Separating E and B types of polarization on an incomplete sky
aa r X i v : . [ a s t r o - ph . C O ] J un Separating E and B types of polarization on an incomplete sky
Wen Zhao
1, 2, 3, ∗ and Deepak Baskaran
1, 2, † School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, U.K. Wales Institute of Mathematical and Computational Sciences, Swansea, SA2 8PP, U.K. Department of Physics, Zhejiang University of Technology, Hangzhou, 310014, P.R.China (Dated: August 31, 2018)
Abstract
Detection of magnetic-type ( B -type) polarization in the Cosmic Microwave Background (CMB) radiationplays a crucial role in probing the relic gravitational wave (RGW) background. In this paper, we propose anew method to deconstruct a polarization map on an incomplete sky in real space into purely electric andmagnetic polarization type maps, E (ˆ γ ) and B (ˆ γ ), respectively. The main properties of our approach are asfollows: Firstly, the fields E (ˆ γ ) and B (ˆ γ ) are constructed in real space with a minimal loss of information.This loss of information arises due to the removal of a narrow edge of the constructed map in order toremove various numerical errors, including those arising from finite pixel size. Secondly, this method isfast and can be efficiently applied to high resolution maps due to the use of the fast spherical harmonicstransformation. Thirdly, the constructed fields, E (ˆ γ ) and B (ˆ γ ), are scalar fields. For this reason varioustechniques developed to deal with temperature anisotropy maps can be directly applied to analyze thesefields. As a concrete example, we construct and analyze an unbiased estimator for the power spectrum ofthe B -mode of polarization C BBℓ . Basing our results on the performance of this estimator, we discuss theRGW detection ability of two future ground-based CMB experiments, QUIET and POLARBEAR.
PACS numbers: 98.70.Vc, 98.80.Cq, 04.30.-w ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The extreme conditions in the very early Universe produce primordial perturbations of twogeneric types, namely, density perturbations (scalar perturbations) and relic gravitational waves(tensor perturbations) [1, 2]. In the simplest scenarios, these perturbations are characterizedby a nearly scale invariant primordial power spectra. The experimental determination of theparameters specifying these power spectra provides an important method to investigating thephysics of the very early Universe. The Cosmic Microwave Background Radiation (CMB) hasproved to be a valuable tool in this respect. Scalar and tensor perturbations leave an observableimprint in the temperature and polarization anisotropies of the CMB. The recent experimentaleffort, including the WMAP satellite [3], QUaD [4], BICEP [5] and so on [6–8][9, 10], has leadto a robust determination of the parameters characterizing the primordial density perturbations.On the other hand, the detection of relic gravitational waves (RGWs) remains an outstandingexperimental challenge, and a key task for current, upcoming and future planned CMB observationson the ground [4, 5, 11–17], on balloons [18–20] and in the space [21–25].Both density perturbations and RGWs contribute to the various CMB anisotropy power spectra,namely
T T (temperature), EE (“electric”-type polarization) and T E (temperature-polarizationcross correlation) [26–35]. In addition, RGWs produce the “magnetic”-type ( BB ) polarizationthat is not produced by density perturbations [36]. In principle, all of these information channels( T T , T E , EE and BB ) shoul be used to infer the RGW signal in the CMB. However, if thecontribution of RGWs to the CMB is small ( r . . BB channel will the best venue fordetecting RGWs [35].The separation of the polarization field into electric and magnetic components is a subtle issue.In practice, in a CMB experiment, one directly observes only the Stokes polarization components Q and U . Given a full sky map of the components Q and U , one can construct the so-called E -modeand B -mode of polarization (sometimes referred to as G and C modes, respectively) using spin-spherical harmonics expansion in an unambiguous manner. It is important to keep in mind that,by virtue of construction, E and B modes of polarization are non-local quantities, and require theinformation on Stokes parameters on the complete sky. However, in realistic cases (ground-based,balloon-borne and satellite experiments), the Stokes parameters are measured only on a fractionalportion of the sky. In this situation, the simplest method for constructing electric and magneticpolarization fields, using the spin-spherical harmonics leads to mutual contamination, often referredto as the EB -mixing. This EB -mixing can become a dominant hinderance for detecting the RGW2ignal [37]. In order to overcome this difficulty, numerous methods have been developed to separate E and B types of polarization on an incomplete sky [38–44]. However, these methods suffer fromone or several of the following drawbacks - they are slow in practice, they are difficult to realize inpixel space, and/or they lead to partial information loss.In the current work, we develop a novel method to separate the electric and magnetic compo-nents of polarization on a partial sky. In contrast to previous works [38–40], in this paper we focusmainly on the construction of pure E and B types of polarization in the real space, as opposedto constructions in harmonics space. Our method is based on a simple redefinition of electric andmagnetic components of polarization, so as to make them only depending on the differential of the Q and U fields. On a full sky, this definition is equivalent to the standard definition. However, dueto the differential nature, our definition is directly extendable to polarization maps given on anincomplete sky. The main advantages of our method are as follows. Firstly, the loss of informationis small, and is only caused by the removal of a narrow edge around the observed portion of thesky to reduce numerical errors. Secondly, this method is easy to realize for pixelized polarizationmaps, and is sufficiently fast so as to be practical for high resolution full sky surveys. Thirdly,the method leads to construction of scalar E and B type fields. For this reason, one can directlyapply the various techniques developed for temperature anisotropy. In particular, along this route,we construct an unbiased estimator for the B -mode of polarization power spectrum and gauge itsperformance in detecting RGWs.The outline of the paper is as follows. In Sec. II, we introduce the basic notations and explainthe general ideas behind our method. In this section we derive the basic equations, and constructthe pure electric and magnetic fields E (ˆ γ ) and B (ˆ γ ), given the polarization map ( Q (ˆ γ ), U (ˆ γ ))on an incomplete sky. In Sec. III, we apply this method in practice for a pixelized polarizationmap. In this section we explicitly construct the B (ˆ γ ) field, and discuss the various sources ofcontamination. We go on to discuss the effect edge removal, size of pixels and experimental beamsize on the resulting contaminations. In Sec. IV, we focus on the applying our method to small skysurveys. Combining our method with the pseudo- C ℓ estimator method we construct an unbiasedestimator for the B -mode power spectrum. We show that, in practice, our estimator performs onlyslightly worse than an estimator in an idealized situation with no loss of information. Based on thisestimator, we analyze the ability to detect relic gravitational waves through their signature in B -mode of polarization in two planned ground-based CMB experiments, QUIET and POLARBEAR.We conclude in Sec. V with a brief summary of our main results.3 I. DECOMPOSITION OF THE POLARIZATION FIELD INTO ELECTRIC AND MAG-NETIC COMPONENTS ON AN INCOMPLETE SKY
Let us first give a brief recount of the standard procedure to construct electric E -mode and mag-netic B -mode of polarization, given a complete sky. For mathematical simplicity, it is convenientto introduce the complex conjugate polarization fields P ± as follows P ± (ˆ γ ) ≡ Q (ˆ γ ) ± iU (ˆ γ ) , (1)where ˆ γ denotes the position on the sky, and Q and U are assumed to be real fields on the sky.The fields P ± , being ± P ± (ˆ γ ) = X ℓm a ± ,ℓm ± Y ℓm (ˆ γ ) , (2)where ± Y ℓm (ˆ γ ) are the spin-weighted spherical harmonics. The multipole coefficients a ± ,ℓm arecalculable by a ± ,ℓm = Z d ˆ γP ± (ˆ γ ) ± Y ∗ ℓm (ˆ γ ) . (3)The E and B mode multipoles are defined in terms of the coefficients a ± ,ℓm in the followingmanner E ℓm ≡ −
