Application of XFASTER power spectrum and likelihood estimator to Planck

Transcription

1 Mon. Not. R. Astron. Soc. 414, (2011) doi: /j x Appication of XFASTER power spectrum and ikeihood estimator to Panck G. Rocha, 1,2 C. R. Contadi, 3 J. R. Bond 4 andk.m.górski 1,5 1 Jet Propusion Laboratory, Caifornia Institute of Technoogy, 4800 Oak Grove Drive, Pasadena, CA 91109, USA 2 Department of Physics, Caifornia Institute of Technoogy, Pasadena, CA 91125, USA 3 Department of Physics, Imperia Coege, University of London, South Kensington Campus, London SW7 2AZ 4 The Canadian Institute for Theoretica Astrophysics, CITA, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 3H8, Canada 5 Warsaw University Observatory, Aeje Ujazdowskie 4, Warszawa, Poand Accepted 2010 November 4. Received 2010 October 28; in origina form 2009 December 15 ABSTRACT We deveop the XFASTER cosmic microwave background (CMB) temperature and poarization anisotropy power spectrum and ikeihood technique for the Panck CMB sateite mission. We give an overview of this estimator and its current impementation, and present the resuts of appying this agorithm to simuated Panck data. We show that it can accuratey extract the power spectrum of Panck data for the high- mutipoes range. We compare the XFASTER approximation for the ikeihood to other high- ikeihood approximations such as Gaussian and Offset Lognorma and a ow- pixe-based ikeihood. We show that the XFASTER ikeihood is not ony accurate at high, but aso performs we at moderatey ow mutipoes. We aso present resuts for cosmoogica parameter Markov chain Monte Caro estimation with the XFASTER ikeihood. As ong as the ow- poarization and temperature power are propery accounted for, e.g. by adding an adequate ow- ikeihood ingredient, the input parameters are recovered to a high eve of accuracy. Key words: methods: data anaysis methods: statistica cosmoogy: observations cosmic microwave background cosmoogica parameters. 1 INTRODUCTION Power spectrum estimation pays a crucia roe in CMB data anaysis. Primordia curvature fuctuations form a homogeneous, isotropic and neary Gaussian random fied in most eary universe scenarios, infationary or otherwise. To the extent that fuctuations are Gaussian, the power spectrum describes their statistica properties fuy. An immediate consequence is that the CMB temperature and poarization primary anisotropies, ineary responding to the primordia fuctuations, form an isotropic neary Gaussian random fied, characterized by their own anguar power spectra. With the advent of arge, high-quaity data sets, especiay that from the Panck mission (Panck Bue Book 2005; Tauber et a. 2010), we can measure the anguar power spectrum accuratey over a wide range of anguar scaes. Power spectrum estimation can be viewed as a significant data compression. For instance, 1 yr of observations from a Panck high-frequency instrument (HFI) channe produces roughy N tod data sampes. This reduces to N map pixes via map-making and finay to N pse vaues in the power spectra. Panck is a fu-sky experiment with beams ranging in size from 30 to 5 arcmin, and with high-resoution maps encompassing tens of miions of pixes. Direct extraction of science from the pixeized maps is computationay expensive and in fact unfeasibe. Accurate estimation of the anguar power spectrum for Panck enabes the extraction of science with minima oss of information. A number of approaches have been deveoped to estimate the anguar power spectrum from CMB data (for a review, see Efstathiou 2004, Ashdown et a., in preparation). Such estimators can be divided into three casses: codes accurate at arge-anguar scaes (ow mutipoes ), incuding codes that evauate or sampe from the ikeihood function directy; codes accurate at sma anguar scaes (high-) that characterize the statistics of an unbiased frequentist estimator for the power spectrum; and hybrid codes that can be appied to both ow and high. The first cass comprises maximum ikeihood estimators (MLEs) either in Fourier space (Górski 1994, 1997; Górski et a. 1994, 1996) or in rea space (Tegmark & Bunn 1995; Hancock et a. 1997) such as MADSPEC (Borri 1999, Bori et a., in preparation), quadratic maximum ikeihood (QML) estimators (Wright et a. 1994, 1996; Hamiton 1997; Tegmark 1997; Knox 1999; Bond, Jaffe & Knox 2000) such as E-mai: graca@its.catech.edu C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged Monthy Notices of the Roya Astronomica Society C 2011 RAS

