Bicep2/keck Array. Vii. Matrix Basede/bseparation Applied to Bicep2 and the Keck Array

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1 Bicep2/keck Array. Vii. Matrix Basede/bseparation Appied to Bicep2 and the Keck Array The Harvard community has made this artice openy avaiabe. Pease share how this access benefits you. Your story matters Citation Ade, P. A. R., Z. Ahmed, R. W. Aikin, K. D. Aexander, D. Barkats, S. J. Benton, C. A. Bischoff, et a Bicep2/ keck Array. Vii. Matrix Basede/bseparation Appied to Bicep2 and the Keck Array. The Astrophysica Journa 825 (1) (June 3): 66. doi:1.3847/4-637x/825/1/66. dx.doi.org/1.3847/4-637x/825/1/66. Pubished Version doi:1.3847/4-637x/825/1/66 Citabe ink Terms of Use This artice was downoaded from Harvard University s DASH repository, and is made avaiabe under the terms and conditions appicabe to Open Access Poicy Artices, as set forth at nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#oap

2 DRAFT VERSION JULY 4, 216 Preprint typeset using LATEX stye emuateapj v. 1/23/15 BICEP2 / Keck Array VII: MATRIX BASED E/B SEPARATION APPLIED TO BICEP2 AND THE Keck Array arxiv: v2 astro-ph.im 1 Ju 216 Keck Array AND BICEP2 COLLABORATIONS: P. A. R. ADE 1, Z. AHMED 2,3, R. W. AIKIN 4, K. D. ALEXANDER 5, D. BARKATS 5, S. J. BENTON 6, C. A. BISCHOFF 5, J. J. BOCK 7,8, R. BOWENS-RUBIN 5, J. A. BREVIK 7, I. BUDER 5, E. BULLOCK 9, V. BUZA 5,1, J. CONNORS 5, B. P. CRILL 8, L. DUBAND 11, C. DVORKIN 1, J. P. FILIPPINI 7,12, S. FLIESCHER 9, J. GRAYSON 2, M. HALPERN 13, S. HARRISON 5, S. R. HILDEBRANDT 7,8, G. C. HILTON 14, H. HUI 7, K. D. IRWIN 2,3, J. KANG 2,3, K. S. KARKARE 5, E. KARPEL 2, J. P. KAUFMAN 15, B. G. KEATING 15, S. KEFELI 7, S. A. KERNASOVSKIY 1, J. M. KOVAC 5,1, C. L. KUO 2,3, E. M. LEITCH 16, M. LUEKER 7, K. G. MEGERIAN 8, T. NAMIKAWA 2,3, C. B. NETTERFIELD 6,17, H. T. NGUYEN 8, R. O BRIENT 7,8, R. W. OGBURN IV 2,3, A. ORLANDO 7, C. PRYKE 9,18, S. RICHTER 5, R. SCHWARZ 18, C. D. SHEEHY 16,18, Z. K. STANISZEWSKI 7,8, B. STEINBACH 7, R. V. SUDIWALA 1, G. P. TEPLY 7,15, K. L. THOMPSON 2,3, J. E. TOLAN 2,21, C. TUCKER 1, A. D. TURNER 8, A. G. VIEREGG 5,16,19, A. C. WEBER 8, D. V. WIEBE 13, J. WILLMERT 18, C. L. WONG 5,1, W. L. K. WU 2,2, AND K. W. YOON 2 Draft version Juy 4, 216 ABSTRACT A inear poarization fied on the sphere can be uniquey decomposed into an E-mode and a B-mode component. These two components are anayticay defined in terms of spin-2 spherica harmonics. Maps that contain fitered modes on a partia sky can aso be decomposed into E-mode and B-mode components. However, the ack of fu sky information prevents orthogonay separating these components using spherica harmonics. In this paper, we present a technique for decomposing an incompete map into E and B-mode components using E and B eigenmodes of the pixe covariance in the observed map. This method is found to orthogonay define E and B in the presence of both partia sky coverage and spatia fitering. This method has been appied to the BICEP2 and the Keck Array maps and resuts in reducing E to B eakage from ΛCDM E-modes to a eve corresponding to a tensor-to-scaar ratio of r < Subject headings: cosmic background radiation cosmoogy: observations gravitationa waves infation poarization 1 Schoo of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom 2 Department of Physics, Stanford University, Stanford, CA 9435, USA 3 Kavi Institute for Partice Astrophysics and Cosmoogy, SLAC Nationa Acceerator Laboratory, 2575 Sand Hi Rd, Meno Park, CA 9425, USA 4 Department of Physics, Caifornia Institute of Technoogy, Pasadena, Caifornia 91125, USA 5 Harvard-Smithsonian Center for Astrophysics, 6 Garden Street MS 42, Cambridge, Massachusetts 2138, USA 6 Department of Physics, University of Toronto, Toronto, Ontario, M5S 1A7, Canada 7 Department of Physics, Caifornia Institute of Technoogy, Pasadena, Caifornia 91125, USA 8 Jet Propusion Laboratory, Pasadena, Caifornia 9119, USA 9 Minnesota Institute for Astrophysics, University of Minnesota, Minneapois, Minnesota 55455, USA 1 Department of Physics, Harvard University, Cambridge, MA 2138, USA 11 Service des Basses Températures, Commissariat à Energie Atomique, 3854 Grenobe, France 12 Department of Physics, University of Iinois at Urbana-Champaign, Urbana, Iinois 6181, USA 13 Department of Physics and Astronomy, University of British Coumbia, Vancouver, British Coumbia, V6T 1Z1, Canada 14 Nationa Institute of Standards and Technoogy, Bouder, Coorado 835, USA 15 Department of Physics, University of Caifornia at San Diego, La Joa, Caifornia 9293, USA 16 Kavi Institute for Cosmoogica Physics, University of Chicago, Chicago, IL 6637, USA 17 Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada 18 Schoo of Physics and Astronomy, University of Minnesota, Minneapois, Minnesota 55455, USA 19 Department of Physics, Enrico Fermi Institute, University of Chicago, Chicago, IL 6637, USA 2 Department of Physics, University of Caifornia, Berkeey, CA 9472, USA 21 Corresponding author: jetoan@stanford.edu 1. INTRODUCTION Current experiments are producing ow noise maps of the poarization of the cosmic microwave background (CMB) radiation abe to constrain modes of infation and measure B- modes from gravitationa ensing. These experiments incude BICEP2, the Keck Array, POLARBEAR, SPTPOL, ACTPo, and Panck (BICEP2 Coaboration I 214; Keck Array and BICEP2 Coaborations VI 216; Hanson et a. 213; Poarbear Coaboration 214; van Engeen et a. 215; Panck Coaboration I 215). These experiments do not measure the CMB over the entire sky for a variety of reasons. Gaactic foregrounds prevent any experiment from producing a map of the CMB over the entire sky. Any ground or baoon based experiment has a imited view of the fu sky. Some experiments, incuding BICEP2 and the Keck Array, choose to observe a imited fied of view to increase map depth over a sma region of sky or choose to fiter their data so that the maps incompetey measure the modes within the fied. The abiity to uniquey separate a inear poarization fied into E and B-modes is critica for measuring gravitationa waves using the B-mode poarization. This separation aows the distinction to be made between E-modes created by scaar perturbations and B-modes coming from tensor perturbations (Kamionkowski et a. 1997; Zadarriaga 1998). Unfortunatey, the unique decomposition into E and B is ony possibe for maps of the fu sky. Maps containing a imited view of the sky, or an incompete measurement of the true sky modes, are said to suffer from E/B eakage. E to B eakage is defined as measured power for a particuar B-mode estimator whose source is true sky E-mode power. B to E eakage is eakage of power in the opposite direction, but in practice it is ess of a concern for CMB measurements due to the much fainter B-mode signa. E/B eakage refers to both