12 [ a ,ℓm + a − ,ℓm ] , B ℓm ≡ − i [ a ,ℓm − a − ,ℓm ] . (4)One can now define the electric polarization sky map E (ˆ γ ), and the magnetic polarization sky map B (ˆ γ ) as E (ˆ γ ) ≡ X ℓm E ℓm Y ℓm (ˆ γ ) , B (ˆ γ ) ≡ X ℓm B ℓm Y ℓm (ˆ γ ) . (5)The power spectra of E and B modes of polarization are defined, in terms of the multipole coeffi-cients E ℓm and B ℓm , as C EEℓ ≡ ℓ + 1 X m h E ℓm E ∗ ℓm i , C BBℓ ≡ ℓ + 1 X m h B ℓm B ∗ ℓm i , (6)where the brackets denote the average over all realizations.It is important to note that, the polarization sky maps E (ˆ γ ) and B (ˆ γ ), are constructed outof underlying Q (ˆ γ ) and U (ˆ γ ) maps in a non local manner. This is to say that, the value of the E or B field at a given point ˆ γ , in virtue of (5), depend on multipole coefficients E ℓm and B ℓm ,4espectively. These coefficients, in turn, depend on integral of P ± (ˆ γ ) over the full sky (see (3) and(4)). Therefore, one requires the knowledge of Q (ˆ γ ) and U (ˆ γ ) (or equivalently P ± (ˆ γ )) over theentire sky in order to construct the E (ˆ γ ) and B (ˆ γ ) fields.As was mentioned previously, in realistic scenarios, one does not have information on Q and U fields on the entire sky. For this reason (3), (4) and (5) cannot be applied directly to construct E and B types of polarization maps on an incomplete sky. In order to avoid this problem, in thepresent paper we adopt a different but related definition for electric and magnetic polarizationmaps E (ˆ γ ) ≡ − (cid:2) ¯ ð ¯ ð P + (ˆ γ ) + ð ð P − (ˆ γ ) (cid:3) , (7) B (ˆ γ ) ≡ − i (cid:2) ¯ ð ¯ ð P + (ˆ γ ) − ð ð P − (ˆ γ ) (cid:3) , (8)where ¯ ð s and ð s ( s = 1 ,
2) are the spin lowering and raising operators, respectively¯ ð s ≡ − (sin θ ) − s (cid:26) ∂∂θ − i sin θ ∂∂φ (cid:27) [sin s θ ] , (9) ð s ≡ − (sin θ ) − s (cid:26) ∂∂θ + i sin θ ∂∂φ (cid:27) [sin s θ ] . (10)The definitions (7) and (8) for E and B have been previously discussed in the literature [46,47][30][41, 42, 44][48]. These have often been denoted as ˜ E and ˜ B , respectively (see for example[47, 48]). The fields E and B are equivalent to the fields E and B introduced in [30], wherethese fields were introduced as two independent invariants constructed out of the second covariantderivatives polarization tensor (see Eq. (36) in [30]). In the present paper, to maintain a cleardistinction from ( E , B ) in (5), we shall use the ( E , B ) notation.The constructed electric and magnetic fields are scalar fields on the sphere. Therefore, assuming E and B given on a full sky, one can determine the spherical harmonics decomposition coefficients E ℓm ≡ Z d ˆ γ E (ˆ γ ) Y ∗ ℓm (ˆ γ ) , B ℓm ≡ Z d ˆ γ B (ˆ γ ) Y ∗ ℓm (ˆ γ ) . (11)These relations can be inverted to give E (ˆ γ ) = X ℓm E ℓm Y ℓm (ˆ γ ) , B (ˆ γ ) = X ℓm B ℓm Y ℓm (ˆ γ ) . (12)The multipole coefficients E ℓm and B ℓm are related to E ℓm and B ℓm defined in (4) by a ℓ -dependentnumerical factor N ℓ ≡ p ( ℓ + 2)! / ( ℓ − E ℓm = N ℓ E ℓm , B ℓm = N ℓ B ℓm . (13)5ne can also define the power spectra of E and B modes of polarization C EE ℓ ≡ ℓ + 1 X m hE ℓm E ∗ ℓm i , C BB ℓ ≡ ℓ + 1 X m hB ℓm B ∗ ℓm i . (14)These are related with the power spectra C EEℓ and C BBℓ through relations C EE ℓ = N ℓ C EEℓ , C BB ℓ = N ℓ C BBℓ . (15)Thus, in comparison with C EEℓ and C BBℓ , the power spectra C EE ℓ and C BB ℓ are “bluer”, due to thefactor N ℓ . Note that, the relations (13) and (15) assume the polarization fields to be given on acomplete sky.It is important to point out that, the quantities E and B defined in (7) and (8) only dependon the differential of the Q and U fields by construction. Therefore, these definitions can be, inprinciple, applied in the case of Q and U given on an incomplete portion of the sky, to construct the E and B fields on this portion. We now proceed to discuss the relevant steps for this constructionon an incomplete sky.In order to describe the partial sky observations, we firstly introduce the mask window function W (ˆ γ ). This mask function is non-zero only in the observational region of the sky. In addition, weshall assume that W (ˆ γ ) is a real function. In particular, the special case with W (ˆ γ ) = 1 in theobservational region corresponds to the widely discussed top-hat window function. In the presentpaper, we denote this special case of a top-hat window function as w (ˆ γ ). With the introduction ofthe window function W (ˆ γ ), the analysis of the polarization field P ± (ˆ γ ) defined on the partial regionof the sky becomes equivalent to studying the masked field P ± (ˆ γ ) W (ˆ γ ) defined on the completesky.In the general case of an arbitrary mask, one can define two full sky maps ˜ E (ˆ γ ) and ˜ B (ˆ γ )constructed out of observational data˜ E (ˆ γ ) ≡ − (cid:2) ¯ ð ¯ ð ( P + (ˆ γ ) W (ˆ γ )) + ð ð ( P − (ˆ γ ) W (ˆ γ )) (cid:3) , (16)˜ B (ˆ γ ) ≡ − i (cid:2) ¯ ð ¯ ð ( P + (ˆ γ ) W (ˆ γ )) − ð ð ( P − (ˆ γ ) W (ˆ γ )) (cid:3) . (17)Due to the presence of the window function W (ˆ γ ), the two maps ˜ E (ˆ γ ) and ˜ B (ˆ γ ) do not correspondto pure electric and magnetic types of polarization. The main task of this work is to constructpure E (ˆ γ ) and B (ˆ γ ) fields out of ˜ E (ˆ γ ) and ˜ B (ˆ γ ).Moving on, we define the multipole decomposition coefficients ˜ E ℓm and ˜ B ℓm as˜ E ℓm = 1 N ℓ Z d ˆ γ ˜ E (ˆ γ ) Y ∗ ℓm (ˆ γ ) , ˜ B ℓm = 1 N ℓ Z d ˆ γ ˜ B (ˆ γ ) Y ∗ ℓm (ˆ γ ) . (18)6ith this definition, the ˜ E and ˜ B fields can be expanded in terms of the multipole coefficients ˜ E ℓm and ˜ B ℓm in the following manner˜ E (ˆ γ ) = X ℓm N ℓ ˜ E ℓm Y ℓm (ˆ γ ) , ˜ B (ˆ γ ) = X ℓm N ℓ ˜ B ℓm Y ℓm (ˆ γ ) . (19)The multipole decomposition coefficients ˜ E ℓm and ˜ B ℓm can be calculated in an alternative manner.One can begin by defining the complex polarization fields ˜ P ± (ˆ γ ) = P ± (ˆ γ ) W (ˆ γ ) and construct themultiple coefficients using (3) and (4) (with tilde placed on all the relevant quantities). It can beverified that the two definitions are equivalent.Before proceeding, let us point out some simplifying relations. Firstly, from the definitions of¯ ð s and ð s , it follows that ¯ ð = ( ð ) ∗ and ¯ ð = ( ð ) ∗ . In light of definitions (7), (8), (16) and (17),it follows that the two sets of fields ( E , B ) and ( ˜ E , ˜ B ) are real. Furthermore, from the definitionsone has ¯ ð ¯ ð P + = −E − i B , ¯ ð ¯ ð ( P + W ) = − ˜ E − i ˜ B . (20)Thus, in order to determine the relation between the two sets of fields ( E , B ) and ( ˜ E , ˜ B ), it sufficesto study the relation between ¯ ð ¯ ð P + and ¯ ð ¯ ð ( P + W ).In order to derive the relation between the two sets, it is convenient to expand the quantity[¯ ð ¯ ð ( P + W )] W , using the definition of ¯ ð s operator (9) in the following way[¯ ð ¯ ð ( P + W )] W = (¯ ð ¯ ð P + ) W + (¯ ð W )(¯ ð P + ) W + (¯ ð W )(¯ ð P + ) W + P + W (¯ ð ¯ ð W ) + cot θ [ W (¯ ð P + ) + P + (¯ ð W )] W + (2 + 2 cot θ ) P + W + 2 cot θW ¯ ð ( P + W ) . (21)Using the following set of relations that follow from (9)(¯ ð P + ) W = ¯ ð ( P + W ) − P + ¯ ð W − θP + W, (¯ ð P + ) W = ¯ ð ( P + W ) − P + ¯ ð W − cot θP + W,W (¯ ð P + ) + P + (¯ ð W ) = ¯ ð ( P + W ) − θP + W, along with expressions in (20) we arrive at the expression[ E + i B ] W = [ ˜ E + i ˜ B ] W + n (¯ ð W )[¯ ð ( P + W ) − P + ¯ ð W − θP + W ]+ (¯ ð W )[¯ ð ( P + W ) − P + ¯ ð W − cot θP + W ] + P + W (¯ ð ¯ ð W )+ cot θ ¯ ð ( P + W ) + 2 W P + + 2 W cot θ ¯ ð ( P + W ) o . (22)7his expression can be rewritten in a compact form[ E + i B ] W = [ ˜ E + i ˜ B ] W + ct , (23)where ct denotes the correction term. This correction term is complex in general. The real andimaginary parts of the correction term are given asRe[ct] = Q [3 cot θW W x + W ( W xx − W yy ) − W x − W y )]+ U [2 cot θW W y + 2 W W xy − W x W y ]+ 2 W x [( QW ) x + ( U W ) y ] + 2 W y [( U W ) x − ( QW ) y ] , (24)and Im[ct] = U [3 cot θW W x + W ( W xx − W yy ) − W x − W y )] − Q [2 cot θW W y + 2 W W xy − W x W y ] − W y [( QW ) x + ( U W ) y ] + 2 W x [( U W ) x − ( QW ) y ] . (25)In the above expressions we have introduced the shorthand notations F x ≡ ∂F∂θ , F y ≡ ∂F sin θ∂φ , F xx ≡ ∂ F∂θ , F yy ≡ ∂ F sin θ∂φ and F xy ≡ ∂ F sin θ∂φ∂θ for an arbitrary function F (ˆ γ ). In Appendix A, wediscuss the question of numerically calculating the various terms in the above expression in pixelspace.Finally, one can construct the pure electric and magnetic fields E and B on the observed portionof the sky (i.e. region of the sky for which W (ˆ γ ) = 0) using expression[ E + i B ] = [ ˜ E + i ˜ B ] W − + ct W − . (26)The construction of the pure electric and magnetic scalar fields E and B is the main result of thispaper. It is worth pointing out that the construction of these fields is independent of the choice ofthe mask function W (ˆ γ ), as long as the mask in non-zero in the observed portion of the sky. Thismethod for recovering the scalar fields E and B is lossless in the real space in following sense. If onewas given the polarization fields Q and U on the entire sphere and constructed the corresponding B field using (8) (or the E field using (7) ) and compared the resulting scalar field in the observedregion with result of above procedure (26), one would find the two fields equal. However, due tothe ill-behaved nature of W − and W − at the edge of observed region, it is difficult to realize theabove construction in practice. In order to circumvent this problem, as will be discussed in thefollowing section, one has to remove the edge of the constructed polarization maps.8n conclusion of this section it is instructive to clarify the issues associated with possible leakagefrom the so-called ambiguous modes. It is known that, on a manifold with a boundary, the decom-position of the polarization field, in addition to pure E and B components, contains ambiguousmodes that satisfy both E -mode and B -mode conditions simultaneously (see [38, 39] for details).In particular, when constructing the power spectrum estimators for B mode one has to ensure thatthere is no leakage from the ambiguous modes. In the current work, the B rec (ˆ γ ) field does notcontain contribution from either E modes of polarization or ambiguous modes, by the virtue ofconstruction (analogous to χ B in [41]). For this reason, the power spectral estimators constructedfrom this field will be free from contaminations from both E -modes and ambiguous modes. III. E / B SEPARATION IN PIXEL SPACE