2 824 G. Rocha et a. BOLPOL (Gruppuso et a. 2009), Gibbs sampers such as COMMANDER (Eriksen et a. 2004; Jewe, Levin & Anderson 2004; Wandet, Larson & Lakshminarayanan 2004), and Importance sampers combined with a Copua-based approximation to the ikeihood such as TEASING (Benabed et a. 2009). The second cass comprises quadratic pseudo-c (PCL) or Master method codes (Wandet, Hivon & Górski 2001; Hivon et a. 2002) such as ROMASTER, CROMASTER (Poenta et a. 2005), XPOL (Tristram et a. 2005a,b) and CROSSPECT (Ashdown et a., in preparation), anguar correation function codes such as SPICE (Szapudi, Prunet & Coombi 2001) and POLSPICE (Chon et a. 2004) and quadratic maximum ikeihood (QML) codes such as XFASTER (Netterfied et a. 2002; Montroy et a. 2006, Contadi et a., in preparation, and this paper, see Section 3.1 for more detais). The third cass consists of hybrid power spectrum estimators such as a QML estimator at ow combined with a PCL estimator at high (Efstathiou 2004, 2005) and, to some extent, the XFASTER method aone. XFASTER (Contadi et a., in preparation) was first deveoped to give rapid and accurate power spectra determinations from boometer data for the Boomerang ong-duration baoon experiments, first for tota anisotropy (Netterfied et a. 2002) and then for poarization (Montroy et a. 2006). XFASTER is a QML estimator formuated in the isotropic, diagona approximation of the Master method (Hivon et a. 2002). The noise becomes a diagonaized Monte Caro-estimated bias and the signa is summed into bands to reduce correations induced by sky cuts. In this sense XFASTER is an extension of the traditiona Master estimators, where the pseudo-c quantity is repaced by the quadratic MLE expression and uncertainties are given by the Fisher matrix. This method has been compared with other high- codes such as POLSPICE, ROMASTER, XPOL, CROSSSPEC within Panck working group C Temperature and Poarization, CTP. A detaied account of this comparison wi be given in a paper by the Panck CTP working group (Ashdown et a., in preparation). A fu and detaied account of XFASTER as a stand-aone method wi be given esewhere (Contadi et a., in preparation). Here we give an overview of the method, but our main goa is to show its adequacy to extract the power spectrum from Panck data. An interesting feature of the method is that it provides a natura expression for the ikeihood based on the assumption that the cut-sky harmonic coefficients a m foow the same distribution as those of the fu-sky harmonics. We compare our approximate ikeihood to the exact fu-sky ikeihood (the inverse Wishart distribution) and to the pixe-based ikeihood (i.e. mutivariate Gaussian of the pixe s I, Q and U Stokes parameters). We show that XFASTER agrees we with the exact ikeihoods at moderate ow- mutipoes as we. Using the XFASTER power spectrum and ikeihood estimator, we show how to go straight from the map to parameters, bypassing the band-power spectrum estimation step. Aternativey, we show how to use the band-power spectrum estimated with XFASTER in combination with any ikeihood approximation to estimate parameters. In particuar, in Rocha et a. (2011) we compare parameters estimated with XFASTER and the Offset Lognorma Bandpower ikeihood to those obtained with the XFASTER ikeihood. From our anaysis we concude that as ong as the ow- poarization is propery accounted for (by adding an adequate ow- ikeihood ingredient), we recover the input parameters accuratey. This paper is organized as foows. Section 2 describes the map and the Monte Caro simuations for two phases of increasing compexity studied in the CTP working group. Section 3 gives an overview of the XFASTER power spectrum and ikeihood estimator, incuding the estimation of kernes, transfer or fiter functions, and window functions. Section 4 shows the resuts of appying XFASTER to Panck simuations in severa different ways. It incudes the impact of beam asymmetries on power spectrum and cosmoogica parameter estimation, comparison of the XFASTER ikeihood to other ikeihood approximations at high and to pixe-based ikeihood at ow, and cosmoogica parameter estimation. Section 5 gives concusions. 2 SIMULATIONS The Panck sateite (Panck Bue Book 2005; Tauber et a. 2010) is a fu-sky experiment with beams ranging in size from 30 to 5 arcmin. The ow-frequency instrument (LFI) covers 30, 44 and 70 GHz; the high-frequency instrument (HFI) covers 100, 143, 216, 353, 545 and 857 GHz. From the second Lagrangian point of the Earth Sun system (L 2 ) Panck scans neary great circes on the sky, covering the fu sky twice over the course of a year (Dupac & Tauber 2005). Panck spins at 1 rpm around an axis that is repointed roughy 30 times per day aong a cycoida path, with the spin axis moving in a 7. 5 circe around the anti-sun direction with a period of 6 months. This ensures that a feeds cover the eciptic poe regions fuy. We aso incude sma perturbations to the pointing, with spin axis nutation and variations in the sateite spin rate. For the anaysis presented here we consider the 70-GHz LFI channe. The simuations used in this work incude CMB and reaistic detector noise ony, and are specified by the scanning strategy (as described above), teescope beams and detector properties. To mimic a more sensitive combination of channes, the white noise eve was taken to be ower than that expected for the rea 70-GHz channe. We used a singe observed map containing CMB and noise as we as Monte Caro simuations of signa and noise. Technica detais of the simuations are given in Ashdown et a., in preparation. We considered a the 12 detectors of the 70-GHz LFI channe. The beams of the detectors have FWHMs of arcmin, so the maps weremadewithn side = 1024, corresponding to a pixe size of 3.4 arcmin. Two sets of maps were provided, one 12-detector map to be used in the autospectrum mode, and three 4-detector maps to be used in the cross-spectrum mode. The input sky signa used to generate the observed map was the CMB map derived from the WMAP 1-yr data used in a previous CTP map-making exercise (Ashdown et a. 2009, Ashdown et a., in preparation). It is derived from the Panck CMB reference C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

3 XFASTER appied to Panck 825 Figure 1. Anguar power spectrum, C,ofthePanck CMB reference sky, obtained from the reference sky spherica harmonics, a m, used to generate the signa of the 70-GHz observed map (green soid ine); C used to generate the Monte Caro simuations for Phase 1 (bue soid ine); C used to generate the Monte Caro simuations for Phase 2 (back soid ine). Figure GHz (I, Q, U) Stokes parameters of the observed map (from eft- to right-hand side) for Phase 2b, generated with a detectors, convoved with the asymmetric beam and with mask (for gaaxy pus missing pixes) appied. sky. 1 Hence the arge-scae structure of the observed map is a WMAP constrained reaization. The anguar power spectrum of the a m is potted in Fig. 1. Simuations in four steps of increasing compexity were used. For historica reasons we refer to these steps as Phases 1a, 1b, 2a and 2b. Phase 1a. Data were simuated with symmetric beams and isotropic white noise. The power spectrum was estimated from the fu sky. Phase 1b. Data were simuated with symmetric beams and anisotropic white noise determined by the scan strategy. A sky cut was appied in the cacuation of the power spectrum to mimic the effects of removing the Gaactic pane from the observed data. For Phases 1a and 1b the input CMB sky was convoved with a Gaussian beam of 14-arcmin FWHM, and the noise reaizations were generated assuming an rms of μk per pixe in temperature and an rms of μk per pixe in the Q and U poarization components. The observed maps were generated in the pixe domain. Monte Caro signa simuations were generated from the WMAP 1-yr best-fitting CDM power spectrum 1 potted in Fig. 1 (bue soid ine). 100 Monte Caro reaizations were generated directy in the pixe domain for both the signa and noise using HEALPIX toos (Górski et a. 2005) and our own simuator. Phase 2a. Data were simuated with both correated 1/f and anisotropic white noise. Symmetric beams were assumed, a Gaussian with FWHM 14 arcmin. Noise effects induced by temperature fuctuations of the 20-K hydrogen sorption cooer were aso incuded. Phase 2b. Data were simuated with both correated 1/f and anisotropic white noise. Asymmetric beams were used, specificay, eiptica Gaussians fit to the centra parts of reaistic beams cacuated by a fu diffraction code for the Panck optica system. For the 12 beams, the geometric mean of the major and minor axis FWHMs ranged from to arcmin. Major axis to minor axis ratio varies from 1.22 to For Phases 2a and 2b the white noise per sampe was μk; the 1/f noise power spectrum had a knee frequency of 0.05 Hz and a sope of 1.7. The observed maps were made from time-ordered data (TOD) using the destriping map-maker SPRINGTIDE (Poutanen 2006; Ashdown et a. 2007a,b, 2009, Ashdown, in preparation). The TOD were generated using modues of the Panck simuator pipeine, LEVELS(Reinecke et a. 2006). Where a sky cut was appied in the anaysis of the maps, the cut was made at the boundary where the tota intensity of the diffuse foregrounds and point sources exceeded twice the CMB sigma. Masks for missing pixes due to the scanning strategy, if any, were aso considered. Fig. 2 shows the observed map for Phase 2b using a the 12 detectors, with the mask for gaaxy pus missing pixes appied. 1 Avaiabe in the webpage panck/reference-sky/cmb/ams/am-cmb-reference-tempate-microkthermodynamic-nside2048.fits. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