3 2 types of eakage. There are severa ways to mitigate the effect of E/B eakage in anaysis. Fu pixe-space ikeihood methods in principe can optimay separate E and B contributions for any given map. These have been appied mainy to maps of reativey modest pixe count, incuding many eary detections of CMB poarization (for exampe, Kovac et a. 22; Readhead et a. 24; Bischoff et a. 28). Current anayses more commony appy fixed estimators of E and B power spectra to observed CMB poarization maps. The simpest way to correct such estimators for eakage is to run an ensembe of simuations through the anaysis and subtract the mean eve of eakage in the anguar power spectrum. However, the sampe variance from the eaked power remains and contributes to the fina uncertainty of measured power in each anguar power spectrum bin, imiting an experiment s abiity to measure B-modes regardess of its instrumenta sensitivity. For many experiments, incuding BICEP2 and the Keck Array, the sampe variance of the eaked E-modes is comparabe to the instrumenta noise and is a significant contribution to the uncertainty in the B- mode power spectrum. Soutions to this probem rey on the fact that for most B- mode science it is not necessary to cassify a the modes in the measured poarization fied. Instead, it is sufficient to find subspaces that are caused by either E or B and ignore the modes whose source cannot be determined. There are a number of pubished methods that attempt this goa. Smith (26) presents an estimator that does not suffer from E/B eakage arising from partia sky coverage. This method has been incorporated into the Xpure and S 2 HAT packages (Grain et a. 29), and the BICEP2 and Keck Array anaysis pipeine contains an option in which this agorithm is impemented. However, many experiments, incuding BICEP2 and the Keck Array, produce maps in which some modes have aso been removed by fitering. The estimator presented in Smith (26) does not prevent fitered modes from creating E/B eakage. Another method, presented in Smith & Zadarriaga (27), accounts for incompete mode measurement in partia sky maps. However, we have found this method to be computationay infeasibe for the BICEP2 and Keck Array observing and fitering strategy. For BICEP2 and the Keck Array, we deveoped a new method for distinguishing true sky B-mode poarization from the eaked E-modes in the observed maps. The method extends the work of Bunn et a. (23) and appies it to a rea data set. It is a standard component of the BICEP2 and Keck Array anaysis pipeine and effectivey eiminates the uncertainty created by E/B eakage. The method reduces the fina uncertainty in the measured BB power spectrum of the BI- CEP2 resuts (BICEP2 Coaboration I 214) by more than a factor of two, compared to anaysis done with the Smith (26) method. The method resuts in a arger improvement for the anaysis of the combined BICEP2 and Keck Array maps (Keck Array and BICEP2 Coaborations VI 216), where the noise eves are ower. The organization of this paper is as foows: Section 2 provides an abbreviated background of a poarization fied on a sphere, decomposition into spin-2 spherica harmonics, and anayticay defines E and B-modes. Section 3 outines the eigenvaue probem used in the matrix based E/B separation. Section 4 describes how an observation matrix is created in the BICEP2 and Keck Array anaysis pipeine. Section 5 describes constructing the signa covariance matrix and Section 6 uses the covariance matrix to sove the eigenvaue probem and find purification matrices. Section 7 prensents resuts of matrix based E/B separation in the BICEP2 data set. Concuding remarks are offered in Section 8. Uness otherwise stated, we adopt the HEALPix poarization convention 2 and work in J2. equatoria coordinates throughout this paper. Bod font etters and symbos represent vectors or matrices, even when containing subscripts, in which case the subscript is meant to designate a new matrix or vector. Norma font etters and symbos represent scaar quantities. 2. E AND B-MODES FROM A POLARIZATION FIELD This section demonstrates the decomposition of a poarization fied on the fu sky into E and B-modes. Much of the discussion foows Zadarriaga & Sejak (1997) and Bunn et a. (23) Fu sky The vaues of the Stokes parameters Q and U for a particuar ocation on the sky are dependent on the choice of coordinate system. By rotating the oca coordinate system, Q is rotated into U and vice versa. Under rotation by an ange φ, the combinations Q + iu and Q iu transform as: (Q + iu) = e 2iφ (Q + iu) (Q iu) = e 2iφ (Q iu). (1) The T, Q, and U fieds can be expressed as sums of spin weighted spherica harmonics. Whie the temperature anisotropies can be broken down into spin- harmonics, the poarization fied of Q and U must be expressed in terms of spin-2 spherica harmonics (Godberg et a. 1967): T (r) = m (Q + iu)(r) = m (Q iu)(r) = m a T m ( Y m (r)) a +2,m ( +2 Y m (r)) a 2,m ( 2 Y m (r)), (2) where ±2 Y m are the spin-2 case of spin weighted spherica harmonics, and the spin- case are the norma spherica harmonics, Y m. Since Q+iU and Q iu are affected by rotations of the coordinate system, it is convenient to express the coefficients of the spin-2 spherica harmonics using a set of coordinate independent scaar a E m coefficients and pseudo-scaar a B m coefficients: a E m (a +2,m + a 2,m )/2 a B m i(a +2,m a 2,m )/2. (3) We aso define two combinations of spin-2 spherica harmonics: X 1,m ( +2 Y m + 2 Y m )/2 X 2,m ( +2 Y m 2 Y m )/2. (4) 2 htm

4 Matrix based E/B Separation 3 We can use the coefficients in Equation 3 and the combinations in Equation 4 to construct rea space forms of T, Q, and U fieds, according to Equation 2: T (r) = m Q(r) = m U(r) = m a T m( Y m (r)) ( a E m X 1,m (r) + ia B mx 2,m (r) ) ( a B m X 1,m (r) ia E mx 2,m (r) ). (5) Using these reations, we can write the poarization fied as a vector: ( ) Q(r) P(r) U(r) = a E m X 1,m (r) + ia B m X 2,m(r) a B m X 1,m(r) ia E m X 2,m(r) m = ( ) ( ) a E X1,m (r) m ix 2,m (r) + a B ix2,m (r) m X 1,m (r) m = a E m Ym(r) E + a B mym(r) B, (6) m where Ym E and Y m B have been introduced and defined in the ast step. On the fu sphere, Ym E and Y m B are orthogona: Ym(r) E Y B m (r)ds =, (7) S 2 for a, and m,m Orthogonaity of pure E and pure B The inner product of two poarization fieds is defined as: P P P P dω, (8) Ω where Ω is the manifod on which the poarization fied is defined: for the fu sky it is the ceestia sphere. In pixeized maps, the vector space of a poarization fied has a finite dimension: twice the number of pixes in the map. As demonstrated in Equation 7, E and B-mode poarization fieds on the fu sky are orthogona. However, experiments produce Q/U maps of portions of the sky, and often fiter spatia modes out of these maps. We define the term observed maps or modes to refer to these incompete measurements of the true sky. The spaces of observed E-modes and B-modes are nonorthogona. The overapping subspace between the two is caed the ambiguous space. We cannot te whether signa in the ambiguous subspace came from fu sky E-modes or fu sky B-modes. The soution is to decompose vector fieds on an observed manifod into three subspaces: pure E-modes, pure B- modes, and ambiguous modes. Pure E and B-modes are subspaces of the poarization vector space of a particuar manifod, defined as: A pure B-mode is orthogona to observed E-modes. A pure E-mode is orthogona to observed B-modes. Therefore, a pure B-mode is one that has no E to B eakage: neither pure E-modes nor ambiguous modes contribute to it. 3. HOW MATRIX BASED E/B SEPARATION FINDS PURE E AND PURE B A pure B-mode on an observed manifod is defined in Section 2.2 as being orthogona to observed E-modes: P E b =. (9) The vector b is any inear combination of modes in the subspace of the pure B-modes. For pixeized maps, b contains Q and U vaues for each of the pixes in the map, and P E is the pixeized version of the E-mode spherica harmonics. It is usefu to mutipy the above equation by its conjugate transpose, and sum over and m, so that we have a scaar representing the degree of orthogonaity: ) b ( m a E m a E my E my E m b =. (1) We have freedom to choose the power spectrum, C EE a E m m ae, which is incuded in the covariance matrix, CE : C E m = C EE Y E my E m. (11) We note that this product is the 2 2 Q,U covariance bock in the signa covariance matrix: C E = P E ( P E) ( Q E = i Q E j Q E i Uj E ) U E U E, (12) i Q E j i U E j where the superscript denotes the E-mode component of the fu sky poarization fied and i, j designate pixes in the map. We can evauate the covariance matrix for a particuar set of pixes and a chosen spectrum. By soving a generaized eigenvaue equation of the form: C B x i = λ i C E x i, (13) and seecting eigenmodes corresponding to the argest eigenvaues, we can find eigenmodes b that are neary orthogona to E-modes and therefore approximate pure B. Eigenmodes corresponding to the smaest eigenvaues approximate pure E. This method is a natura extension to the signa to noise truncation discussed in Bond et a. (1998) and Bunn & White (1997) and appied in Kuo et a. (24). The specific appication to E and B-modes was first discussed in Bunn et a. (23). We say the modes approximate pure E and pure B-modes because the degree of orthogonaity is proportiona to the magnitude of the eigenvaues. The eve of orthogonaity is discussed further in Section 6. However, for the remainder of the paper, we wi use the terms pure B and pure E to refer to the argest and smaest eigenmodes of Equation 13, despite the fact that their inner product is not identicay zero. Now suppose that the true sky poarization fied, P, is transformed into an observed poarization fied, P, by a rea space inear operation, R: P E = RP E = R m a E my E m. (14)