In this section, we shall discuss the issues related to separation of electric and magnetic polar-izations E (ˆ γ ) and B (ˆ γ ) in the pixel space using the results of the previous section, in particularexpression (26). We shall discuss this procedure using a toy model. For this toy model, we assumethat an experiment will only observe the Stokes parameters Q and U in the northern hemisphere.Following [37, 40], we adopt the following axially-symmetric form for the mask window function W (ˆ γ ) W (ˆ γ ) = θ < θ − θ , − cos( θ − θ θ π ) θ − θ < θ < θ , θ > θ . (27)In the above expression, θ corresponds to edge of the observational area, and θ ≥ θ = 0, corresponds to the top-hat window function ( w (ˆ γ ) = 1for θ < θ and w (ˆ γ ) = 0 for θ > θ ). However, the top-hat function is discontinuous at θ = θ , whichmakes quantities W xx , W xy and W yy ill-defined at the edge. In order to avoid these difficulties,it is convenient to use a window function with θ = 0. Throughout the present section we usevalues θ = 90 o and θ = 30 o . It is important to point out that the formalism outlined in Sec. IIis applicable for arbitrary mask window functions, not necessarily axially symmetric. The simplesymmetric form (27) for the mask was chosen for simplicity and clarity of presentation. In realisticscenarios one will have to use a more complex mask, that will take into account the non-symmetricform of the observed region and various point source contaminations.For an axially symmetric window function W (ˆ γ ) (i.e. when W (ˆ γ ) is independent of φ ), such as9he one considered in the present section, the correction term ct in (24) and (25) simplify toRe[ct] = Q [3 cot θW W x + W W xx − W x ] + 2 W x [( QW ) x + ( U W ) y ] , (28)Im[ct] = U [3 cot θW W x + W W xx − W x ] + 2 W x [( U W ) x − ( QW ) y ] . (29)In order to demonstrate the E / B separation we shall work with simulated polarization maps.We use the synfast subroutine included in the HEALPix package to generate a full sky map of Q (ˆ γ )and U (ˆ γ ) fields. In order to generate this map we use the best-fit WMAP5 values for cosmologicalparameters [49] Ω b h = 0 . , Ω c h = 0 . , Ω Λ = 0 . ,τ reion = 0 . , A s = 2 . × − , n s = 0 . , (30)and assume no contribution from gravitational waves and cosmic lensing to the B -mode of po-larization, i.e. C BBℓ = 0. We adopt the pixelization with N side = 512. We set the full width athalf maximum (FWHM) for the Gaussian beam to θ F = 30 ′ . In what follows we shall be solelyinterested in the determination of B polarization field, and studying the possible contaminationsto it. In this context, since C BBℓ = 0, a non-zero field B would be wholly attributed to the residual EB -mixing contamination.Before proceeding to construct the pure magnetic field B , we shall construct and analyze theauxiliary ˜ B field. From the simulated Q (ˆ γ ) and U (ˆ γ ) maps we construct the ˜ B (ˆ γ ) in the followingmanner. Firstly, we construct the multipole coefficients ˜ B ℓm . This is done by building multipolecoefficients ˜ a ± ,ℓm = R d ˆ γ P ± (ˆ γ ) W (ˆ γ ) ± Y ∗ ℓm (ˆ γ ) using the simulated ( Q (ˆ γ ), U (ˆ γ )) maps along withthe window function W (ˆ γ ) given in (27), and then calculating ˜ B ℓm = − i [˜ a ,ℓm − ˜ a − ,ℓm ]. Thesesteps were performed numerically using the anafast routine from the HEALPix package. Followingthis, we use (19) to construct ˜ B from the multipole coefficients ˜ B ℓm . The resulting ˜ B map isillustrated in Fig. 1.One can see that although the map was generated with C BBℓ = 0, ˜
B 6 = 0 in the region θ − θ <θ < θ (i.e. 60 o < θ < o ). This can be viewed as a of leakage of E -type of polarization into˜ B field, due to the presence of the window function W . In order to quantify this leakage in theharmonic (multipole) space, we construct the pseudo power spectrum as D ˜ B ˜ B ℓ ≡ ℓ + 1 X m ˜ B ℓm ˜ B ∗ ℓm , where ˜ B ℓ m ≡ Z ˜ B (ˆ γ )Y ∗ ℓ m (ˆ γ )dˆ γ. (31)The resulting pseudo power spectrum is plotted a the black line in Fig. 3.10 IG. 1: The ˜ B map constructed from an input model with no magnetic polarization (in µ K).
We can now reconstruct pure magnetic type field B using (26) in the following manner B rec (ˆ γ ) ≡ ˜ B (ˆ γ ) W − (ˆ γ ) + Im(ct(ˆ γ )) W − (ˆ γ ) . (32)We use the subscript rec to indicate that this field was reconstructed field from ˜ B and the ( Q , U )fields. The results of reconstruction are presented in Fig. 2. Since the input cosmological modelassumes no contribution from gravitational waves, one can expect B rec (ˆ γ ) = 0. A visual comparisonof Fig. 1 and Fig. 2 shows that we have been able to remove much of the leakage that was presentin ˜ B . The remaining residual contamination in B rec is shown (with a magnified scale) in the rightpanel of Fig. 2. This residuals are a small fraction ( ∼ B with B rec in (31). The resulting pseudo power spectrum is plotted as the red curve in Fig. 3. It canbe seen that the resulting leakage power for B rec is significantly lower that the corresponding powerfor ˜ B . In particular, in the practically interesting range of multipoles ℓ ∈ (50 , B rec field is roughly four order of magnitude lower than the spectrum for ˜ B .The remaining residuals in B rec are attributed to numerical errors.We believe that the small remaining residuals in B rec are a result of two types of numericalerrors that cannot be avoided in practice. The first reason for errors is pixelization. In [39], it wasargued that pixelization can lead to the mixing of electric and magnetic modes. This point can beintuitively understood in the following way. Imagine a survey that observes polarization on a smallsquare region of the sky. Pixelization introduces a Nyquist wavenumber k N , such that all modeswith wavenumbers greater than k N are aliased to modes with wavenumbers less than the Nyquistvalue. This aliasing completely shuffles the direction of wavenumbers, thus essentially leading toa complete mixing of electric and magnetic modes. Although the complete avoidance of the errorsdue to pixelization is impossible, these numerical errors can be reduced by using a larger value for11 IG. 2: The pure magnetic field B rec (ˆ γ ). The edge of the map with W (ˆ γ ) < .
03 has been removed. Theleft panel is scaled similarly to Fig. 1, while the right panel has the scaling magnified in order to show theresidual leakage. Both the panels use the units µ K. N side (see Sec. III B for details). In the present evaluation, with N side = 512, finite pixelizationseems to be the main reason for residual power spectrum for ℓ > B rec (ˆ γ ) at the pole in the real space (see theright panel in Figure 2).The second reason for numerical errors is the steep growth of power spectrum C EE ℓ with in-creasing ℓ . Due to this, even a small relative numerical error at higher multipoles seeps through tolower multipoles. In other words, these errors occur due to fact that the various sources of noiseand E mode signal are not band limited. We believe that these types of errors mainly account forthe residual power spectrum of B rec at multipoles ℓ < B rec (ˆ γ ) around the observed edge θ = θ in the real space (see the right panel in Figure2). A. Dependence of residual leakage on edge removal
Figure 2 shows that much of the residual leakage occurs around the edge of the observationregion. It is therefore instructive to study the edge effects in more detail. The expression (32) for B rec depends on Im(ct) correction term given by (29). This correction term contains two terms,(3 cot θW W x + W W xx − W x ) /W and 2 W x /W , which have the following asymptotic at the edgeof the map as θ → θ θW W x + W W xx − W x W → − θ − θ ) , W x W → − θ π ( θ − θ ) . Thus, the two functions are divergent for θ → θ . This implies that the signal-to-noise will tendto zero for data as the boundary of the observed region is approached. Because of this divergence,12 -5 -4 -3 -2 -1 line 2line 1 P se udo po w e r s p ec t r u m D l ( K ) Multipole: l
FIG. 3: The pseudo power spectrum D ˜ B ˜ B ℓ (black line, i.e. line 1) of the ˜ B field and pseudo power spectrum D ℓ (red line, i.e. line 2) of the B rec field. in numerical calculations, one must remove the edge of the map in order to avoid numerical errorsassociated with these divergences, thereby introducing a small loss in information.In order to investigate the dependence of residual leakage on the edge removal, in Fig. 4 weplot the pseudo power spectrum D ℓ of residual B rec constructed for the same simulated data butfor two different edge removals. The first case (red line) corresponds to portion of the sky with W < .
03 removed (this corresponds to removal of data with θ − θ < . W < . θ − θ < . ℓ < B rec becomes smaller. Onthe other hand, in the region of higher multipoles, where the dominant contribution comes fromfinite pixelization errors, the two power spectra are comparable.13 -5 -4 -3 -2 -1 line 3 line 2 line 1 P se udo po w e r s p ec t r u m D l ( K ) Multipole: l
FIG. 4: The red line (i.e. line 2) shows the pseudo power spectrum D ℓ calculated for the edge removal at W = 0 .