4 826 G. Rocha et a. As with Phases 1a and 1b, 100 Monte Caro signa simuations were generated from the first-year WMAP+CBI+ACBAR best-fitting CDM power spectrum 2 with BB-mode power set to zero (see Fig. 1). For the symmetric beam case ony the noise TOD were generated, whie the signa was simuated in the map domain. For the asymmetric beam case both signa and noise simuations were generated in the time domain. As mentioned above, the arge-scae structure of the observed map is derived from rea observations, i.e. a WMAP constrained reaization, hence it is not necessariy consistent with the best-fitting spectrum at ow mutipoes. This discrepancy wi become evident ater when comparing the power spectrum estimated from the observed map with the best-fitting theoretica spectrum, as we as when comparing the cosmoogica parameters estimated with XFASTER power spectrum and ikeihood and the theoretica best-fitting parameters. As the Monte Caro simuations are reaizations of the WMAP 1-yr best-fitting CDM power spectrum for Phases 1a and 1b, and the first-year WMAP+CBI+ACBAR best-fitting CDM power spectrum for Phases 2a and 2b, such discrepancy is no onger present. Parameters estimated from these Monte Caro simuation maps are now cose to the WMAP best-fitting parameters. The choice of 70 GHz for the simuations was driven by practica matters of computationa resources having to do with the size of the TOD, the number of pixes in the maps and the number of mutipoes that had to be cacuated. The HFI channes have higher anguar resoution and sensitivity, and wi extend to smaer anguar scaes with reduced error bars. Increases in computationa capabiity over time make it possibe now to generate thousands of Monte Caro simuations at the higher frequencies as we. Resuts wi be presented in a future pubication (Ashdown et a., in preparation; Rocha et a. 2011). 3 XFASTER POWER SPECTRUM AND LIKELIHOOD ESTIMATOR 3.1 XFASTER power spectrum estimator XFASTER (Netterfied et a. 2002; Montroy et a. 2006, Contadi et a., in preparation) is an iterative, maximum ikeihood, quadratic bandpower estimator (Hamiton 1997; Tegmark 1997; Knox 1999; Bond et a. 2000; Tegmark & de Oiveira-Costa 2001) based on a diagona approximation to the quadratic Fisher matrix estimator. It is a QML estimator formuated in the isotropic, diagona approximation of the Master method (Hivon et a. 2002). It is common to expand the pixe temperature fuctuations (Stokes I), T (ˆn), on the ceestia sphere in terms of spherica harmonic functions, Y m,as T (ˆn) = a m Y m (ˆn) (1) m with coefficients a m. The maximum ikeihood estimator (MLE) is based on a Gaussian assumption for the ikeihood of the observed data (e.g., the pixe temperature, T, or its spherica harmonic transform a m ): 1 L(d p) = exp ( 12 ) (2π) N/2 C d 1/2 C 1 d t, (2) where C is the covariance of the data, and p is the set of mode parameters. C( p) = S( p) + N, wheres is the sky signa and N is the noise. For singe-dish, fu-sky observations, an isotropic signa is diagona in the spherica harmonic space and can be described by an m-averaged power spectrum C on each mutipoe, i.e. S m, m = δ δ mm C. The noise is generay not diagona. A high- codes assume the data to be Gaussian distributed; however, they differ from XFASTER on the devised unbiased frequentist power spectrum estimator. A agorithms form quadratic functions of the data. Pseudo-C (PCL) codes estimate C = 1 m= a m m= 4π using fast spherica transforms. Exampe of such agorithms are a Master-type codes (Wandet et a. 2001; Hivon et a. 2002) such as ROMASTER, CROMASTER (Poenta et a. 2005), XPOL (Tristram et a. 2005a,b), and CROSSPECT (Ashdown et a., in preparation). Others cacuate the anguar correation function C(θ) = 1 4π 2 (2 + 1)C P (cos θ) [wherep (cos θ) is the Legendre poynomia and θ is an anguar separation on the sky] using fast evauation of the two-point correation function such as SPICE (Szapudi et a. 2001) and POLSPICE (Chon et a. 2004), though this is achieved using fast spherica transforms. XFASTER (Netterfied et a. 2002; Montroy et a. 2006, Contadi et a., in preparation), instead uses the QML expression as derived beow. It can be shown that the(, m) space maximum ikeihood soution for the power spectrum is given by (Hamiton 1997; Tegmark 1997; Bond et a. 2000; Tegmark & de Oiveira-Costa 2001): C = 1 [ F 1 2 Tr C 1 S C 1 (C obs ] N), (3) C where C obs m, m = aobs m aobs m is the quadratic in the coefficients of the expansion of the observed map and F 3 is the Fisher information matrix for the C parameters (= curvature matrix in the ensembe average imit), given by F = 1 [ S 2 Tr C 1 S ] C 1. (4) C C 2 Avaiabe at 3 It is traditiona in the iterature to use F to represent the Fisher matrix and F its ensembe average; however, to avoid confusion with F, which denotes the fiter or transfer function, we use F for the ensembe average of the Fisher matrix. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

5 XFASTER appied to Panck 827 An iterative scheme can be empoyed to reach the maximum ikeihood estimate for the C : start with an initia guess; compute F; evauate equation (3). However, matrix operations become prohibitive for dimensions arger than a few thousand. To circumvent this probem XFASTER recasts the estimator in the isotropic, diagona approximations of the Master methods (Wandet et a. 2001; Hivon et a. 2002) simpifying the cacuations above. In this case the noise becomes a diagonaized Monte Caro estimated bias and the signa is summed into bands to average down the correations induced by any reduced sky coverage. In this case for a singe mode, say temperature aone, the covariance of the observed cut-sky modes is approximated by C m, m = δ δ mm ( C + Ñ ), (5) where C is the cut-sky mode power spectrum. In our case the cut-sky power spectrum is parametrized through a set of deviations q from a tempate fu-sky shape spectrum C (S), C = K B 2 F C(S) q, (6) where K is the couping matrix due to the cut-sky observations (see Section 3.1.1), F is a transfer or fiter function accounting for the effect of pre-fitering the data both in time and spatia domain (see Section 3.1.2) and B expresses the effect of a finite beam. For the case where the spectrum is parametrized in bands, we consider band-power deviations q b : C = q b C S b = q b K B 2 F CS χ b( ), (7) b b where χ b () is a binning function. Assume for simpicity fat binning with χ b () = 1 within the band and zero outside. The ML soution for the q b is q b = 1 C F 1 S ( b bb 2 (2 + 1)g C ( C obs b + Ñ ) 2 Ñ ), (8) where isotropy reduces the trace as Tr (2 + 1)g,andgdescribes the effective degrees of freedom in the maps (which may be reduced by additiona weighting of the modes such as fitering or pixe weighting), and is reated to the moments of the pixe weighting and the sky coverage, see Hivon et a. (2002), and it can be further impacted by the binning of the power spectrum: g = f sky w2 2, where f w sky w i = 1 W i (ˆn)dn, (9) 4 4π 4π where W (ˆn) is the window or mask appied to the data, f sky w i is the i-th moment of the arbitrary weighting scheme and is the width of the mutipoe bins. The expression for the Fisher matrix is now given by F bb = 1 C S (2 + 1)g C b S b 2 ( C + Ñ ). (10) 2 For poarization-sensitive observations, the data incude the I, Q and U Stokes parameters. As mentioned before, the I map is expanded in terms of spherica harmonics whie the Q and U maps are expanded in spin-2 spherica harmonics, 2 Y m,toobtaine and B (grad-type or cur-type) poarization coefficients: (Q ± iu)(ˆn) = ( a E m ± ) iab m Y ±2 m(ˆn). (11) m C TT There are six spectra representing the six independent eements of the 3 3 covariance matrix of the (ã T m, ãe m, ãb m )vector: = b q TT b C (S)TT b + Ñ TT, (12) C EE C BB = b = b ( ( q EE (S)EE b + C b q BB (S)BB b + C b + q BB b + q EE b ) (S)BB C b + Ñ EE, ) (S)EE C b + Ñ BB, (13) (14) C TE = b q TE b C (S)TE b + Ñ TE, (15) C TB = b q TB b C (S)TB b + Ñ TB, (16) C EB = b q EB b C (S)EB b + Ñ EB. (17) The tempate shape matrices are defined using the various couping kernes for the different poarization types. The transfer functions are distinct for each poarization type: C (S)TT b = K F TT B2 C(S)TT χ b(), (18) C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

