Efficiency of pseudo-spectrum methods for estimation of Cosmic Microwave Background B-mode power spectrum

Transcription

1 Efficiency of pseudo-spectrum methods for estimation of Cosmic Microwave Background B-mode power spectrum For these reasons, many poarization experiments targeting B-modes have been buit or proposed, incuding ground based observatories, those aready operating e.g., poarbear [1] or sptpo [13] or those, which are being deveoped, e.g., qubic [14], actpo [15], baoon borne experiments such as spider [17] or ebex [16], which few in winter 01/13, or to even a potentia sateite mission itebird [18], core[19], pixie [0]. With an exceparxiv: v1 [astro-ph.co] 31 May 013 A. Ferté, 1,, J. Grain,, 1, M. Tristram,, 3, and R. Stompor 4, 1 Université Paris-Sud 11, Institut d Astrophysique Spatiae, UMR8617, Orsay, France, F CNRS, Orsay, France, F Université Paris-Sud 11, Laboratoire de Accéérateur Linéaire, Bâtiment 00, Orsay Cedex, France 4 AstroParticue et Cosmoogie, Université Paris Diderot, CNRS/INP3, CEA/Irfu, Obs. de Paris, Sorbonne Paris Cité, France Estimation of the B-mode anguar power spectrum of poarized anisotropies of the cosmic microwave background (CMB) is a key step towards a fu expoitation of the scientific potentia of this probe. In the context of pseudo-spectrum methods the major chaenge is reated to a contamination of the B-mode spectrum estimate with residua power of much arger E-mode. This so-caed E-to-B eakage is unavoidaby present whenever an incompete sky map is ony avaiabe, as is the case for any reaistic observation. The eakage has to be then minimized or removed and ideay in such a way that neither a bias nor extra variance is introduced. In this paper, we compare from these two perspectives three different methods proposed recenty in this context [5, 8, 9], which we first introduce within a common agebraic framework of the so-caed χ-fieds and then study their performance on two different experimenta configurations one corresponding to a sma-scae experiment covering 1% of the sky motivated by current ground-based or baoon-borne experiments and another to a neary fu-sky experiment, e.g., a possibe CMB B-mode sateite mission. We find that though a these methods aow to reduce significanty the eve of the E-to-B eakage, it is the method of [5], which at the same time ensures the smaest error bars in a experimenta configurations studied here, owing to the fact that it permits straightforwardy for an optimization of the sky apodization of the poarization maps used for the estimation. For a sateite-ike experiment, this method enabes a detection of B-mode power spectrum at arge anguar scaes but ony after appropriate binning. The method of [8] is a cose runner-up in the case of a neary fu sky coverage. PACS numbers: k; Vc; Kf Keywords: Cosmoogy: cosmic background radiation Cosmoogy: observation I. INTRODUCTION Poarized anisotropies of the cosmic microwave background (CMB) radiation come in two favors: gradientike, E, component, and cur-ike, B, [1, ]. Ten years ago, the first detection of the E-mode anisotropies was announced by the dasi team [3]. Since then many subsequent experiments e.g. wmap [4], quad [5], or bicep [6] have detected the E-mode anisotropies with high significance deepening and confirming our understanding of the Universe s evoution and structure formation. panck [7] is widey expected to provide shorty most comprehensive and precise constraints on the E-mode poarization properties in a range of anguar scaes extending from the argest down to few arc minutes. In contrast, no B-mode anisotropy has been detected yet ony some upper imits are currenty avaiabe, e.g., [4 6]. This is expected given minute ampitudes predicted for this signa. At the same time the scientific potentia of the B-mode probe has been generay recog- Eectronic address: agnes.ferte@ias.u-psud.fr Eectronic address: juien.grain@ias.u-psud.fr Eectronic address: tristram@a.inp3.fr Eectronic address: radek@apc.univ-paris-diderot.fr nized as extremey promising. For instance, on the inear eve the B-modes can be sourced by the primordia gravitationa waves [8, 9] and not by the scaar fuctuations, thought to be argey responsibe for the observed tota intensity and E-mode anisotropies. Consequenty, a detection of the B-mode anisotropy at arge anguar scaes ( 100) in excess of what is expected from the gravitationa ensing signa, see beow, coud be seen as a direct vaidation of infationary theories, as the atter are considered to be the most ikey source of the gravity waves, and coud aow for discrimination between different infationary modes. It coud aso set usefu constraints on the reionization period [10]. At smaer anguar scaes, B-modes are expected to be mainy due to gravitationa ensing of CMB photons which converts E-modes into B-modes [11] and therefore their detection a source of constraints on the matter perturbation evoution at redshift z 1 when ight massive neutrinos and eusive dark energy both pay potentiay visibe roes.