03, while the green line (i.e. line 3) shows the D ℓ for edge removal at W = 0 .
1. For comparison, theblack line (i.e. line 1) shows the pseudo power spectrum D ˜ B ˜ B ℓ . Note that, the black line (i.e. line 1) and redline (i.e. line 2) are identical to those in Fig. 3. B. Dependence of residual leakage on pixelization number N side As was pointed out earlier, one of the reasons for residual leakage of power into B rec is finitepixelization of the sky map. In order to demonstrate the effect of pixelization on the residualpower, in Fig. 5 we show the pseudo power spectrum D ℓ calculated for two different pixelizationnumbers N side = 512 (red line) and N side = 1024 (green line). As one might expect, the increasein the pixelization number reduces the leakage power spectrum. This reduction is most dramaticat higher multipoles ℓ > ℓ <
150 the reduction is not as dramatic, and is roughly by a factor 3. These resultsare consistent with our previous statements about the cause of numerical errors. Indeed, at highermultipoles the main cause of errors seems to be finite pixelization, whereas at lower multipoles theerrors are generated by a combination of factors.14 -5 -4 -3 -2 -1 line 3 line 2 line 1 P se udo po w e r s p ec t r u m D l ( K ) Multipole: l
FIG. 5: The red line (i.e. line 2) shows the pseudo power spectrum D ℓ calculated for N side = 512, while thegreen line (i.e. line 3) shows D ℓ calculated for N side = 1024. For comparison, the black line (line 1) showsthe pseudo power spectrum D ˜ B ˜ B ℓ . Note that, the black line (i.e. line 1) and red line (i.e. line 2) are identicalto those in Fig. 3. C. Dependence of residual leakage on θ F The full width at half maximum parameter θ F has an important effect on the residual leakageof power into B rec . In order to understand the reason for this, one has to remember that one of thetwo reasons for residual leakage is the steep growth of power spectrum C EE ℓ with increasing ℓ . Theparameter θ F regulates the exponential damping of this power spectrum at multipoles ℓ ≃ θ − F ,and therefore limits the propagation of the power in C EE ℓ into D ℓ .The various contributions to the spectrum C EE ℓ are illustrated in Fig. 6. The main contributionto power spectrum comes from density perturbations (dashed blue line). For comparison, on thisfigure, we show the contribution to C EE ℓ (solid blue line) and C BB ℓ (solid red line) from gravitationalwaves (characterized by tensor-to-scalar ratio r = 0 . C BB ℓ from lensing(red dashed line). The spectrum C EE ℓ from density perturbations at high multipoles acts as themain source for the residual leakage into D ℓ . 15 -5 -4 -3 -2 -1 XX= (lens) XX= (g.w.)XX= (g.w.)XX= (d.p.) P o w e r s p ec t r a C l XX ( K ) Multipole: l
FIG. 6: The polarization power spectra generated by density perturbations (d.p.), gravitational waves (g.w.)with r = 0 .
1, and cosmic lensing (lens).
The contribution to the various spectra at high multipoles are effectively damped by a choice ofan appropriate θ F . This parameter leads to the damping of power spectrum C EE ℓ proportional toexp (cid:16) − ℓ ( ℓ +1) θ F (cid:17) . In Fig. 7 we show the residual leakage for two choices of the FWHM parameter θ F = 30 ′ (red line) and θ F = 10 ′ (green line). For comparison, in this figure we also show thepseudo spectrum C ˜ B ˜ B ℓ calculated θ F = 30 ′ (black line) and θ F = 10 ′ (blue line). As one mightexpect, the residual power spectrum reduces significantly with an increase in θ F . For this reason,for the purposes of extracting the magnetic pattern of polarization in experiments with small θ F (for example POLARBEAR experiment discussed in Sec. IV E), it becomes necessary to artificiallyincrease θ F in order to reduce residual leakage in B rec . In Appendix D, we suggest a ‘map smoothing’technique to achieve this goal. 16 -5 -4 -3 -2 -1 line 2line 4line 3line 1 P se udo po w e r s p ec t r u m D l ( K ) Multipole: l
FIG. 7: The red line (i.e. line 2) shows the pseudo power spectrum D ℓ , calculated for a map with θ F = 30 ′ ,while the green line (i.e. line 4) shows D ℓ calculated with θ F = 10 ′ . The black line (i.e. line 1) shows D ˜ B ˜ B ℓ calculated for θ F = 30 ′ , and the blue line (i.e. line 3) shows D ˜ B ˜ B ℓ for θ F = 10 ′ . Note that, the black line(i.e. line 1) and red line (i.e. line 2) are identical to those in Fig. 3. IV. E / B SEPARATION AND POWER SPECTRUM ESTIMATION FOR SMALL SKYSURVEYS
In Sec. II and Sec. III we developed a method to construct pure electric E rec and magnetic fields B rec out of the original Stokes parameter fields Q and U on a fractional portion of the sky. Ignoringthe small numerical errors, it was shown that the resulting fields did not exhibit mixing. A crucialpoint about the constructed fields is that they are scalar fields. For this reason, one can use all ofthe robust techniques developed for studying CMB temperature anisotropy to the fields E rec and B rec . Based on appropriation of these techniques, in this section, we shall focus on an importantpractical application, namely constructing the estimator for the power spectrum of the B -mode ofpolarization C BBℓ . For this reason, as in the previous sections, we shall restrict our analysis to justthe magnetic field B rec .The question of constructing an estimator for the power spectrum C BBℓ from the field B rec
17s analogous to the problem of construction an estimator for the temperature anisotropy powerspectrum C T Tℓ given a temperature map on a partial sky. Fortunately, there are a large number ofmethods that have been developed for this purpose [50–54]. Amongst these, a popular method isthe so-called ‘pseudo- C ℓ ’ estimator method [53]. This method can be easily realized in pixel spaceusing fast spherical harmonics transformation, and has been applied to various CMB observationsincluding WMAP data [55]. However, it is well known that pseudo- C ℓ estimators are sub-optimal,particularly for low multipoles. For this reason, many authors have developed alternative estimatorsthat are optimal, in particular the maximum likelihood estimators in pixel realization [50, 52]. Thefundamental problem with the maximum likelihood estimators is that these methods are veryslow, especially for larger multipoles. For large sky surveys, such as the Planck satellite, the use ofhybrid estimators, which combine the two methods, has been suggested [54]. The hybrid estimatorcombines the best of two worlds, it is nearly optimal and can be realized of a laptop computer evenfor large sky surveys such as Planck.In the present section we shall focus on small sky polarization surveys, corresponding to variousground-based CMB experiments [4, 5, 12–14]. Since these surveys will be primarily sensitive torelatively large multipoles ℓ &
20, we shall restrict our analysis to pseudo- C ℓ type estimators, whichare nearly optimal for large multipoles. Before proceeding we would like to point out that a hybridtype of estimator could be potentially used to construct an estimator for C BBℓ from B rec in thecase of large sky surveys such as Planck. We leave this exercise for future.Below we shall work with a small fraction of the sky characterized by a window function (27) with θ = 20 o and θ = 10 o , corresponding to a 3% sky survey. In an ideal case, neglecting numericalerrors, the reconstructed field B rec (ˆ γ ) would be related to the underlying full sky field B (ˆ γ ) through B rec (ˆ γ ) = B (ˆ γ ) w (ˆ γ ), where w (ˆ γ ) is the corresponding top-hat window function. However, as waspointed out in Sec. III A, one needs to remove a narrow edge from the observational area in orderto avoid excessive numerical errors. For this reason, in practice, we remove the region θ − θ < . θ > o ) from the analysis. Below we shall use the notation w ′ (ˆ γ ) to denote thetop-hat window function for data with edge removal.18 . Pseudo estimators The first step in constructing the pseudo estimator is the definition of spherical harmonicscoefficients a ℓm of the scalar field B rec (ˆ γ ) as follows a ℓm = Z d ˆ γ B rec (ˆ γ ) W (ˆ γ ) Y ∗ ℓm (ˆ γ ) , (33)where W (ˆ γ ) is the weight function. In principle, one can choose an arbitrary form for the weightfunction. In particular, the choice W (ˆ γ ) = 1 corresponds to the widely discussed pseudo- C ℓ estimator introduced in [53]. This choice will be the main focus of our attention in the presentwork. An alternative choice W (ˆ γ ) = W (ˆ γ ) (where W (ˆ γ ) is the mask window function in Eq. (27))corresponds to the analysis in [41], where it was shown that the resulting a ℓm lead to the pure B -mode estimators defined in [40]. The comparison of this choice for the weight function with ourmain choice W (ˆ γ ) = 1 is discussed in Appendix C. The optimal choice of the weight function invarious cases has been discussed in [37, 41, 56]. In [41] the authors suggest a general method tobuild the weight function W (ˆ γ ) for different multipole ℓ in order to optimize the estimator. At thispoint it is important to emphasize that although B rec (ˆ γ ) preserves the available information in realspace, a non-optimal power