6 828 G. Rocha et a. ± C (S)EE b = ±K F EE B2 C(S)EE χ b(), (19) ± C (S)BB b = ±K F BB B2 C(S)BB χ b(), (20) C (S)TE b = K F TE B2 C(S)TE χ b(), (21) C (S)TB b = K F TB B2 C(S)TB χ b(), (22) C (S)EB b = ( + K K )F EB B2 C(S)TB χ b(). (23) For simpicity the beam B is assumed to be independent of poarization. (In principe, it coud aso be treated distincty for each of the T, E and B modes.) The mask-couping kernes, K, and the two additiona poarization mask-couping kernes, ± K, K,aredefinedin Section Extending the above formaism to poarization, the XFASTER estimator takes a matricia form, impemented triviay since the matrix C is now bock diagona: C diag( D min, D min +1,..., D max ), where each mutipoe s covariance is a 3 3matrix: D = C TT C TE C TB C TE C EE C EB C TB C EB C BB, Simiary, its inverse is a bock diagona of the inverses of D matrices and therefore simpe to compute. The noise covariance matrix is aso of this form, the Ñ XY in each bock diagona is obtained by noise-ony Monte Caro simuations. The band-power deviations, q b,takethe foowing form now: q b = 1 [ 2 b F 1 bb (2 + 1)g Tr D 1 S ( ) ] D 1 D obs Ñ, (25) q b and the Fisher matrix is now given by F bb = 1 [ ] (2 + 1)g Tr D 1 S D 1 S. (26) 2 q b q b where the band index, b, spans bands in a poarization types. The derivatives of the signa matrices with respect to the deviations q b are given by C (S)TT b C (S)TE S b::tt = q b 0 0 0, b 0 S b::te = q b C (S)TE b 0 0, (27) S b::ee = (S)EE q b 0 + C b S b::tb = q b 0 0 C (S)TB b C (S)TB b 0 0 C (S)EE b 0 0 0, S b::bb = (S)BB q b 0 C b , S C (S)BB b (24), (28) b::eb = q b 0 0 C (S)EB b. (29) 0 C (S)EB b 0 These derivatives incude contributions from both (+) and( ) kernes in the cases EE and BB due to the geometrica eakage. This estimator makes use of the Monte Caro pseudo-c formaism of Master methods to estimate noise bias and inear fiter functions. It requires noise-ony Monte Caro simuations to estimate the noise bias and signa-ony Monte Caro simuations to estimate the fiter function. Contrary to the conventiona pseudo-c -based methods, it does not require signa+noise Monte Caro simuations to estimate the uncertainty (variance) of the band-powers, which is given by the Fisher matrix instead, a by-product of the method. The Fisher matrix is computed sef-consistenty and runs over a band-powers and poarizations. It takes into account a the correations in the approximation used (couping kerne and diagona noise bias) and can therefore be used to compute, sef-consistenty, a anciary information required in the estimation process, correations, window functions, etc. In Fig. 3 we pot the inverse of the Fisher matrix for Phase 2, symmetric beam case. Note that the Fisher matrix is not diagona. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

7 XFASTER appied to Panck 829 Figure 3. Logarithm of the absoute vaue of the normaized (i.e. set to 1 at the maximum vaue) inverse of the Fisher matrix (covariance matrix) for Phase 2, symmetric beam case. TT, EE, BB and TE modes are dispayed sequentiay from bottom eft-hand corner to the upper right-hand corner aong the diagona. Furthermore, XFASTER can estimate both autospectra and cross-spectra jointy, using the fu covariance of the a m s, via a mutipe-map anaysis. As mentioned above, the noise is generay not diagona. For Panck, the noise is white to good approximation at sma anguar scaes; however, at arge anguar scaes instrumenta characteristics such as 1/f noise and therma fuctuations combine with the scan strategy to produce significant off-diagona correations in the noise. Therefore, the XFASTER approximation is not optima at ow- mutipoe range. We show in Sections 4.2 and 4.3 that XFASTER is a very good approximation for >30. We have no intention of using XFASTER for Panck at ow-. Instead, we wi combine one of the codes adequate at ow- (isted in Section 1) with XFASTER (or another high- estimator) into a hybrid estimator of the power spectrum that covers the entire mutipoe range. Previous CMB experiments have customariy binned power spectra in mutipoe bands. The main reason for this is that it enhances the signa-to-noise ratio of the spectra. It aso averages down correations due to the reduced sky coverage at the same time as the couping matrices due to the cut-sky attempt to correct the cut-sky effect. The XFASTER power spectrum can be computed mutipoe by mutipoe, i.e. for each, or in mutipoe bands. The band-power spectra are given in Section 4.1. To estimate cosmoogica parameters, we can use band-power spectra with any high- ikeihood approximation, e.g. with the Offset Lognorma Bandpower ikeihood presented in Rocha et a. (2011). However, as the XFASTER ikeihood is estimated mutipoe by mutipoe, we can bypass the band-power spectrum estimation step and estimate parameters directy from the maps (via its raw pseudo-c ). Sices of the XFASTER ikeihood and parameter constraints are given in Sections 4.2 and 4.3. As kernes and transfer (fiter) functions are an important ingredient in power spectrum estimation, we describe next how they are computed within the XFASTER approach. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

8 830 G. Rocha et a Kernes The effect of masking the sky is to produce a power spectrum that is a inear combination of the fu sky power spectrum mutipoes on the sky. The couping matrix due to the cut sky observations, K, encodes this effect, it ony depends on the geometry of the mask or window and is easiy computabe. Considering a window function W (ˆn), and ignoring the effects of beam convoution and fitering effects due to any pre-processing of the time-ines, the ensembe averages of the cut-sky C and the fu-sky anguar power spectrum C can be reated by C = K C, (30) with couping matrix, K given by K = J (,, ;0, 0, 0) 2 W 2 4π, (31) ( ) where J(,, ;0, 0, 0) = is the 3j symbo, and W is the power spectrum of the window function W (ˆn), that is W 2 = (2 + 1)W with: W = 1 W m 2 and W m = dnw (ˆn)Y,m (ˆn), (32) m W 0 = W 2 0 = 4πf 2 sky w2 1 Hence K = π and 0 W 2 = 0 (2 + 1)W ( ) L (2 + 1)W = 4πf sky w 2. (33) Extending the above to poarized data, we consider the additiona poarization mask couping kernes defined as foows: ( ) ±K = L (2L + 1)W L (1 ± ( 1) + +L ), (35) 16π K = π ( )( ) L L (2L + 1)W L (1 + ( 1) + +L ). (36) L These kernes account for the eakage of power between E and B modes induced by the usage of the fu-sky ±2 Y m (ˆn) basis on a cut-sky. In Fig. 4 we pot the masks for temperature and poarization used in Phase 2 (see Section 2). (34) Transfer or fiter functions To compute F we start with equation (7) for the signa-ony Monte Caro simuations, and as F is assumed to be smooth over each mutipoe bin we move F out of the summation for each bin b,toget C = q b C S b = q b F b K B 2 CS χ b( ) (37) b b Figure 4. Mask for temperature (eft-hand side) and poarization (right-hand side) used in Phase 2. The f sky 0.85 for temperature and f sky 0.73 for poarization. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