2 tion of the qubic experiment, a these experiments scan the sky with one or more dishes and therefore most directy produce maps of the poarized Stokes parameters, Q and U. Cacuation of the E and B signas from the Q and U maps is a non-oca operation [1] and can be done uniquey ony if the fu sky maps are avaiabe. However, this can be hardy the case even for the sateite missions due to the presence of heavy non-cosmoogica contamination due to the Gaactic emissions, which typicay have to be masked out even after advanced and compex ceaning procedures have been appied. In the context of the pseudo-spectrum methods [ 4] the incompete sky coverage eads to the so caed E-to-B eakage, when the signa from E-modes is present in the reconstruction of B-modes power spectrum C B and more probematic, in the B-modes uncertainties. Though no bias is directy introduced, the eakage is a probem due to the much higher ampitudes of the E-modes signa, which then infates the overa uncertainty of the estimated B-modes signa potentiay precuding its detection. Severa extensions of the standard pseudo-spectrum methods have been recenty proposed designed to aeviate the E-to-B eakage probem. In this work we focus on the technique presented in Refs. [5 7] and working in the harmonic domain and referred to as the sz-method hereafter, and on two other techniques operating in the pixe domain presented in Ref. [8] and Refs. [9, 30], and referred to as the zb- and kn-techniques 1, respectivey. A these methods consist in fitering E-modes eaking into B-modes for each specific reaization of the poarized anisotropies and thus potentiay resove the excessive variance probem referred to earier. In this artice, we first describe each of these methods within a common framework of so-caed χ-fieds and then our impementations of them, emphasizing differences and simiarities with those proposed in the origina papers. Throughout this work we compute spatia derivatives of the sky maps in the harmonic domain. This is in agreement with the origina impementations of the considered techniques. We note however that an interesting, pixe-domain aternative has been recenty proposed in Ref. [31] and coud be expoited in future work. For spectrum estimators we use consistenty cross-spectra [3], rather than auto-spectra, avoiding therefore a need for estimating the instrumenta noise spectrum. We use numerica experiments to test the efficiency of each of these methods in terms of quaity of the C B reconstruction and above a of the resuting uncertainty. The numerica experiments invove two experimenta setups: one mimicking a sateite mission (oosey based on epic [33]), and, the other, a baoon-borne instrument (inspired by ebex [34]). We note that this kind of anayses of sateite-mission-ike set-ups are argey absent in 1 The methods names are based on the first etters of the names of the authors of the corresponding papers. the iterature, which predominanty has focused on smasky cases ony. Though other techniques, e.g., maximumikeihood based power spectrum estimators, may address better some of the probems faced by neary-fu sky observations, performance of the pseudo-spectrum methods in this regime is ceary of practica importance. The genera pseudo-spectrum formaism, as we as its standard and extended renditions reevant for this work, are introduced in section II. An overview of the methods and their impementations can be found in section III. The numerica resuts are given in section IV, which aso presents the case for the sz-method as the one which gives the smaest variances whie avoiding a bias. More extensive concusions are then given in section VII, whie technica detais are deferred to appendices, with App. C treating the probem of the noise bias for the zb and kn methods. II. PSEUDO-SPECTRUM POLARIZED POWER SPECTRUM ESTIMATORS A. Genera considerations The ineary poarized CMB poarization fied is competey described by a spin-() and a spin-(-) fieds, P ± ( n) = Q( n) ± iu( n), with Q and U denoting two Stokes parameters. Pseudo-spectrum methods disti the observed information into a set of harmonic coefficients, ã E m and ãb m, referred to as pseudo-mutipoes. These are reated to true mutipoes, a E m and ab m as foows, ã E m = [ ] H (+) m, m a E m + ih( ) m, m a B m, (1) m ã B m = [ ] ik ( ) m, m a E m + K(+) m, m a B m, () m where H (±) and K (±) are kernes, which in genera can be a different, non-vanishing, and non-diagona in both and m. Noise terms have been negected in these equations for shortness. The kernes are typicay singuar and it is not in genera possibe to sove the inverse probem to recover the true mutipoes, a X m, directy. Instead the pseudospectrum approaches attempt to do so ony on the power spectrum eve. This is achieved in two steps. First, owing to the statistica isotropy of CMB fuctuations, we can rewrite Eqs. (1) and () on the power spectrum eve as, CE CB = [ H (+) C E ( ) ] + H C B, (3) = [ K ( ) C E + K (+) C B ]. (4)

3 3 where the new kernes, X (±) X (±) 1 = + 1 m = are given by (X = K, H), m= and... denotes an ensembe average and, C X m= X (±) m, m, (5) ãx m. (6) The kernes obtained on the power spectrum eve are ceary more manageabe and easier to cacuate, nevertheess, they sti wi be singuar. To avoid this issue, the inverse probem defined in Eqs. (3) and (4) is soved ony for binned spectra [4], C X b C X b P b CX P b C X, where the binning operators are defined as, S P b = b max b, [ b min, b max] min 0, / [ b min, b max] 1, [ b Q b = S min, b max] 0, / [ b min, b max], (7) satisfying therefore the reation Q b P b = δ bb. Here, we have introduced a shape function, S. Its roe is to minimize possibe binning effects by making S C neary fat within the bin. Hereafter, we wi adopt the standard choice for it, i.e., S = ( + 1)/π. The binned version of Eqs. (3) and (4) now reads, ( CE b C B b ) b ( where, for X = K or H, H (+) bb H ( ) bb K ( )) bb K (+) bb ) ( C E b C B b ), (8) X bb P b X Q b. (9), To incude a correction for the presence of the instrumenta noise, the pseudo-power spectrum on the right hand side of the first of Eqs. (7) needs be corrected for the noise pseudo spectrum prior to the binning operations. The estimates of the true spectra, C X, can be then obtained by directy soving the fu system in Eq. (8). We note that by construction, and negecting the binning effects, which are argey controabe, these wi be unbiased estimates of the true binned spectra. However, as ong as the poarization mode mixing kerne, K ( ), does not vanish the power contained in the E-poarization component wi contribute to the overa variance of the B-spectrum estimate an effect referred to as the E-to- B eakage. To avoid that one shoud resort to methods for which K ( ) is either zero or neary so. We aso note that if K ( ) = 0 then the estimate of the B-mode spectrum can be derived independenty on the E one. This coud be aso the method of choice even if K ( ) vanishes ony approximatey. In this case a sma bias in the B spectrum estimate is however to be expected. B. Standard pseudo spectrum approach If the poarization fieds are known on the entire ceestia sphere, their E- and B-representation can be easiy obtained in the harmonic domain using the spin-weighted spherica harmonics 3, a E m = 1 [P ( n) Y m( n) + P ( n) Ym( n)] d n, a B m = i (10) [P ( n) Y m( n) P ( n) Ym( n)] d n. If the poarization fied is measured on a fraction of the sky ony, the above decomposition can be most straightforwardy appied to such a case by positing that the signa over the unobserved part of the sky vanishes.. This choice defines the standard pseudo-spectrum method, in which the resuting pseudo-mutipoes, ã X m, X = E, B, can be expressed as foows, ã E m 1 M [P ( n) Y m( n) + P ( n) Y m( n)] d n = [ ] K (+) m, m a E m + ik( ) m, m a B m, (11) m ã B m i M [P ( n) Y m( n) P ( n) Ym( n)] d n = [ ] ik ( ) m, m a E m + K(+) m, m a B m, (1) m where M is a binary mask defining observed patch, and where we introduced the convoution kernes, K (±) m, m, expicit expressions for which are we-known and can be found esewhere, e.g., [7]. We see that for the standard technique both the H (±) and K (±) kernes, Eqs. (1) & (), coincide and that the poarization-mode mixing kerne, K ( ) m, m, does not vanish and therefore Stricty speaking what is required is that the mutipoe kerne, K ( ) m, m vanishes but if Eq. (5) is satisfied, exacty or approximatey, it is equivaent to requiring the power spectrum kerne, K ( ), to be (neary) zero. 3 A the integras in this paper are taken over the entire ceestia sphere. We therefore do not specify that the integration domain is S.