spectrum estimation will lead to loss of some of this information. Thismakes the study of the optimal choice of weight function particularly important. However, in thecurrent paper we concentrate mainly on the simplistic case W (ˆ γ ) = 1, leaving the important butcomplicated question of optimal choice of weight function for future work.For the choice W (ˆ γ ) = 1, the spherical harmonics coefficients a ℓm in (33) take the simplifiedform a ℓm = Z d ˆ γ B rec (ˆ γ ) Y ∗ ℓm (ˆ γ ) . (34)These are related to the coefficients B ℓm (which were defined in (11) in terms of the underlying fullsky map B (ˆ γ )) through the coupling matrix K ℓmℓ ′ m ′ (see for instant [54]) a ℓm = X ℓ ′ m ′ B ℓ ′ B ℓ ′ m ′ K ℓmℓ ′ m ′ = X ℓ ′ m ′ B ℓ ′ N ℓ ′ B ℓ ′ m ′ K ℓmℓ ′ m ′ , (35)where B ℓ is a window function describing the combined smoothing effects of the beam and thefinite pixel size. The coupling matrix K can be expressed in terms of the function w ′ (ˆ γ ) as K ℓ m ℓ m = Z d ˆ γ w ′ (ˆ γ ) Y ℓ m (ˆ γ ) Y ∗ ℓ m (ˆ γ ) . (36)19he pseudo estimator D ℓ is defined analogous to (31) in terms of the multipole coefficients (34)as D ℓ = 12 ℓ + 1 X m a ℓm a ∗ ℓm . (37)Using relations (6), (15) and (35), one obtains that the expectation value of this estimator D ℓ isrelated to the true power spectrum C BBℓ by the following convolution h D ℓ i = X ℓ ′ M ℓℓ ′ B ℓ ′ C BB ℓ ′ = X ℓ ′ M ℓℓ ′ N ℓ ′ B ℓ ′ C BBℓ ′ . (38)The coupling matrix M in the above expression can be expressed in terms of 3 j symbols as M ℓ ℓ = (2 ℓ + 1) X ℓ (2 ℓ + 1)4 π w ′ ℓ ℓ ℓ ℓ , (39)where w ′ ℓ is the power spectrum of the window function w ′ (ˆ γ ) defined in an analogous manner to(31).It can be shown that the covariance matrix for the pseudo estimator D ℓ has the form h ∆ D ℓ ∆ D ℓ ′ i = 2(2 ℓ + 1)(2 ℓ ′ + 1) X mm ′ X ℓ m X ℓ m B ℓ N ℓ C BBℓ B ℓ N ℓ C BBℓ ×× K ℓmℓ m K ∗ ℓ ′ m ′ ℓ m K ∗ ℓmℓ m K ℓ ′ m ′ ℓ m . (40)As it stands, this formula is not useful due to the high cost of computation. However, for highmultipoles, this formula simplifies to [54] h ∆ D ℓ ∆ D ℓ ′ i ≈ B ℓ N ℓ C BBℓ B ℓ ′ N ℓ ′ C BBℓ ′ M ℓℓ ′ / (2 ℓ ′ + 1) . (41)In order to implement and verify the above analytical results we have conducted numericalcalculations using simulated data. In the first instance, we generate 1000 random full sky ( Q , U ) maps with no contribution from gravitational waves (i.e. r = 0) and no lensing. For eachrealization, we reconstruct the magnetic field B rec (ˆ γ ) and evaluate the pseudo estimator D ℓ . Theaverage over 1000 realizations D ℓ is plotted (green line) in Fig. 8. Note that, here and below weuse the over-line to denote averaging over simulated realizations, as opposed to the angle bracketswhich denote ensemble averaging. The average for the uncleaned spectrum D ˜ B ˜ B ℓ (defined in (31))is plotted (black line) for comparison on the same figure. For next calculations, we simulate 1000random full sky maps with contribution from gravitational waves characterized by r = 0 . B -mode of polarization from cosmic lensing. The average value of the estimator D ℓ is plotted (red line) in Fig. 8. The comparison of curves in Fig. 8 shows that the residual noisecontribution to the pseudo estimator due to numerical errors (green line) is negligible in comparisonwith the contribution to the estimator from the signal (red line). One can therefore conclude thatthe resulting pseudo estimator D ℓ is effectively free from EB -mixing.In Fig. 9, in order to verify (38), we plot the left-hand side (solid blue line) of this equationfor C BBℓ a model with r = 0 . D ℓ over 1000 realizations for the same model (solid red line). Asexpected the two lines are close to each other, being practically indistinguishable for multipoles ℓ &
20. For comparison, in Fig. 9, we also plot the individual contributions from gravitationalwaves (solid magenta line) and cosmic lensing (solid green line). Finally, in Fig. 9, we plot thesquare root of the average over 1000 realizations of the diagonal terms in the covariance matrix (cid:0) ∆ D ℓ ∆ D ℓ (cid:1) / (dashed red line). In order to check the analytical approximation (41), we alsoplot the diagonal term h ∆ D ℓ ∆ D ℓ i / evaluated using the right side of expression (41) (dashed blueline). As expected the two curves for the covariance matrix practically coincide for large multipoles ℓ &
80, which corresponds to the region of applicability of the approximation (41).
B. Unbiased estimators for C BBℓ
Having constructed the pseudo estimator D ℓ , we are one step away from constructing an un-biased estimator for the B -mode power spectrum C BBℓ . In this subsection, we shall discuss thisconstruction. Let us for the moment assume that the coupling matrix M ℓℓ ′ in (38) is invertible. Inthis case, using relation (38), one can immediately verify that the estimator defined as D BBℓ = N − ℓ B − ℓ X ℓ ′ M − ℓℓ ′ D ℓ ′ , (42)is an unbiased estimator of the power spectrum C BBℓ . In practice, this simple estimator can be usedin large sky surveys such as Planck, where M ℓℓ ′ is indeed invertible. However, in the case of smallsky surveys, one cannot construct this estimator since the matrix M ℓℓ ′ becomes singular. In thiscase it is possible to bin the pseudo estimator data into multipole bins, and construct an unbiasedestimator for the binned power spectrum. Following the analysis for temperature anisotropy [58],we build the so-called full-sky CMB bandpowers P BBb asP
BBb = X b ′ M − bb ′ X ℓ p b ′ ℓ D ℓ , (43)21 -5 -4 -3 -2 -1 (line 3) line 1(line 2)r=0, no lenr=0.1 and len Th e ave r a g e d p se odu es t i m a t o r D l ( K ) Multipole: l
FIG. 8: The averaged pseudo estimator D ℓ from 1000 simulations. The red line (i.e. line 2) shows the resultfor an input model with r = 0 . C BBℓ = 0). For comparison, the black line (i.e.line 1) shows the averaged estimator D ˜ B ˜ B ℓ for an input model with no magnetic polarization. where the subscript b denotes the multipole bands. p bℓ is a binning operator in ℓ -space defined as p bℓ = ℓ ( ℓ +1)2 πN ℓ ( ℓ ( b +1)low − ℓ ( b )low ) , if ℓ (b)low ≤ ℓ < ℓ (b+1)low . (44)The non-singular binned coupling matrix M bb ′ participating in (43) is constructed from the couplingmatrix M ℓℓ ′ M bb ′ = X ℓ p bℓ X ℓ ′ M ℓℓ ′ B ℓ ′ q ℓ ′ b ′ . (45)The function B ℓ ′ takes into account the effects arising due to finite beam size and finite pixelization.In the above expression, q ℓb is the reciprocal operator of p bℓ q ℓb = πN ℓ ℓ ( ℓ +1) , if ℓ (b)low ≤ ℓ < ℓ (b+1)low . (46)22
10 1000.1110100 line 6line 5 line 4line 3line 2 line 1 P o w e r s p ec t r u m ( K ) multipole: l FIG. 9: All of the lines are a result of averaging over 1000 simulated samples. The red line (i.e. line 2)shows the averaged pseudo estimator D ℓ for an input model with r = 0 . (cid:0) ∆ D ℓ ∆ D ℓ (cid:1) / . The blue solid (i.e. line 1) line shows the analytical result of h D ℓ i , for amodel with r = 0 . h ∆ D ℓ ∆ D ℓ i / . It is straightforward to verify that P
BBb is an unbiased estimator of the B -mode of polarizationpower spectrum ℓ ( ℓ + 1) C BBℓ / π , i.e. h P BBb i = ℓ ( ℓ + 1)2 π C BBℓ . The covariance matrix of the bandpowers is related to the covariance matrix h ∆ D ℓ ∆ D ℓ ′ i in (40)by [59] h ∆P BBb ∆P BBb ′ i = M − bb p b ℓ h ∆ D ℓ ∆ D ℓ ′ i ( p b ℓ ′ ) T ( M − b ′ b ) T . (47)In Fig. 10 we plot the value of the bandpower P BBb (red dots) averaged over 1000 realizations.23he realizations were generated for a model including contribution from gravitational waves with r = 0 . ℓ = 10 for each bin. Theanalysis shows that (up to discrepancies that can be attributed to finite number of realizations)the average of the power spectrum estimators coincide with the theoretical (input) spectrum. Theerror bars (cid:16) ∆P BBb ∆P BBb ′ (cid:17) / (red error bars) were calculated using (47), with ensemble averagereplaced by an average over realizations. As can be expected, the error bars are large for thefirst three data points, due to the small sky coverage. In addition to evaluating error bars, it wasverified that the correlation between various multipole bins is QUIET weak (all of the correlationcoefficients are smaller than 0 . . ℓ . ℓ ∼ S/N = sX bb ′ h P BBb (gw) i (Cov − ) bb ′ h P BBb ′ (gw) i , (48)where Cov bb ′ ≡ h ∆P BBb ∆P BBb ′ i is the covariance matrix of the bandpower estimator (47). For theexample considered above with r = 0 .
1, we find
S/N = 8 . C ℓ estimator is QUIET different from the pseudo- C ℓ polarization estimators suggested in [37, 61], or an equivalent estimator suggested in [62]. In[37, 61], the unbiased estimators are constructed directly from the pseudo estimators ˜ C EEℓ and˜ C BBℓ , both of which are a mixture of electric and magnetic types of polarization. The resultantmixing increases the magnitude of the covariance matrix for the unbiased estimator, and becomesone of the main contaminations for the detection of gravitational waves. In [37], the authors foundthat, for small sky surveys covering one or two percent of the sky, the mixing contamination tothe covariance matrix of B -mode power spectrum estimator typically limits the tensor-to-scalarratio that can be probed to r & .
