QuickPol: Fast calculation of effective beam matrices for CMB polarization
AAstronomy & Astrophysics manuscript no. quickpol˙v2 c (cid:13)
ESO 2018June 26, 2018
QuickPol : Fast calculation of effective beam matrices for CMBpolarization
Eric Hivon , Sylvain Mottet & Nicolas Ponthieu , Sorbonne Universit´es, UPMC Univ. Paris 6 & CNRS (UMR7095): Institut d’Astrophysique de Paris, 98 bis Boulevard Arago,F-75014, Paris, France Institut de Plan´etologie et d’Astrophysique de Grenoble, Universit´e Grenoble Alpes, CNRS (UMR5274), F-38000, Grenoble,France Institut d’Astrophysique Spatiale, CNRS (UMR8617) Universit´e Paris-Sud 11, Bˆatiment 121, F-91405, Orsay, FranceReceived Aug 31, 2016 / Accepted Oct 03, 2016
ABSTRACT
Current and planned observations of the cosmic microwave background (CMB) polarization anisotropies, with their ever increasingnumber of detectors, have reached a potential accuracy that requires a very demanding control of systematic e ff ects. While some ofthese systematics can be reduced in the design of the instruments, others will have to be modeled and hopefully accounted for orcorrected a posteriori . We propose QuickPol , a quick and accurate calculation of the full e ff ective beam transfer function and oftemperature to polarization leakage at the power spectra level, as induced by beam imperfections and mismatches between detectoroptical and electronic responses. All the observation details such as exact scanning strategy, imperfect polarization measurements, andflagged samples are accounted for. Our results are validated on Planck high frequency instrument (HFI) simulations. We show howthe pipeline can be used to propagate instrumental uncertainties up to the final science products, and could be applied to experimentswith rotating half-wave plates.
Key words.
Cosmology, polarization, systematic e ff ects
1. Introduction
We are now entering an era of precise measurements of the cosmic microwave background (CMB) polarization, with po-tentially enough sensitivity to detect or even characterize the primordial tensorial B modes, the smoking gun of inflation(e.g., Zaldarriaga & Seljak (1997) and references therein). This raises expectations about the control and the correction of con-taminations by astrophysical foregrounds, observational features, and instrumental imperfections. As it has in the past, progresswill come from the synergy between instrumentation and data analysis. Improvements in instrumentation call for improved pre-cision in final results, which are made possible by improved algorithms and the ability to deal with more and more massive datasets. In turn, expertise gained in data processing allows for better simulations that lead to new instrument designs and better suitedobservations. An example of such joint developments is the study of the impact of optics- and electronics-related imperfectionson the measured CMB temperature and polarization angular power spectra and their statistical isotropy. Systematic e ff ects suchas beam non-circularity, response mismatch in dual polarization measurements and scanning strategy imperfections, as well ashow they can be mitigated, have been extensively studied in the preparation of forthcoming instruments (including, but not limitedto Souradeep & Ratra 2001; Fosalba et al. 2002; Hu et al. 2003; Mitra et al. 2004, 2009; O’Dea et al. 2007; Rosset et al. 2007;Shimon et al. 2008; Miller et al. 2009a,b; Hanson et al. 2010; Leahy et al. 2010; Rosset et al. 2010; Ramamonjisoa et al. 2013;Rathaus & Kovetz 2014; Wallis et al. 2014; Pant et al. 2016), and during the analysis of data collected by WMAP (Smith et al.2007; Hinshaw et al. 2007; Page et al. 2007) or Planck (Planck 2013-VII 2014; Planck 2013-XVII 2014; Planck 2015-XI 2016)satellite missions.At the same time, several deconvolution algorithms and codes have been proposed to clean up the CMB maps from suchbeam-related e ff ects prior to the computation of the power spectra, like PreBeam (Armitage-Caplan & Wandelt 2009),
ArtDeco (Keih¨anen & Reinecke 2012), and in Bennett et al. (2013) and Wallis et al. (2015).Finally, in a related e ff ort, the FEBeCoP pipeline, described in Mitra et al. (2011) and used in
Planck data analysis (Planck 2013-IV2014; Planck 2013-VII 2014), can be seen as a convolution facility, by providing, at arbitrary locations on the sky, the e ff ective beammaps and point spread functions of a detector set, which, in turn, can be used for a Monte-Carlo based description of the e ff ectivebeam window functions for a given sky model.In this paper, we introduce the QuickPol pipeline, an extension to polarization of the
Quickbeam algorithm used inPlanck 2013-VII (2014). It allows a quick and accurate computation of the leakage and cross-talk between the various tempera-ture and polarization power spectra (
T T , EE , BB , T E , etc.) taking into account the exact scanning, sample flags, relative weights,
Send o ff print requests to : [email protected] Wilkinson microwave anisotropy probe: http://map.gfsc.nasa.gov . . 1 a r X i v : . [ a s t r o - ph . C O ] N ov . Hivon, S. Mottet & N. Ponthieu: QuickPol and scanning beams of the considered set(s) of detectors. The end results are e ff ective beam matrices describing, for each multipole (cid:96) , the mixing of the various spectra, independently of the actual value of the spectra. As we shall see, the impact of changing anytime-independent feature of the instrument, such as its beam maps, relative gain calibrations, detector orientations, and polarizatione ffi ciencies can be propagated within seconds to the final beam matrices products, allowing extremely fast Monte-Carlo explorationof the experimental features. QuickPol is thus a powerful tool for both real data analysis and forthcoming experiments, simulationsand design.The paper is organized as follows. The mathematical formalism is exposed in Section 2 and analytical results are given inSection 3. The numerical implementation is detailed in Section 4 and compared to the results of
Planck simulations in Section 5.Section 6 shows the propagation of instrumental uncertainties. We discuss briefly the case of rotating half-wave plates in Section 7and conclude in Section 8.
2. Formalism
As usual in the study of polarization measurement, we will use Jones’ formalism to study the evolution of the electric componentof an electro-magnetic radiation in the optical system. Let us consider a quasi monochromatic radiation propagating along the z axis, and hitting the optical system at a position r = (cid:32) xy (cid:33) . The incoming electric field e ( r ) = (cid:32) e x e y (cid:33) e ik ( z − ct ) will be turned into e (cid:48) ( r ) = J ( r ) . e ( r ), where J ( r ) is the 2x2 complex Jones matrix of the system.A rotation of the optical system by α around the z axis can be seen as a rotation of both the orientation and location of theincoming radiation by − α in the detector reference frame, and the same input radiation is now received as e (cid:48) ( α, r ) = J ( r α ) . R † α . e ( r ) , (1)with r α = R † α . r , (2) R α = (cid:32) cos α − sin α sin α cos α (cid:33) , (3)and the † sign representing the adjoint operation, which for a real rotation matrix, simply amounts to the matrix transpose. Themeasured signal is d ( α ) = (cid:90) d r d ( α, r ) (4)with d ( α, r ) = (cid:68) e (cid:48)† . e (cid:48) (cid:69) = (cid:68) Tr (cid:16) e (cid:48) . e (cid:48)† (cid:17)(cid:69) = Tr (cid:16) J ( r α ) . R † α . (cid:68) e . e † (cid:69) . R α . J † ( r α ) (cid:17) . (5)We now introduce the Stokes parameters of the input signal (dropping the dependence on r ) (cid:68) e . e † (cid:69) = (cid:32) T + Q U + iVU − iV T − Q (cid:33) (6)and of the (un-rotated) instrument response J † . J = (cid:32) (cid:101) I + (cid:101) Q (cid:101) U − i (cid:101) V (cid:101) U + i (cid:101) V (cid:101) I − (cid:101) Q (cid:33) , (7)to obtain d ( α ) = (cid:90) d r (cid:104)(cid:101) I ( α, r ) T ( r ) + (cid:101) Q ( α, r ) Q ( r ) + (cid:101) U ( α, r ) U ( r ) − (cid:101) V ( α, r ) V ( r ) (cid:105) . (8)With the rotated instrument response: (cid:101) I ( α, r ) = (cid:101) I ( r α ) , (9a) (cid:101) Q ( α, r ) = (cid:101) Q ( r α ) cos 2 α − (cid:101) U ( r α ) sin 2 α, (9b) (cid:101) U ( α, r ) = (cid:101) Q ( r α ) sin 2 α + (cid:101) U ( r α ) cos 2 α, (9c) (cid:101) V ( α, r ) = (cid:101) V ( r α ) . (9d) Although it is important when trying to disentangle sky signals with di ff erent electromagnetic spectra (Planck 2013-VI 2014), the finitebandwidth of the actual detectors only plays a minor role in the problem considered here, and will be ignored in this paper.2. Hivon, S. Mottet & N. Ponthieu: QuickPol
Following Rosset et al. (2010), we can specify the instrument as being a beam forming optics, followed by an imperfect polarimeterin the direction x , with 0 ≤ η ≤
1, and having an overall optical e ffi ciency 0 ≤ τ ≤ J ( r ) = √ τ (cid:32) √ η (cid:33) (cid:32) b xx ( r ) b xy ( r ) b yx ( r ) b yy ( r ) (cid:33) , (10)with (cid:32) b ∗ ax b ∗ ay (cid:33) . (cid:16) b ax b ay (cid:17) = (cid:32) (cid:101) I a + (cid:101) Q a (cid:101) U a − i (cid:101) V a (cid:101) U a + i (cid:101) V a (cid:101) I a − (cid:101) Q a (cid:33) (11)for a = x , y . The Stokes parameters of the instrument are then (cid:101) S = τ ( (cid:101) S x + η (cid:101) S y ) for (cid:101) S = (cid:101) I , (cid:101) Q , (cid:101) U , (cid:101) V .If the beam is assumed to be perfectly co-polarized, that is, it does not alter at all the polarization of the incoming radiation, with b xy = b yx = b xx = b yy , then (cid:101) U x = (cid:101) U y = (cid:101) V x = (cid:101) V y = (cid:101) I x = (cid:101) I y = (cid:101) Q x = − (cid:101) Q y , and (cid:101) I = (1 + η ) (cid:101) I x , (cid:101) Q = (1 − η ) (cid:101) Q x , (cid:101) U = (cid:101) V = d ( α ) = + η τ (cid:90) d r (cid:101) I x ( r α ) (cid:2) T ( r ) + ρ ( Q ( r ) cos 2 α + U ( r ) sin 2 α ) (cid:3) , (12)where ρ = − η + η (13)is the polar e ffi ciency, such that 0 ≤ ρ ≤ ρ = ρ = Planck high frequency instrument (HFI), Rosset et al. (2010) showed the measured polarization e ffi ciencies to di ff erby ∆ ρ (cid:48) =
1% to 16% from their ideal values, with an absolute statistical uncertainty generally below 1%. The particular case ofco-polarized beams is important because in most experimental setups, such as
Planck , the beam response calibration is done onastronomical or artificial far field sources. Well known, compact, and polarized sources are generally not available to measure (cid:101) Q and (cid:101) U and only the intensity beam response (cid:101) I is measured. In the absence of reliable physical optics modeling of the beam response,one therefore has to assume (cid:101) Q and (cid:101) U to be perfectly co-polarized.So far, we have only considered the optical beam response. We should also take into account the scanning beam, which is theconvolution of the optical beam with the finite time response of the instrument (or its imperfect correction) as it moves aroundthe sky, as described in Planck 2013-VII (2014) and Planck 2015-VII (2016). These time related e ff ects can be a major source ofelongation of the scanning beams, and can increase the beam mismatch among sibling detectors. If one assumes the motion of thedetectors on the sky to be nearly uniform, as was the case for Planck , then optical beams can readily be replaced by scanning beamsin the
QuickPol formalism.