4 4 though unbiased, the standard pseudo-power spectrum estimator suffers from the E-to-B eakage. This can be quite severe. For instance, an experiment covering around 1% of the sky essentiay unabe to detect a power at the scaes arger than < 140 (see Fig. 16 of [7]). The above formuae can be extended to incude an arbitrary weighting of the observed sky pixes as given by a window function, W. This can be done by inserting W M instead of M in a the equations above, incuding those for the kernes. If we further assume that the window function is aways zero outside of the observed sky, i.e. if M = 0 then aso W = 0, then, as a consequence, W M = W and M can be dropped from the equations in favor of W. The mask, M, is then assumed to be defined impicity by W. We wi use this simpification in the foowing. Aso for definiteness hereafter, we assume that a fied defined on the sphere, e.g., P ±, is known on the fu sky and wi appy a mask or an apodization expicity to such a fied to emphasize that it is known ony over a imited sky area, e.g., W P ±. C. Leakage-free pseudo-power spectrum approaches To aeviate the eakage probem within the pseudospectrum methods one woud need to adapt a different definition of the pseudo-mutipoes than the one used in the standard approach. Such a new definition shoud not rey directy on the poarization fieds, as does the standard approach, as those unavoidaby incorporate contributions from both types of poarized mutipoes. Instead it shoud based on some other fieds, which depend ony on one set of the mutipoe coefficients, and which woud therefore ensure that the poarisation mode mixing kernes, K ( ) m, m and H ( ) m, m, indeed vanish, resoving the eakage issue. Such a construction has been indeed proposed by [1] and the corresponding fieds are caed χ-fieds. They can be derived from the poarization fieds as foows, χ E ( n) = 1 [ P ( n) + P ( n) ], (13) χ B ( n) = i [ P ( n) P ( n) ], (14) In the fu-sky case, Eq. (15) can be readiy inverted giving, χ X m = χ X ( n)y m( n )d n = N, a X m, (16) what in turn can be adapted for cases of partia sky experiments in a usua manner, rendering the foowing definition of the pseudo mutipoes, ã X m 1 N, M( n) χ X ( n)y m( n) d n. (17) This definition can be then used in the genera pseudospectrum formaism as deveoped in Sect. II and though it wi resut in a mixing of different -modes, it wi not cause any eakage between the poarization modes as by construction the off-diagona kernes, H ( ) and K ( ) in Eqs. (1) and (), vanish. The major difficuty of this approach is the computation of the χ X -fieds. Indeed, Eqs. (13) & (14), as they are, require in principe knowedge of the fu sky poarization fieds. As we wi see in the next section a three methods designed to resove the eakage probem and studied in this work rey on the χ X fied cacuation, impicity or expicity, and circumvent the probem of having ony a imited sky coverage differenty. We note that if the χ X fieds were known exacty on the cut sky, the inverse probem in Eq. (8), coud be soved separatey for E and B spectra, as the off-diagona kernes woud, by construction, vanish. In more reaistic circumstances the χ X fieds, actuay estimated on the cut sky, may be imperfect giving, at east in principe, rise to non-zero off-diagona contributions. These, if not corrected for, coud ead to a bias of the estimated power spectra. Soving the fu system, accounting for the nondiagona kernes, coud hep to trade the bias for an extra, but presumaby sma variance of the spectrum estimate. Though this indeed coud be possibe at east for some of the methods, for others, the difficuty in cacuating the off-diagona kernes, either anayticay or numericay, e.g., via Monte Caro simuations, can be prohibitive, and an approach favored in practice is often simpy to accept the bias, once it is found to be sufficienty sma. where ( ) denotes the spin-raising(owering) operator [1]. These χ fieds invove indeed either E-, in the case of χ E, or B-, for χ B, modes. This can be seen directy by noting that the χ X -fieds, X = E, B, are scaar and given by, III. SPECIFIC APPROACHES A. sz-approach χ X ( n) =,m N, a X m Y m ( n), (15) 1. Theoretica description where for the future convenience we have introduced, ( + s)! N,s ( s)!. Let us start from the pseudo-mutipoes for B-modes defined as in Eq. (17) with the binary mask, M, repaced by an arbitrary window, W. By performing an integration by parts twice [5, 6], we can rewrite this equation

5 5 as, [ ã B i m = d n P ( n) ( W ( n)y m ( n)) (18) N, P ( n) ( W ( n)ym ( n) ) ], where a the boundary terms are omitted corresponding to an assumption that the apodization window, W ( n) and its first derivative, W, vanish at the observed patch boundaries. This atter equation has an advantage over the former, Eq. (17), as it does not invove any expicit cacuation of derivatives of noisy sky maps. Instead, the differentiation needs to be ony appied to a presumaby smooth window function, W. We can therefore use Eq. (18) as a definition of the pseudo-mutipoes, which we wi appy from now on aso in cases when the apodization does not conform with the boundary conditions. Note that in these atter cases there wi be no assurance that no E-to-B eakage is present. Hereafter we wi refer to this technique as a pure pseudo-spectrum estimator, as Eq. (18) can be interpreted as projecting the poarization fied P ± onto a basis of pure functions representing ony B-ike poarization modes on a cut sky [5, 6, 35].. Numerica impementation Our impementation of the approach foows cosey that proposed in [7] and proceeds in four steps. Step 1: We compute spin-0, spin-1 and spin- renditions of the window function, W, given by, W 0 = W, W 1 = W, W = W. (19) Because W is rea, then W s = W s for a spin s = 1,. Step : We compute pure pseudo-mutipoes by constructing first three apodized maps, P ± = W 0 P ±, P ±1 = W 1 P ±, P ±0 = W P ±. (0) and then cacuating pure ã B m as, ã B m = 1 N, (B 0,m + N,1 B 1,m + N, B,m ), (1) where B s,m is a B-type mutipoe of P ±s defined as B s,m = i [ P +s( n) s Y m( n) () ] ( 1) s P s ( n) s Ym( n) d n. Step 3: On this step we compute the convoution kernes for pseudo-c as defined in Eqs. (3) & (4). This can be done using, e.g., Eqs. (A13) and (A14) of [7]. If the appied apodization does not