05. On the other hand, the pseudo- C ℓ method suggested in thepresent work explicitly separates the electric and magnetic types of polarization up to very smallnumerical errors. For this reason, the effects of mixing of electric and magnetic modes, which are24ompletely removed (reduced to negligible levels) in our case, do not put a limit on the ability todetect gravitational waves. This is the main advantage of our method, and is the main motivationfor this paper.At the end of the subsection, we would like to point out that, if one proceeds to constructan unbiased estimator for C BBℓ using the B rec (ˆ γ ) W (ˆ γ ) field (instead of B rec (ˆ γ ) field used above),the resulting unbiased estimator will be equivalent to the “pure B-mode” estimator defined in[40]. This has been discussed in Appendix C. On the other hand, if one constructs the unbiasedestimator for C BBℓ using ˜ B (ˆ γ ) (instead of B rec (ˆ γ )) adopting a top-hat window function (instead of W (ˆ γ )), one would return to the B -mode estimator defined in [44]. As was pointed out in [44], theresulting estimator suffers from large EB -mixing at the edge of the observed field. C. Information loss due to edge removal
As was emphasized in Sec. III, in practical calculations, the edge of the partial sky map hasto be removed in order to reduce numerical errors. The edge removal leads to the partial loss ofinformation. In this subsection we study the impact of this information loss on the performance of B -mode of polarization power spectrum estimator P BBb .In order to study the performance of the estimator P
BBb in an ideal case with no edge removalwe perform the following steps:1. We generate 1000 full sky ( Q , U ) maps, for a cosmological model with r = 0 . B ℓm using (2), (3) and (4).2. With the multipole coefficients B ℓm we construct the full sky map B (ˆ γ ) using (12).3. We construct the top-hat mask window function w (ˆ γ ) equal to unity for θ < θ = 20 o andzero otherwise. We now construct the masked magnetic field B rec (ˆ γ ) = B (ˆ γ ) w (ˆ γ ). Themasked field B rec (ˆ γ ) constructed in this manner corresponds to a reconstructed magneticfield map in an idealized case with no edge removal in the absence of numerical errors.4. Working with the field B rec (ˆ γ ), following the steps outlined in Sec. IV A and Sec. IV B, webuild the unbiased estimator P BBb and calculate its covariance matrix. The resulting estima-tor is equivalent to one that could be constructed in an ideal situation, without numericalerrors, in which we could have worked without edge removal.25
100 200 300 400-0.020-0.015-0.010-0.0050.0000.0050.0100.0150.020 l ( l + ) C l BB / ( K ) ( g . w . ) Multipole: l l ( l + ) C l BB / ( K ) Multipole: l
FIG. 10: The averaged values (red larger dots) and error bars (red larger error bars) following fromsimulations of the unbiased estimators of the power spectrum ℓ ( ℓ + 1) C BBℓ (total) / π (left panel) and ℓ ( ℓ + 1) C BBℓ (gw) / π (right panel). In both panels, the black solid line denotes the theoretical values ofthe underlying power spectra. For comparison, in both panels, we plot the averaged values (blue smallerdots) and simulated error bars (blue smaller error bars) of the unbiased estimators for an ideal case withoutinformation loss (see text for the details). In both the panels, we have considered a case with no instrumentalnoise. The resulting averaged value for the estimator (blue dots) and the corresponding error bars (blueerror bars) are plotted in Fig. 10. Once again, we find that the average values of the estimatorsare practically coincident with the theoretical (input) values. In both the panels, the blue errorbars are slightly smaller than the red ones for all multipole bins. The difference reflects the loss ofinformation due to edge removal. In Fig. 11, we plot the ratio of the two error bars as a functionof the multipole bin. This ratio in almost everywhere less than 1 .
2. The signal-to-noise ratio (48)calculated for the ideal is
S/N = 9 .
59, which is less than 15% higher than the practically relevantexample considered in the previous subsection. The results of this section demonstrate that theloss of information, gauged by the increase in the error bars of the spectral estimator, is sufficientlysmall .
100 200 300 4001.101.121.141.161.181.201.221.241.26 e rr o r b a r r a t i o Multipole: l
FIG. 11: The ratio of practically achievable error bars (red error bars in Fig. 10) to the corresponding errorbars in an information lossless case (blue error bars in Fig. 10), as a function of multipole ℓ . D. Power spectrum estimators in the presence of instrumental noise
In the previous subsections we have considered a situation in which the magnetic type of polar-ization was generated solely by gravitational waves and cosmic lensing. In realistic observations,in addition to these two contributions, there are contaminating contributions from various othersources like instrumental noise and astrophysical foregrounds. This said, it is reasonable to assumethat, for an appropriate choice of observed sky region, the astrophysical foregrounds are typicallyexpected to be small in comparison with instrumental noises [5]. For this reason, we shall ignorethe foreground contaminations, and restrict our analysis to the study of power spectrum estimatorsin the presence of only instrumental noises.The pseudo estimator D ℓ (37) has the following expectation in the presence of noise (comparewith no noise case (38)) h D ℓ i = X ℓ ′ M ℓℓ ′ N ℓ ′ B ℓ ′ C BBℓ ′ + hN BBℓ i , (49)where N BBℓ is the pseudo estimator for the full sky noise power spectrum N BBℓ . The expectation27alue of this noise estimator is hN BBℓ i = X ℓ ′ M ℓℓ ′ N ℓ ′ N BBℓ ′ . (50)The presence of noise leads to a redefinition of the unbiased estimator P BBb P BBb = X b ′ M − bb ′ X ℓ p b ′ ℓ ( D ℓ − hN BBℓ i ) , (51)with matrices p b ′ ℓ and M bb ′ given in (44) and (45). The covariance matrix for this estimator hasthe form given by (47), where h ∆ D ℓ ∆ D ℓ ′ i are calculated from the right side of (40) with B ℓ C BBℓ terms replaced by (cid:0) B ℓ C BBℓ + N BBℓ (cid:1) .The estimator P
BBb defined in (51) is an unbiased estimator for the B -mode of polarizationpower spectrum ℓ ( ℓ + 1) C BBℓ / π , where C BBℓ contains contribution from both gravitational waves(gw) and cosmic lensing (lens) C BBℓ = C BBℓ (gw) + C BBℓ (lens) . However, if we are primarily interested in detection of gravitational waves, we can treat the cosmiclensing contribution as effective noise, and define an unbiased estimator for the B -mode powerspectrum due to gravitational waves ℓ ( ℓ + 1) C BBℓ (gw) / π asP BBb (gw) = X b ′ M − bb ′ X ℓ p b ′ ℓ ( D ℓ − h ˜ N BBℓ i ) , (52)where the effective noise term h ˜ N BBℓ i contains contribution from instrumental noises and cosmiclensing h ˜ N BBℓ i = X ℓ ′ M ℓℓ ′ N ℓ ′ ( B ℓ ′ C BBℓ ′ (len) + N BBℓ ′ ) . (53)The covariance matrix for this estimator is same as that calculated for estimator (51). E. Expected performance of ground-based CMB experiments
In this subsection we shall investigate the prospects of detecting the B -mode signature fromrelic gravitational waves by two future ground based experiments, QUIET [14] and POLARBEAR[13].Let us firstly consider the QUIET experiment. We shall restrict our analysis to the 40GHzfrequency channel. The FWHM for the Gaussian beam at this channel is θ F = 23 ′ , and the28xpected instrumental noise is N BBℓ = 2 . × − µ K [14] (see also [60]). We shall assume thatexperiment will observe f sky = 3% fraction of the sky, corresponding to θ = 20 o . followingthe steps outlined in Sec. IV A and Sec. IV B, using the experimental characteristics for QUIETexperiment, we construct the unbiased estimators P BBb and P
BBb (gw) and their covariance matricesfor 1000 realizations with r = 0 . r = 0 .
01. The average values for the estimators and theircorresponding error bars are plotted in Fig. 12, for r = 0 . r = 0 .
01 (rightpanel). The error bars in this case are larger than the error bars in Fig. 10 due to the inclusion ofinstrumental noises. The total signal-to-noise ratio in (48) is
S/N = 7 .
05 for r = 0 . S/N = 1 .
25 for the model with r = 0 . θ F = 4 ′ , and the expected instrumental noise is N BBℓ = 4 . × − µ K [13]. As above, we assume f sky = 3%. In order to study the performance of POLARBEAR, we simulate 1000 realizations of ( Q , U ) maps with r = 0 .
1. Before proceeding to construct the power spectrum estimators, one shouldnotice that, in comparison with QUIET, the value of θ F = 4 ′ for POLARBEAR is substantiallysmaller. Thus, in order to avoid leakage from higher multipole electric type polarization, wefirstly apply the ‘map smoothing’ procedure outline in Appendix D. Following this, we constructthe estimators P BBb and P
BBb (gw) and their covariance matrices following the steps explained inSec. IV A and Sec. IV B. In Fig. 13 we plot the average values of the estimators and their errorbars. The error bars for POLARBEAR experiment are considerably larger than those in Fig. 12(and Fig. 10) due to larger instrumental noise in comparison with QUIET. Finally, we calculatethe signal-to-noise for the POLARBEAR experiment to be
S/N = 4 .