We now define the tools that are required to extend the above results to the full celestial sphere. The temperature T is a scalarquantity, while the linear polarization Q ± iU is of spin ±
2, and the circular polarization V is generally assumed to vanish. They canbe written as linear combinations of spherical harmonics (SH): T ( r ) = (cid:88) (cid:96) m a T (cid:96) m Y (cid:96) m ( r ) , (14) Q ( r ) ± iU ( r ) = (cid:88) (cid:96) m ± a (cid:96) m ± Y (cid:96) m ( r ) , (15)although one usually prefers the scalar and fixed parity E and B components a E (cid:96) m ± ia B (cid:96) m = − ± a (cid:96) m (16)such that a X ∗ (cid:96) m = ( − m a X (cid:96) − m for X = T , E , B . In other terms a (cid:96) m a (cid:96) m − a (cid:96) m = R . a T (cid:96) m a E (cid:96) m a B (cid:96) m (17)with R = − − i − i . (18)The sign convention used in Eq. (16) is consistent with Zaldarriaga & Seljak (1997) and the HEALPix library (G´orski et al. 2005). http://healpix.sourceforge.net . 3. Hivon, S. Mottet & N. Ponthieu: QuickPol
The response of a beam centered on the North pole can also be decomposed in SH coe ffi cients b (cid:96) m = (cid:90) d r (cid:101) I ( r ) Y ∗ (cid:96) m ( r ) , (19) ± b (cid:96) m = (cid:90) d r (cid:16) (cid:101) Q ( r ) ± i (cid:101) U ( r ) (cid:17) ± Y ∗ (cid:96) m ( r ) , (20)while the coe ffi cients of a rotated beam can be computed by noting that under a rotation of angle α around the direction r , the SHof spin s transform as s Y (cid:96) m ( r (cid:48) ) −→ (cid:88) m (cid:48) s Y (cid:96) m (cid:48) ( r (cid:48) ) D (cid:96) m (cid:48) m ( r , α ) . (21)The elements of Wigner rotation matrices D are related to the SH via (Challinor et al. 2000) D (cid:96) m (cid:48) m ( r , α ) = ( − m q (cid:96) − m Y ∗ (cid:96) m (cid:48) ( r ) e − im α , (22)with q (cid:96) = (cid:113) π (cid:96) + .If the beam is assumed to be co-polarized and coupled with a perfect polarimeter rotated by an angle γ , such that (cid:101) Q + i (cid:101) U = (cid:101) Ie i γ in cartesian coordinates (or (cid:101) Q + i (cid:101) U = (cid:101) Ie i ( γ − φ ) in ( θ, φ ) polar coordinates), simple relations between b (cid:96) m and ± b (cid:96), m can beestablished. For a Gaussian circular beam of full width half maximum (FWHM) θ FWHM = σ √ ≈ . σ and of throughput (cid:82) d r (cid:101) I ( r ) = √ π b = , Challinor et al. (2000) found b (cid:96) m = (cid:114) (cid:96) + π e − (cid:96) ( (cid:96) + σ δ m , , (23a) ± b (cid:96), m = b (cid:96), m ± e σ e ± i γ . (23b)The factor c = e σ in Eq. (23b) is such that c − < . − for θ FWHM ≤ ◦ and c − < . − for θ FWHM ≤ (cid:48) , and will beassumed to be c = ± b (cid:96), m = b (cid:96), m ± e ± i γ , (24)while we show in Appendix G that Eq. (24) is true for arbitrarily shaped co-polarized beams. This result can also be obtained bynoting that an arbitrary beam is the sum of Gaussian circular beams with di ff erent FWHM and center (Tristram et al. 2004), each ofthem obeying Eq. (23b).The detector associated to a beam is an imperfect polarimeter with a polarization e ffi ciency ρ (cid:48) and the overall polarized responseof the detector, in a referential aligned with its direction of polarization (the so-called Pxx coordinates in Planck parlance), reads (cid:101) Q = ρ (cid:48) (cid:101) I , (25)so that ± b (cid:96), m = ρ (cid:48) b (cid:96), m ± . (26)We introduced ρ (cid:48) to distinguish it from the ρ value used in the map-making, as described below. A polarized detector pointing, at time t , in the direction r t on the sky, and being sensitive to the polarization with angle α t withrespect to the local meridian, measures d ( r t , α t ) = (cid:90) d r (cid:48) (cid:104)(cid:101) I ( r t , α t ; r (cid:48) ) T ( r (cid:48) ) + (cid:101) Q ( r t , α t ; r (cid:48) ) Q ( r (cid:48) ) + (cid:101) U ( r t , α t ; r (cid:48) ) U ( r (cid:48) ) (cid:105) . (27)The factor 1 / V to vanish. With the definitionsintroduced in Section 2.2, this becomes d ( r t , α t ) = (cid:88) (cid:96) ms (cid:104) a (cid:96) m b ∗ (cid:96) s + / (cid:16) a (cid:96) m b ∗ (cid:96) s + − a (cid:96) m − b ∗ (cid:96) s (cid:17)(cid:105) ( − s q (cid:96) e is α t − s Y (cid:96) m ( r t ) . (28)The map-making formalism is set ignoring the beam e ff ects, assuming a perfectly co-polarized detector and an instrumentalnoise n (Tristram et al. 2011, and references therein), so that, for a detector j , Eq. (12) becomes d j ( t ) = T ( p ) + ρ j Q ( p ) cos 2 α ( j ) t + ρ j U ( p ) sin 2 α ( j ) t + n j ( t ) , (29)
4. Hivon, S. Mottet & N. Ponthieu:
QuickPol where the leading prefactors are here again absorbed in the gain calibration. Let us rewrite it as d j ( t ) = A ( j ) t , p m ( p ) + n j ( t ) , (30)with (Shimon et al. 2008) A ( j ) t , p = (cid:18) , ρ j e − i α ( j ) t , ρ j e i α ( j ) t (cid:19) , (31) m ( p ) = ( T , P / , P ∗ / T , (32)and P = Q + iU . Assuming the noise to be uncorrelated between detectors, with covariance matrix N j = (cid:68) n j . n Tj (cid:69) for detector j , thegeneralized least square solution of Eq. (29) for a set of detectors is (cid:101) m = (cid:88) k A ( k ) † . N − k . A ( k ) − . (cid:88) j A ( j ) † . N − j . d j . (33)Let us now replace the ideal data stream (Eq. 29) with the one obtained for arbitrary beams (Eq. 27) and further assume that thenoise is white and stationary with variance σ j , so that N − j = /σ j = w j . Let us also introduce the binary flag f j , t used to rejectindividual time samples from the map-making process; Eq. (33) then becomes (cid:101) m ( p ) ≡ (cid:101) m (0; p ) (cid:101) m (2; p ) / (cid:101) m ( − p ) / , (34) = (cid:88) k (cid:88) t ∈ p A ( k ) † p , t w k f k , t A ( k ) t , p − (cid:88) j (cid:88) t ∈ p A ( j ) † p , t w j f j , t d j , t . (35)We have assumed here the pixels to be infinitely small, so that, starting with Eq. (28), the location of all samples in a pixel coincideswith the pixel center. The e ff ect of the pixel’s finite size and the so-called sub-pixel e ff ects will be considered in Section 3.5. To compute the cross-power spectrum of any two spin v and v maps, we first project each polarized component v of (cid:101) m ( p ) on theappropriate spin weighted sets of spherical harmonics, x (cid:101) m (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( v ) = (cid:90) d r (cid:101) m ( v ; r ) x Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( r ) , (36)and average these terms according to (cid:101) C v v (cid:96) (cid:48)(cid:48) ≡ (cid:96) (cid:48)(cid:48) + (cid:88) m (cid:48)(cid:48) (cid:68) v (cid:101) m (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( v ) v (cid:101) m ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( v ) (cid:69) , (37) = (cid:88) u u j j (cid:96) s s ( − s + s + v + v C u u (cid:96) (cid:96) + π u ˆ b ( j ) ∗ (cid:96) s u ˆ b ( j ) (cid:96) s × k u k u k v k v (cid:88) (cid:96) (cid:48) m (cid:48) ρ j , v ρ j , v s + v (cid:101) ω ( j ) (cid:96) (cid:48) m (cid:48) s + v (cid:101) ω ( j ) ∗ (cid:96) (cid:48) m (cid:48) (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − s s + v − v (cid:33) (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − s s + v − v (cid:33) , (38)where Eq. (C.6) was used. The detailed derivation of this relation and its associated terms is given in Appendix A. Su ffi ce itto say here that k u terms are either 1 or 1 / u ˆ b ( j ) (cid:96) s terms are inverse noise-weighted beam multipoles, and (cid:101) ω ( j ) terms are e ff ectiveweights describing the scanning and depending on the direction of polarization, hit redundancy (both from sky coverage and flaggedsamples), and noise level of detector j .Equation (38) is therefore a generalization to non-circular beams of the pseudo-power spectra measured on a masked or weightedmap (Hivon et al. 2002; Hansen & G´orski 2003), and extends to polarization the Quickbeam non-circular beam formalism used inthe data analysis conducted by Planck 2013-VII (2014). It also formally agrees with Hu et al. (2003)’s results on the impact ofsystematic e ff ects on the polarization power spectra, with the functions u ˆ b ( j ) ∗ (cid:96) s ρ j , v s + v (cid:101) ω ( j ) (cid:96) (cid:48) m (cid:48) absorbing the systematic e ff ect parametersrelative to detector j . In the next sections, we present the numerical results implied by this result and compare them on full-fledgedPlanck-HFI simulations.
3. Results
We now apply the
QuickPol formalism to configurations representative of current or forthcoming CMB experiments, and to acouple of idealized test cases for which the expected result is already known, as a sanity check. The e ff ect of the finite pixel size isalso studied.
5. Hivon, S. Mottet & N. Ponthieu:
QuickPol -1.0 1.0 h cos 2 α i -1.0 1.0 h sin 2 α i Fig. 1.
Orientation of polarization measurements in
Planck . The two left panels show, for an actual
Planck detector, the mapsof (cid:104) cos 2 α (cid:105) and (cid:104) sin 2 α (cid:105) respectively, where α is the direction of the polarizer with respect to the local Galactic meridian, whichcontributes to the spin 2 term ω ( j )2 defined in Eq. (A.3). The right panel shows the power spectrum C (cid:96) of (cid:68) e i α (cid:69) = ω ( j )2 /ω ( j )0 ,multiplied by (cid:96) ( (cid:96) + / π . -1.0 1.0 LB: h cos 2 α i -1.0 1.0 LB: h sin 2 α i Fig. 2.
Same as Fig. 1 for an hypothetical detector of a LiteBIRD-like mission, except for the right panel plot which has a di ff erent y -range. ‘ W X Y , TT ‘ → TTTT → TETT → TBTT → ETTT → EETT → EBTT → BTTT → BETT → BB ‘ x W X Y , TT ‘ / W TT , TT ‘ → TETT → TBTT → ETTT → EETT → EBTT → BTTT → BETT → BB Fig. 3. E ff ective beam window matrix W XY , TT (cid:96) introduced in Eq. (41) and detailed in Eq. (E.8a), for the cross-spectra of twosimulated Planck maps discussed in Section 5. Left panel: raw elements of W XY , TT (cid:96) , showing for each (cid:96) how the measured XY mapangular power spectrum is impacted by the input T T spectrum, because of the observation of the sky with the beams. Right panel:blown-up ratio of the non-diagonal elements to the diagonal ones: 100 W XY , TT (cid:96) / W TT , TT .(cid:96) To begin with, let us consider the scanning strategy of
Planck and of another satellite mission optimized for the measurement ofCMB polarization.Figure 1 illustrates the orientation of the polarization measurements achieved in
Planck . It shows, for an actual
Planck detector,the maps of (cid:104) cos 2 α (cid:105) and (cid:104) sin 2 α (cid:105) respectively, where α is the direction of the polarizer with respect to the local Galactic meridian.These quantities contribute to the spin 2 term ω ( j )2 defined in Eq. (A.3). The large amplitude of these two maps is consistent with thefact that for a given detector, the orientation of the polarization measurements is mostly α and − α , as expected when detectors move
6. Hivon, S. Mottet & N. Ponthieu:
QuickPol on almost great circles with very little precession. Another striking feature is the relative smoothness of the maps, which translateinto the power spectrum C (cid:96) of (cid:68) e i α (cid:69) = ω ( j )2 /ω ( j )0 peaking at low (cid:96) values.Figure 2 shows the same information for an hypothetical LiteBIRD like detector (but without half-wave plate modulation) inwhich we assumed the detector to cover a circle of 45 ◦ in radius in one minute, with its spin axis precessing with a period of fourdays at 50 ◦ from the anti-sun direction. As expected for such a scanning strategy, the values of α are pretty uniformly distributedover the range [0 , π ], which translates into a low amplitude of the (cid:104) cos 2 α (cid:105) and (cid:104) sin 2 α (cid:105) maps. Even if those maps do not look assmooth as those of Planck , their power spectra peak at fairly low multipole values.