9 XFASTER appied to Panck 831 and proceed with the iterative scheme as one woud to estimate q b but now we estimate the transfer function F b instead. This is achieved by fixing q b = 1, varying F b and considering the signa-ony Monte Caro simuations: F b = 1 C F 1 S b bb 2 (2 + 1)g ( C b + Ñ ) S 2, (38) where S is the average of the signa-ony Monte Caro simuations. Extending to poarization, we have F b = 1 [ F 1 bb 2 (2 + 1)gT r D 1 S q b b ] D 1 S. (39) 3.2 XFASTER ikeihood estimator A very attractive feature of the XFASTER power spectrum estimator is that it naturay provides a ikeihood, i.e. the probabiity of the observed cut-sky data given the mode. In the XFASTER approximation, considering ony one mode (say, temperature aone), the ikeihood takes the foowing form, up to a constant (where à means A estimated on the cut-sky): ( n L = 1 g(2 + 1) 2 obs C ) ( C + N ) + n( C + N ), (40) where C is the cut-sky mode power spectrum given by equation (7) in Section 3.1, for the case where the spectrum is parametrized in bands we consider band-power deviations q b. Extending to temperature and poarization, we have n L = 1 { [ g(2 + 1) Tr D obs ] } ( D + N ) 1 + n D + N, (41) 2 where N and D are given in Section 3.1. An interesting point to note is that XFASTER ikeihood foows intuitivey from the usua fu-sky idea case exact ikeihood [an inverse Gamma distribution for temperature aone and an inverse Wishart distribution for temperature + poarization (see e.g. Rocha et a. 2011a)]. Here we use one-dimensiona sices as an approximation to investigate the non-gaussianity of the ikeihood. One sampes in each deviation q b direction individuay around the maximum ikeihood soution q b. This approximation is adequate if the band-powers are not heaviy correated. Note that the ikeihood sices are estimated aong the bands and not aong each, and hence wi be affected by the binning procedure. To compare XFASTER ikeihood to other approximations, we make use of sices computed aong the band-power spectrum deviations, q b. Such ikeihood sices for the 70-GHz observed map are potted in Section 4.2. When estimating parameters with XFASTER ikeihood, by defaut estimated mutipoe by mutipoe, we do not make use of the bandpower spectra. It is in this sense that XFASTER ikeihood aows going straight from the maps to parameters bypassing the band-power spectrum step. It ony requires the raw pseudo-c of the observations pus the kerne and transfer function to reate the cut-sky pseudo-c to the fu-sky C Window functions To compare the theoretica power spectrum to the observed power spectrum and to estimate parameters, we must construct an operator for obtaining theoretica band-powers from mode power spectra C T. Foowing Bond et a. (2000) we define a ogarithmic integra I[f ] = ( + 1) f, (42) which is used to cacuate the expectation vaues for the deviations q b (when a shape mode, C S, is considered), or band-powers C b (when C S is assumed to be fat): q b = I [ W bc ] [ ] C b = I [ W bc ] I W b C (S) I [ ], (43) W b where W b is the band-power window function and C (S) = ( + 1)C (S) /2π. We define normaized window functions to be [ I W b C(S) ] = 1. By taking the ensembe average imit of equation (8) and using the fact that ( C obs Ñ ) C, C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS (44) (45)

10 832 G. Rocha et a. we obtain W b = 4π (2 + 1) b C (S) F 1 bb g(2 b + 1) ( C + Ñ ) K 2 F B 2. (46) Extending to poarization, W b = 4π F 1 bb (2 + 1) g(2 + 1)Tr [W b K ], (47) b where W b = D 1 S D 1,andK gives the cut-sky response to the individua fu-sky mutipoes: q b K F TT B 2 K F TE B 2 K F TB B 2 K = K F TE B 2 +K F EE B 2 + K F BB B 2 ( + K K ) F EB K F TB B 2 ( + K K ) F EB B 2 +K F BB B 2 B 2 + K F EE B 2. (48) These window functions were used and compared to the top hat window functions in Rocha et a. (2011) using the XFASTER band-power spectra and the Offset Lognorma Bandpower ikeihood. However, as the XFASTER ikeihood is estimated for each, a comparison of observed to theoretica power spectrum does not make use of such windows. Instead the raw pseudo-c of the observations, the kernes, and transfer or fiter functions, are a that is required for such comparison (see Section 4.3). 3.3 The agorithm The power spectrum is estimated by the foowing procedure. (i) Generate Monte Caro simuations of TOD for both signa and noise. The noise must have the same characteristics as the observed data, and in practice must be determined from the observed data. The simuated signa, on the other hand, can be amost anything, as it is simpy a tracer of the effects of time and spatia domain fitering in the process, and used to cacuate the transfer function F. In practice, it is convenient to use an approximate mode of the CMB to generate the signa. (ii) Make maps of the TOD using the same map-making code as used for the observations. (iii) Estimate the pseudo-c spectra and the spherica harmonic coefficients a m from the signa-ony maps to get the transfer function F. (iv) Estimate the pseudo-c spectra from the noise-ony maps to compute the noise bias Ñ. The pseudo-spectra can be computed with the anafast function of HEALPIX package (Górski et a. 2005) when the masks are the same for temperature and poarization, otherwise we use a specific code from the suite of XFASTER modues. (v) Iterate equations (8) and (10) to obtain an estimate of q b. In the diagona, isotropic approximation of XFASTER, the computationa cost of the iterative estimator is very sma compared with that of the TOD generation and map-making stages. (vi) The iterative estimator yieds the Fisher information matrix automaticay. An estimate of the band-power covariance is given by F 1, therefore we automaticay get the uncertainty on the estimator. Large ensembes of signa+noise simuations are not required to estimate the band-power covariance matrix as in the Master procedure, cutting the cost of Monte Caro simuations by 1/3. Furthermore, the covariance is not biased by an assumed mode (which at very east requires the Master procedure to be run twice to be cose to unbiased errors). As mentioned in Section 3.1, XFASTER can estimate both autospectra and cross-spectra jointy, using the fu covariance of the a m, via a mutipe-map anaysis Computationa scaing The overa scaing for XFASTER without accounting for the signa and noise Monte Caro simuations shoud go as max (n maps n po ) 3 for the interna Fisher cacuation, where n po is either 1 or 3, with a further scaing of (n bins ) 3 for the outer iteration step. Currenty the code is not optimized for speed. It coud be sped up substantiay by paraeizing the Fisher computation and woud then scae ineary with number of processors. Approximate times for a singe CPU for Phase 2, with CTP binning and using 30 Fisher iterations are as foows: (i) a m and C from the 100 Monte Caro simuated maps: 8h( 5 min each); (ii) kerne: 30 min; (iii) transfer function: 30 min (ess if one reaxes the binning); (iv) power spectrum: 1 h; (v) average mode to check for possibe bias: min. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