31 for a model with r = 0 . S/N ∝ p f sky , where f sky is the sky cut factor. This29pproximation follows from following considerations. In the case of full sky coverage, one canconstruct an unbiased estimator D XXℓ (where X = T, E or B ) for the various power spectra C XXℓ in a straightforward manner (see for example [35, 65] for details). In this case the covariance matrixis diagonal with q h ∆ D XXℓ ∆ D XXℓ i = r ℓ + 1 ( C XXℓ + N XXℓ B − ℓ ) , with (2 ℓ + 1) in the denominator on the right side playing the role of number of degrees of free-dom for a given multipole ℓ . The above expression was elegantly extrapolated for temperatureanisotropy power spectrum estimator D T Tℓ to partial sky surveys in [66]. The author proposed toreplace (2 ℓ + 1) with the effective number of degrees of freedom (2 ℓ + 1) f sky in the above expres-sion, to account for the loss of information that arrises due to partial sky coverage. This simpleconsideration was extended to B -mode power spectrum estimator in [67], and was further extendedto account for multipole binning [68]. These approximation lead to h ∆P BBb (gw) i = s ℓ + 1)∆ ℓf sky (cid:18) ℓ ( ℓ + 1)2 π (cid:19) (cid:0) C BBℓ + N BBℓ B − ℓ (cid:1) , (54)with ℓ being the central multipole in each bin. In this approximation, the total signal-to-noise ratiofor gravitational wave signal in the B -mode of polarization takes the form S/N = vuutX b (cid:18) h P BBb (gw) ih ∆P BBb (gw) i (cid:19) . (55)In order to guage the performance of this approximation, in Fig. 11 and Fig. 12, we plot theerror bars calculated using (54) (grey error bars). For this calculation we have set f sky = 0 . θ = 18 o . One can see that, in both thefigures, the analytical approximation leads to smaller error bars than those obtained from numericalsimulations. We use (55) to calculate the analytical signal-to-noise ratio for the two consideredexperiments. The results for signal-to-noise ratio are summarized in Table I. It can be seen thatthe analytical approximation for signal-to-noise ratio (55) considerably overestimates the detectionability, particularly for smaller values of actual S/N . Several works [39, 69] have pointed out thatthe analytical approximation (55) exaggerates the detection ability. However, these paper arguedthat the primary reason for overestimation is due to the omission of possible contaminations from EB -mixing. However, our approach shows that the analytical approximation (55) with an effectivesky-cut factor also overvalues the S/N in comparison with the case with no EB -mixing. One shouldtherefore use this approximation with caution [75]. At the same time, it is very important to point30
100 200 300 400-0.02-0.010.000.010.020.030.040.050.06 l ( l + ) C l BB / ( K ) Multipole: lr=0.01 r=0.1 l ( l + ) C l BB / ( K ) Multipole: l
FIG. 12: The averaged values and simulated error bars of the unbiased estimators for the power spectrum ℓ ( ℓ + 1) C BBℓ / π (green dots and green largers error bars) and ℓ ( ℓ + 1) C BBℓ (gw) / π (red dots and red largererror bars). In both panels, the solid lines denote the theoretical values for these power spectra. In thisfigure, we have considered the instrumental noise for QUIET experiment. The left panel shows the resultsfor an input cosmological model with r = 0 .
1, while the right panel shows the results for an r = 0 .
01 model.In both panels, the smaller error bars calculated using the analytical approximation (54) are plotted in grey. out that this conclusion about overestimation is based on analysis of small sky coverage and theuse of pseudo- C ℓ estimators with the uniform weight function W (ˆ γ ). In contrast, for large scalesurveys [54] or the small scale surveys by using the pseudo- C ℓ estimators with the optimal choice ofthe weight function W (ˆ γ ) [41], the conclusion might change. Especially, for the large scale surveysand maximum likelihood estimators, the discussed analytical approximation may underestimatethe true S/N , as was shown for temperature anisotropy in [54].31
100 200 300 400-0.02-0.010.000.010.020.030.040.050.06 l ( l + ) C l BB / ( K ) Multipole: lr=0.1
FIG. 13: The results for POLARBEAR experiment. The averaged values and simulated error bars of theunbiased estimators for the power spectrum ℓ ( ℓ + 1) C BBℓ / π (green dots and green larger error bars) and ℓ ( ℓ + 1) C BBℓ (gw) / π (red dots and red larger error bars), calculated for an input model with r = 0 .
1. Thesolid lines denote the theoretical values for these power spectra. The smaller error bars calculated using theanalytical approximation (54) are plotted in grey.TABLE I: The total signal-to-noise
S/N for the gravitational waves signal in the B -mode of polarizationfor the various cases considered in the textideal no noise case QUIET noise QUIET noise POLARBEAR noise r = 0 . r = 0 . r = 0 . r = 0 . S/N
S/N
V. CONCLUSION
In this paper we have proposed a new method to construct pure electric and magnetic typefields E (ˆ γ ) and B (ˆ γ ) from polarization field given on an incomplete sky. Due to the differentialdefinitions of these fields, we avoid the so-called EB -mixing problem. In practice when working32ith pixelized maps, residual leakages from various numerical errors require the removal of datafrom a narrow edge on the boundary of the observed sky. This leads to a minor loss of informationin comparison with the idealized lossless case considered in Sec. IV C.A major advantage of our approach is that the constructed fields E (ˆ γ ) and B (ˆ γ ) are scalar. Forthis reason, the various techniques developed for the analysis of temperature anisotropy maps canbe directly applied to these fields. As an important and motivating application, we discuss theconstruction of an unbiased estimator for the B -mode power spectrum C BBℓ , using the pseudo- C ℓ estimator approach. We find that our method is computationally feasible even in the case ofhigh resolution maps. In particular, it takes 2 . B (ˆ γ ) in pixel spacewith N side = 512, and the construction of unbiased estimators for C BBℓ .With the help of the constructed unbiased estimator, we have investigated the ability to detectgravitational waves through the B -mode of polarization in CMB experiment covering 3% of the sky.In the absence of instrumental noise, we find S/N = 8 .
26 for a model with r = 0 .
1. This value is 14%smaller than an idealized situation with no information loss. In the case of realistic experiments,the signal to noise reduces to
S/N = 7 .
05 for QUIET and
S/N = 4 .
31 for POLARBEAR.In conclusion, we would like to point out that, a similar analysis can be applied to large skysurveys. In particular, for Planck satellite and the planned CMBPol experiment, one can constructunbiased estimators for the polarization power spectra C EEℓ and C BBℓ , by synthesizing the approachoutlined in this paper together with the hybrid estimator method suggested in [54]. We leave thistask for future work.
Acknowledgements
The authors appreciate help from E. Hivon, L. Cao and S. Gupta in using the HEALPix package.The authors thank L. P. Grishchuk for stimulating discussions. WZ is partially supported byChinese NSF Grants No. 10703005, No. 10775119, and the Foundation for University YoungTeaching Excellence of the Ministry of Education, Zhejiang Province. In this paper, we have usedthe CAMB package [70] and HEALPix package [71].33 ppendix A: Numerical calculation of the correction term ct in pixel space In order to calculate the correction term ct in (24) and (25), one needs to be calculate the terms( QW ) x , ( QW ) y , ( U W ) x and ( U W ) y . Below, we discuss the calculation of ( QW ) x . The otherterms are calculated in a similar manner.We expand the polarization fields ( Q + iU ) W and ( Q − iU ) W in terms of spin-weighted harmonics( Q (ˆ γ ) ± iU (ˆ γ )) W (ˆ γ ) = X ℓm ˜ a ± ,ℓm ± Y ℓm (ˆ γ ) . (A1)It follows that QW (ˆ γ ) = − X ℓm ˜ E ℓm X ,ℓm (ˆ γ ) + i ˜ B ℓm X ,ℓm (ˆ γ ) , (A2)where ˜ E ℓm ≡ − (˜ a ,ℓm + ˜ a − ,ℓm ) / , ˜ B ℓm ≡ − (˜ a ,ℓm − ˜ a − ,ℓm ) / i,X ,ℓm = ( Y ℓm + − Y ℓm ) / , X ,ℓm = ( Y ℓm − − Y ℓm ) / . The quantity ( QW ) x can be numerically calculated using( QW ) x ≡ ∂ ( QW ) /∂θ = − X ℓm ˜ E ℓm ( ∂X ,ℓm /∂θ ) + i ˜ B ℓm ( ∂X ,ℓm /∂θ ) . (A3)Thus, using the expansion coefficients in (A1) and expression (A3) one can calculate the quantity( QW ) x in terms of quantities QW , U W and functions ( ∂X n,ℓm /∂θ ). We would like mention herethat in the HEALPix version 1.23, the subroutine alm map template.f90 had a bug, that led toerroneous results for ( QW ) x and ( U W ) x [72]. This problem has been fixed in the latest HEALPixversion 1.24.In the present paper we use a simple analytical form (27) for the mask window function W (ˆ γ ).For this window function, the various derivatives, such as W x (ˆ γ ) and W xx (ˆ γ ), can be calculatedanalytically. However, in practical situations, the window function does not have such a simpleform (see for instant [73]). For this reason, one would need to calculate the various derivativeterms, W x , W y , W xx , W yy and W xy , numerically. This can be done in the following way. Onefirstly defines the multiple expansion coefficients W ℓm in the standard way W ℓm ≡ Z W (ˆ γ ) Y ∗ ℓm (ˆ γ ) d ˆ γ. W x (ˆ γ ) ≡ ∂W∂θ = X ℓm W ℓm (cid:18) ∂∂θ Y ℓm (ˆ γ ) (cid:19) = X ℓm W ℓm ℓ tan θ Y ℓm (ˆ γ ) − θ r ℓ + 12 ℓ − ℓ − m ) Y ℓ − m (ˆ γ ) ! . The other quantities can be calculated in an anologous manner. It is important to point out thatthe steps mentioned above can be realized in a straightforward manner using the anafast and synfast routines in the HEALPix package.