If one assumes that ω s ( p ) and (cid:101) ω s ( p ) vary slowly across the sky, as we just saw in the case of Planck and LiteBIRD - and probablya wider class of orbital and sub-orbital missions - then s (cid:101) ω (cid:96) (cid:48) m (cid:48) is dominated by low (cid:96) (cid:48) values and one expects (cid:96) (cid:39) (cid:96) (cid:48)(cid:48) because of thetriangle relation imposed by the 3J symbols (see Appendix C). If one further assumes C (cid:96) and b (cid:96) to vary slowly in (cid:96) , then Eqs. (C.5)and (C.9) can be used to impose s + v = s + v = s in Eq. (38) and provide (cid:101) C v v (cid:96) = (cid:88) u u C u u (cid:96) k u k u k v k v (cid:88) j j (cid:88) s u ˆ b ( j ) ∗ (cid:96), s − v u ˆ b ( j ) (cid:96), s − v (cid:101) Ω ( j j ) v , v , s , (39)with (cid:101) Ω ( j j ) v , v , s ≡ ρ j , v ρ j , v π (cid:88) (cid:96) (cid:48) m (cid:48) s (cid:101) ω ( j ) (cid:96) (cid:48) m (cid:48) [ v ] s (cid:101) ω ( j ) ∗ (cid:96) (cid:48) m (cid:48) [ v ] , (40a) = ρ j , v ρ j , v N pix (cid:88) p (cid:101) ω ( j ) s [ v ]( p ) (cid:101) ω ( j ) ∗ s [ v ]( p ) , (40b) = (cid:101) Ω ( j j ) ∗− v , − v , − s . (40c)As derived in Appendix E.1, Eq. (39) reduces to a mixing equation relating the observed cross-power spectra to the true ones: (cid:101) C XY (cid:96) = (cid:88) X (cid:48) Y (cid:48) W XY , X (cid:48) Y (cid:48) (cid:96) C X (cid:48) Y (cid:48) (cid:96) (41)with X , Y , X (cid:48) , Y (cid:48) ∈ { T , E , B } .In the smooth scanning case representative of past and forthcoming satellite missions, the e ff ect of observing the sky with non-ideal beams is therefore to couple the temperature and polarization power spectra C X (cid:48) Y (cid:48) (cid:96) at the same multipole (cid:96) through an extendedbeam window matrix W XY , X (cid:48) Y (cid:48) (cid:96) , as illustrated on Fig. 3. If the scanning beams are now assumed to all be circular and identical, the measured (cid:101) C ( (cid:96) ) will not depend on the details of thescanning strategy, orientation of the detectors, or relative weights of the detectors. We are indeed exactly in the ideal hypotheses ofthe map making formalism (Eq. 29) and get the well known and simple result that the e ff ect of the beam can be factored out.If one considers detectors with identical circular copolarized beams, and whose actual polarization e ffi ciency was used duringthe map making: ρ j = ρ (cid:48) j , such that u ˆ b ( j ) (cid:96), s ≡ w j q (cid:96) u b ( j ) (cid:96), s = w j q (cid:96) ρ j , u b (cid:96) δ s , − u , (42)then Eqs. (39) and (40b) feature terms like (cid:80) j u ˆ b ( j ) ∗ (cid:96), s − v ρ j , v (cid:101) ω ( j ) s [ v ], which when written in a matrix form, verify the equality q (cid:96) b (cid:96) (cid:88) j w j (cid:101) ω ( j )0 [0] ρ j (cid:101) ω ( j ) − [0] ρ j (cid:101) ω ( j )2 [0] ρ j (cid:101) ω ( j )2 [2] ρ j (cid:101) ω ( j )0 [2] ρ j (cid:101) ω ( j )4 [2] ρ j (cid:101) ω ( j ) − [ − ρ j (cid:101) ω ( j ) − [ − ρ j (cid:101) ω ( j )0 [ − = q (cid:96) b (cid:96) , (43)according to Eq. (B.9). The measured power spectra are then (cid:101) C XY (cid:96) = ˆ b (cid:96) C XY (cid:96) = q (cid:96) b (cid:96) C XY (cid:96) , (44)and (cid:101) C XY (cid:96) = exp (cid:16) − (cid:96) ( (cid:96) + σ (cid:17) C XY (cid:96) for the Gaussian circular beam introduced in Eq. (23a). Obviously, these very simple resultsassume that the whole sky is observed. If not, the cut-sky induced (cid:96) − (cid:96) and E − B coupling e ff ects mentioned at the end of Section2.4 have to be accounted for, as described, for example, in Chon et al. (2004), Mitra et al. (2009), Grain et al. (2009), and referencestherein. http://litebird.jp/eng/ . 7. Hivon, S. Mottet & N. Ponthieu: QuickPol
Let us now consider the case of an ideal scanning of the sky, for which in any pixel p , the number of valid (unflagged) samples isthe same for all detectors h j ( p ) = h ( p ), and each detector j covers uniformly all possible orientations within that pixel along theduration of the mission. This constitutes the ideal limit aimed at by the scanning strategy illustrated in Fig. 2. The assumption ofsmooth scanning is then perfectly valid, and details of the calculations can be found in Appendix E.2. We find for instance that thematrix describing how the measured temperature and polarization power spectra are a ff ected by the input T T spectrum reads W XY , TT (cid:96) ≡ W TT , TT (cid:96) W EE , TT (cid:96) W BB , TT (cid:96) W TE , TT (cid:96) W T B , TT (cid:96) W EB , TT (cid:96) W ET , TT (cid:96) W BT , TT (cid:96) W BE , TT (cid:96) = (cid:88) j j ˆ b ( j ) (cid:96), ˆ b ( j ) ∗ (cid:96), ξ (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ − (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) ˆ b ( j ) ∗ (cid:96), ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) ˆ b ( j ) ∗ (cid:96), ρ j ξ i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ − ˆ b ( j ) (cid:96), (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ i ˆ b ( j ) (cid:96), (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ , (45)with the normalization factors ξ − = (cid:88) k k w k w k , ξ − = (cid:88) k k w k w k ρ k , ξ − = (cid:88) k k w k w k ρ k , ξ − = (cid:88) k k w k w k ρ k ρ k . (46)This confirms that in this ideal case, as expected and discussed previously (e.g., Wallis et al. 2014, and references therein), theleakage from temperature to polarization (Eq. 45) is driven by the beam ellipticity (ˆ b ( j ) l , ± terms) which has the same spin ± E and B spectra by T are swapped (e.g., W EE , TT (cid:96) ←→ W BB , TT (cid:96) ) when thebeams are rotated with respect to the polarimeter direction by 45 ◦ (ˆ b (cid:96), ± −→ ± i ˆ b (cid:96), ± ), as shown in Shimon et al. (2008). e I e Q e U e I − e Q e I e Q e U e I − e Q -10 -10 -10 -10 -10 -10 -1 0 1 10 10 Fig. 4.
Computer simulated beam maps ( (cid:101) I , (cid:101) Q , (cid:101) I − (cid:101) Q and (cid:101) U clockwise from top-left) for two of the Planck -HFI detectors (100-1aand 217-5a) used in the validation of
QuickPol . Each panel is 1 ◦ x1 ◦ in size, and the units are arbitrary.
8. Hivon, S. Mottet & N. Ponthieu:
QuickPol
As shown in Planck 2013-VII (2014), in the case of temperature fluctuations, the e ff ect of the finite pixel size is twofold. First, ineach pixel, the distance between the nominal pixel center and the center of mass of the observations couples to the local gradient ofthe Stokes parameters to induce noise terms. Second, there is a smearing e ff ect due to the integration of the signal over the surfaceof the pixel. Equation (41) then becomes (cid:101) C XY (cid:96) = W pix (cid:96) (cid:88) X (cid:48) Y (cid:48) W XY , X (cid:48) Y (cid:48) (cid:96) C X (cid:48) Y (cid:48) (cid:96) + N XY (cid:96) (47)with W pix (cid:96) = − (cid:96) ( (cid:96) + σ / + O (cid:16) ( σ(cid:96) ) (cid:17) , and σ = (cid:68) d r (cid:69) the squared displacement averaged over the hits in the pixels and overthe set of considered pixels. As shown in Appendix F, the additive noise term, sourced by the temperature gradient within the pixel,a ff ects both temperature and polarization measurements, with N EE (cid:96) = N BB (cid:96) and (cid:12)(cid:12)(cid:12) N EE (cid:96) (cid:12)(cid:12)(cid:12) < ∼ (cid:12)(cid:12)(cid:12) N TT (cid:96) (cid:12)(cid:12)(cid:12) , while the other spectra are much lessa ff ected, that is, (cid:12)(cid:12)(cid:12) N TE (cid:96) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) N T B (cid:96) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) N EB (cid:96) (cid:12)(cid:12)(cid:12) (cid:28) (cid:12)(cid:12)(cid:12) N TT (cid:96) (cid:12)(cid:12)(cid:12) . The sign of this noise term is arbitrary and can be negative when cross-correlatingmaps with a di ff erent sampling of the pixels.
4. Numerical implementation
Numerical implementations of this formalism are performed in three steps, assuming that the individual beam b ( j ) (cid:96) s is already com-puted for 0 ≤ s ≤ s max + ≤ (cid:96) ≤ (cid:96) max :1. For each involved detector j , and for 0 ≤ s ≤ s max , one computes the s -th complex moment of its direction of polarizationin pixel p : ω ( j ) s ( p ) defined in Eq. (A.3). Since this requires processing the whole scanning data stream, this step can be timeconsuming. However it has to be computed only once for all cases, independently of the choices made elsewhere on the beammodels, calibrations, noise weighting, and other factors. As we shall see below, it may not even be necessary to compute it, orstore it, for every sky pixel.2. The ω ( j ) s ( p ) computed above are weighted with the assumed inverse noise variance weights w j and polar e ffi ciencies ρ j to buildthe hit matrix H in each pixel, which is then inverted to compute the (cid:101) ω ( j ) s ( p ), defined in Eq. (A.6). Those are then multipliedtogether to build the scanning information matrix (cid:101) Ω using its pixel space definition (Eq. 40b). The resulting complex matrixcontains 9 n n (2 s max +
1) elements, where n and n are the number of detectors in each of the two detector assemblies whosecross-spectra are considered. This step can be parallelized to a large extent, and can be dramatically sped up by building thismatrix out of a representative subset of pixels. In our comparison to simulations, described in Section 5, and performed onHEALPix map with n side = N pix = n =
50 10 pixels, we checked that using only N pix /
64 pixels evenly spreadon the sky gave final results almost identical to those of the full calculations.3. Finally, using Eqs. (E.1-41) we note that W XY , X (cid:48) Y (cid:48) (cid:96) = ∂ (cid:101) C XY (cid:96) /∂ C X (cid:48) Y (cid:48) (cid:96) , so that, for instance, for a given (cid:96) , the 3x3 W XY , TE (cid:96) matrix iscomputed by replacing in Eq. (E.1) its central term C (cid:96) with its partial derivative, such as ∂∂ C TE (cid:96) C (cid:96) = R . ∂∂ C TE (cid:96) C TT (cid:96) C TE (cid:96) C T B (cid:96) C ET (cid:96) C EE (cid:96) C EB (cid:96) C BT (cid:96) C BE (cid:96) C BB (cid:96) . R † (48) = R . . R † , (49)where we assumed in Eq. (49) that, on the sky, C TE (cid:96) = C ET (cid:96) and generally C X (cid:48) Y (cid:48) (cid:96) = C Y (cid:48) X (cid:48) (cid:96) , like for CMB anisotropies.On the other hand, when dealing with arbitrary foregrounds cross-frequency spectra, we would have to assume C X (cid:48) Y (cid:48) (cid:96) (cid:44) C Y (cid:48) X (cid:48) (cid:96) when X (cid:48) (cid:44) Y (cid:48) , and compute W XY , X (cid:48) Y (cid:48) (cid:96) and W XY , Y (cid:48) X (cid:48) (cid:96) separately. As we shall see in Section 6, this final and fastest step is the onlyone that needs to be repeated in a Monte-Carlo analysis of instrumental errors, and it can be sped up. Indeed, since the input b (cid:96) m and output W (cid:96) are generally very smooth functions of (cid:96) , it is not necessary to do this calculation for every single (cid:96) , but rather fora sparse subset of them, for instance regularly interspaced by δ(cid:96) . The resulting W (cid:96) matrix is then B-spline interpolated. In ourtest cases with θ FWHM =
10 to 5 (cid:48) , using δ(cid:96) =
10 leads to relative errors on the final product below 10 − for each (cid:96) .In our tests, with s max = (cid:96) max = n = n = , and all proposed speed-ups in place, Step 2 took about ten minutes, dominatedby IO, while Step 3 took less than a minute on one core of a 3GHz Intel Xeon CPU. The final product is a set of six (or nine) realmatrices W XY , X (cid:48) Y (cid:48) (cid:96) , each with 9( (cid:96) max +
1) elements.