11 XFASTER appied to Panck 833 Figure 5. Power spectrum estimated with XFASTER and 1σ error bars. Top row Phase 1a (eft-hand side) and Phase 1b (right-hand side) for map generated with a quadrupet of detectors. Bottom row Phase 2a symmetric beam (eft-hand side) and Phase 2b asymmetric beam (right-hand side) for map generated with a the 12 detectors,. The pot dispays the estimated power spectrum (bue) of the observed map, overpotted with the C fiducia mode used as input in Phase 2 signa simuations, first-year WMAP+CBI+ACBAR best-fitt mode (back). 4 RESULTS 4.1 Resuts: power spectrum We estimated transfer (fiter) functions, kernes and the power spectrum for the observed map described in Section 2. We aso computed the power spectrum for the average of the signa+noise simuated maps. This average mode run checks for possibe biases of the power spectrum estimator itsef. In principe, the estimator if unbiased shoud foow cosey the input signa C mode used to generate the signa simuations. Fig. 5 shows the power spectra estimated for the observed map for Phase 1 and Phase 2 and their 1σ error bars. The power spectra recover accuratey the input power spectrum in the midde range of mutipoes, At high ( 1000), they are impacted by the noise. At ow-, as the arge-scae structure of the observed map is a WMAP constrained reaization, the estimated power spectrum is not necessariy consistent with, and exhibits a dispersion around, the best-fitting spectrum. Comparing the power spectra for the diverse phases we concude that the cut-sky anisotropic noise case (Phase 1b) exhibits sighty greater uncertainties and sighty arger mutipoe-to-mutipoe variations at ow- than the fu-sky, isotropic noise case (Phase 1a). This is expected, as the cut-sky wi induce correations at ow-. The kernes correct these correations; however, there is sti a sma residua dispersion. On the other hand, the anisotropic noise wi enhance the overa white noise eve, increasing the power spectrum uncertainty. For Phase 2 the dispersion of the power spectrum and uncertainties at ow- are enhanced due to the residuas of correated 1/f noise. As in Fig. 5, the power spectrum for Phase 1 is estimated from maps generated with a smaer number of detectors (four) than those for Phase 2 (12), and is therefore noisier by a factor of the order of 1/ 3. Therefore, the error bars of the Phase 2 power spectrum are smaer than those of Phase 1. The right thing to do though is to compare the power spectra estimated for the same number of detectors. In Fig. 6 we compare the power spectra for Phase 1a (midde pot) and Phase 2 (right-hand side), both estimated on maps generated with a the 12 detectors. As expected, the uncertainty for Phase 1a is now smaer than that of Phase 2. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

12 834 G. Rocha et a. Figure 6. Power spectrum estimated with XFASTER and 1σ error bars, considering transfer function=1 for Phase 1a, for map generated with a quadrupet of detectors (eft-hand side), with a the 12 detectors (midde) and for Phase 2a, map generated with a the 12 detectors and convoved with a symmetric beam (right-hand side). This pot dispays the estimated power spectrum (bue) of the observed map, overpotted with the C fiducia mode used as input in Phase 2 signa simuations, first-year WMAP+CBI+ACBAR best-fit mode (back). The power spectra for Phase 2 for both the symmetric and asymmetric beams are highy consistent. Therefore, we concude that the beam asymmetry is reasonaby we handed by XFASTER (see Section for more detais). Fig. 7 shows the transfer functions for Phase 1 and Phase 2. Except for Phase 2b, a are cose to 1. This is expected, as we do not pre-fiter the TOD and the ony effect at ow- is that due to the map-making step and to the imited number of Monte Caros of the signa maps avaiabe. This means there is remaining sampe variance on arge scaes athough not very significant. We incuded the transfer function estimates because they are an integra part of the method and in rea ife they wi not be equa to 1. However, to investigate and show that the significance of these sma departures from 1 are not significant, we estimated the power spectrum with transfer function=1 as potted in Fig. 6. We aso highighted the fact that the transfer functions are estimated consistenty between poarization types (i.e. taking into account cross-correation between poarization modes in any reaization). Note that in this work we did not take into account the remaining MC error in the transfer function in the fina estimate of the power spectrum since any production run used in the rea case wi incude many more reaizations than used in this work (they coud easiy be incuded by adding to the fina Fisher matrix if needed). However, for the asymmetric beam case the transfer function exhibits an upturn at high. This upturn tries to correct the mismatch between the rea asymmetric beam and our assumed symmetric beam (as discussed in Section 4.1.1). As the input BB power spectrum mode of the signa Monte Caro simuations for Phase 2 is set to zero, we cannot constrain the BB transfer function. The transfer function obtained refects the inadequacy of the input mode and hence is cose to zero. When estimating the power spectrum we repace the BB transfer function by the EE transfer function. Fig. 8 shows the power spectrum estimated for the average mode run. Whereas Fig. 9 shows the difference between the power spectrum obtained for the average mode run (with error bars) and the fiducia mode, C, used as input, for Phase 2, asymmetric beam case. The stepwise decreases of the ampitude of the error bars are caused by the changes of the C s bins size. These pots show that the power spectrum foows the input signa C, confirming that XFASTER is an unbiased estimator. These resuts were obtained using 100 Monte Caro simuations. Considering 500 simuations for Phase 1a, the sma departures of the transfer function from 1 reduces sighty. Due to computationa constraints, this was not feasibe for Phase 2. However, increases in computationa capabiity over time make it now possibe to generate thousands of simuations. Resuts for a 143-GHz channe wi be presented in our upcoming papers (Ashdown et a., in preparation; Rocha et a. 2011). Fig. 10 shows a variation of the above pots in which the BB power spectrum repaces the noise Monte Caro simuations. These pots show that whenever the noise Monte Caro simuations exhibit issues in the sense that they do not refect accuratey the noise characteristics of the observations, repacing them by the BB power spectrum is an adequate procedure, as for Panck at 70 GHz the BB power spectrum is mosty dominated by noise. The power spectrum estimated for the observed map has been compared with those from severa other methods (Ashdown et a., in preparation). To further assess the power spectrum estimator, we propagated this anaysis to cosmoogica parameter estimation using the new XFASTER ikeihood as described in Section Symmetric and asymmetric beams To study the impact of beam asymmetries on power spectrum and cosmoogica parameter estimation, we took a minima, non-informative approach. When computing the power spectrum for the observed maps convoved with symmetric and asymmetric beams, we assumed an FWHM = 14 arcmin symmetric beam for both cases. We started by investigating the effect of this assumption on the shape of the transfer function, F, in equations (6) and (7), or F b in equations (37), (38) and (39). In effect we compute not ony F,butasoF δb,whereδb is the correction to the beam transfer function B. This correction arises from assuming a symmetric beam when estimating the power spectrum of an observed map that in fact has been convoved with an asymmetric beam. Ca this function the generaized transfer function, (BF). As mentioned in Section 4.1, since we do not pre-fiter the TOD the ony effect at ow- is that due to the map-making step. Therefore, the transfer function shoud be very cose to C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