5 4 Throughout this paper, transformations into observed quantities are indicated by the incusion of a tide over the variabe, in the above equation, P P. The operator R wi typicay represent fitering operations necessary to suppress noise and/or systematics pus an apodization of the resuting observed maps. The condition for pure E and pure B must be the same after mutipying by R. We sti demand that the vectors of pure B be orthogona to a those in the E space, which incudes both the pure E-modes and the ambiguous modes: ( ) R m a E my E m b =. (15) We create a basis of pure E and pure B-modes by soving the eigenvaue probem with the covariances of the form: ) C E = R ( m C B = R ( m so that Equation 13 becomes: C EE Y E my E m C BB Y B my B m ) R R, (16) C B x i = λ i C E x i. (17) In the simpest case, the matrix R is an apodization window and fied ony on its diagona. However, Equation 15 does not necessitate that the rea space operator be a diagona matrix. Any anaysis steps that can be expressed as inear operations can be incuded. In BICEP2 and Keck Array anaysis a number of fitering operations are typicay performed during the map making process. In the next section the matrix R corresponding to these operations is derived. The practica impementation of a soution to the eigenvaue equation is discussed in Section OBSERVATION MATRIX The matrix R transforms an input map, m, a vector of the true sky poarization fied, into a vector of the observed map, m. If the matrix R represents the apodization and inear fitering of an anaysis pipeine, it is defined to be the observation matrix for a particuar experiment. This choice of R ensures the eigenspaces of Equation 17 are pure E and B for the observed map. This section describes how the observation matrix is computed for BICEP2 and the Keck Array. The steps in constructing the observation matrix mirror functions in the data reduction pipeine that was originay deveoped for QUAD (Pryke et a. 29) and ater used in the BICEP1 (BICEP1 Coaboration 214), BICEP2 (BICEP2 Coaboration I 214), and Keck Array (Keck Array and BI- CEP2 Coaborations V 215) anayses. This pipeine consists of a MATLAB ibrary of procedures which constructs maps, incuding severa fitering steps, from rea data or simuated timestream data for a given input sky map. The fitering operations performed sequentiay in the standard pipeine incude data seection, poynomia fitering, scan-synchronous signa subtraction, weighting, binning into map pixes, and deprojection of eaked temperature signa. To construct the observation matrix, matrices representing each of these steps are mutipied together to form a fina matrix that performs a of the operations at once. Since each of the operations is inear, the observation matrix is independent of the input map. Therefore, the same matrix can be used on any input map and wi perform the same operations as the standard pipeine. If the combined matrix of timestream operations is V, then transforming a timestream, d, into an observed map, m, is simpy: m = Vd. (18) The signa component of a timestream can be generated from an input map, m, using a matrix that contains information about the pointing and orientations of the detectors, according to the equation d = Am. The observation matrix, R, is given by the product of V and A: m = VAm (19) = Rm. (2) It is not necessary for the input maps and observed maps to share the same pixeization scheme, since the observation matrix can easiy be made to transform between the two Input HEALPix maps We choose a HEALPix pixeization scheme (Gorski et a. 25) for the input maps, m, because it has equa area pixes on the sphere and is widey used in the cosmoogy community. A true sky signa is represented by the map m o = Txy,Q o o xy,uxy, o where (x,y) are the (RA,Dec) coordinates of the map. Using synfast 3, the unobserved input map is convoved with the array averaged beam function, B, constructed from measurements of the beam function of a detectors in the array: Txy Q xy U xy = B m o = B xy T xy o. (21) Q o xy Uxy o The input map vector is found by reforming the beam convoved two dimensiona map into a one dimensiona vector, m, of ength 3 j, where j = 1...n p, for n p pixes in the input map: m Tj Q j U j. (22) 4.2. BICEP2 and Keck Array scan strategy The observing strategies for BICEP2 and the Keck Array are very simiar and borrow heaviy from BICEP1. A three experiments target a region of sky centered at a right ascension of degrees and decination of degrees. A detaied description of the scan strategy is contained in BICEP2 Coaboration II (214). Hafscans: During norma observations, the teescope scans in azimuth at a constant eevation. The scan speed of 2.8 deg s 1 in azimuth paces the targeted mutipoes of 2 < < 2 at tempora frequencies ess than 1 Hz. Each scan covers 64.2 degrees in azimuth, at the end of which the teescope stops and reverses direction in azimuth and scans back across the fied center. A scan in a singe direction is known as a hafscan. 3 synfast is a program in the HEALPix suite that renders sky maps from sets of input a m s.

6 Matrix based E/B Separation 5 Scansets: Hafscans are grouped into sets of 1 hafscans, which are known as scansets. The scan pattern deiberatey covers a fixed range in azimuth within each scanset, rather than a fixed range in right ascension. Over the course of the 5 minute scanset, Earth s rotation resuts in a reative drift of azimutha coordinates and right ascension of about 12.5 degrees. At the end of each scanset the eevation is offset by.25 degrees, and a new scanset commences. The teescope steps in.25 degree eevation increments between each scanset. A observations take pace at 2 eevation steps, with a boresight pointing ranging in eevation between 55 and degrees. The geographic ocation of the teescope, near the South Poe, means that eevation and decination are approximatey interchangeabe. Phases: Scansets are grouped together into sets known as phases. For BICEP2 and the Keck Array, CMB phases consist of ten scansets, comprising 9 hours of observations. CMB phases are grouped into seven types and each type has a unique combination of eevation offset and azimutha position. Schedues: The third degree of freedom in the BICEP2 and Keck Array teescope mounts is a rotation about the boresight, referred to as deck rotation. The poarization anges reative to the cryostats are fixed, so rotating in deck ange aows detector pairs to observe at mutipe poarization anges. A schedue typicay consists of a set of phases at a particuar deck ange. The deck ange is rotated between schedues. There is typicay one schedue per fridge cyce, occurring every 3 days for BICEP2 and 2 days for the Keck Array Reationship between timestreams and T,Q,U The BICEP2 and Keck Array detectors consist of pairs of nominay co-pointed, orthogona, poarization sensitive phased array antennas couped to TES boometers (BI- CEP2/Keck and Spider Coaborations 215). The signa in the timestream from detector A is: τ A t = T t + cos(2ψ A t )Q t + sin(2ψ A t )U t, (23) where T t,q t,u t are the Stokes parameters of the beam convoved sky signa for timestream sampe t. A timestream consists of n t time ordered measurements of the sky, t = 1...n t. Ψ A is the ange the A antenna makes with the Q,U axis on the sky. For the HEALPix poarization convention, this axis is a vector pointing towards the north ceestia poe. The reative gain normaized A timestream is summed and differenced with the normaized timestream from its orthogona partner B : s t = 1 2 (τ t A + τt B ) = T t + α t + Q t + β t + U t d t = 1 2 (τ t A τt B ) = αt Q t + βt U t. (24) The variabes α and β are defined by: α ± t 1 2 β ± t 1 2 cos(2ψ A t ) ± cos(2ψ B t ) sin(2ψ A t ) ± sin(2ψ B t ), (25) where Ψ B is the ange the B antenna makes with the Q,U axis on the sky. Assuming that A and B are perfecty copointed and orthogona, the signa portion of the timestream vectors can be described with a transformation, A t j, from the input map pixe (with index j) to the timestream sampe (with index t): s t = A t j T j d t = α t A t j Q j + β t A t j U j, (26) where the terms α + and β + in the pair sum timestream cance due to the orthogona orientation of the A and B detectors. The signa-pus-noise timestreams, in vector notation, are: s = 1 2 (n A + n B ) + AT (27) d = 1 2 (n A n B ) + α β A Q A U. (28) where n A and n B are the time ordered noise components of detector A and detector B, assuming the noise is uncorreated with the pointing of the detector pair. For signa ony simuations, n A and n B can be ignored. The matrix α β contains the information about the orientation of a pair s antennas reative to Q and U defined on the sky. We ca it the detector orientation matrix. The combination: α β A A (29) transforms input Q,U maps into a pair difference timestream. α β is constructed from two diagona matrices, α and β, which are fied with the sine and cosine of the detector orientations at each time sampe. A graphica representation of the detector orientation matrix is shown in Figure 1. (Additiona steps accounting for poarization efficiency and pair non-orthogonaity are absorbed into a normaization correction to the pair difference timestream.) n t n t FIG. 1. Detector orientation matrix, α β. The matrix is ony fied on the diagonas of the two sub-bocks, α and β. n t 4.4. Timestream forming matrix, A The matrix A = A t j represents the timestream forming matrix for a detector pair. It transforms the input temperature map, T j, into the signa component of the pair sum timestream, s t. A graphica representation of the timestream forming matrix is shown in Figure 2. To create timestreams with smooth transitions at pixe boundary crossings, the input maps shoud have a resoution higher than the spatia band imit imposed by the beam function. For this reason, N side =512 HEALPix maps are used,

