in which the B -maps were generated from full-sky Q and U observables, and the real case, wherewe generated the B -maps after applying the smoothed Planck polarization mask UT78 to the Q and U sky by means of the Smithmethod described in [32]. Different from the real case, the ideal case is free from the E -to- B leakage residual caused by the E / B separations. So, the difference between these two cases reflects the imprint of E -to- B leakage.First, we compare the power spectra of B -mode polarization maps in these two cases, and found that they are nearly same,in particular in the low-multipole range, and the contribution of E -to- B leakage is tiny. Second, we did not find any imprintof the E -to- B leakage in the simulated B -maps when considering both skewness and kurtosis statistics, which shows that theseone-point statistics are not sensitive to the leakage. These results confirm the effectiveness of the E / B separation method.However, by comparing the MFs and Betti numbers applied to the B -mode maps generated in the ideal and real cases, we findthat the effect of leakage residuals decreases quickly with the increasing of θ s , which significantly shows that the residuals aredominant by the higher multipoles. This result is consistent with the one derived from the power spectrum analysis. In addition,we also found that the leakage cannot be ignored when combining the results of all the smoothing scales, θ s , in both models, r = 0 and r = 0 . . Considering individual smoothing scales leads to the mistaken conclusion that the significance of the leakageis small and that it can be safely neglected. It is then important to point out that the large correlation of the MFs for differentsmoothing scales is expected since the leakage is not randomly distributed in the sky. The E -to- B leakage plays an importantrole in the final B -map and must be taken into account to avoid misinterpretation of the data.4 FIG. 9: The value for the skewness for 500 simulations considering r = 0 for the ideal and real cases as specified in the figure. From top tobottom: θ s = 10 (cid:48) , (cid:48) , (cid:48)(cid:48)
The CMB primordial B -type polarization is the main target of future CMB observations since it provides the unique op-portunity to directly probe the evolution of the universe in the inflationary stage. To avoid misunderstanding of the data, onemust take into account the different sources of the observed B -mode signal, as for example, CMB weak lensing, astrophysicalforegrounds and instrumental noises. It is well known that the unavoidable partial sky CMB analysis lead to a leakage from E to B mode, acting as an extra noise for the primordial signal. In this paper, we analyzed the effect of the E -to- B leakage in themeasurement of the CMB B -mode. In order to clearly show the effect of leakage residuals, and exclude the effects of the otherfactors, throughout this paper, we do not consider the instrumental noises (depending on the specific experiment, and the survey3 FIG. 8: The difference between the mean values of real and ideal case for the Betti numbers considering r = 0 . over 500 simulations. Fromtop to bottom: θ s = 10 (cid:48) , (cid:48) , (cid:48) , respectively. From left to right: the first and the second, respectively. scheme) and the possible foreground radiations.A number of practical methods have been developed for the E / B decomposition in the incomplete sky surveys. In this paper,we adopted the method suggested by Smith et al., which ensures the small error bars in all experimental configurations and leadsto the smallest leakage residuals. In addition, this method is based on the algebraic framework of χ -field, which avoids the highcomputational cost and can be easily applied to the high resolution CMB maps. However, even if this separation method isused, the residual of the E -to- B leakage is left in the constructed B -maps. In the present article, we employed the MFs, Bettinumbers, skewness and kurtosis statistics to study the morphological imprint of the leakage in the B -mode polarization maps.We compared the ideal case, in which the B -maps were generated from full-sky Q and U observables, and the real case, wherewe generated the B -maps after applying the smoothed Planck polarization mask UT78 to the Q and U sky by means of the Smithmethod described in [32]. Different from the real case, the ideal case is free from the E -to- B leakage residual caused by the E / B separations. So, the difference between these two cases reflects the imprint of E -to- B leakage.First, we compare the power spectra of B -mode polarization maps in these two cases, and found that they are nearly same,in particular in the low-multipole range, and the contribution of E -to- B leakage is tiny. Second, we did not find any imprintof the E -to- B leakage in the simulated B -maps when considering both skewness and kurtosis statistics, which shows that theseone-point statistics are not sensitive to the leakage. These results confirm the effectiveness of the E / B separation method.However, by comparing the MFs and Betti numbers applied to the B -mode maps generated in the ideal and real cases, we findthat the effect of leakage residuals decreases quickly with the increasing of θ s , which significantly shows that the residuals aredominant by the higher multipoles. This result is consistent with the one derived from the power spectrum analysis. In addition,we also found that the leakage cannot be ignored when combining the results of all the smoothing scales, θ s , in both models, r = 0 and r = 0 . . Considering individual smoothing scales leads to the mistaken conclusion that the significance of the leakageis small and that it can be safely neglected. It is then important to point out that the large correlation of the MFs for differentsmoothing scales is expected since the leakage is not randomly distributed in the sky. The E -to- B leakage plays an importantrole in the final B -map and must be taken into account to avoid misinterpretation of the data.4 FIG. 9: The value for the skewness for 500 simulations considering r = 0 for the ideal and real cases as specified in the figure. From top tobottom: θ s = 10 (cid:48) , (cid:48) , (cid:48)(cid:48) , respectively. Acknowledgements
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