5. Comparison to
Planck -HFI simulations
The di ff erential nature of the polarization measurements, in the absence of modulating devices such as rotating half-wave plates,means that any mismatch between the responses of the two (or more) detectors being used will leak a fraction of temperature intopolarization. This was observed in Planck , even though pairs of polarized orthogonal detectors observed the sky through the samehorn, therefore with almost identical optical beams. Optical mismatches within pairs of detectors were enhanced by residuals ofthe electronic time response deconvolution which could a ff ect their respective scanning beams di ff erently (Planck 2013-IV 2014;
9. Hivon, S. Mottet & N. Ponthieu:
QuickPol ‘ ( ‘ + ) ∆ C ‘ / π [ µ K ] ∆ TT TT / 5000 ∆ EE EE / 100 ∆ BB BB / 10Sim: GRASP fullSim: GRASP copolarQP: GRASP copolar ‘ ( ‘ + ) ∆ C ‘ / π [ µ K ] ∆ TE TE / 50 ∆ TB ∆ EB
500 1000 1500
Multipole ‘ ‘ ( ‘ + ) ∆ C ‘ / π [ µ K ] ∆ ET TE / 10
500 1000 1500
Multipole ‘ ∆ BT
500 1000 1500
Multipole ‘ ∆ BE100ds1x217ds1 ‘ ( ‘ + ) ∆ C ‘ / π [ µ K ] ∆ TT TT / 5000 ∆ EE EE / 100 ∆ BB BB / 10Sim: GRASP fullSim: GRASP copolarQP: GRASP copolar ‘ ( ‘ + ) ∆ C ‘ / π [ µ K ] ∆ TE TE / 50 ∆ TB ∆ EB
500 1000 1500
Multipole ‘ ‘ ( ‘ + ) ∆ C ‘ / π [ µ K ] ∆ ET TE / 10
500 1000 1500
Multipole ‘ ∆ BT
500 1000 1500
Multipole ‘ ∆ BE143ds1x217ds1
Fig. 5.
Comparison to simulations for 100ds1x217ds1 (lhs panels) and 143ds1x217ds1 (rhs panels) cross power spectra, for com-puter simulated beams. In each panel is shown the discrepancy between the actual (cid:96) ( (cid:96) + C (cid:96) / π and the one in input, smoothedon ∆ (cid:96) =
31. Results obtained on simulations with either the full beam model (green curves) or the co-polarized beam model (bluedashes) are to be compared to
QuickPol analytical results (red long dashes). In panels where it does not vanish, a small fraction ofthe input power spectrum is also shown as black dots for comparison.Planck 2013-VII 2014). Other sources of mismatch included their di ff erent noise levels and thus their respective statistical weighton the maps, which could reach relative di ff erences of up to 80%, and the number of valid samples which could vary by up to20% between detectors. As seen previously, these detector-specific features can be included in the QuickPol pipeline in order todescribe as closely as possible the actual instrument. In this section, we show how we actually did it and how
QuickPol comparesto full-fledged simulations of
Planck -HFI observations.Noiseless simulations of
Planck -HFI observations of a pure CMB sky were run for quadruplets of polarized detectors at threedi ff erent frequencies (100, 143, and 217GHz), and identified as 100ds1, 143ds1, and 217ds1 respectively. The input CMB powerspectrum C XY (cid:96) was assumed to contain no primordial tensorial modes, with the traditional C T B (cid:96) = C EB (cid:96) = C XY (cid:96) = C YX (cid:96) . The samemission duration, pointing, polarization orientations ( γ j ) and e ffi ciencies ( ρ j ), flagged samples, and discarded pointing periods ( f j )were used as in the actual observations, with computer simulated polarized optical beams for the relevant detectors produced withthe GRASP physical optics code (Rosset et al. 2007, and references therein) as illustrated on Fig. 4. Data streams were generatedwith the LevelS simulation pipeline (Reinecke et al. 2006), using the
Conviqt code (Pr´ezeau & Reinecke 2010) to perform theconvolution of the sky with the beams, including the b (cid:96) s for | s | ≤ s max =
14 and (cid:96) ≤ (cid:96) max = Polkapix destriping code (Tristram et al. 2011), assuming the same noise-based relative weights ( w j ) asthe actual data, and their cross spectra were computed over the whole sky with HEALPix anafast routine to produce the empiricalpower spectra ˆ C XY (cid:96) .The same exercise was reproduced replacing the initial (cid:101) I , (cid:101) Q , (cid:101) U beam maps with a purely co-polarized beam based on the same (cid:101) I , inorder to test the validity of the co-polarized assumption in Planck .Figure 5 shows how the empirical power spectra are di ff erent from the input ones, ∆ ˆ C XY (cid:96) = ˆ C XY (cid:96) W pix (cid:96) W XY , XY (cid:96) − C XY (cid:96) , (50)after correction from the pixel and (scalar) beam window functions, and compares those to the QuickPol predictions ∆ (cid:101) C XY (cid:96) = W pix (cid:96) (cid:88) X (cid:48) Y (cid:48) W XY , X (cid:48) Y (cid:48) (cid:96) C X (cid:48) Y (cid:48) (cid:96) W pix (cid:96) W XY , XY (cid:96) − C XY (cid:96) , (51)for all nine possible values of XY for the cross-spectra of detector sets 100ds1x217ds1 and 143ds1x217ds1. The results are actuallymultiplied by the usual (cid:96) ( (cid:96) + / π factor, and smoothed on ∆ (cid:96) =
31. The empirical results are shown both for the full-fledgedbeam model (green curves) and the purely co-polarized model (blue dashes). One sees that the change, mostly visible in the EE case, is very small, validating the co-polarized beam assumption, at least within the limits of this computer simulated Planck optics. The
QuickPol predictions, only shown in the co-polarized case for clarity (long red dashes), agree extremely well with thecorresponding numerical simulations. We have checked that this agreement to simulations remains true in the full beam model. TICRA: .10. Hivon, S. Mottet & N. Ponthieu:
QuickPol
6. Propagation of instrumental uncertainties
We assumed so far the instrument to be non-ideal, but exactly known. In practice, however, the instrument is only known withlimited accuracy and the final beam matrix will be a ff ected by at least four types of uncertainties: – limited knowledge of the beam angular response, which a ff ects the b ( j ) (cid:96) m , replacing them with b (cid:48) ( j ) (cid:96) m while preserving the beam totalthroughput after calibration (see below) b (cid:48) ( j )00 = b ( j )00 . We therefore assume the beam power spectrum W (cid:96) = (cid:80) m | b (cid:96) m | / (2 (cid:96) +
1) tobe the same at (cid:96) =
0, where the beam throughput is defined, and at (cid:96) =
1, where the detector gain calibration is usually doneusing the CMB dipole. – error on the gain calibration of detector j , which translates into b ( j ) (cid:96) m −→ (1 + δ c j ) b ( j ) (cid:96) m , with | δ c j | (cid:28) – error on the polar e ffi ciency of detector j , which translates into ρ (cid:48) j = ρ j (1 + δρ j /ρ j ). As discussed in Section 2.1, we expect inthe case of Planck -HFI a relative uncertainty | δρ j /ρ j | < – error on the actual direction of polarization: for each detector j , the direction of polarization measured in a common referentialbecomes γ j −→ γ j + δγ j . In the case of Planck -HFI, Rosset et al. (2010) found the pre-flight measurement of this angle tobe dominated by systematic errors of the order of 1 ◦ for polarization sensitive bolometers (PSBs). These uncertainties can belarger for spider web bolometers (SWBs), but as we shall see below, the coupling with the low polarization e ffi ciency ρ j of thosedetectors makes them somewhat irrelevant.All these uncertainties can be inserted in Eq. (E.1) by substituting Eq. (E.4)ˆ B ( j ) (cid:96), s = ˆ b ( j ) (cid:96), s ˆ b ( j ) (cid:96), s − ˆ b ( j ) (cid:96), s + ρ j ˆ b ( j ) (cid:96), s + ρ j ˆ b ( j ) (cid:96), s ρ j ˆ b ( j ) (cid:96), s + ρ j ˆ b ( j ) (cid:96), s − ρ j ˆ b ( j ) (cid:96), s − ρ j ˆ b ( j ) (cid:96), s , with ˆ B (cid:48) ( j ) (cid:96), s = (1 + δ c j ) ρ j x j
00 0 ρ j x ∗ j . ˆ b (cid:48) ( j ) (cid:96), s ˆ b (cid:48) ( j ) (cid:96), s − ˆ b (cid:48) ( j ) (cid:96), s + ˆ b (cid:48) ( j ) (cid:96), s + ˆ b (cid:48) ( j ) (cid:96), s ˆ b (cid:48) ( j ) (cid:96), s + ˆ b (cid:48) ( j ) (cid:96), s − ˆ b (cid:48) ( j ) (cid:96), s − ˆ b (cid:48) ( j ) (cid:96), s , (52)where x j = (1 + δρ j /ρ j ) e i δγ j .As mentioned in Section 3, such substitutions are done in Step 3 of the QuickPol pipeline. A new set of numerical values forthe instrument model can therefore be turned rapidly into a beam window matrix (Eq. 41), allowing, for instance, a Monte-Carloexploration at the power spectrum level of the instrumental uncertainties.
7. About rotating half-wave plates
In the previous sections we have focused on experiments that rely on the rotation of the full instrument with respect to the skyto have the angular redundancy required to measure the Stokes parameters. An alternative way is to rotate the incoming polar-ization at the entrance of the instrument while leaving the rest fixed. This is most conveniently achieved with a rotating half-wave plate (rHWP). The rotation is either stepped (Polarbear Collaboration: Ade et al. 2014) or continuous (Chapman et al. 2014;Essinger-Hileman et al. 2016; Ritacco et al. 2016a). The advantages of this system are numerous, the first of which is the decouplingbetween the optimization of the scanning strategy in terms of “pure” redundancy and its optimization in terms of “angular” redun-dancy. It is much easier to control the rotation of a rHWP than of a full instrument and therefore ensure an optimal angular coveragewhatever the observation scene is. If the rotation is continuous and fast, typically of the order of 1 Hz, it has the extra advantageof modulating polarization at frequencies larger than the atmospheric and electronics 1 / f noise knee frequency, hence ensuring anatural rejection of these low frequency noises. Furthermore, this allows us to build I , Q , and U maps per detector, without needingto combine di ff erent detectors with their associated bandpass mismatch or other di ff erential systematic e ff ects mentioned in theprevious sections. Individual detector systematics therefore tend to average out rather than combine to induce leakage between skycomponents. On the down side, this comes at the price of moving a piece of hardware in the instrument and all its associated system-atic e ff ects, starting with a signal that is synchronous with the rHWP rotation as observed in Johnson et al. (2007), Chapman et al.(2014), and Ritacco et al. (2016b).Such trade-o ff s are being investigated by current experiments using rHWPs and will certainly be studied in more details in prepa-ration of future CMB orbital and sub-orbital missions, such as CMB-S4 network (CMB-S4 collaboration 2016). We here brieflycomment on how the addition of a rHWP to an instrument can be coped with in QuickPol .The Jones matrix of a HWP (which shifts the y -axis electric field by a half period) rotated by an angle ψ is (O’Dea et al. 2007) J rHWP ( ψ ) = R ψ . (cid:32) − (cid:33) . R † ψ , (53) = (cid:32) cos 2 ψ sin 2 ψ sin 2 ψ − cos 2 ψ (cid:33) . (54)If a rotating HWP is installed at the entrance of the optical system, the Jones matrix of the system becomes J ( r α ) −→ J ( r α , ψ ) = J ( r α ) J rHWP ( ψ ), and the signal observed in the presence of arbitrary beams (Eq. 8) becomes (after dropping the circular polarization
11. Hivon, S. Mottet & N. Ponthieu:
QuickPol V terms) d ( α, ψ ) = (cid:90) d r (cid:104)(cid:101) I ( α, ψ, r ) T ( r ) + (cid:101) Q ( α, ψ, r ) Q ( r ) + (cid:101) U ( α, ψ, r ) U ( r ) (cid:105) , (55)with (cid:101) I ( α, ψ, r ) = (cid:101) I ( r α ) , (56a) (cid:101) Q ( α, ψ, r ) = (cid:101) Q ( r α ) cos(2 α + ψ ) + (cid:101) U ( r α ) sin(2 α + ψ ) , (56b) (cid:101) U ( α, ψ, r ) = (cid:101) Q ( r α ) sin(2 α + ψ ) − (cid:101) U ( r α ) cos(2 α + ψ ) . (56c)These new beams can then be passed to Eq. (30) and propagated through the rest of QuickPol . Together with Eq. (9), we seethat, if ψ is correctly chosen, the modulation of Q and U , by 2 α + ψ , is now clearly di ff erent from that of T which depends only on α via r α , even for non-circular (cid:101) I beams. The leakages from temperature to polarization are therefore expected to be much smaller thanwhen the polarization modulation is performed only by a rotation of the whole instrument, and O’Dea et al. (2007) showed, thateven for non-ideal rHWP, the induced systematic e ff ects are limited to polarization cross-talks without temperature to polarizationleakage.As previously mentioned, specific systematic e ff ects such as the rotation synchronous signal must be treated with care. Oncesuch time domain systematic e ff ects are identified and modeled, they, together with realistic optical properties of the instrument,can be integrated in the QuickPol formalism in order to be taken into account, quantified, and / or marginalized over at the powerspectrum level.