7 6 whose pixes have a Nyquist frequency 2 the band imit of the BICEP2 and Keck Array 15 GHz beam function. The current BICEP2 and Keck Array CMB observations fa within the region of sky bounded in right ascension by 3 h 4 m < α < 3 h 4 m and in decination by 7 < δ < 45. This region contains n p = 111,593 pixes in an N side =512 HEALPix map. The number of sampes in a scanset is typicay n t 43,. The simpest form of A performs nearest neighbor interpoation of the HEALPix maps, in which case A is (n t n p ) and is fied with ones where the detector pair is pointed and zeros otherwise. A more sophisticated form of A performs Tayor interpoation on the HEALPix map, in which case A is (n t λ(λ+1) 2 n p ), where λ is the order of the Tayor poynomia used in interpoation. In this case, A is a matrix that performs Tayor interpoation, aowing sub-pixe accuracy to be recovered from the input map, and m must aso contain derivatives of the true sky temperature and poarization fied. This matrix is used to buid the deprojection tempates in Section 4.1 but is not used for forming timestreams because it increases the dimensions of the observation matrix, making the computation of the observation matrix more difficut. FIG. 2. Timestream forming matrix, A: fied eements of the matrix that takes HEALPix maps to timestreams. This matrix contains the pointing of a singe detector pair over one scanset within a N side =512 HEALPix map. The pattern of the fied eements is determined by the particuar HEALPix pixe indexing scheme. There are n t fied entries, consisting of a 1 for each timestream sampe. Note that athough the above image appears to have mutipe pointing ocations for a singe timestream sampe, n t, this is merey a resut of imited resoution in the image. The timestream forming matrix contains ony one HEALPix pixe ocation for each time sampe Poynomia fitering matrix, F To remove ow frequency atmospheric noise from the data, a third order poynomia is fit and subtracted from each hafscan in the timestreams. Since each hafscan traces an approximatey constant eevation trajectory across the target fied, the poynomia fiter removes power ony in the right ascension direction of the maps. In mutipoe,, the third order poynomia fiter typicay ros off power beow < 4. This can be represented by a fitering matrix, F, which is bock diagona with the bock size being the tempora ength of a hafscan. Each bock is composed of a matrix: F = I V(V V) 1 V, (3) where I is the identity matrix and V is the same third order Vandermonde matrix for each hafscan of equa ength. The Vandermonde matrix is defined as: V t j = x j 1 t, (31) where j = 4 for a third order fiter and x t are the coordinate ocations. For BICEP2 and the Keck Array, x t is a vector of the reative azimutha ocation of each sampe in the hafscan. A representation of the poynomia fitering matrix is shown in Figure 3. FIG. 3. Poynomia fitering matrix, F, showing the fied eements of the matrix. The matrix is very sparse and is bock diagona with bocks the size of a hafscan ( 44 sampes) Scan-synchronous signa remova matrix, G Scan-synchronous subtraction removes signa in the timestreams that is fixed reative to the ground rather than moving with the sky. These azimuthay fixed signas are decouped from signas rotating with the sky by the scan strategy, which observes over a fixed range in azimuth as the sky sides by (as described in Section 4.2). A tempate of the mean azimutha signa is subtracted from the timestreams for each scan direction. This procedure can be represented as a matrix operator, referred to as a scan-synchronous signa matrix. The mean azimutha signa can be found using a matrix X=X tt, for which each row is ony fied for entries containing the same azimutha pointing as the diagona entry. The scansynchronous signa matrix subtracts off the mean azimutha signa: G = I X, (32) where I is the identity matrix. A graphica representation of the scan-synchronous signa remova matrix is shown in Figure 4. Note that whie the F matrix is bock diagona and sparse, and the G matrix is sparse, once the two are combined, the resuting fiter matrix is neither sparse nor bock diagona, making matrix operations more computationay demanding. FIG. 4. Scan-synchronous signa remova matrix, G, showing the fied eements. The scan-synchronous signa matrix is sparse Toepitz, with off diagona components that subtract the average scan-synchronous signa for one of the two scan directions in a scanset Inverse variance weighting matrices, w ± The timestreams are weighted based on the measured inverse variance of each scanset. Pair sum and pair difference are weighted separatey from weights cacuated from the two

8 Matrix based E/B Separation 7 timestreams, w + and w. The scheme assigns ower weight to particuary noisy channes and periods of bad weather. This choice of weighting is not a fuy optima map maker (Tegmark 1997), but instead represents a practica soution that avoids cacuating and inverting a arge noise covariance matrix. The weighting is represented by a matrix whose diagona is fied with the vector w + =w + tt for pair sum and w =w tt for pair difference, shown in Figure 5. be accounted for. Figure 2 shows A for a singe detector over a scanset and Figure 7 shows Λ for a singe detector over a scanset. n t FIG. 5. Weighting matrices w ±, showing the fied eements. The weighting matrices are zero except on the diagona, where they contain the weights based on the inverse variance of the timestream Fitered signa timestream generation n t Ignoring noise and combining a the operators of Sections 4.5, 4.6 and 4.7, the sum and difference 1 timestreams in Equation 26 are tranformed to the fitered timestreams: s = w + GFAT d = w GF α β A Q A U, the second of which is graphicay represented in Figure 6. (33) FIG. 6. Matrix generation of simuated timestreams corresponding to Equation Pointing matrix, Λ The timestream quantities s and d are converted to maps by the pointing matrix, Λ = Λ it. If the pixeization of the input maps were identica to the output maps, the pointing matrix woud be the transpose of the timestream forming matrix: Λ = A. As discussed in Section 4.1, the input maps are HEALPix N side =512. However, the BICEP maps instead use a simpe rectanguar grid of pixes in RA and Dec: the size of the pixes is.25 degrees in Dec, with the pixe size in RA set to be equivaent to.25 degrees of arc at the mid-decination of the map, resuting in = 23, 6 pixes. If the BICEP maps are naivey used as fat maps then projection distortions are inherent. However, note that the A and Λ matrices together fuy encode the mapping from the underying curved sky to the observed map pixes, aowing such distortions to FIG. 7. Pointing matrix, Λ: fied eements of the pointing matrix that transforms timestreams to an observed map in the BICEP pixeization. This matrix contains the mapping between the pointing of a singe detector pair over one scanset and the output map pixes. There are 23,6 pixes in a BICEP map, denoted as ñ p. There are n t fied entries, consisting of a 1 for each timestream sampe. Each eg of the zigzag pattern corresponds to a hafsan within the scanset, where the teescope is scanning back and forth at a fixed eevation. The pointing matrix for a singe detector pair can be used to construct a pair sum pairmap: m T = Λs = Λw + GFAT. (34) The pair difference timestream is converted into pairmaps using two copies of the pointing matrix. The two pair difference pairmaps correspond to inear combinations of Stokes Q and U: mα m β = Λ α Λ β w GF α β A Q A U. (35) For ater convenience in abbreviating this equation, we define: Λ α P Λ β w GF α β A A. (36) 4.1. Deprojection matrix, D A potentia systematic concerning poarization measurements is the eakage of unpoarized signa into poarized signa. In the case of CMB poarization, this takes the form of the reativey bright temperature anisotropy eaking into the much fainter poarization anisotropy. The eakage is caused by imperfect differencing between the orthogona pairs of detectors. The beam functions can be we approximated by eiptica Gaussians, the difference of which correspond to gain, pointing, width and eipticity (Hu et a. 23; Shimon et a. 28). The BICEP2 and Keck Array pipeine removes eaked temperature signa from the poarization signa using inear regression to fit eakage tempates to the poarization data. This method aows the beam mismatch parameters to be fitted directy from the CMB data itsef, rather than reying on externa caibration data sets, and is robust to tempora variations of the beam mismatch. The tempates used in the regression are constructed from Panck 143 GHz temperature maps 4. These maps contain both CMB and foreground emission at approximatey the BICEP2 band. The noise in Panck 143 GHz is significanty subdominant to the CMB temperature anisotropy. For a fu description and derivation of the deprojection technique, see Aikin 4 For the Keck Array 95 GHz and 22 GHz bands, we use Panck 1 GHz and 217 GHz maps.