Appendix B: Construction of magnetic map B rec (ˆ γ ) from simulated polarization maps In this appendix, we outline the steps which were used to simulate the polarization maps andconstruct the pure magnetic map B rec (ˆ γ ).1. We generate the mask window function W (ˆ γ ) using (27) in pixel space using the standardpixelization scheme used in HEALPix with N side = 512 (or N side = 1024 in the example inSec. III B).2. Using synfast HEALPix routine, we generate full sky ( Q (ˆ γ ), U (ˆ γ )) maps with N side =512 or 1024, using cosmological parameters (30) and appropriate value of tensor-to-scalarratio r as input. Using the window function W (ˆ γ ), we build the masked ( ˜ Q (ˆ γ ), ˜ U (ˆ γ )) maps(where ˜ Q = QW and ˜ U = U W ).3. Using anafast
HEALPix routine, we calculate the coefficients ( ˜ E ℓm , ˜ B ℓm ). The field ˜ B (ˆ γ ) iscalculated from ˜ B ℓm according to (19) using synfast routine.4. With coefficients ( ˜ E ℓm , ˜ B ℓm ) we construct the fields QW (ˆ γ ), U W (ˆ γ ), ( U W ) x (ˆ γ ) and( QW ) y (ˆ γ ), using the 5 th option in synfast routine.5. Using the fields U W (ˆ γ ), ( U W ) x (ˆ γ ) and ( QW ) y (ˆ γ ) constructed in the previous step andanalytical expressions for W , W x and W xx (derived by differentiating (27)), we calculateIm(ct) in (29).6. The pure magnetic field B rec (ˆ γ ) is now constructed from ˜ B , W and Im(ct) using (32). Thepure magnetic field B rec (ˆ γ ) is truncated at the edges in order to remove residual leakagesassociated with numerical errors. 35 ppendix C: Pseudo Estimators for a special choice of weight function W (ˆ γ ) = W (ˆ γ ) In Sec. IV A it was pointed out that, in principle one can construct pseudo estimators of thepower spectrum by adopting an arbitrary weight function W (ˆ γ ) in (33). Above, in the main text,we have focused on a specific case corresponding to a uniform weight function W (ˆ γ ) = 1 . Thischoice is nearly optimal for high multipoles. However, this choice becomes sub-optimal at lowermultipoles [41]. In this appendix we study another possible choice for the weight function, namely W (ˆ γ ) = W (ˆ γ ), where W (ˆ γ ) is the mask window function in (27) with θ = 20 o and θ = 10 o . Notethat the function W (ˆ γ ) is the same window function that was used for constructing B rec (ˆ γ ). Withthis choice, the resulting pseudo estimator is equivalent to the pure B -mode estimator studied in[40].The construction of the pseudo estimators and the corresponding unbiased estimators followsclosely the discussion in Sec. IV. The only difference is that the definition of coefficients a ℓm in(34) are modified to a ℓm = Z d ˆ γ B rec (ˆ γ ) W (ˆ γ ) Y ∗ ℓm (ˆ γ ) , (C1)and the quantities w ′ (ˆ γ ) and w ′ ℓ in (36) and (39) would now be replaced by W and its powerspectrum, respectively.In Fig. 14, we plot the unbiased estimators for the power spectra ℓ ( ℓ + 1) C BBℓ (total) / π (leftpanel) and ℓ ( ℓ + 1) C BBℓ (gw) / π (right panel) together with the corresponding error bars (thin blueerror bars). It can be seen that, in comparison with the estimators in the case of a uniform weightfunction, the error bars of the new estimators are larger at high multipoles, but are smaller at lowmultipoles. This result is consistent with findings in [41] that the optimal weight functions for highmultipoles tend to the top-hat function. On the other hand, for low multipoles, the optimal weightfunction tend to smooth out (see the right panel of Fig.2 in [41] for a concrete example). Appendix D: Smoothing the polarization maps
The high value of power spectrum C EE ℓ of the electric component at large values of multipoles(due to the presence of N ℓ factor) leads to substantial leakage of power into the reconstructed puremagnetic field B rec . This leakage seeps through to low multipoles playing a role of residual effectivenoise. In order to reduce this contamination, below we introduce a map smoothing procedure forpolarization maps. The idea behind this method is similar to the ‘prewhitening’ method suggestedin [74]. 36
100 200 300 400-0.020-0.015-0.010-0.0050.0000.0050.0100.0150.020 l ( l + ) C l BB / ( K ) ( g . w . ) Multipole: l l ( l + ) C l BB / ( K ) Multipole: l
FIG. 14: The averaged values (blue smaller and red larger dots) and error bars (blue smaller and red larger er-ror bars) following from simulations of the unbiased estimators of the power spectrum ℓ ( ℓ + 1) C BBℓ (total) / π (left panel) and ℓ ( ℓ +1) C BBℓ (gw) / π (right panel). In both panels, the black solid line denotes the theoreticalvalues of the underlying power spectra. The blue smaller dots and error bars denote the result by adoptingthe weight function W (ˆ γ ) = W (ˆ γ ), and the red larger dots and error bars denote the result by adopting auniform weight function W (ˆ γ ). Note that, in both panels, we have considered a case with no instrumentalnoise. The red larger dots and error bars are identical to those in Fig. 10. The smoothing procedure is simple and straightforward in the case of full sky coverage. Giventhe ( Q , U ) polarization maps on a full sky, one can calculate the multipole coefficients E ℓm and B ℓm using (1)-(4). In order to smooth the polarization maps we firstly use a damping function tomodify the multipole coefficients E ′ ℓm ≡ E ℓm e − ℓ θ F , B ′ ℓm ≡ B ℓm e − ℓ θ F . (D1)Following this, we reconstruct the smoothed polarization fields Q ′ and U ′ using the modifiedcoefficients E ′ ℓm and B ′ ℓm in the standard way. The reconstructed maps can be thought of as thethe result of observing the original ( Q , U ) polarization field in an experiment with FWHM of theGaussian beam equal to θ F . Overall, the smoothing has effect of exponentially damping the powerin high multipoles. 37e can extend the smoothing procedure to the case of partial sky coverage. Given the ( Q , U )polarization maps on a partial sky, we calculate the coefficients ˜ E ℓm and ˜ B ℓm . These coefficientsare smoothed in analogy with full sky case˜ E ′ ℓm ≡ ˜ E ℓm e − ℓ θ F , ˜ B ′ ℓm ≡ ˜ B ℓm e − ℓ θ F . (D2)The smoothed polarization maps ( Q ′ , U ′ ) are reconstructed from the modified multipole coefficients˜ E ′ ℓm and ˜ B ′ ℓm .It is important to point out that, in the case of partial sky coverage, the smoothing procedureoutlined above introduces mixture of electric and magnetic polarizations. In particular, even if theoriginal ( Q , U ) did not contain magnetic type of polarization, the smoothed map ( Q ′ , U ′ ) wouldcontain it. We have verified numerically that in practically interesting cases the resulting mixtureis very small, and would not significantly affect the ability to detect gravitational waves.In order to verify the small of the resulting mixing, we have performed the following calculation.Using an input model with no B -mode of polarization (i.e. C BBℓ = 0) and θ F = 30 ′ we generated afull sky ( Q , U ) map. Following this, we truncate the map to keep the data from only the northernhemisphere. Using the procedure outlined in Sec. II and Sec. III we construct the field B rec , whichis expected to be equal to zero except for the residual leakage. The psedo power spectrum D ℓ forthis field (red line) is plotted in Fig. 15. Following this, for the same input model, we generate the( Q , U ) map with θ F = 10 ′ , once again restricting the data to just the northern hemisphere. Wenow smooth this map with θ F = 30 ′ using the anafast , alteralm and synfast HEALPix routines. Weconstruct B ′ rec from the smoothed ( Q ′ , U ′ ) map and plot the corresponding pseudo power spectrum D ′ ℓ (green line) in Fig. 15. The difference between the two spectra D ℓ and D ′ ℓ can be interpretedas the result of mixing introduced by map smoothing (blue line in Fig. 15). It can be seen that,the mixing due to smoothing is QUIET small, comparable to leakage due to numerical errors, atall the relevant multipoles. The power spectrum of the leakage due to smoothing peaks at ℓ ∼ ℓ . E ℓm , ˜ B ℓm ) and ( E ℓm , B ℓm ) are related by the following expression (see for instant[37]) ˜ E ℓm + i ˜ B ℓm = X ℓ ′ m ′ I ( ℓm )( ℓ ′ m ′ ) [ E ℓ ′ m ′ + iB ℓ ′ ,m ′ ] , (D3)where I ( ℓm )( ℓ ′ m ′ ) is the coupling matrix, which depends only on the mask window function. From38his relation, it formally follows that E ℓm + iB ℓm = X ℓ ′ m ′ ( I − ) ( ℓm )( ℓ ′ m ′ ) [ ˜ E ℓ ′ m ′ + i ˜ B ℓ ′ ,m ′ ] . (D4)For the smoothed multipole coefficients one has E ′ ℓm + iB ′ ℓm ≡ X ℓ ′ m ′ ( I − ) ( ℓm )( ℓ ′ m ′ ) [ ˜ E ′ ℓ ′ m ′ + i ˜ B ′ ℓ ′ ,m ′ ]= X ℓ ′ m ′ ( I − ) ( ℓm )( ℓ ′ m ′ ) [ ˜ E ℓ ′ m ′ + i ˜ B ℓ ′ ,m ′ ] e − ℓ ′ θ F . Since the coupling matrix I ( ℓm )( ℓ ′ m ′ ) is sharply peaked at ℓ = ℓ ′ [37], the above expression can beapproximated by E ′ ℓm + iB ′ ℓm ≈ X ℓ ′ m ′ ( I − ) ( ℓm )( ℓ ′ m ′ ) [ ˜ E ℓ ′ m ′ + i ˜ B ℓ ′ ,m ′ ] e − ℓ θ F = ( E ℓm + iB ℓm ) e − ℓ θ F . It therefore follows that E ′ ℓm ≈ E ℓm e − ℓ θ F , B ′ ℓm ≈ B ℓm e − ℓ θ F . 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