8. Conclusions
Polarization measurements are mostly obtained by di ff erencing observations by di ff erent detectors. Mismatch in their optical beams,time responses, bandpasses, and so on induces systematic e ff ects, for example, temperature to polarization leakage. The QuickPol formalism allows us to compute accurately and e ffi ciently the induced cross-talk between temperature and polarization power spec-tra. It also provides a fast and easy way to propagate instrumental modeling uncertainties down to the final angular power spectraand is thus a powerful tool to simulate observations and to help with the design and specifications of future experiments, suchas acceptable beam distortions, polarization modulation optimization, and observation redundancy. It can cope with time varyinginstrumental parameters, realistic sample flagging, and rejection. The method was validated through comparison to numerical sim-ulations of realistic Planck observations. The hypotheses required on the instrument and survey, described in Sections 2 and 3, areextremely general and apply to
Planck and to forthcoming CMB experiments such as PIXIE, LiteBIRD, COrE, and others. Contraryto Monte-Carlo based methods, such as
FEBeCoP , the impact of the beam related imperfections on the measured power spectra areobtained without having to assume any prior knowledge of the sky power spectra.Of course, the beam matrices provided by
QuickPol can be used in the cosmological analysis of a CMB survey. Indeed, the skypower spectra can be modeled as functions of cosmological parameters { θ C } , foreground modeling { θ F } , and nuisance parameters { θ n } . These C XY (cid:96) ( { θ C } , { θ F } , { θ n } ) can then be generated, multiplied with the beam matrices W X (cid:48) Y (cid:48) , XY (cid:96) for the set of detectors beinganalyzed, and compared to the measured (cid:101) C X (cid:48) Y (cid:48) (cid:96) in a maximum likelihood sense, in the presence of instrumental noise. The parameters { θ C } , { θ F } , { θ n } can be iterated or integrated upon, with statistical priors, until a posterior distribution is built. In this kind of forwardapproach, it is not necessary to correct the observations from possibly singular transfer functions, nor to back-propagate the noise.At least some of the instrumental uncertainties { θ I } a ff ecting the e ff ective beam via W X (cid:48) Y (cid:48) , XY (cid:96) ( { θ I } ) could be included in the overallanalysis, and marginalized over, thanks to the fast calculation times by QuickPol of the impact of changes in the gain calibrations,polarization angles, and e ffi ciencies, as discussed in Section 6. While QuickPol has been originally developed and tested in thecase of experiments without a rotating half-wave plate, it is straightforward to add one to the current pipeline and assess its impacton the aforementioned systematics. Specific additional e ff ects such as a HWP rotation synchronous signal or the e ff ect of a tiltedHWP are expected to show up in real experiments. As long as these can be physically modeled, they can be inserted in QuickPol as well.
Acknowledgements.
Thank you to the
Planck collaboration, and in particular to D. Hanson, K. Benabed, and F. R. Bouchet for fruitful discussions. Some resultspresented here were obtained with the
HEALPix library.
References
Armitage-Caplan, C. & Wandelt, B. D., PReBeaM for Planck: A Polarized Regularized Beam Deconvolution Map-Making Method. 2009, ApJS, 181, 533,arXiv:0807.4179 1Bennett, C. L., Larson, D., Weiland, J. L., et al., Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results. 2013, ApJS,208, 20, arXiv:1212.5225v1 1Bunn, E. F., Zaldarriaga, M., Tegmark, M., & de Oliveira-Costa, A., E / B decomposition of finite pixelized CMB maps. 2003, Phys. Rev. D, 67, 023501, arXiv:astro-ph / / /
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CMB-S4 collaboration, CMB-S4 Science Book, First Edition. 2016, arXiv:1610.02743 7Edmonds, A. R. 1957, Angular Momentum in Quantum Mechanics (Princeton University Press) CEssinger-Hileman, T., Kusaka, A., Appel, J. W., et al., Systematic e ff ects from an ambient-temperature, continuously-rotating half-wave plate. 2016, ArXiv e-prints,arXiv:1601.05901 7Fosalba, P., Dor´e, O., & Bouchet, F. R., Elliptical beams in CMB temperature and polarization anisotropy experiments: An analytic approach. 2002, Phys. Rev. D,65, 063003, arXiv:astro-ph / / / / / / / / / / / / / / / polarized camera at the IRAM 30 m telescope and its prototype. 2016a, preprint,arXiv:1602.01605 7Ritacco, A., Ponthieu, N., Catalano, A., et al., Polarimetry at millimeter wavelength with NIKA: calibration and performance. 2016b, preprint, arXiv:1609.02042 7Rosset, C., Tristram, M., Ponthieu, N., et al., Planck pre-launch status: High Frequency Instrument polarization calibration. 2010, A&A, 520, A13 + , arXiv:1004.25951, 2.1, 2.1, 6Rosset, C., Yurchenko, V., Delabrouille, J., et al., Beam mismatch e ff ects in Cosmic Microwave Background polarization measurements. 2007, A&A, 464, 405,arXiv:astro-ph / / / /
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Appendix A: Projection of maps on spherical harmonics
Here we give more details on the steps required to go from Eq. (35) to Eq. (38). Let us recall Eq. (35) and explain it further: (cid:101) m ( p ) ≡ (cid:101) m (0; p ) (cid:101) m (2; p ) / (cid:101) m ( − p ) / , (A.1) = (cid:88) k (cid:88) t ∈ p A ( k ) † p , t w k f k , t A ( k ) t , p − (cid:88) j (cid:88) t ∈ p A ( j ) † p , t w j f j , t d j , t , = (cid:88) k w k ω ( k )0 ρ k ω ( k ) − ρ k ω ( k )2 ρ k ω ( k )2 ρ k ω ( k )0 ρ k ω ( k )4 ρ k ω ( k ) − ρ k ω ( k ) − ρ k ω ( k )0 − (cid:88) j (cid:96) ms w j ( − s q (cid:96) − s Y (cid:96) m ( p ) ω ( j ) s ρ j ω ( j ) s + ρ j ω ( j ) s − b ( j ) ∗ (cid:96) s b ( j ) ∗ (cid:96) s − b ( j ) ∗ (cid:96) s T a (cid:96) m a (cid:96) m / − a (cid:96) m / , = (cid:88) j (cid:96) ms ( − s − s Y (cid:96) m ( p ) (cid:101) ω ( j ) s ρ j (cid:101) ω ( j ) s + ρ j (cid:101) ω ( j ) s − ˆ b ( j ) ∗ (cid:96) s ˆ b ( j ) ∗ (cid:96) s − ˆ b ( j ) ∗ (cid:96) s T a (cid:96) m a (cid:96) m / − a (cid:96) m / , (A.2)where we introduced the s -th complex moment of the direction of polarization for detector j , ω ( j ) s ( p ) ≡ (cid:88) t ∈ p f j , t e is α ( j ) t , (A.3)the hit matrix H defined for ( u , v ) ∈ { , , − } as H vu ( p ) ≡ (cid:88) j w j ω ( j ) v − u ( p ) ρ j , v ρ j , u , (A.4)with ρ j , v ≡ δ v , + ρ j (cid:0) δ v , − + δ v , (cid:1) , (A.5)the hit normalized moments (cid:101) ω ( j ) s ( p ) ρ j (cid:101) ω ( j ) s + ( p ) ρ j (cid:101) ω ( j ) s − ( p ) ≡ H ( p ) − ω ( j ) s ( p ) ρ j ω ( j ) s + ( p ) ρ j ω ( j ) s − ( p ) (A.6)which are described in Appendix B, and finally the inverse noise variance weighted beam spherical harmonics (SH) coe ffi cients u ˆ b ( j ) (cid:96), s ≡ w j q (cid:96) u b ( j ) (cid:96), s , (A.7) = ρ (cid:48) j ˆ b ( j ) (cid:96), s + u . (A.8)Since the solution of Eq. (33) remains the same when all the noise covariances are rescaled simultaneously by an arbitrary factor a : N j −→ a N j , one can also rescale the weights w j appearing in Eqs (A.4) and (A.7), with for instance w j −→ w j / (cid:80) k w k withoutaltering the final result.The components of the observed polarized map are then (cid:101) m ( v ; p ) = (cid:88) u k u k v (cid:88) j (cid:88) s ρ j , v (cid:101) ω ( j ) s + v ( p ) (cid:88) (cid:96) m u a (cid:96) m u ˆ b ( j ) ∗ (cid:96) s ( − s − s Y (cid:96) m ( p ) , (A.9)with k = , k ± = / . (A.10)After expansion of the hit normalized moments (Eq. A.6) in spherical harmonics: (cid:101) ω ( j ) s + v ( p ) = (cid:88) (cid:96) (cid:48) m (cid:48) s + v (cid:101) ω ( j ) (cid:96) (cid:48) m (cid:48) s + v Y (cid:96) (cid:48) m (cid:48) ( p ) , (A.11)the polarized map reads (cid:101) m ( v ; p ) = (cid:88) u k u k v (cid:88) j (cid:96) ms (cid:96) (cid:48) m (cid:48) ( − s − s Y (cid:96) m ( p ) s + v Y (cid:96) (cid:48) m (cid:48) ( p ) u a (cid:96) m u ˆ b ( j ) ∗ (cid:96) s ρ j , v s + v (cid:101) ω ( j ) (cid:96) (cid:48) m (cid:48) , (A.12)
14. Hivon, S. Mottet & N. Ponthieu:
QuickPol and the SH coe ffi cients of spin x of map (cid:101) m ( v ; p ) are, for pixels of area Ω p , x (cid:101) m (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( v ) ≡ (cid:88) p Ω p (cid:101) m ( v ; p ) x Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( p ) = (cid:90) d r (cid:101) m ( v ; r ) x Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( r ) (A.13) = (cid:88) u k u k v (cid:88) j (cid:88) (cid:96) ms (cid:96) (cid:48) m (cid:48) ( − s u a (cid:96) m u ˆ b ( j ) ∗ (cid:96) s ρ j , v s + v (cid:101) ω ( j ) (cid:96) (cid:48) m (cid:48) (cid:90) d r − s Y (cid:96) m ( r ) s + v Y (cid:96) (cid:48) m (cid:48) ( r ) x Y ∗ (cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) ( r ) = (cid:88) u k u k v (cid:88) j (cid:96) ms (cid:96) (cid:48) m (cid:48) u a (cid:96) m u ˆ b ( j ) ∗ (cid:96) s ρ j , v s + v (cid:101) ω ( j ) (cid:96) (cid:48) m (cid:48) ( − s + x + m (cid:48)(cid:48) + (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:34) (2 (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + π (cid:35) / (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) m m (cid:48) − m (cid:48)(cid:48) (cid:33) (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − s s + v − x (cid:33) (A.14)which are only non-zero when x = v . The cross power spectrum of spin v and v maps is then given by Eq. (38). Appendix B: Hit matrix