9 8 (213), Sheehy (213), BICEP1 Coaboration (214), and BICEP2 Coaboration III (215). In this section, the entire deprojection agorithm is re-cast as a matrix operation. For the purposes of generating deprojection tempates, we use a timestream forming matrix that performs Tayor expansion around the nearest pixe center to the detector pointing ocation. The Tayor interpoating matrix produces higher fideity timestreams than a nearest neighbor matrix. This is important for the deprojection agorithm since sma dispacements in beam position are responsibe for the systematic effect that is removed. Without Tayor interpoation, pixe boundary discontinuities introduce noise and imit the effectiveness of deprojection. A Tayor poynomia of order λ has λ(λ+1) 2 terms, so the dimensions of the input map vector for second order interpoation is 1 6n p. Using Equation 21, the input maps are convoved with the array averaged beam function. The smoothing is done using synfast, which contains the abiity to output derivatives of the temperature (and poarization) fied. Because the beam is appied first, the output derivatives are ess noisy than they woud be in the raw maps. The maps are of the form: T θ T Θ = φ T θθ T, (37) φφ T θφ T where θ and φ are the HEALPix map s atitude and ongitude. Using the temperature map and its derivatives, we can find the Tayor interpoated temperature timestream by repacing A with a Tayor interpoating matrix, A : A = A A θ A φ A θ2 2 A φ2 2 A θ φ, (38) where θ and φ are diagona matrices giving the difference between the detector pair s pointing and the nearest HEALPix pixe center. A differentia beam generating operator is appied to the timestreams to create differentia beam timestreams. For exampe, the differentia gain timestream is just the beam convoved temperature fied: d δg = δ g A Θ, (39) where the fit coefficient for the gain mismatch is δ g. The differentia pointing components are found from the first derivatives of the temperature fied with respect to the foca pane coordinates, x and y: d δx = δ x x A Θ (4) d δy = δ y y A Θ, (41) where δ x and δ y are the differentia beam coefficients and x and y are partia differentia operators with respect to the foca pane coordinates. Further detais of this cacuation and derivations for other beam modes are discussed in Appendix C of BICEP2 Coaboration III (215). The differentia beam timestreams are transformed into maps anaogousy to Equation 35, creating a pairmap tempate, T j, for each differentia beam mode, j. The tempate pairmaps for each scanset, S, are then coadded over phases. For instance, the tempate pairmap for differentia gain, is T 1 : T 1 = S phase S ( Λ α Λ β w GF α β ) A Θ A Θ. S (42) A matrix performing weighted inear east-squares regression against rea pairmaps produces the fitted coefficients for each of the differentia beam modes, c δ g,δ x,δ y,...: where c = ( T (W ) 1 T mα m β ) 1 T (W ) 1 mα m β, (43) is the rea data pairmap coadded over a phase, T is a vector of pairmap tempates, and W is the pair difference weight map, created from the weight matrix according to: W = S phase S ( ) Λα α w Λ Λβ β w Λ. (44) The pairmap tempates weighted by c are then subtracted from the rea data pairmap: mα mα m = T c. (45) β m β This process takes the form of a matrix operator that incudes each of the beam systematics, giving the deprojection matrix: ( ) D I T T 1 (W ) 1 T T (W ) 1. (46) Deprojected pairmaps are then found according to: S phase mα Q m = D P S β U. (47) The regression in Equation 43 operates simutaneousy over a the modes to be deprojected. Because the tempates for different modes are not in genera orthogona, the coefficient for each mode depends on the fu set of modes. Therefore, the subtraction in Equation 46 must incude the same mode ist used in the regression in Equation 43. If the regression incuded more modes than the subtraction step, the regression woud have extra degrees of freedom. This coud resut in incompete remova of eakage signa. We avoid this possibiity by deferring the regression step unti immediatey before the subtraction step, and expicity using the same mode seection for both. As in the standard pipeine, we deproject for each detector pair, after coadding scanset to phases. To reduce the computationa demands, the matrix deprojection pairmaps have additionay been coadded over scan direction, whereas the standard pipeine performs regression separatey for eft going and right going scans. This is the ony difference between simuations run with the standard pipeine and those cacuated from the observation matrix and eads to a negigibe difference, see Figure 11 and Figure 12. The deprojection matrix made for a phase is ess sparse than one made for a scanset because over the course of a phase a particuar pair wi observe a arger range of eevation than it S S

10 Matrix based E/B Separation 9 woud in a scanset. The fied eements of the matrix D, for one pair across one phase, is shown in Figure 8. ñp ñp ñ p FIG. 8. Deprojection matrix, D: fied eements of the deprojection matrix for one pair, for one phase of data. The overa dimensions are 2ñ p 2ñ p, twice the number of pixes in a BICEP map Coadding over scansets and detector pairs to form the observation matrix An observed temperature map, T, can be found by summing the pair sum pairmaps temporay over scansets (S) and over detector pairs (P): T = P,S The matrix performing this transformation 1 is defined as R TT, ñ p Λw + GFAT. (48) where the prime indicates the apodization comes from the inverse variance of the pair sum timestream, w +. The fina apodization is appied in Section The transformation from pair difference pairmaps to Q,U maps depends on the detector orientations during the observations. This transformation reies on an inversion of a 2 2 detector orientation matrix. We wi now derive the matrix that performs this transformation. Ignoring fitering, the pair difference timestream is found using the timestream forming matrix, A t j : d t = 1 2 (τ t A τt B ) = αt A t j Q j + βt A t j U j. (49) Forming inear combinations of the pair difference timestream, α t d t α βt d = t αt A t j αt βt A t j Q j t αt βt A t j βt βt A t j U, (5) j and appying the pointing matrix, Λ, the vectors α d and β d are binned into map pixes, i. At this point we coadd over scansets and detector pairs, and appy a weighting, w, equa to the inverse of the variance of the timestreams during a scanset: where the t index has been summed over to find each of the eements in the 2 2 matrix on the right hand side and A t j has been repaced by Λ ti so the equation now determines the Q,U vaues in the observed map, Q i,u i. There is one 2 2 matrix inversion performed for each pixe, i, in the observed map. In other words, one vaue of e i, f i, and g i is computed for each pixe in the observed map, and fied into the i-th diagona eement of e, f and g. The pairmaps m α and m β are transformed into Stokes e f Q,U by mutipying by f g. Observed Q,U maps are found according to: Q Ũ = e f Q f g P P,S U, (53) P,S where P was defined in Equation 36. If the sum of P matrices coud be inverted, it woud be possibe to use this inverse to recover an unbiased estimate of the origina Q and U. However, P is singuar because it incudes poynomia fitering and scan-synchronous signa subtraction, which competey remove some modes that were present in the origina maps. We therefore use instead the matrix defined by the matrix inversion in Equation 52, which does not incude these fitering operations. Even the inversion in Equation 52 is singuar uness the coadded data contains observations at mutipe detector anges, Ψ t. Observations at mutipe detector orientations are made through deck rotations or by coadding over receivers in different orientations. As described in Section 4.2, deck rotations occur between phases, so coadding over phases makes the matrix invertibe. Incuding deprojection, Equation 53 becomes: Q Ũ = e f f g ( D P S phase S P S ) P Q U. (54) This represents the entire Q,U map making process for signa simuations: from input maps to observed maps, incuding fitering operations. It can be summarized as: Q R = QQ R QU Q R UQ UU R U. (55) Ũ Non-apodized observation matrix As constructed, the matrix, R, contains an apodization based on the inverse variance of the timestreams, w + and w. We can, however, choose to remove this apodization, producing maps with equa weight across the fied in units of µk. We construct the quantities: W + = P,S ( Λw + Λ ), (56) P,S P,S Λit wt αt d t Λ it wt βt d = t ( Λit wt αt αt A t j Λ it wt αt βt A t j P,S Λ it w t α t β t A t j ) Q j Λ it wt βt βt A t j U. j (51) We invert the matrix on the right hand side of Equation 51 to compute a matrix that generates Q and U maps: ( ei f i Λit w f i g t αt αt Λ ti Λ it wt αt βt Λ ) 1 ti i Λ it wt αt βt Λ ti Λ it wt βt βt Λ, (52) ti P,S W e f ( ) Λα = α w Λ f g Λβ β w Λ P,S P,S, (57) and use these to remove the apodization from the observation matrix, soving for the non-apodized observation matrix, R: R T T = ( W +) 1 R TT (58)