fufi the boundary conditions then the off-diagona bock, K ( ), has to be aso incuded. In practice, the off-diagona couping between the poarization components wi aso resut due to pixeization effects. Though such effects are not accounted for in the anaytic formuae for the kernes, they can be corrected for, to some extent, by a procedure described in [7], eading to a remova of the majority of sma bias induced by the residua, pixe-induced, E-to-B eakage. We note that typicay, if the method is appied consistenty to both E and B-modes the corresponding H and K kernes are identica. However, in some circumstances it may be advantageous and possibe to appy hybrid approaches in which both kinds of spectra are treated differenty. Such cases have been discussed recenty in [36]. Step 4: This step consists of standard operations invoved in any pseudo-spectrum method as summarized by Eqs. (7) & (8) and discussed in Sect. II. 3. Sky apodization As emphasized in [5 7], an appropriate sky apodization is a key eement of any such a construction. In the specific method discussed here the degree to which the apodization fufis the boundary conditions wi be a principa factor determining the eve of a suppression of the E-to-B eakage. At the same any apodization appied to reaistic, meaning noisy, data wi have a direct impact on the resuting uncertainties of the spectrum estimate. In the context of the pure pseudo-spectrum method, systematic approaches have been deveoped and studied in detai, which aow for a numerica optimisation of sky apodizations in order to ensure a neary minima vaue of the fina spectrum uncertainty [5 7]. These are either based on MC simuations or semi-anaytic techniques. In the former case, MC simuations are used to tune the apodization ength of the sky apodization given by some anaytic formuæ. In this work, we wi use the so-caed C function as given by equation (31) of [7]. In the atter case, the optimized sky apodization can be computed by soving a arge inear system as proposed in [6]. We refer to these atter windows as variance-optimized apodization. In both cases the optimization coud, and shoud, be appied bin-by-bin to ensure the best resuts. As discussed at ength in [7] both these approaches require some prior assumptions concerning, for instance, the anguar power spectra of E- and B-modes, however, the resuts of the optimisation are found to be ony midy dependent on detais of the assumed B-mode spectrum. It has been shown via numerica experiments [7] that the variance-optimized apodizations ead systematicay to the owest error bars on the reconstructed C B s and

6 6 therefore wi be used them in this work. Those varianceoptimized apodizations can be computed in two ways, depending on the domain (harmonic domain or pixe domain) in which the inear system is soved. For the pecuiar case of homogeneous noise, resoution can be vasty done in the harmonic domain. In such a case, the derivative reationship W s=1, = s W 0 and the boundary conditions W 0 (C) = W 1 (C) = 0 on the contour of the observed region are fufied (up to pixeization effects). For more genera cases, the inear system providing the variance-optimized apodization is soved in the pixe domain. In such a setting, both the derivative reationship and the boundary conditions are reaxed (W 0, W 1 and W are considered as independent). As a consequence, the fina sky apodizations does not stricty satisfy these conditions and the resuting pseudo-mutipoes wi not be stricty equa to the pure pseudo-mutipoes. However, it has been shown in [6, 7] that the anguar power spectra recovered in such cases consistenty achieve smaer uncertainties than those of other apodization choices. B. zb-approach 1. Theoretica description In this approach the χ X fieds are computed directy in the pixe domain and for the cut-sky. This is made possibe thanks to a formua derived in [8], which reads, W ( n)χ B ( n) = i [ (W P ) (W P ) ] (3) [ W i W (W P ) W ] W (W P ) i [( ) ] W P ( W ) P. ) ] [( W + i W P ( W ) W P. As usua here W is assumed to be zero outside the observed region. Moreover, if we assume that it and its first derivative vanish at the edges of the observed region, a the operations on the right hand side of this equation can be performed with ony knowedge of the poarization fied on the cut-sky. Consequenty, we coud estimate the fied, χ B consistenty on the cut-sky by first computing the rhs of Eq. (3), then dividing it by the window, W, and ater use it to cacuate pseudo-mutipoes via Eq. (17) as proposed in [8] or use some apodized rendition of the χ B fied to derive the pseudo-mutipoes, which are then corrected on the power spectrum eve as proposed here 4. In either case the pseudo-mutipoes 4 Stricty speaking, the pseudo-mutipoe are not divided by N, in the impementation of Ref. [8]. Instead, the pseudo-spectrum are in principe free of any E-to-B eakage due to cut sky effects and the K ( ) kerne shoud vanish. However, as underined by [8], both pixeization and convoution by the beam ead to some residua E-to-B eakage and ideay one woud ike to sove the fu inear system, Eq. (8), to get the fina, unbiased power spectrum estimation.. Numerica impementation An impementation of this technique is proposed in [8] and invoves four steps. The impementation used in this work foows that of the origina authors with an exception of the second step as detaied beow. Step 1: We compute the χ B fied on the observed patch of the sky using Eq. (3). This in turn requires a numerica cacuation of derivatives of noisy fieds, which constitutes the principa difficuty of this technique. These in our impementation, as we as that of [8] are performed in the harmonic domain. We emphasize that with such a choice this method becomes effectivey a harmonic space approach. Yet another potentia probem is reated to the cacuation of the terms, which invove expicit mutipication by W 1 W, as W itsef becomes very sma at the boundary. This probem cannot be avoided by imposing more boundary conditions on W as W 1 W θ θ c 1, at the boundary, θ c, and therefore necessariy diverges at the boundary 5. This can be however deat with on Step. Step : We compute the pseudo-mutipoes, ã B m, of the newy constructed χ B map. This requires effectivey dividing by the window, W. Though straightforward a priori a care has to be exercised whie doing so because of W vanishing at the observed area edges. One option, adopted in [8], reies on simpe trimming the troubesome, boundary ayer, eaving ony those pixes for which the division is numericay reiabe. This eads to some oss of the information but soves simutaneousy the divergence probem appearing on step 1. The amount ost due to trimming wi depend on the detais of how the trimming is done, a practica compication, which needs to be addressed in this approach. An aternative way of resoving both these issues at the same time, which we propose here and which is free of such practica compications, is to define pseudomutipoes using the fied, W χ B, and then to correct for are divided by N, in the binning process. The two choices are however competey equivaent. 