13 XFASTER appied to Panck 835 Figure 7. Transfer function: Top row for Phase 1a (eft-hand side) and Phase 1b (right-hand side); Bottom row for Phase 2a symmetric beam (eft-hand side) and Phase 2b asymmetric beam (right-hand side). unity, particuary for the symmetric beam. Fig. 7 and ater in Fig. 13 show (BF) for both cases. For the symmetric case this function is very cose to 1 as expected, with tiny osciations around 1 at ow- consistent with our expectations. Apart from the transfer function for the BB mode for Phase 2 which is cose to zero. Since the input BB power spectrum mode of the signa Monte Caro simuations for Phase 2 is set to zero, we cannot constrain the transfer function for the BB mode. We resort to using the transfer function for the EE mode to estimate the BB power spectrum instead. The symmetric and asymmetric (BF) differ. In the asymmetric case, the function exhibits an upturn at high. This upturn tries to correct the mismatch between the rea asymmetric beam and our assumption. Hence the resuting power spectrum is pretty consistent for both cases, as shown in Fig. 11. Considering the input symmetric beam with FWHM = 14 arcmin and the estimated FWHM 13 arcmin for the C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

14 836 G. Rocha et a. Figure 8. Power spectrum obtained with XFASTER in average mode, meaning that we repaced the observed map by the average of the signa+noise simuated maps (bue). Top row Phase 1a (eft-hand side) and Phase 1b (right-hand side) for map generated with a quadrupet of detectors. Bottom row Phase 2a symmetric beam (eft-hand side) and Phase 2b asymmetric beam (right-hand side) for map generated with a the 12 detectors, overpotted with the C fiducia mode used as input in our Phase 1 signa simuations, first-year WMAP best-fit mode (back) for Phase 1 and first-year WMAP+CBI+ACBAR best-fit mode (back) for Phase 2. It serves the purpose of checking for possibe biases of the power spectrum estimator in principe the power spectrum estimated in average mode shoud foow cosey the input signa C mode used to generate the signa simuations (back). Figure 9. Difference of the power spectrum obtained with XFASTER in average mode (meaning that we repaced the observed map by the average of the signa+noise simuated maps) and the C fiducia mode used as input (first-year WMAP+CBI+ACBAR best-fit mode), for Phase 2, asymmetric beam case. The stepwise decreases of the ampitude of the error bars are caused by the changes of the C bins size. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

15 XFASTER appied to Panck 837 Figure 10. Power spectrum estimated with XFASTER and 1σ error bars for Phase 2, for map generated with a the 12 detectors, for symmetric (top row) and asymmetric (bottom row) beams. Left-hand pot dispays the estimated power spectrum (bue) of the observed map, overpotted is the C fiducia mode used as input in our signa simuations, first-year WMAP+CBI+ACBAR best-fitt mode (back); Right-hand pot dispays the estimated power spectrum with noise Monte Caro simuations repaced by the BB power spectrum. Figure 11. Power spectrum estimated with XFASTER and 1σ error bars for Phase 2. The eft-hand pot dispays the power spectrum for a map generated with a the 12 detectors, for both symmetric (red) and asymmetric (bue) beam cases. For both runs a symmetric beam with FWHM = 14 arcmin is assumed. The resuting power spectra are highy consistent, the compensation is achieved via the generaized transfer function (BF). The right-hand pot dispays the power spectrum estimated for map generated with a quadrupet of detectors, with two different sets of Monte Caro simuations. In one case we use the Monte Caro simuations convoved with the symmetric beam (bue); in the other we use the correct Monte Caro simuations convoved with the asymmetric beam (red). Making use of the Monte Caro simuations for the symmetric case gives rise to a bias high at high. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

16 838 G. Rocha et a. Figure 12. Power spectrum estimated with XFASTER and 1σ error bars for Phase 2, for map generated with a the 12 detectors, for both symmetric (red) and asymmetric (bue) beam cases. For both runs a symmetric beam with FWHM = 14 arcmin is assumed, the resuting power spectrum is consistent with each other, the mismatch is corrected via the generaized transfer function, (BF). There is sti a sight bias for the asymmetric case in agreement with the sma differences of the estimated cosmoogica parameters (see Section 4.3). asymmetric beam, this effect shoud be approximatey (14/13) 2, athough anisotropic pixe fitering wi add an extra component of aiasing in the maps at high. Athough the power spectra ook consistent, the parameter estimation shows departures of the order of σ/2 for some of the parameters, as shown in Section 4.3. To investigate this further, we enhanced the previous pot into Fig. 12. Though there is very good agreement between the two power spectra, there is sti a sight bias for the asymmetric beam case. This bias is consistent with the sma differences in the estimated parameters, in particuar for n s, σ 8 and og [10 10 A s ] (see Section 4.3). In Fig. 13 we overpot (BF) for both symmetric and asymmetric beam cases. This pot shows ceary the differences of both (BF) s previousy potted separatey in Fig. 7. Next we study the impact of using two different sets of Monte Caro simuations on the estimation of the power spectrum for the asymmetric beam case, Phase 2b. One set is the Monte Caro simuations for the symmetric beam (Phase 2a) and the other the correct Monte Caro simuations for the asymmetric beam (Phase 2b) study. On the right-hand side of Fig. 11, we pot the power spectrum estimated using both sets of simuations. The power spectrum of the observed map of Phase 2b (convoved with an asymmetric beam), estimated using the Monte Caro simuations for Phase 2a (convoved with the symmetric beam), is biased high at high, as expected. 4.2 Resuts: ikeihood Foowing Section 3.2 we use one-dimensiona sices as an approximation to investigate the non-gaussianity of the ikeihood, samping in each q b direction around the maximum ikeihood soution q b. This approximation is adequate if the band-powers are not heaviy correated. Note that the ikeihood sices are estimated aong the q b band-power deviations and not aong the q power deviations for each mutipoe, and hence are affected by the binning procedure. These sices are potted in Figs Figs 14 and 15 compare the XFASTER ikeihood to four other ikeihood approximations, Gaussian, Lognorma, Offset Lognorma and Equa Variance (Bond et a. 2000, Rocha et a. 2011). A thorough account of these ikeihood approximations is given esewhere C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

17 XFASTER appied to Panck 839 Figure 13. Comparison of transfer (fiter) functions for Phase 2, for map generated with a the 12 detectors, for symmetric beam (red soid ine) and for asymmetric beam case (bue soid ine). Figure 14. Likeihoods for Phase 2b (12-detector map convoved with an asymmetric beam): Likeihood functions are samped in each band-power direction whie fixing the other bands at the maximum ikeihood vaues. The back (dotted) curve is the Gaussian approximation given by the Fisher matrix. The bue (dashed) curve is the offset ognorma approximation using the noise q N b. The magenta (dash dotted) curve is the equa variance approximation. The red (dashed) curve is ognorma distribution. The back (soid) curve is the XFASTER ikeihood estimated for temperature and poarization. The numbers in the right upper corner indicate the mutipoe or effective of the binned mutipoes. As for this set of mutipoes = 1 these numbers are the singe mutipoe. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