11 1 RQQ R QU R UQ R UU = (W ) 1 R QQ R QU R UQ R UU Seecting the observation matrix s apodization (59) Using the non-apodized observation matrix of Section 4.12, we can create an observation matix with an arbitrary apodization, Z. The matrix R is constructed as foows: RQQ R QU R UQ R = UU R TT = ZR T T (6) Z RQQ R QU Z R UQ R UU (61) A sensibe choice for the apodization, Z, may be the inverse variance mask we removed in Section 4.12, or a smoothed version thereof. However, there is freedom to choose any apodization at this point, and this may prove usefu in joint anayses with other experiments, where the anaysis combines maps with ow noise regions in sighty different regions of the sky Summary We have constructed a matrix, R, which performs the inear operations of poynomia fitering, scan-synchronous signa subtraction, deprojection, weighting and pointing. R has dimensions (3ñ p,3n p ) where ñ p is the number of pixes in the BICEP map and n p is the number of pixes in the input HEALPix map. Using the observation matrix, the entire process of generating a signa simuation from an input map is: T Q = Ũ R TT R QQ R QU R UQ R UU T Q. (62) U Here the off diagona terms, R QU and R UQ, exist because the fitering operations are performed on pair difference timestreams, which are a combination of Q and U. The deprojection operator contains regression against a temperature map that must be chosen before constructing the observation matrix. The deprojection operator is a inear fitering operation, and it ony removes beam systematics arising from one particuar temperature fied. One coud in principe appy the deprojection operator to Q,U maps corresponding to a different temperature fied. The operator woud remove the same modes from the poarization fied, but these modes woud not correspond to those which had been mixed between T and Q,U by beam systematics (or T E correation). When constructing an ensembe of E-mode reaizations for use in Monte Caro power spectrum anaysis, the T E correation and the fixed temperature sky force us to buid constrained reaizations. The ensembe of simuations a contain identica temperature fieds, so we cannot use them for anaysis of temperature, which is acceptabe because the focus of our anaysis is poarization. The ensembe does contain different reaizations of Q,U, constrained for the given temperature fied, and these can be used in Monte Caro anaysis of poarization. The detais of constructing these constrained input maps is the subject of Appendix A. Because the construction of R QQ, R QU, and R UU depends on a fixed temperature fied, the deprojection tempates can be thought of as numerica constants. R TT performs a separate fitering on the temperature fied that is argey decouped from the fitering of Q and U. To incude systematics that eak temperature to poarization, the terms R TQ and R TU woud in principe need to be non zero. However, so ong as the eakage corresponded to modes being removed by the deprojection matrix D the the deprojection eements in the R QQ, R QU, and R UU bocks woud ensure that the output Q,U maps were identica. Athough the nomina dimensions of R are arge, our constant eevation scan strategy means that R is ony fied for pixes at roughy the same decination. This means that R is argey sparse, as shown in Figure 9. FIG. 9. Observation matrix, R: fied eements of the observation matrix for the BICEP2 3-year data set. T Q, TU, UT, and QT are empty because no T P eakage is simuated. The horizonta axis corresponds to the HEALPix pixeization and has 3 111,593 eements. The vertica axis corresponds to the BICEP pixeization, and has 3 23,6 eements. The matrix has ony 5% of its eements fied. Some intuition about the operations the observation matrix performs can be gained by potting a coumn of the matrix reshaped as maps see Figure 1. The coumn chosen in this case corresponds to a centra pixe in the observed fied. It shows how Q and U vaues in the observed map are sourced from a Q pixe in the HEALPix map. The bright pixe in the Q observed map corresponds to the ocation of the input Q. The effects of poynomia and scan-synchronous signa subtraction are visibe to the eft and right of the bright Q pixe. These two types of fitering are performed on scansets and are therefore confined to a row of pixes. Deprojection operates on phases, creating the effects seen at other decinations. Because a of these fitering operations are performed on pair difference data, which contains inear combinations of Q and U, signa in the observed U map can be created by signa in the input HEALPix Q map. This is why the U map in Figure 1 is non-zero Forming maps from rea timestreams We can form observed maps from the rea timestreams using the matrices constructed above: T rea = Z ( W +) 1 ( Λw + GFs ) (63) P,S P,S Q rea Ũ rea = Z (W ) 1 e f Z f g ( S phase ( Λ α D Λ β P S w GFd ) ). (64) It is important to note that the exact same matrices are used to process the rea data in Equation 64 as are used to construct the simuated maps in Equation 35. S P

12 Matrix based E/B Separation 11 Q 4 3 Standard Pipeine µk Decination deg U Right ascension deg asinh(µk) Observation Matrix µk FIG. 1. A singe coumn of the observation matrix R, for a HEALPix Q pixe near the center of our fied. The vaue of a singe input Q pixe affects both Q and U vaues in the observed map over the range of decinations covered in a phase Equivaence of observation matrix with standard pipeine The matrix formaism described above is sef-contained and compete in the sense that it contains the toos necessary to create rea data maps and simuated maps. We demand that the map making and fitering operations be identica between the standard pipeine and the observation matrix. It is straightforward to test this equivaence: simuated maps run through the standard pipeine must be identica to the maps found with the observation matrix. Figure 11 and Figure 12 show that the two match quite we, within a few percent over the mutipoes of 5 < < 35. The ack of a perfect match is due to the difference in deprojection timescae and because the standard pipeine uses N side =248 HEALPix input maps that are Tayor interpoated, whereas the observation matrix uses N side =512 input HEALPix maps with nearest neighbor interpoation. 5. SIGNAL COVARIANCE MATRIX, C The signa covariance matrix contains the pixe-pixe covariances of a map for a given spectrum of Gaussian fuctuations. The diagona entries contain the variance of each pixe, and each row describes the covariance of a given pixe with the other pixes in the map. For T,Q,U maps, the covariance matrix contains nine sub-matrices for the correations between T,Q, and U True sky signa covariance matrix A pixe on the sky at ocation i, has vaues of the Stokes parameters: x i T i Q i. (65) U i The 3 3 pixe-pixe covariance between two ocations on the sky, i and j, is given by: C i, j x i x j = R(α)M(r i r j )R(α). (66) The covariance matrix, M, is defined with the Q,U convention referenced to the great circe connecting the two points, i, j. For a particuar spectrum M depends ony on the dot product Difference 5 5 FIG. 11. Comparison of observed Q maps created by the observation matrix and the standard pipeine. The input map for both is from the same simuation reaization. There are sma differences due the difference in deprojection timescaes, input HEALPix map resoution, and interpoation. / 2π (+1)C BB / 2π (+1)C EE Mutipoe,.8 Theory.6 Standard Pipeine Observation Matrix.4 Fractiona diff Mutipoe, FIG. 12. Comparison of power spectra of maps created by the observation matrix and the standard pipeine. The input map for both is from the same simuation reaization, which differs from the theory curve for this particuar reaization in the BICEP fied. The two methods are fractionay the same to within a few percent over the mutipoes of interest, 5 < < 35. between the pixes, r i r j. M contains nine symmetric sub- µk