5 Constraining W together with its first derivative W, both to be continuous on the entire ceestia sphere but zero outside the observed part of the sky necessary eads to W θ θ c n with n 1, cose to the boundary.

7 7 the presence of the apodization on the binned spectrum estimation step, Eq. (8). It is cear from Eq. (3) that the estimation of the W χ B fied does not suffer of any singuarities at the edges. This method is the method of choice in this work. We note that this method is not ossess either, as the apodization it invokes wi unavoidaby compromise some information. Nevertheess, the information oss in this case is expected to be smaer than in the former one. For instance, it is argued in Sec. IV of [8] that to anayze a map covering 3% of the sky (a spherica cap with a radius of 0 degrees is assumed as the observed part of the sky), it is necessary to remove an externa ayer with a width of degrees; thus reducing the effective sky coverage from 3% to.4% (assuming a binary mask to weight the resuting χ B map). As shown hereafter, by focusing on W χ B, we are abe to sove for the E-to-B eakage by using an apodization ength of 1 degree. As a consequence, for a spherica cap with a radius of 0 degrees, the effective sky coverage is reduced from 3% to.9% (an expicit expression for the effective sky coverage assuming non-binary mask can be found in [36]). Step 3: Kerne K (+) is computed taking advantage of the fact that the χ B fied is a scaar, ike temperature, made of B-modes. The expicit expression of K (+) is given by Eq. (39) of [8] (foowing what was derived for temperature [, 4, 3, 37]), i.e., K (+) = ( + 1)N, 4πN, m ( w () m with w () m the mutipoes of the W function 6. Step 4: ), (4) The inear system in Eq. (8) is inverted negecting the off-diagona bock, K ( ), and therefore aso the residua E-to-B eakage. 3. Sky apodization In this approach we coud either use anaytic windows or the variance-optimized windows obtained from the optimization procedure deveoped within the framework of the sz-method. In this former case, we wi aways use the C famiy of windows from Ref. [7] and use MC simuations to determine their optima apodization ength. In the case of the variance-optimized apodizations computed in the harmonic domain, it may appear that to ensure their optimaity, we shoud use a window given 6 We stress that the mutipoes of W are not equa to the square of the mutipoes of W. by a square root of the actua optimized one, i.e., W ZB WSZ, to compensate for the fact that it is a square of the window which is used as the apodization in our impementation of the zb-approach. Whether such a window coud be a viabe option, wi depend whether it does not cause any probems in the cacuation of the rhs of Eq. (3) at the patch edges. It is straightforward to show that this is aways the case for windows, which are forced to obey the boundary conditions stricty. This is because such windows scae at the boundary as W SZ θ θ c n, with n >, [6] therefore both quantities, W ZB and W ZB, (where W ZB = W SZ ), needed to compute the rhs of Eq. (3) are we-behaved for θ θ c. However, the variance-optimized windows fufi the boundary condition ony approximatey, what may ead to singuarities of the derivatives of W ZB. To avoid that, we further mutipy the variance-optimised windows by some anaytic window, with a narrow apodizaton ength. This is designed to affect as itte as possibe the properties of the initia window but enforce the boundary conditions stricty and therefore ensure proper behaviour of the resuting window at the boundary. In practice, we have found that using either the corrected W ZB window or directy W SZ eads to comparabe resuts and numerica resuts presented hereafter are using the atter ones. It is important to notice that in such settings, the variance-optimized windows computed in the pixe domain cannot be directy appied. Indeed, such windows do not conform typicay with the derivative reationship between the different windows, i.e., W s=1, s W 0 or the boundary conditions, i.e., W 0 (C) W 1 (C) = 0. However, these conditions are essentiay mandatory for the zb-method for two reasons. First, the method requires that W χ B is reated to W s=0,1, P ± and M P ±, as e.g., it is in Eq. (3), that however without the assumptions about the windows properties is at east tedious. Second, even if such an expression is found, this wi ead to mixing kernes, which wi not be numericay computabe from the first principes, as in e.g., Eq. (4), as they wi invove the product of three functions : P ± mutipied by either W 0 or M, and by W s=1,, therefore eaving time consuming Monte Caros as the ony viabe option for their estimation. C. kn-approach 1. Theoretica description Another way of estimating the χ B fied is by generaizing its definition to the cut-sky case. This can be done straightforwardy by modifying Eq. (14) as foows, χ B ( n) = i [ M P ( n) M P ( n) ], (5) where as usua M stands for a binary mask and the tide over the χ symbo is used to emphasize that at east in principe this is a different object than the true χ B fied