18 840 G. Rocha et a. Figure 15. Likeihoods for Phase 2b (12-detector map convoved with an asymmetric beam): Likeihood functions are samped in each band-power direction whie fixing the other bands at the maximum ikeihood vaues. The back (dotted) curve is the Gaussian approximation given by the Fisher matrix. The bue (dashed) curve is the offset ognorma approximation using the noise q N b. The magenta (dash dotted) curve is the equa variance approximation. The red (dashed) curve is ognorma distribution. The back (soid) curve is the XFASTER ikeihood estimated for temperature and poarization, for TT (first coumn), EE (second coumn) and TE (third coumn); for = 5 (first row), = 7 (second row), = 10 for TT and TE and for bin with in [10, 13] for EE mode (third row). (see for instance Rocha et a. 2011); here we give a brief account of their definitions. In what foows Ĉ means the measured or observed quantity C. The Gaussian approximation (Bond et a. 2000) is a ikeihood that is Gaussian in the Ĉ,i.e. { P (Ĉ C) exp 1 } 2 (Ĉ C)T S 1 (Ĉ C), (49) where C is a vector of C vaues (and simiary Ĉ) ands 1 is the inverse signa covariance matrix. The Offset Lognorma ikeihood, (Bond et a. 2000), is given by { P LN (Ĉ C) exp 1 } 2 (ẑ z)t M(ẑ z), where z = n (C + x ) and the matrix M is reated to the inverse covariance matrix by M = (C + x )S 1 (C + x ). (The offset factors x are simpy a function of the noise and beam of the experiment.) The Equa Variance ikeihood (Bond et a. 2000) is given by n L = 1 2 G { e (z ẑ) [1 (z ẑ)] } (50) (51) (52) with z = n ( ) q b + q N b and G = [ e σz (1 σ z ) ] 1 F 1 bb with σ z = ( ). qb + qb N (53) (54) The noise offset q N b is estimated using the equation of the maximum ikeihood soution for the q b repacing the observed map with the average of the noise Monte Caro simuation power spectra N. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

19 XFASTER appied to Panck 841 Figure 16. Likeihoods for Phase 2a (12-detector map convoved with a symmetric beam) for TT mode. XFASTER ikeihood (Phase 2 cut sky) (bue soid ine) versus Exact Fu Sky ikeihood (Phase 1a fu sky) (red soid ine). At ow- the XFASTER ikeihood is wider due to the correations induced by the cut-sky whie at high- XFASTER ikeihood is narrower due to the binning effect. The numbers in the right upper corner indicate the mutipoe or the bin number. Up to 10 the bin number is the singe mutipoe as = 1, whereas bin = 50 corresponds to in [61, 62] and bin = 100 corresponds to in [257, 263] Figure 17. Likeihoods for Phase 2a (12-detector map convoved with a symmetric beam) for TT mode. Left-hand pot: XFASTER ikeihood sices (bue soid ine) versus BFLIKE, Pixe based ikeihood sices (red soid ine), both curves are estimated on the cut-sky map of Phase 2. This pot can be miseading as the widths of both distributions depend on their peaks ocations. Right-hand pot: XFASTER ikeihood sices (bue soid ine) versus BFLIKE, Pixe-based ikeihood sices (red soid ine), both curves are estimated on the cut-sky map of Phase 2, assuming they both peak at the same vaue, i.e. after dividing both distributions by their peak vaues. The agreement is aready apparent at as ow as = 16, 32. The numbers in the right upper corner indicate the mutipoe or effective of the binned mutipoes. As for this set of mutipoes = 1 these numbers are the singe mutipoe. We use the vaues of the power spectrum and Fisher errors estimated with XFASTER for Phase 2b (map generated with a the 12 detectors and convoved with the asymmetric beam). As we are primariy interested in the shape of the ikeihoods not on the actua vaue of the peaks, the comparison of the sices is done assuming they a peak at the same vaue, i.e. we use the band-power spectra estimated with XFASTER and the functiona shape of the other high- ikeihoods. This is the same to say that we first compute the band-power spectra and appy the functiona forms whereas XFASTER ikeihood sices are computed when estimating the band-power spectra. Fig. 14 shows temperature sices of the XFASTER joint temperature and poarization ikeihood. Fig. 15 shows sices for TT (first coumn), EE (second coumn) and TE (third coumn) (more precisey n L). At the owest mutipoes the approximations differ, but as we move towards higher a but the Gaussian ikeihood converge to the same functiona form. For EE, however, the approximations differ noticeaby for the Gaussian and Lognorma ikeihood approximations (at 10, for instance). We further compare the XFASTER ikeihood to the exact ikeihoods at ow mutipoes. The purpose is two-fod. On one hand, we want to vaidate the XFASTER approximation, on the other, we want to determine the range at which the approximations used for the high- power spectrum estimator breaks down. Fig. 16 shows XFASTER ikeihood sices for the Phase 2 binned power spectra and the exact fu-sky ikeihood estimated for Phase 1a. At ow- the correations induced by the cut-sky widen the XFASTER ikeihood, whie the binning effect at high- resuts in a narrower C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS

20 842 G. Rocha et a. distribution (when compared to the fu-sky exact ikeihood). Both effects are given by 2 σ = (2 + 1)f sky C, (55) where f sky is the fraction of the sky observed and is the width of the mutipoe bins. To make a direct comparison of the XFASTER ikeihood with the exact ikeihood at ow- estimated in the same cut-sky map (Phase 2), which accounts for the correations induced by the sky cut, Fig. 17 shows the XFASTER ikeihood versus the pixe-based ikeihood sices for temperature aone (Phase 2b, asymmetric beam case). The pixe-based ikeihood, BFLike, is a brute-force ikeihood evauation of the mutivariate Gaussian in pixe domain for a ow-resoution map. The ow- data set of the CTP Phase 2 simuations was generated directy at N side = 16. In computing the sices, we conditioned on the remaining TT mutipoes, C TT with, and on a mutipoes of the TE and EE spectra (for detais see Rocha et a. 2011). As for this case the BFLIKE estimates its own peak by computing a brute-force pixe-based ikeihood on a downgraded map, we pot in Fig. 17 both ikeihoods with its own peaks ocations (eft-hand side) and assuming they peak at the same vaue (right-hand side), i.e. after dividing both distributions by their peak vaues. The pot on the eft-hand side might be miseading Figure 18. Parameter constraints from Phase 2a (12-detector map convoved with a symmetric beam). The one-dimensiona marginaized posteriors are from XFASTER ikeihood (with incusion of a modes, TT, EE, BB, TE): for ensembe average of signa+noise Monte Caro simuations, i.e. average run (soid back ines), for severa singe signa+noise Monte Caro simuations, and for vaues of the fiducia best-fit input parameters (back vertica ines). The parameters for the average power spectrum recover the true input parameters. Furthermore, this pot shows that there is no systematic bias for each Monte Caro simuation. This is to be expected as the Monte Caro simuations are reaizations of the WMAP best-fit mode. C 2011 Caifornia Institute of Technoogy. US government sponsorship acknowedged, MNRAS 414, Monthy Notices of the Roya Astronomica Society C 2011 RAS