13 12 matrices: M(r i r j ) = T it j T i Q j T i U j Q i T j Q i Q j Q i U j U i T j U i Q j U i U j (67) The 3 3 matrix, R, is appied to rotate from this oca reference frame to a goba frame where Q,U are referenced to the North-South meridians. The ange between the great circe connecting any two points and the goba frame is given by the parameter α. R(α) = 1 cos 2α sin 2α sin2α cos2α (68) Changing the sign of α aows us to change the poarization convention from IAU to HEALPix (U to U), see Hamaker & Bregman (1996). We have chosen to use the IAU convention for BICEP2 and Keck Array covariance matrices. The true sky pixe-pixe signa covariance matrix for the Stokes Q,U parameters is derived in Kamionkowski et a. (1997); Zadarriaga (1998). To cacuate the covariances, we foow some of the suggestions in Appendix A of Tegmark & de Oiveira-Costa (21). We use the HEALPix ring pixeization, which aows the covariance to be cacuated simutaneousy for a pixes at a particuar atitude that are separated by the same distance. 5 This shortcut is expoited by simutaneousy cacuating a equidistant pixes for rows in the map that have the same atitude to within degrees. This approximation is much smaer than the 7 arcminute pixes in N side =512 maps, and the rounding error has been found to be insignificant Observed signa covariance matrix, C The observed signa covariance matrix contains the pixepixe covariance in the observed BICEP pixeized maps. Theoreticay, modifying the true sky signa covariance matrix is simpe: take the unobserved signa covariance of C of Section 5.1 and the observation matrix R from Section 4 and form the product: C = RCR. (69) This equation resuts in a symmetric positive definite matrix, which is rank deficient because of the fitering steps in the observing process. Unfortunatey, performing this mutipication is computationay demanding: The input C is a square matrix, with 3 111,593 eements on a side, corresponding to the eements of T, Q, and U. To reduce the memory requirements of the cacuation, we divide the covariance matrix, C, into row subsets and cacuate in parae. Once a row subset is cacuated, the observation matrix is immediatey appied to transform the HEALPix covariance to the observed map covariance, which reduces the dimensions of the covariance to the 23,6 pixes of the observed maps. The covariance and observed covariance shoud both be symmetric, which provides a good check on our math. Usuay the output is sighty (fractionay, 1/1 7 ) nonsymmetric due to rounding errors in the mutipication, and we force the fina matrix to be symmetric to numerica precision by averaging across the diagona before moving to the 5 This quaity of the HEALPix maps is by design, see Gorski et a. (25). next steps, since symmetric matrices often aow the use of faster agorithms. A row of the observed covariance matrix can be reshaped into a map, which reveas the structure of the covariance for a particuar pixe, see Figure / 2 E 1/ 2 B ΛCDM Right Ascension deg r =.2 5 FIG. 13. Maps showing a row of the observed covariance matrix C. The row seected corresponds to the covariance of an individua Q pixe at the center of the map. The top row shows the covariance used to cacuate the pure E and B-modes described in Section 6. The bottom row shows the covariance for an input spectrum corresponding to ΛCDM eft, and r =.2 tensors right. 6. E/B SEPARATION USING A PURIFICATION MATRIX The observed covariance matrix contains expected pixepixe covariance in our observed maps given an initia spectrum. The observation matrix, R, has made the E and B-mode spaces of the observed covariance non-orthogona. The resut of Section 3 is that we can find the orthogona pure E and pure B spaces by soving the eigenvaue probem from Equation 17: C B x i = λ i C E x i Construction of purification matrix As written, Equation 17 is not sovabe: C B has a nu space that is the set of pure E-modes. Simiary, the space of pure B-modes is the nu space of C E. By adding the identity matrix mutipied by a constant, σ 2 I, to the covariance matrices we reguarize the probem to find approximate soutions and eiminate the nu spaces: ( C B + σ 2 I)x i = λ i ( C E + σ 2 I)x i. (7) The ampitude of σ 2 sets the reative magnitude of the ambiguous mode eigenvaues versus the pure E and pure B-mode eigenvaues. The eigenvaues are shown in Figure 14. In our anaysis, we choose σ 2 to be 1/1th the mean of the diagona eements of the covariance matrices, C E and C B. In Equation 7, C E is an observed covariance matrix for Q,U that is constructed according to Equations 11 and 69. The input spectrum is set to a steepy red E-mode spectrum, C EE = 1/ 2, C BB =. C B is the same except for an input spectrum with ony B-modes, C BB = 1/ 2, C EE =. The eigenmodes in x i with the argest eigenvaues comprise a set of vectors that span a space of pure B-modes, b i. The pure B quaity of these vectors can be seen in the fact that the product C E x is much smaer than the product C B x. The eigenmodes in x i

14 Matrix based E/B Separation 13 Eigenvaue Emodes Ambiguous Bmodes This section describes the appication of the matrix based E/B separation to the BICEP2 data set. The technique reies on the existence of the observation matrix and purification matrix from the previous sections Motivation for matrix based E/B separation in BICEP2 The BICEP2 and Keck Array anaysis pipeine contains the foowing attributes that can eak E-modes to B-modes: partia sky coverage, timestream fitering (incuding deprojection), and choice of map projection+estimator. A simuation demonstrating the eaked B-mode maps for each of these effects is shown in Figure 16. These maps are created by appying the standard E and B estimator in Fourier space and then an inversion back to map space. The observations and data reduction produce three casses of E/B eakage: Eigenmode sorted by eigenvaue FIG. 14. The generaized eigenvaues for the BICEP2 observed covariance matrix, sorted by magnitude. Eigenvaues near one correspond to ambiguous modes: the modes that are simutaneousy E and B in the observed space and must be thrown out. By seecting eigenmodes with eigenvaues that are the argest and smaest 1/4 of the set of eigenvaues (shown to the eft and right of the dashed red ines), we can construct subspaces that span the spaces of B-modes and E-modes that can be effectivey observed using our scan strategy and anaysis. with the smaest eigenvaues comprise a set of vectors that span a space of pure E-modes, e i. Using the reddened input spectrum 1/ 2 causes the magnitude of the eigenvaues to be proportiona to the band of mutipoe,, that each mode contains. The steepness of the spectrum ensures each mode contains power from a imited range of. The particuar choice of 1/ 2 is arbitrary. A basis constructed from a subset of eigenmodes with arge eigenvaues spans a subspace of pure B-modes. We arbitrariy choose the pure B subspace to consist of eigenmodes whose corresponding eigenvaues are the argest 1/4 of the set. We find modes contained in this subset adequate for preserving power up to 7. The pure E subspace is simiary constructed with eigenmodes corresponding to the 1/4 smaest eigenvaues. Figure 14 shows the sorted eigenvaues, and Figure 15 shows four eigenmodes of the BICEP2 observed covariance. Using the set of pure E and pure B-mode basis vectors, two projection matrices are constructed from the outer products: Π E = i Π B = i e i e i b i b i, (71) which we ca the purification matrices for pure E and pure B. Operating the purification matrices on an input map projects onto the space of pure E and pure B: m puree = Π E m m pureb = Π B m. (72) From the construction of the pure B basis, b i, one can see that the vector m pureb vanishes for arbitrary input containing ony E-modes: m = R a E my E m, as desired for a purified B map. 7. MATRIX E/B SEPARATION APPLIED TO BICEP2 Apodization: The first obvious deviation from the idea fu sky map is the partia sky coverage of BICEP2 and Keck Array maps. Once a boundary is imposed E/B eakage is created. Map boundary effects are reduced by an apodization window which tapers the maps smoothy to zero near the edges. Use of apodization windows is common practice in any Fourier transform anaysis of finite regions to prevent ringing near boundaries. For sma regions of sky, effects of the map boundary dominate the eakage even after apodization is appied. The apodization window used in BI- CEP2 and the Keck Array are modified inverse variance maps. We appy a smoothing Gaussian with a width of σ =.5 to the inverse variance map, and, for the combined anayses of BICEP2 and the Keck Array, we use a geometric mean of the individua experiments apodization maps. Map projection: The tota extent of the BICEP2 and Keck Array maps is about 5 degrees on the sky in the direction of RA, over a decination of 7 < δ < 45. The chosen projection is a simpe rectanguar grid of pixes in RA and Dec. Taking standard discrete Fourier transforms of such maps resuts in significant E/B eakage. Whie we note that other map projections wi have significanty ower distortion, such effects wi be present for a projections when subjected to Fourier transform. However, note that the A and Λ matrices together fuy encode the mapping from the underying curved sky to the fat sky of the observed map pixes, and therefore so does the observing matrix R derived from these. Linear fitering effects: There are three main anaytic fiters appied in the standard pipeine: poynomia fiter, scan-synchronous subtraction, and deprojection. With respect to E to B eakage, a three fiters are simiar: by removing modes in the Q,U maps, an E-mode can be turned into an observed B-mode. Poynomia fitering and scan-synchronous signa subtraction create comparabe eakage, both in ampitude and morphoogy. Deprojection creates more power in the eaked B maps, and at smaer anguar scaes, than either poynomia fitering or scan-synchronous signa subtraction. If matrix purification is not used, the sampe variance of E/B eakage in the BICEP2 BB power spectrum is comparabe to the uncertainty due to instrumenta noise. This is ceary