Introducing, for detector j , H ( j ) s = ω ( j ) s ρ j ω ( j ) s − ρ j ω ( j ) s + ρ j ω ( j ) s + ρ j ω ( j ) s ρ j ω ( j ) s + ρ j ω ( j ) s − ρ j ω ( j ) s − ρ j ω ( j ) s , (B.1)the Hermitian hit matrix for a weighted combination of detectors is H ≡ (cid:88) j w j H ( j )0 , (B.2) = h z z z x z ¯ z ¯ z x , (B.3)with h , x real and z , z complex numbers, and has for inverse H − = h ∆ x − | z | z ¯ z − x ¯ z ¯ z z − xz ¯ z z − xz x − | z | z − z z ¯ z − x ¯ z ¯ z − ¯ z x − | z | , (B.4)with ∆ = x − x | z | − | z | + z ¯ z + ¯ z z . (B.5)In Eq. (A.6) we defined (cid:101) ω ( j ) s [0] ρ j (cid:101) ω ( j ) s + [2] ρ j (cid:101) ω ( j ) s − [ − ≡ H − ω ( j ) s ρ j ω ( j ) s + ρ j ω ( j ) s − (B.6)for any value of s , which provides (cid:101) ω ( j ) s [0] ρ j (cid:101) ω ( j ) s [2] ρ j (cid:101) ω ( j ) s [ − = h ∆ ( x − | z | ) ω ( j ) s + ( z ¯ z − x ¯ z ) ρ j ω ( j ) s + + (¯ z z − xz ) ρ j ω ( j ) s − ( x − | z | ) ρ j ω ( j ) s + (¯ z z − xz ) ω ( j ) s − + ( z − z ) ρ j ω ( j ) s − ( x − | z | ) ρ j ω ( j ) s + ( z ¯ z − x ¯ z ) ω ( j ) s + + (¯ z − ¯ z ) ρ j ω ( j ) s + , (B.7)so that (cid:101) ω ( j ) s is of spin s , provided z and z are of spin 2 and 4 respectively. Since ω ( j ) s = ω ( j ) ∗− s , we get (cid:101) ω ( j ) ∗ s [2] = (cid:101) ω ( j ) − s [ − (cid:101) ω ( j ) s [0] ρ j (cid:101) ω ( j ) s − [0] ρ j (cid:101) ω ( j ) s + [0] ρ j (cid:101) ω ( j ) s + [2] ρ j (cid:101) ω ( j ) s [2] ρ j (cid:101) ω ( j ) s + [2] ρ j (cid:101) ω ( j ) s − [ − ρ j (cid:101) ω ( j ) s − [ − ρ j (cid:101) ω ( j ) s [ − = H − . H ( j ) s (B.8)so that (cid:88) j w j (cid:101) ω ( j )0 [0] ρ j (cid:101) ω ( j ) − [0] ρ j (cid:101) ω ( j )2 [0] ρ j (cid:101) ω ( j )2 [2] ρ j (cid:101) ω ( j )0 [2] ρ j (cid:101) ω ( j )4 [2] ρ j (cid:101) ω ( j ) − [ − ρ j (cid:101) ω ( j ) − [ − ρ j (cid:101) ω ( j )0 [ − = . (B.9)
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Appendix C: Wigner 3J symbols
The Wigner 3J symbols describe the coupling between di ff erent spin weighted spherical harmonics at the same location: s Y (cid:96) m ( r ) s Y (cid:96) m ( r ) = (cid:88) (cid:96) s m (cid:32) (2 (cid:96) + (cid:96) + (cid:96) + π (cid:33) / (cid:32) (cid:96) (cid:96) (cid:96) m m m (cid:33) (cid:32) (cid:96) (cid:96) (cid:96) − s − s − s (cid:33) s Y ∗ (cid:96) m ( r ) (C.1)and the symbol (cid:32) (cid:96) (cid:96) (cid:96) m m m (cid:33) is non-zero only when, | m i | ≤ (cid:96) i for i = , , m + m + m = | (cid:96) − (cid:96) | ≤ (cid:96) ≤ (cid:96) + (cid:96) . (C.2)They obey the relations (cid:32) (cid:96) (cid:96) (cid:96) − m − m − m (cid:33) = ( − (cid:96) + (cid:96) + (cid:96) (cid:32) (cid:96) (cid:96) (cid:96) m m m (cid:33) , (C.3)and (cid:32) (cid:96) (cid:96) m − m (cid:33) = ( − (cid:96) − m √ (cid:96) + . (C.4)Their standard orthogonality relations are (cid:88) (cid:96) (2 (cid:96) + (cid:32) (cid:96) (cid:96) (cid:96) m m m (cid:33) (cid:32) (cid:96) (cid:96) (cid:96) m (cid:48) m (cid:48) m (cid:48) (cid:33) = δ m m (cid:48) δ m m (cid:48) , (C.5)and (cid:88) m m (cid:32) (cid:96) (cid:96) (cid:96) m m m (cid:33) (cid:32) (cid:96) (cid:96) (cid:96) (cid:48) m m m (cid:48) (cid:33) = δ (cid:96) (cid:96) (cid:48) δ m m (cid:48) δ ( (cid:96) , (cid:96) , (cid:96) )2 (cid:96) + , (C.6)where δ ( (cid:96) , (cid:96) , (cid:96) ) = (cid:96) , (cid:96) , (cid:96) obey the triangle relation of Eq. (C.2) and vanishes otherwise.For (cid:96) (cid:28) (cid:96) , (cid:96) (Edmonds 1957, Eq. A2.1) (cid:32) (cid:96) (cid:96) (cid:96) m m − m − m (cid:33) (cid:39) ( − (cid:96) + m + m √ (cid:96) + d (cid:96) (cid:96) − (cid:96) , m ( θ ) , (C.7)where d is the Wigner rotation matrix and cos θ = m / (2 (cid:96) + | m | (cid:28) (cid:96) (cid:32) (cid:96) (cid:96) (cid:96) m m − m − m (cid:33) (cid:39) ( − m − m (cid:48) (cid:32) (cid:96) (cid:96) (cid:96) m m (cid:48) − m − m (cid:48) (cid:33) , (C.8)and an approximate orthogonality relation can therefore be written, for (cid:96) , | m | , | m | (cid:28) (cid:96) , (cid:96) (cid:88) (cid:96) (2 (cid:96) + (cid:32) (cid:96) (cid:96) (cid:96) m m m (cid:33) (cid:32) (cid:96) (cid:96) (cid:96) m (cid:48) m (cid:48) m (cid:48) (cid:33) (cid:39) ( − m − m (cid:48) δ m m (cid:48) . (C.9) Appendix D: Spin weighted power spectra
Since a complex field of spin s can be written as C s = R s + iI s where R s and I s are real, with R s ± iI s = (cid:88) (cid:96) m ± s a (cid:96) m ± s Y (cid:96) m (D.1)and, with the Condon-Shortley phase convention s Y ∗ (cid:96) m = ( − s + m − s Y (cid:96) − m , then s a ∗ (cid:96) m = ( − s + m − s a (cid:96) − m . (D.2)When s =
2, one defines a E (cid:96) m = − ( a (cid:96) m + − a (cid:96) m ) / a B (cid:96) m = − ( a (cid:96) m − − a (cid:96) m ) / (2 i ) (D.3b)such that a X ∗ (cid:96) m = ( − m a X (cid:96) − m , with X = E , B , and C EE (cid:96) = (cid:16) C (cid:96) + C − (cid:96) + C − (cid:96) + C − − (cid:96) (cid:17) / , (D.4a) C BB (cid:96) = (cid:16) C (cid:96) − C − (cid:96) − C − (cid:96) + C − − (cid:96) (cid:17) / , (D.4b) C EB (cid:96) = − (cid:16) C (cid:96) − C − (cid:96) + C − (cid:96) − C − − (cid:96) (cid:17) / (4 i ) . (D.4c)
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QuickPol
When s =
1, one defines a G (cid:96) m = − ( a (cid:96) m − − a (cid:96) m ) / a C (cid:96) m = − ( a (cid:96) m + − a (cid:96) m ) / (2 i ) (D.5b)such that a X ∗ (cid:96) m = ( − m a X (cid:96) − m , with X = G , C , and C GG (cid:96) = (cid:16) C (cid:96) − C − (cid:96) − C − (cid:96) + C − − (cid:96) (cid:17) / , (D.6a) C CC (cid:96) = (cid:16) C (cid:96) + C − (cid:96) + C − (cid:96) + C − − (cid:96) (cid:17) / , (D.6b) C GC (cid:96) = − (cid:16) C (cid:96) + C − (cid:96) − C − (cid:96) − C − − (cid:96) (cid:17) / (4 i ) . (D.6c) Appendix E: Window matrices W XY , X (cid:48) Y (cid:48) (cid:96) E.1. Arbitrary beams, smooth scanning case
Let us come back to Eqs. (39) and (40). These can be cast in a more compact matrix form (cid:101) C (cid:96) = (cid:88) j j (cid:88) s (cid:110)(cid:104) D − . ˆ B ( j ) † (cid:96), s . D . C (cid:96) . D . ˆ B ( j ) (cid:96), s . D − (cid:105) ∗ (cid:101) Ω ( j j ) s (cid:111) (E.1)where C (cid:96) ≡ C (cid:96) C (cid:96) C − (cid:96) C (cid:96) C (cid:96) C − (cid:96) C − (cid:96) C − (cid:96) C − − (cid:96) , (E.2) D ≡ / / , (E.3)ˆ B ( j ) (cid:96), s ≡ ˆ b ( j ) (cid:96), s ˆ b ( j ) (cid:96), s − ˆ b ( j ) (cid:96), s + ˆ b ( j ) (cid:96), s ˆ b ( j ) (cid:96), s − ˆ b ( j ) (cid:96), s + − ˆ b ( j ) (cid:96), s − ˆ b ( j ) (cid:96), s − − ˆ b ( j ) (cid:96), s + = ˆ b ( j ) (cid:96), s ˆ b ( j ) (cid:96), s − ˆ b ( j ) (cid:96), s + ρ (cid:48) j ˆ b ( j ) (cid:96), s + ρ (cid:48) j ˆ b ( j ) (cid:96), s ρ (cid:48) j ˆ b ( j ) (cid:96), s + ρ (cid:48) j ˆ b ( j ) (cid:96), s − ρ (cid:48) j ˆ b ( j ) (cid:96), s − ρ (cid:48) j ˆ b ( j ) (cid:96), s , (E.4)and X ∗ Y denotes the elementwise product (also known as Hadamard or Schur product) of arrays X and Y . Noting that C (cid:96) C (cid:96) C − (cid:96) C (cid:96) C (cid:96) C − (cid:96) C − (cid:96) C − (cid:96) C − − (cid:96) = R . C TT (cid:96) C TE (cid:96) C T B (cid:96) C ET (cid:96) C EE (cid:96) C EB (cid:96) C BT (cid:96) C BE (cid:96) C BB (cid:96) . R † (E.5)where R was introduced in Eq. (18), which leads to Eq. (41) that we recall here for convenience: (cid:101) C XY (cid:96) = (cid:88) X (cid:48) Y (cid:48) W XY , X (cid:48) Y (cid:48) (cid:96) C X (cid:48) Y (cid:48) (cid:96) . (E.6)Introducing the short-hand ˆ Ω sv v ≡ (cid:101) Ω ( j j ) v , v , s , (E.7)describing the coupled moments of the polarized detectors j and j orientation, and assuming in Eq. (E.4) the beams to be perfectlyco-polarized, with polar e ffi ciencies ρ (cid:48) j , one gets, for XY = T T , EE , BB , T E , T B , EB , ET , BT , BE : W XY , TT (cid:96) = (cid:88) s (cid:88) j j ˆ Ω s ˆ b ( j ) ∗ (cid:96), s ˆ b ( j ) (cid:96), s ˆ b ( j ) ∗ (cid:96), s + (cid:16) ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:17) + ˆ b ( j ) ∗ (cid:96), s − (cid:16) ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:17) ˆ b ( j ) ∗ (cid:96), s + (cid:16) ˆ Ω s − − ˆ b ( j ) (cid:96), s + − ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:17) + ˆ b ( j ) ∗ (cid:96), s − (cid:16) ˆ Ω s ˆ b ( j ) (cid:96), s − − ˆ Ω s − ˆ b ( j ) (cid:96), s + (cid:17) − ˆ b ( j ) ∗ (cid:96), s (cid:16) ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:17) − i ˆ b ( j ) ∗ (cid:96), s (cid:16) ˆ Ω s ˆ b ( j ) (cid:96), s − − ˆ Ω s − ˆ b ( j ) (cid:96), s + (cid:17) i ˆ b ( j ) ∗ (cid:96), s + (cid:16) ˆ Ω s − ˆ b ( j ) (cid:96), s − − ˆ Ω s − − ˆ b ( j ) (cid:96), s + (cid:17) + i ˆ b ( j ) ∗ (cid:96), s − (cid:16) ˆ Ω s ˆ b ( j ) (cid:96), s − − ˆ Ω s − ˆ b ( j ) (cid:96), s + (cid:17) − ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ b ( j ) ∗ (cid:96), s + + ˆ Ω s ˆ b ( j ) ∗ (cid:96), s − (cid:17) − i ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ b ( j ) ∗ (cid:96), s + − ˆ Ω s ˆ b ( j ) ∗ (cid:96), s − (cid:17) i ˆ b ( j ) ∗ (cid:96), s + (cid:16) ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:17) − i ˆ b ( j ) ∗ (cid:96), s − (cid:16) ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:17) , (E.8a)
17. Hivon, S. Mottet & N. Ponthieu:
QuickPol which is illustrated in Fig. 3; W XY , EE (cid:96) = (cid:88) s (cid:88) j j ρ (cid:48) j ρ (cid:48) j ˆ Ω s (cid:16) ˆ b ( j ) ∗ (cid:96), s − + ˆ b ( j ) ∗ (cid:96), s + (cid:17) (cid:16) ˆ b ( j ) (cid:96), s − + ˆ b ( j ) (cid:96), s + (cid:17) ˆ b ( j ) ∗ (cid:96), s (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − + ˆ Ω s − + ˆ Ω s − + ˆ Ω s (cid:17) + ˆ b ( j ) (cid:96), s + (cid:16) ˆ Ω s − − + ˆ Ω s − (cid:17) + ˆ b ( j ) (cid:96), s − (cid:16) ˆ Ω s − + ˆ Ω s (cid:17)(cid:105) + ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − + ˆ Ω s − (cid:17) + ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) + ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − + ˆ Ω s (cid:17) + ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) ˆ b ( j ) ∗ (cid:96), s (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − − ˆ Ω s − − ˆ Ω s − + ˆ Ω s (cid:17) + ˆ b ( j ) (cid:96), s + (cid:16) ˆ Ω s − − − ˆ Ω s − (cid:17) + ˆ b ( j ) (cid:96), s − (cid:16) ˆ Ω s − ˆ Ω s − (cid:17)(cid:105) + ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − − ˆ Ω s − (cid:17) + ˆ Ω s − − ˆ b ( j ) (cid:96), s + − ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) + ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ Ω s − (cid:17) − ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) − (cid:16) ˆ b ( j ) ∗ (cid:96), s − + ˆ b ( j ) ∗ (cid:96), s + (cid:17) (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − + ˆ Ω s (cid:17) + ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) − i (cid:16) ˆ b ( j ) ∗ (cid:96), s − + ˆ b ( j ) ∗ (cid:96), s + (cid:17) (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ Ω s − (cid:17) − ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) i ˆ b ( j ) ∗ (cid:96), s (cid:104) ˆ b ( j ) (cid:96), s (cid:16) − ˆ Ω s − − + ˆ Ω s − − ˆ Ω s − + ˆ Ω s (cid:17) − ˆ b ( j ) (cid:96), s + (cid:16) ˆ Ω s − − + ˆ Ω s − (cid:17) + ˆ b ( j ) (cid:96), s − (cid:16) ˆ Ω s − + ˆ Ω s (cid:17)(cid:105) + i ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − ˆ Ω s − − (cid:17) − ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) + i ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ Ω s − (cid:17) − ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) − (cid:16) ˆ b ( j ) (cid:96), s − + ˆ b ( j ) (cid:96), s + (cid:17) (cid:104)(cid:16) ˆ Ω s − + ˆ Ω s (cid:17) ˆ b ( j ) ∗ (cid:96), s + ˆ Ω s − ˆ b ( j ) ∗ (cid:96), s + + ˆ Ω s ˆ b ( j ) ∗ (cid:96), s − (cid:105) i (cid:16) ˆ b ( j ) (cid:96), s − + ˆ b ( j ) (cid:96), s + (cid:17) (cid:104)(cid:16) ˆ Ω s − ˆ Ω s − (cid:17) ˆ b ( j ) ∗ (cid:96), s − ˆ Ω s − ˆ b ( j ) ∗ (cid:96), s + + ˆ Ω s ˆ b ( j ) ∗ (cid:96), s − (cid:105) i ˆ b ( j ) ∗ (cid:96), s (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − + ˆ Ω s − − ˆ Ω s − − ˆ Ω s (cid:17) + ˆ b ( j ) (cid:96), s + (cid:16) ˆ Ω s − − − ˆ Ω s − (cid:17) + ˆ b ( j ) (cid:96), s − (cid:16) ˆ Ω s − − ˆ Ω s (cid:17)(cid:105) + i ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − + ˆ Ω s − (cid:17) + ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) − i ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − + ˆ Ω s (cid:17) + ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) , (E.8b) W XY , TE (cid:96) = (cid:88) s (cid:88) j j ρ (cid:48) j − ˆ Ω s ˆ b ( j ) ∗ (cid:96), s (cid:16) ˆ b ( j ) (cid:96), s − + ˆ b ( j ) (cid:96), s + (cid:17) − ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − + ˆ Ω s − (cid:17) + ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) − ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − + ˆ Ω s (cid:17) + ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) − ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − − ˆ Ω s − (cid:17) + ˆ Ω s − − ˆ b ( j ) (cid:96), s + − ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) − ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ Ω s − (cid:17) − ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) ˆ b ( j ) ∗ (cid:96), s (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − + ˆ Ω s (cid:17) + ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) i ˆ b ( j ) ∗ (cid:96), s (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ Ω s − (cid:17) − ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) − i ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − ˆ Ω s − − (cid:17) − ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) − i ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − ˆ Ω s − (cid:17) − ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105)(cid:16) ˆ b ( j ) (cid:96), s − + ˆ b ( j ) (cid:96), s + (cid:17) (cid:16) ˆ Ω s − ˆ b ( j ) ∗ (cid:96), s + + ˆ Ω s ˆ b ( j ) ∗ (cid:96), s − (cid:17) − i (cid:16) ˆ b ( j ) (cid:96), s − + ˆ b ( j ) (cid:96), s + (cid:17) (cid:16) ˆ Ω s ˆ b ( j ) ∗ (cid:96), s − − ˆ Ω s − ˆ b ( j ) ∗ (cid:96), s + (cid:17) − i ˆ b ( j ) ∗ (cid:96), s + (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − − + ˆ Ω s − (cid:17) + ˆ Ω s − − ˆ b ( j ) (cid:96), s + + ˆ Ω s − ˆ b ( j ) (cid:96), s − (cid:105) + i ˆ b ( j ) ∗ (cid:96), s − (cid:104) ˆ b ( j ) (cid:96), s (cid:16) ˆ Ω s − + ˆ Ω s (cid:17) + ˆ Ω s − ˆ b ( j ) (cid:96), s + + ˆ Ω s ˆ b ( j ) (cid:96), s − (cid:105) . (E.8c)Since, by definition (Eqs. A.7 and E.7), ˆ b ( j ) ∗ (cid:96), s = ( − s ˆ b ( j ) (cid:96), − s and ˆ Ω s ∗ v , v = ˆ Ω − s − v , − v , one can check that each term of W XY , X (cid:48) Y (cid:48) (cid:96) is real, asexpected. E.2. Arbitrary beams, ideal scanning
In the case of ideal scanning described in Section 3.4, one gets ω ( j ) s ( p ) = δ s , h ( p ), so that the hit matrix is diagonal: H ( p ) = h ( p ) (cid:88) j w j w j ρ j
00 0 w j ρ j , (E.9)and the orientation moments (cid:101) ω ( j ) s [0] = δ s , (cid:88) k w k − , (cid:101) ω ( j ) s [ ± = δ s , (cid:88) k w k ρ k − , (E.10)are such that ˆ Ω sv v = ξ ρ j ξ ρ j ξ ρ j ξ ρ j ρ j ξ ρ j ρ j ξ ρ j ξ ρ j ρ j ξ ρ j ρ j ξ δ s , , (E.11)
18. Hivon, S. Mottet & N. Ponthieu:
QuickPol with ξ − = (cid:88) k k w k w k , ξ − = (cid:88) k k w k w k ρ k , ξ − = (cid:88) k k w k w k ρ k , ξ − = (cid:88) k k w k w k ρ k ρ k . (E.12)One then obtains the beam matrices W XY , TT (cid:96) = (cid:88) j j ˆ b ( j ) (cid:96), ˆ b ( j ) ∗ (cid:96), ξ (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ − (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) ˆ b ( j ) ∗ (cid:96), ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) ˆ b ( j ) ∗ (cid:96), ρ j ξ i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ − ˆ b ( j ) (cid:96), (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ i ˆ b ( j ) (cid:96), (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ , (E.13a) W XY , EE (cid:96) = (cid:88) j j ρ (cid:48) j ρ (cid:48) j (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ξ (cid:16) ˆ b ( j ) (cid:96), − + b ( j ) (cid:96), + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + b ( j ) ∗ (cid:96), + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ − (cid:16) ˆ b ( j ) (cid:96), − + b ( j ) (cid:96), + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + b ( j ) ∗ (cid:96), + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ − (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + b ( j ) ∗ (cid:96), + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ i (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − + b ( j ) (cid:96), + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ , (E.13b) W XY , TE (cid:96) = (cid:88) j j ρ (cid:48) j − (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) ˆ b ( j ) ∗ (cid:96), ξ − (cid:16) ˆ b ( j ) (cid:96), − + b ( j ) (cid:96), + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ − (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ (cid:16) ˆ b ( j ) (cid:96), − + b ( j ) (cid:96), + ˆ b ( j ) (cid:96), (cid:17) ˆ b ( j ) ∗ (cid:96), ρ j ξ i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) ˆ b ( j ) ∗ (cid:96), ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − − ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − + ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ − i (cid:16) ˆ b ( j ) (cid:96), − + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ξ i (cid:16) ˆ b ( j ) (cid:96), − + b ( j ) (cid:96), + ˆ b ( j ) (cid:96), (cid:17) (cid:16) ˆ b ( j ) ∗ (cid:96), − − ˆ b ( j ) ∗ (cid:96), (cid:17) ρ j ρ j ξ . (E.13c)The implications of Eq. (E.13) are discussed in Section 3.4. Appendix F: Finite pixel size
Introducing the spin raising and lowering di ff erential operators applied to a function f of spin s , (Zaldarriaga & Seljak 1997;Bunn et al. 2003, and references therein) ð f = − sin s θ (cid:32) ∂∂θ + i sin θ ∂∂ϕ (cid:33) (cid:2) sin − s θ f (cid:3) = s cot θ f − ∂ f ∂θ − i sin θ ∂ f ∂ϕ (F.1)¯ ð f = − sin − s θ (cid:32) ∂∂θ − i sin θ ∂∂ϕ (cid:33) (cid:2) sin s θ f (cid:3) = − s cot θ f − ∂ f ∂θ + i sin θ ∂ f ∂ϕ (F.2)the spin weighed spherical harmonics are defined as s Y (cid:96) m ≡ (cid:115) ( (cid:96) − s )!( (cid:96) + s )! ð s Y (cid:96) m , ≤ s ≤ (cid:96) ; (F.3) s Y (cid:96) m ≡ ( − s (cid:115) ( (cid:96) + s )!( (cid:96) − s )! ¯ ð − s Y (cid:96) m , − (cid:96) ≤ s ≤
0; (F.4)
19. Hivon, S. Mottet & N. Ponthieu:
QuickPol such that ð s Y (cid:96) m = f ( (cid:96), s ) s + Y (cid:96) m , (F.5)¯ ð s Y (cid:96) m = − f ( (cid:96), − s ) s − Y (cid:96) m , (F.6)with f ( (cid:96), s ) = √ ( (cid:96) − s )( (cid:96) + s + = √ (cid:96) ( (cid:96) + − s ( s + ff ect is very close to the oneof gravitational lensing described in Hu (2000) and Lewis & Challinor (2006).For r = (1 , θ, ϕ ) = e r and d r = (0 , d θ, d ϕ ) = d θ e θ + sin θ d ϕ e ϕ , s Y (cid:96) m ( r + d r ) = s Y (cid:96) m ( r ) + d r . ∇ s Y (cid:96) m ( r ) + (cid:88) i j d r i d r j ∇ i ∇ j s Y (cid:96) m ( r ) (F.7) = s Y (cid:96) m ( r ) − (cid:16) ¯dr ð + dr ¯ ð (cid:17) s Y (cid:96) m ( r ) + (cid:16) ¯dr ¯dr ðð + ¯drdr ð ¯ ð + dr ¯dr ¯ ðð + drdr ¯ ð ¯ ð (cid:17) s Y (cid:96) m ( r ) (F.8) = s Y (cid:96) m ( r ) − (cid:16) ¯dr f ( (cid:96), s ) s + Y (cid:96) m ( r ) − dr f ( (cid:96), − s ) s − Y (cid:96) m ( r ) (cid:17) −
14 dr ¯dr (cid:16) (cid:96) ( (cid:96) + − s (cid:17) s Y (cid:96) m ( r ) + (cid:16) ¯dr ¯dr g ( (cid:96), s ) s + Y (cid:96) m ( r ) + drdr g ( (cid:96), − s ) s − Y (cid:96) m ( r ) (cid:17) (F.9)with dr = d r . ( e θ + i e ϕ ) = d θ + i sin θ d ϕ , ¯dr = d θ − i sin θ d ϕ and g ( (cid:96), s ) = f ( (cid:96), s ) f ( (cid:96), s + r to the position of a measurement relative to the nominal center r of the pixel to which it is attributed, thisexpansion of s Y (cid:96) m can be injected into Eqs. (28) and (A.14). Assuming d r to be uncorrelated with the orientation of the detector,two extra terms, both quadratic in d r , will appear in the final power spectra.The first term involves the scalar product of the gradient of the signal in the pixel, assumed to be totally dominated by thetemperature, with the weighted sum of d r over all samples in that pixel. Introducing ρ j , v (cid:101) ω ( j ) ± s + v ( p ) = (cid:88) v (cid:48) ( H − ( p )) vv (cid:48) ρ j , v (cid:48) (cid:88) t ∈ p (d θ t ± i sin θ t d ϕ t ) f j , t e i ( s + v (cid:48) ) α ( j ) t (F.10)which is of spin s + v ± (cid:16) ρ j , v (cid:101) ω ( j ) + s + v (cid:17) ∗ = ρ j , − v (cid:101) ω ( j ) −− s − v , one finds ∆ (cid:101) C v v (cid:96) (cid:48)(cid:48) = k v k v (cid:88) s s ( − s + s (cid:88) j j (cid:96) (cid:96) ( (cid:96) +
1) 2 (cid:96) + π C TT (cid:96) ˆ b ( j ) ∗ (cid:96) s ˆ b ( j ) (cid:96) s × (cid:88) (cid:96) (cid:48) (cid:96) (cid:48) + (cid:104) D ( j j ) ++ s + v , s + v ,(cid:96) (cid:48) J v , v s + , s + + D ( j j ) −− s + v , s + v ,(cid:96) (cid:48) J v , v s − , s − − D ( j j ) + − s + v , s + v ,(cid:96) (cid:48) J v , v s + , s − − D ( j j ) − + s + v , s + v ,(cid:96) (cid:48) J v , v s − , s + (cid:105) (F.11)with D ( j j ) σ σ s + v , s + v ,(cid:96) (cid:48) = ρ j , v ρ j , v (cid:96) (cid:48) + (cid:88) m (cid:48) s + v (cid:101) ω ( j ) σ (cid:96) (cid:48) m (cid:48) s + v (cid:101) ω ( j ) σ ∗ (cid:96) (cid:48) m (cid:48) with { σ , σ } ∈ { + , −} , (F.12) J v , v s + σ , s + σ = (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − s − σ s + σ + v − v (cid:33) (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − s − σ s + σ + v − v (cid:33) with { σ , σ } ∈ { + , − } . (F.13)In the case of temperature, and assuming the beams to be circular, this simplifies to ∆ (cid:101) C TT (cid:96) (cid:48)(cid:48) = (cid:88) j j (cid:96) (cid:96) ( (cid:96) +
1) 2 (cid:96) + π C TT (cid:96) ˆ b ( j ) ∗ (cid:96) ˆ b ( j ) (cid:96) × (cid:88) (cid:96) (cid:48) (cid:96) (cid:48) + (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − (cid:33) (cid:104)(cid:16) D ( j j ) ++ ,(cid:96) (cid:48) + D ( j j ) −− ,(cid:96) (cid:48) (cid:17) − ( − (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) (cid:16) D ( j j ) + − ,(cid:96) (cid:48) + D ( j j ) − + ,(cid:96) (cid:48) (cid:17)(cid:105) (F.14)in agreement with Planck 2013-VII (2014), once one identifies (cid:16) D ( j j ) ++ ,(cid:96) (cid:48) + D ( j j ) −− ,(cid:96) (cid:48) (cid:17) / − (cid:16) D ( j j ) + − ,(cid:96) (cid:48) + D ( j j ) − + ,(cid:96) (cid:48) (cid:17) / ff erence (see Section D).It is instructive to further assume the relative location of the hit’s center of mass to be only weakly correlated between pixels, sothat all its derived power spectra can be assumed to be white: D ( j j ) σ σ s + v , s + v ,(cid:96) (cid:48) = D ( j j ) σ σ s + v , s + v (i.e., with a variance D ( j j ) σ σ s + v , s + v / Ω pix inpixels of solid angle Ω pix = π/ N pix ). Equation (C.5) then ensures that the sub-pixel noise of Eq. (F.11) is also white, with constantpolarized spectra ∆ (cid:101) C XY (cid:96) (cid:48)(cid:48) = N XY = (cid:88) (cid:96) (cid:96) ( (cid:96) +
1) 2 (cid:96) + π C TT (cid:96) W XY (cid:96) (F.15)
20. Hivon, S. Mottet & N. Ponthieu:
QuickPol with W TT (cid:96) = (cid:88) j j s (cid:16) ˆ b ( j ) ∗ (cid:96) s ˆ b ( j ) (cid:96) s (cid:104) D ( j j ) ++ ss + D ( j j ) −− ss (cid:105) + ˆ b ( j ) ∗ (cid:96) s ˆ b ( j ) (cid:96), s + D ( j j ) + − s , s + + ˆ b ( j ) ∗ (cid:96), s + ˆ b ( j ) (cid:96) s D ( j j ) − + s + , s (cid:17) , (F.16) (cid:39) (cid:88) j j ˆ b ( j ) ∗ (cid:96) ˆ b ( j ) (cid:96) (cid:104) D ( j j ) ++ + D ( j j ) −− (cid:105) ; (F.17) W EE (cid:96) = W BB (cid:96) = (cid:88) j j s (cid:88) v = − , (cid:16) ˆ b ( j ) ∗ (cid:96) s ˆ b ( j ) (cid:96) s (cid:104) D ( j j ) ++ s + v , s + v + D ( j j ) −− s + v , s + v (cid:105) + ˆ b ( j ) ∗ (cid:96) s ˆ b ( j ) (cid:96), s + D ( j j ) + − s + v , s + + v + ˆ b ( j ) ∗ (cid:96), s + ˆ b ( j ) (cid:96) s D ( j j ) − + s + + v , s + v (cid:17) , (F.18) (cid:39) (cid:88) j j ˆ b ( j ) ∗ (cid:96) ˆ b ( j ) (cid:96) (cid:104) D ( j j ) ++ + D ( j j ) −− + D ( j j ) ++ − − + D ( j j ) −−− − (cid:105) ; (F.19) W XY (cid:96) = X (cid:44) Y (F.20)where the approximate results are obtained for circular beams.Even for more realistic hypotheses on the hits locations, the sub-pixel contributions to the respective power spectra follow thehierarchy ∆ (cid:101) C TT (cid:96) ∼ ∆ (cid:101) C EE (cid:96) ∼ ∆ (cid:101) C BB (cid:96) (cid:29) ∆ (cid:101) C TE (cid:96) ∼ ∆ (cid:101) C T B (cid:96) ∼ ∆ (cid:101) C EB (cid:96) . (F.21)Let us consider now the other extra contribution to the power spectrum, involving the Laplacian of the sky signal and thequadratic norm of d r . Introducing ρ j , v (cid:101) ω ( j ) (cid:48) v ( p ) = (cid:88) v (cid:48) ( H − ( p )) vv (cid:48) ρ j , v (cid:48) (cid:88) t ∈ p (d θ t + sin θ t d ϕ t ) f j , t e iv (cid:48) α ( j ) t (F.22) (cid:39) σ p ρ j , v (cid:101) ω ( j ) v ( p ) (F.23)where σ p is the second order moment of the hit location in pixel p . If we assume this and (cid:101) ω ( j ) v ( p ) to be slowly varying functions of p , and consider (cid:96) (cid:29) s , the power spectra become C (cid:96) −→ (cid:32) − (cid:96) ( (cid:96) + σ (cid:33) C (cid:96) (F.24)which describes to leading order, the smoothing e ff ect of the integration of the signal on the pixel. Appendix G: Co-polarized beam
For an arbitrarily shaped beam having the intensity harmonics coe ffi cients b (cid:96) m = (cid:90) d r (cid:101) I ( r ) Y ∗ (cid:96) m ( r ) , (G.1)and assumed to be perfectly co-polarized in direction γ , its polarized harmonics’ content will be ± b (cid:96) m = (cid:90) d r (cid:16) (cid:101) Q ( r ) ± i (cid:101) U ( r ) (cid:17) ± Y ∗ (cid:96) m ( r ) = (cid:90) d r (cid:101) I ( r ) e ± i ( γ − ϕ r ) ± Y ∗ (cid:96) m ( r ) (G.2) = e ± i γ (cid:88) (cid:96) (cid:48) m (cid:48) b (cid:96) (cid:48) m (cid:48) (cid:90) π d ϕ (cid:90) π d θ sin θ e ∓ i ϕ r Y (cid:96) (cid:48) m (cid:48) ( θ, ϕ ) ± Y ∗ (cid:96) m ( θ, ϕ ) (G.3) = e ± i γ π (cid:88) (cid:96) (cid:48) b (cid:96) (cid:48) , m ± ( − m (cid:88) (cid:96) (cid:48)(cid:48) ≥ (cid:32) (2 (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + π (cid:33) / (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − m m ± ∓ (cid:33) (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) ± ∓ (cid:33) I (cid:96) (cid:48)(cid:48) (G.4)where we used Eq. (C.1) and introduced, for (cid:96) (cid:48)(cid:48) ≥ I (cid:96) (cid:48)(cid:48) ≡ (cid:90) π d θ sin θ ± Y ∗ (cid:96) (cid:48)(cid:48) , ∓ ( θ ) , (G.5) = (cid:114) (cid:96) (cid:48)(cid:48) + π (cid:90) π d θ sin θ d (cid:96) (cid:48)(cid:48) ∓ , ∓ ( θ ) , (G.6) = (cid:114) (cid:96) (cid:48)(cid:48) + π (cid:96) (cid:48)(cid:48) ( (cid:96) (cid:48)(cid:48) +
1) ( − (cid:96) (cid:48)(cid:48) . (G.7)
21. Hivon, S. Mottet & N. Ponthieu:
QuickPol
Since I (cid:96) (cid:48)(cid:48) peaks at (cid:96) (cid:48)(cid:48) =
2, the 3J symbols will enforce (cid:96) (cid:48)(cid:48) (cid:28) (cid:96) (cid:39) (cid:96) (cid:48) in Eq. (G.4). If the beam is narrow enough in real space, b (cid:96) (cid:48) , m will be almost constant over the allowed range (cid:96) − (cid:96) (cid:48)(cid:48) ≤ (cid:96) (cid:48) ≤ (cid:96) + (cid:96) (cid:48)(cid:48) , and we use the approximate orthogonality relation of Eq. (C.9)to write (cid:88) (cid:96) (cid:48) b (cid:96) (cid:48) , m ± ( − m (cid:112) (2 (cid:96) + (cid:96) (cid:48) + (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − m m ± ∓ (cid:33) (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) ± ∓ (cid:33) (cid:39) b (cid:96), m ± (cid:88) (cid:96) (cid:48) ( − m (2 (cid:96) (cid:48) + − m ± (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) ± ∓ (cid:33) , = b (cid:96), m ± . (G.8)Finally, we note that in Eq. (G.4), the sum2 π n + (cid:88) (cid:96) (cid:48)(cid:48) = (cid:114) (cid:96) (cid:48)(cid:48) + π I (cid:96) (cid:48)(cid:48) = n + (cid:88) (cid:96) (cid:48)(cid:48) = ( − (cid:96) (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) + (cid:96) (cid:48)(cid:48) ( (cid:96) (cid:48)(cid:48) + , (G.9) = n (cid:88) p = p ( p + = n (cid:88) p = p − p + , (G.10) = − n + , (G.11)to obtain ± b (cid:96) m = e ± i γ b (cid:96), m ± , (G.12)which is valid for any (narrow) co-polarized beam.(G.12)which is valid for any (narrow) co-polarized beam.