8 8 defined on the cut sky, i.e., M χ B. We note however that as ong as M is constant (and for simpicity assumed to be equa to 1), i.e., in the interior of the observed patch, the two fieds are indeed identica χ B ( n) = χ B ( n). In principe the ony probem arises therefore at the patch edges. As proposed in Ref. [9] one coud use this observation to reconstruct the true χ B fied everywhere with an exception of the boundary ayer. The probem becomes then technica and bois down to a question how to cacuate the derivatives required by such a procedure. [9] propose to do it in the harmonic domain and use semi-anaytic formuae of [1] to represent the derivatives via convoutions of some geometrica kernes. Given that the mask is abrupty faing from 1 to 0 at the edges, it is not surprising, that such a procedure eads to significant osciatory behavior at the edges, which extends we within the center part of the observed patch. This is a resut of the necessity of imposing a finite band-imit on a harmonic decompositions performed as part of this procedure, even if the considered functions, with an abrupt jump does not have such a imit. Such a band-imit is directy reated to the pixeization used to represent the poarization fieds. This has two practica consequences. First, a robust criterion has to be found deciding which pixes are to be retained, i.e, which are sufficienty cean of any E-mode contamination, second, the oss of area is expected to be rather significant. We refer the reader to [9] for more detais of this specific impementation. A more robust approach woud either invoke different ways of cacuating the derivatives, e.g., as proposed by [31], or introducing in Eq. (5) a smooth apodization, W, in pace of the binary mask, M. This second option was proposed by [30] and this is the one we impement in this work. The apodization coud aeviate the pixeization effects described earier by truncating the band imit of the apodized poarization fied, so the harmonic domain derivatives perform better. Such a window woud need to have a centra region, where W is constant (and equa to 1) before smoothy roing off at the edges. As in the case of the binary mask ony in this centra region the reconstructed χ B fied woud coincide with the true one and woud be used for the power spectrum estimation. The main advantage of such a technique is that it provides a cear criterion which pixes to retain or to reject. Nevertheess, it does not sove competey the pixeization effects as pixes inside the centra area can be affected by the pixe-induced eakage but this time originating from the contour around this centra area. However, and as numerica resuts shown in [30] suggest, the pixeization effects at the inner contour are mitigated by the fact that W is continuous as compared to the pixeization effects induced by considering the non-continuous binary mask. Hereafter, we wi use this second approach and appy a sky apodization to the poarization fied. We wi then use Eq. (5) but with a mask, M, repaced by a window, W, to cacuate χ B and ater, the true χ B = χ B Mχ B where, M χ B is the binary mask buit from the kept-inthe-anaysis pixes, i.e., pixes for which W is essentiay constant.. Numerica impementation The numerica impementation of this approach consists then in two main steps, which need to be first appied to simuated and ater actua data. The Monte Caro simuations are empoyed to seect optima windows for a given probem. Step 1: We cacuate the apodized χ B fied for a seected window, W. This invoves performing numerica derivatives of the avaiabe poarization fieds, P ± and those are performed in the harmonic domain. In this work we use a famiy of arch-sine windows as defined in [7] with an apodization ength which is to be tuned via Monte Caro simuations. The criteria we use in the apodization ength optimization process are the eve of the B-spectrum bias and variance. Step : We compute the B-mode power spectrum from the precomputed χ B fied. The spectrum is computed using ony the trimmed, centra part of the avaiabe patch, χ B Mχ, which can be further apodized, if needed, and B foows the genera pseudo-spectrum method framework. Hereafter, foowing [9] we wi negect possibe eakages from the E-spectrum and use the scaar kerne as aso used in the zb-approach, Eq. (4). We note however that unike in the zb-method the eakage in this approach can be more pervasive affecting even the most centra areas of the patch and therefore never fuy removed via simpe area trimming. For this reason one may ponder whether a more appropriate kernes can not be derived, which coud account for these effects. The answer, which we discuss in more detai in Appendix A, is that such kernes woud need to be evauated numericay and be necessariy very costy. We wi therefore ony consider the simpified case in this work. 3. Sky apodization The sky apodization and masking needs to be performed on three different stages in this approach. First, we need to apodize the maps before computing the χ B fied. Then we need to mask pixes, which are expected to be contaminated by the residua E-to-B eakage. Finay, we may want to apodize the reduced χ B maps to ocaize better bin-to-bin correations of the recovered B spectrum. Unike in the case of the sz- and zb-techniques, one cannot derive here some optima windows from first principes. Instead for the sky apodization required for the computation of χ B we use a famiy of the arch-sine anaytic windows, proposed in [7], and resort to Monte

9 9 Caro simuations to optimize their apodization ength. In this optimization procedure we aways trim a the pixes within the boundary ayer of W, i.e., where it is not constant, as these are the pixes, which are unavoidaby affected by the E-to-B eakage, and we use ony the remaining ones for the spectrum estimation. Ceary, there wi be sti some eve of the E-mode power in the map eft over after such a trimming procedure, mosty due to pixe induced E-to-B eakage. The eve of this eakage depends on the assumed apodization ength, becoming sower for its arger vaues, and the MC simuations are then used to find the smaest vaue of the atter ensuring a sufficienty ow eve of the eakage. This wi at the same time maximize the sky area, given the acceptabe eakage requirement, eft for the fina spectrum determination and therefore ensure that the spectrum variance is the smaest. D. Brief appraisa The three methods considered in this work can be introduced within a common framework based on the χ B fied concept as has been done in this Section and demonstrated to be a rather cosey reated. The fact, which may be potentiay somewhat surprising given their origina derivations. The two first methods, sz and zb, in the