15 14 B 5th Eigenmodes Decination deg 1st Eigenmodes E Right Ascension deg F IG. 15. Eigenmodes of the B ICEP 2 observed covariance matrix. Shown are the modes corresponding to the argest and 5th argest eigenvaues of Equation 7. Coormap shows ampitude of E and B-modes. The eigenvaues are shown graphicay in Figure 14. Projection + Apodization Third order poynomia fiter.5µk.3µk Scan synchronous subtraction Deprojection Decination deg.3µk.4µk Right Ascension deg F IG. 16. E to B eakage maps: exampes of eaked B-modes in the B ICEP 2 maps. Top row, eft: Leaked B-modes due to map projection and apodization. Top row, right: Leaked B-modes due to third order poynomia subtraction. Bottom row, eft: Leaked B-modes due to scan-synchronous signa subtraction. Bottom row, right: Leaked B-modes due to deprojection of beam systematics.

16 Matrix based E/B Separation 15 highy undesirabe, and ed to the deveopment of the purification matrix described in this paper. The purification matrix knows about a of the E/B mixing effects and how to dea with them Effectiveness of purification matrix The effectiveness of the purification matrix given by Equation 72 can be immediatey tested by appying the operator to a vector of Q,U maps simuated with the standard pipeine. The upper eft map of Figure 17 shows an observed map whose input is unensed-λcdm E-modes, and the next two rows in that coumn show the resuting E and B maps after projection onto the pure E and B spaces. The right coumn of Figure 17 shows an observed input map with both unensed- ΛCDM and r =.1 projected onto pure E and B-modes. Matrix purification is integrated with the existing BICEP anaysis code by appying the purification operator to maps before cacuating the power spectra. Since the observation matrix is ony used for this purification step, the purification matrix need ony work we enough to resut in E/B eakage ess than the noise eve of the experiment. Therefore, it is acceptabe to use an approximate observation matrix constructed from a subset of the fu observation ist, as ong as it is representative of the fu scan strategy. This shortcut was empoyed in Keck Array and BICEP2 Coaborations V (215), but the resuts shown in this section from BICEP2 are from the fu set of observations. Figure 18 compares the spectra for purified maps to the spectra from maps without purification, and to the spectra found using the improved estimator suggested in Smith (26). Both the purified maps and unpurified maps use the standard E and B estimator in Fourier space, and we refer to the unpurified maps processed this way as the norma method. Figure 18 shows the spectra for 2 noiseess unensed-λcdm simuations passed through the three estimators. The eaked power is roughy three orders of magnitudes smaer when using the matrix purification than when using the norma method or Smith estimator. Whie the Smith estimator improves over the norma estimator by eiminating E/B eakage from apodization, it does not account for spatia fitering, which is a significant source of E/B eakage in the anaysis pipeine. The mean of the eaked power is de-biased from the fina power spectra, so what matters is the variance of the eaked spectra. Computing the 95% confidence imits based on the variance in each of the three methods, we find that in the absence of B-mode signa or instrumenta noise, the matrix estimator achieves a imit on the tensor-to-scaar ratio of r < , whie the norma method and Smith estimator achieve imits of r <.17 and r <.74 respectivey. Figure 19 shows the spectra from the three estimators for input maps containing ony input B-modes at the eve of r =.1. The spectra for a three estimators show beam ro off at high. At the owest, the fitering prevents arge anguar scae modes from being measured. For mutipoes around 1, the matrix estimator recovers sighty ess signa than the other two methods. However, the extra power measured by the other methods near 1 argey comes from the ambiguous modes. On the eft of Figure 18, these ambiguous modes are seen as the bump in the norma and Smith method near 1. The tota number of degrees of freedom in each band power TABLE 1 DEGREES OF FREEDOM IN BINNED BB POWER SPECTRA FOR DIFFERENT ESTIMATORS Degrees of Freedom Bin center, Norma Smith Matrix can be estimated according to the formua: N = 2 (m )2 σ 2, (73) where m is the mean of the simuations in band power and σ 2 is the variance of the simuations in the band power. Tabe 1 shows the number of degrees of freedom for the three estimators for a tensor B-mode. The fewer degrees of freedom at ow for the matrix estimator are consistent with the decrease in recovered power on the eft side of Figure 18. The highest bins in Tabe 1 aso show fewer degrees of freedom in the matrix estimator because the purification matrix incudes a imited number of pure eigenmodes, as shown in Figure 14. Figure 2 shows signa pus noise spectra for a set of 2 unensed-λcdm+noise spectra. The noise simuations are the standard BICEP2 sign fip reaizations discussed in BI- CEP2 Coaboration I (214). In Figure 2, the mean noise eve and eaked BB power are de-biased. The resuting ensembe of simuations is used to construct the errorbars in the fina spectra. The tighter distribution of the matrix estimator simuations is a resut of the decrease in E/B eakage. Using the matrix estimator resuts in an improvement in the r imit, for BICEP2 noise eve and fitering in the absence of B-mode signa, of about a factor of two over the Smith method. The remaining variance in the BB spectrum of the matrix estimator is instrumenta noise Transfer functions The observation matrix transforms an input HEALPix map into an observed map with a simpe matrix mutipication. The speed of the operation faciitates the cacuation of anaysis transfer functions, which are a necessary component of the pseudo-c MASTER agorithm (Hivon et a. 22). We start with input maps, m, which are deta functions in a particuar mutipoe. These maps are then observed using the matrix R. The spectra cacuated from these maps represent the response in our anaysis pipeine to the input deta function, in a manner conceptuay anaogous to Green s functions. Our procedure uses two sets of HEALPix maps, one set corresponding to T T = T E = EE = 1 and one set with T T = BB = 1. The observation matrix is used to create maps for =1 through 7, with 1 random reaizations for each. Processing the 14, maps woud be infeasibe without the observation matrix, but using the observation matrix it can be accompished in a few hours.

17 16 Scaars ony Scaars + tensors (r=.1) µk 5.µK 5.µK µk 5 Tota Poarization 5 5.µK 5.µK 5 pure E 5 µk Decination deg µK.3µK.3 pure B.3 5 Right Ascension deg FIG. 17. Poarization maps showing the effectiveness of the BICEP2 purification matrix at separating noiseess simuations into pure E and pure B. Left coumn: on a scaar ony unensed-λcdm BICEP2 simuation. Right Coumn: on the same simuation with the addition of a sma tensor component. Top Row: the tota poarization of the BICEP2 observed map, containing E and ambiguous modes. Center Row: pure E-mode map, constructed by projecting the tota poarization map onto the E eigenmodes found in Equation 7. Bottom Row: pure B-mode map, constructed by projecting the tota poarization map onto the B eigenmodes of Equation 7..1 norma: r <.17.1 / 2π µk Smith: r <.74 / 2π µk (+1)C BB.1 1e 5 1e 6 matrix: r < 8.3x1 5 ens+r=.1 norma Smith matrix Mutipoe, (+1)C BB 1e 5 1e 6 Input: r=.1 norma Smith matrix Mutipoe, FIG. 18. BB power spectra of noiseess unensed-λcdm (r = ) simuations, estimated using various methods, demonstrating the effectiveness of the BICEP2 purification matrix. The E/B eakage using the matrix estimator is 3 orders of magnitude ower than other methods. FIG. 19. BB power spectra of noiseess unensed (r =.1) tensor ony simuations, estimated using various methods. A methods suffer from oss of power due to fitering and beam effects. The remova of ambiguous modes at ow resuts in a further decrease in power for the matrix method. Note that the spectra in this pot have not been corrected for the beam and fiter suppression factors, but in Figure 18 the correction is appied Band power window functions The power spectra of the output maps for a particuar are averaged over the N = 1 reaizations. The averaged spectra are used to form a band power window function, M XX, for a particuar band power,, which is a function of the input