renditions as considered in this paper are ceary equivaent on the anaytica eve, if the apodizations empoyed in both these cases are reated to each other as, W ZB = W SZ, and W SZ fufis stricty the boundary conditions. The differences between these two approaches are therefore ony in their numerica impementations and approximations which they impy. Both these methods suffer due to pixeization issues, in particuar arising due to a need to compute numerica derivatives, and which give rise to a residua contamination of the B-spectrum with the E- mode power. The sz-method requires ony derivatives of the window functions, therefore, at east in the cases when these are given anayticay, it is possibe to estimate the non-diagona couping kerne, K ( ), and correct for some of those effects. Such corrections are more difficut in the case of the zb-approach, where the nondiagona kerne woud have to be estimated competey numericay. The sz-method can potentiay offer more freedom for an optimization of the B-spectrum variance as estimated for reaistic noisy maps as the boundary conditions on the appied apodizations can be reaxed eading to an increase of the signa variance reated to aowing for some E-to-B eakage but a decrease of the tota, signa+noise, one. At the same the off-diagona, poarization mode couping kernes can be readiy cacuated and the estimated B-spectrum unbiased. The kn-approach can be ooked at as an approximation of the zb-method. Indeed the first term on the rhs of Eq. (14) used by the zb-method coincides with the rhs of Eq. (5) (repacing M by a sky apodization W ), which defines the first step of the kn-approach. We refer to App. B for a detaied discussion. The contributions of the extra three terms in Eq. (14) are ocaised around the patch boundary and removed in the kn-method by trimming the boundary ayer, which is retained and used for the power spectrum estimation in the case of the former method. For this reason we may expect that the performance of the kn-method shoud be inferior to both the zb- and sz-approaches, which in turn we coud expect to be neary equivaent. In turn, the kn-method may appear as the most straightforward on the impementation eve and therefore attractive at east at first stages of the anaysis. IV. NUMERICAL EXPERIMENTS A. Experimenta set-ups For numerica investigations, we define two fiducia experimenta setups. Though ideaized, they are chosen to refect the genera characteristics of forthcoming CMB experiments dedicated to B-modes detection. Those characteristics which cruciay impact on the anguar power spectrum reconstruction are the noise eve, the beam width and a pecuiar sky coverage. We first consider the case of a possibe sateite experiment aimed at B-mode detection. For such an experiment, we reied on the epic m [33] specifications for the noise eve and the beam width, setting these to. µkarcmin for the noise eve and 8 arcmin for the beam width. For the pecuiar sky coverage of such a neary fu-sky experiment, we consider the gaactic mask R9 used for poarized data in wmap 7yrs reease (see [38]) adding the point-sources cataog mask. So we obtain a 71% sky coverage patch showed in the ower pane of Fig. 1. Throughout this work we use Heapix pixeization scheme [44]. Here the pixe size is 7 arc minutes, i.e. N side = 51. Second, we consider the case of baoon-borne experiment inspired by the ongoing ebex experiment [34]. The noise eve and the beam width are respectivey set equa to 5.75 µk-arcmin and 8 arcmin. The observed part of the sky covers 1% of the tota ceestia sphere and its pecuiar shape is dispayed on the upper pane of Fig. 1. It consists of a square patch of an area of 400 square degrees incuding hoes to mimic poarized point-sources remova. In such a case, we choose N side = 104 corresponding to a pixe size of 3.5 arc minutes. B. Simuations We numericay impement the three techniques described in the previous section and test their respective efficiency with Monte-Caro simuations. We investigated the fu performances of those approaches from the perspective of B-mode power spectrum reconstruction and

10 10 two maps and assumes that the noise of the two maps is uncorreated and its eve per pixe is given by σ p. This naïve mode-counting is bound to underestimate the variance in our study cases and is therefore used ony as a ower imit. An effective, observed fraction of the sky, f sky, depends on an assumed apodization and therefore wi be in genera different for each of the methods considered here and may vary from a bin to a bin. For definiteness hereafter as a reference we wi use its vaue computed assuming ony binary mask, M. Such a choice, in terms of the Fisher errors eads to the owest variances. V. RESULTS: SATELLITE CASE A. Standard pseudo-spectrum method FIG. 1: Sky areas as observed by the fiducia sateite-ike experiment (upper pane) and for the baoon-borne, smascae experiment (ower pane) as considered in this work. The sky coverages are respectivey 71% and 1% of the tota ceestia sphere. For the sateite experiment, the mask is a combination of the gaactic mask R9 and the point-sources cataog used for poarized data in wmap 7yr reease. Ony the atter mask is used for the baoon-borne case. therefore incorporate noise with the eve as stated in Sec. IV A. To simuate the CMB sky, the input E-mode signa is that of the cosmoogica mode with parameters as given by the WMAP 7yrs data [39] and the input B-mode incudes ensing and primordia B-modes with r = 0.05 (Our convention for r foows the WMAP convention: r = P T (k 0 )/P S (k 0 ) with P S(T), the primordia scaar(tensor) power spectrum and k 0 = 0.00 Mpc 1 the pivot scae). We wi assume that two identica maps are aways avaiabe with the same eve of the homogeneous noise in each of them, which is taken to be uncorreated between the two maps and use their cross-spectra and their variance to compare different approaches. We cacuate the atter with hep of Monte Caro simuations and use as a common reference an estimation of the variance based on simpe mode-counting and given by δ, Σ = ( ) C B ) + (C B + 4π σp ( + 1)f sky N pix B, (6) where B is the beam function and σ p the noise per pixe. This formua appies to a cross-spectrum between The major advantage of the sateite experiments is their abiity to measure the sky signas on the argest anguar scaes, and therefore having potentia to constrain their power spectra a the way to the owest mutipoes. Indeed, the simpe Fisher variance formua introduced earier seems to suggest that this shoud be possibe if ony the sky coverage is sufficienty arge. Though this formua negects the eakage it seems ony natura to expect that it shoud be sma for neary fu sky maps, and therefore shoud ead to subdominant effects as compared to other uncertainties, e.g., cosmic variance. In this section we confront these expectations against reaistic simuations within the paradigm of pseudospectrum methods. In this context, if the eakage is indeed sma, we may expect that even the standard pseudo-spectrum technique coud perform sufficienty we assuring precision comparabe to that of the other methods, which expicity invoke some eakage correction, and not that far off the Fisher predictions. Beow we therefore start from a discussion of the standard pseudospectrum technique. 1. Leakage We quantify the eve of the E-to-B eakage using standard pseudo-spectra cacuated in the case of simuations with no input B-mode power and, which woud have been zero had there been no eakage at a. These are denoted E B hereafter as C. We compare these pseudo-spectra with those cacuated assuming no input E-mode power, B B denoted C, and therefore expressing the pseudopower of the genuine B-modes. These pseudo-spectra E B are shown in Fig., which dispays C, upper curve, B B and C, ower curves, computed for three different vaues of r = 0.1, 0.05, Ceary, the eaked power, C E B, dominates over the true B-modes at east up to 700. We therefore concude that the eakage is by far not insignificant even in the sateite case.

11 11 C E B C B B Furthermore, if we take a ratio of and as a measure of the magnitude of the eakage we find that its vaues are within a factor of from those obtained for the sma-scae experiment considered ater on, indicating that the eakage amount in both cases is in fact comparabe, even if the atter experiment covers roughy 71 smaer sky area than the former. This demonstrates that it is not merey sky area which matters as far as the eakage is concerned. In fact, the gain in the sky area in the case of the sateite experiment considered here comes at the price of a significanty more compex and onger perimeter, effects of which, [e.g. 35] offset the sky area advantage. We note that though we may attempt to simpify the boundary of the Gaactic mask to suppress the eakage, this is more difficut to be done with the point sources, which indeed seem to provide the major contribution to the observed eve of the eakage.. Variance The arge eakage found present on the pseudospectrum eve wi inevitaby ead to excess variance of the B-mode spectrum estimate. These are depicted in Fig. 3, where variances computed assuming three different apodizations are shown. We see that in either case no meaningfu constraints on the owest mutipoes, < 30, can be set at east as ong as no binning is appied. These resuts demonstrate that for reaistic observations the standard pseudo-spectrum method can not ensure sufficient precision for the argest anguar scaes and some aternatives, expicity correcting for the eakage, need to be considered instead, as we do so in the next section. Fig. 3 aso shows a B-mode spectrum averaged over a performed MC simuations. It is unbiased, as expected, given that we incude expicity in the cacuations the off-diagona couping kerne, K ( ), correcting the spectra on average for the E-mode power eaked to B. In practice, we find however that a specia care needs to be taken whie cacuating this kerne to ensure the absence of the bias. This is because the eaked power is indeed grossy dominant over that genuine B-mode, see Fig., setting very demanding constraints on the precision of the kerne. For instance, the good agreement shown in Fig. 3 has been ony obtained, when we minimized the spurious contributions due to the pixeization coming specificay from the poar caps by rotating the sky map so those have been hidden in the regions excuded by the empoyed mask. The residua scatter at its ow- end is just a resut of the insufficient number of simuations and the huge variance dispayed by the standard pseudo-spectrum estimator on these scaes. The good overa agreement of the averaged spectrum with the theoretica spectrum used for the simuations vaidates our MC-based predictions for the variances. FIG. : Contribution of E-modes (back curve) and B-modes (coored curves) to the B-modes pseudo-c for the case of a sateite mission. This measures the reative amount of E- mode eaking into B if one does not correct for such eakages. The corresponding mask is depicted in the upper pane of figure 1. B. Leakage-correcting methods 1. Apodization The resuts described above demonstrate that the standard approach is not suitabe for the ow- recovery of the B-mode spectrum even for the neary fu sky experiments. Therefore, if such a goa is achievabe at a with a pseudo-spectrum method, it woud have to be a method, which tackes the eakage probem case-by-case, as do the three methods discussed earier. It is important however to emphasize that the suppression of the E-to-B eakage in these methods comes at a price as the corrections they invoke may affect the variance of the recovered spectrum. Consequenty, this variance wi not be in genera cose to the variance of the B-mode spectrum as obtainabe in the standard pseudo-spectrum approach in a case, when the CMB E-mode power, and therefore the eakage, is set artificiay to zero, as one coud ideay hope for. Instead there wi be typicay an extra contribution to the variance, not due to the eakage anymore, as it is expicity treated for, but from remova of part of the information as resuting from the eakage correction procedure. This in principe cas for some optimization procedure between the eve of the eakage and the bias (at east for some of the methods studied here) and the variance of the recovered B-mode power spectrum. As the oss of the information is reated to the apodization and/or masking appied in these methods, and used sometimes on mutipe stages, such an optimization coud be in genera rather cumbersome to formaize and to date has been impemented in a systematic way ony in the case of the sz-