Systematic Effects in Interferometric Observations of the CMB Polarization
Ata Karakci, Le Zhang, P. M. Sutter, Emory F. Bunn, Andrei Korotkov, Peter Timbie, Gregory S. Tucker, Benjamin D. Wandelt
aa r X i v : . [ a s t r o - ph . C O ] F e b Draft version October 15, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
SYSTEMATIC EFFECTS IN INTERFEROMETRIC OBSERVATIONS OF THE CMB POLARIZATION
Ata Karakci , Le Zhang , P. M. Sutter , , , , Emory F. Bunn , Andrei Korotkov , Peter Timbie ,Gregory S. Tucker , and Benjamin D. Wandelt , , , Department of Physics, Brown University, 182 Hope Street, Providence, RI 02912, USA Department of Physics, University of Wisconsin, Madison, WI 53706, USA Department of Physics, 1110 W Green Street, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, 98 bis, boulevard Arago, 75014 Paris,France CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis, boulevard Arago, 75014 Paris, France Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA Physics Department, University of Richmond, Richmond, Virginia 23173, USA Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Draft version October 15, 2018
ABSTRACTThe detection of the primordial B -mode spectrum of the polarized cosmic microwave background(CMB) signal may provide a probe of inflation. However, observation of such a faint signal requiresexcellent control of systematic errors. Interferometry proves to be a promising approach for over-coming such a challenge. In this paper we present a complete simulation pipeline of interferometricobservations of CMB polarization, including systematic errors. We employ two different methodsfor obtaining the power spectra from mock data produced by simulated observations: the maximumlikelihood method and the method of Gibbs sampling. We show that the results from both methodsare consistent with each other, as well as, within a factor of 6, with analytical estimates. Severalcategories of systematic errors are considered: instrumental errors, consisting of antenna gain and an-tenna coupling errors, and beam errors, consisting of antenna pointing errors, beam cross-polarizationand beam shape (and size) errors. In order to recover the tensor-to-scalar ratio, r , within a 10%tolerance level, which ensures the experiment is sensitive enough to detect the B -signal at r = 0 .
01 inthe multipole range 28 < ℓ < | g rms | = 0 . | ǫ rms | = 5 × − for antenna coupling, δ rms ≈ . ◦ for pointing, ζ rms ≈ . ◦ for beam shape, and µ rms = 5 × − forbeam cross-polarization. Subject headings : cosmic background radiation - cosmology:observations - instrumentation:interferometers - methods:data analysis - techniques: polarimetric INTRODUCTION
The Cosmic Microwave Background (CMB) has be-come one of the most fundamental tools for cosmology.High-precision measurements of the CMB polarization,especially detecting the primordial “B-mode” polariza-tion signals (Kamionkowski et al. 1997), will represent amajor step towards understanding the extremely earlyuniverse. These B -modes are generated by primordialgravitational waves. A detection of these signals wouldprobe the epoch of inflation and place an important con-straint on the inflationary energy scale (Hu & Dodelson2002). In addition, the secondary B -modes induced bygravitational lensing encode information about the dis-tribution of dark matter. However, the B -mode signalsare expected to be extremely small and current experi-ments can only place upper limits (Hinshaw et al. 2012)on the tensor-to-scalar ratio; the quest for the B -modesis a tremendous experimental challenge.Due to the weakness of the B -mode signals – the largestsignal of the primordial B -modes is predicted to be less Email: ata [email protected] than 0 . µ K – exquisite systematic error control is cru-cial for detecting and characterizing them. Comparedto imaging systems, interferometers offer certain advan-tages for controlling systematic effects because: (1) aninterferometer does not require rapid chopping and scan-ning (Timbie et al. 2006) and, with simple optics, inter-ferometric beam patterns have extremely low sidelobesand can be well understood; (2) interferometers are in-sensitive to any uniform sky brightness or fluctuations inatmospheric emissions on scales larger than the beamwidth; (3) without differencing the signal from sepa-rate detectors, interferometers measure the Stokes pa-rameters directly and inherently avoid the leakage fromtemperature into polarization (Bunn 2007) caused bymismatched beams and pointing errors, which are se-rious problems for B -mode detection with imaging ex-periments (Hu et al. 2003; Su et al. 2011; Miller et al.2009; O’Dea et al. 2007; Shimon et al. 2008; Yadav et al.2010; Takahashi et al. 2010); (4) for observations of smallpatches of sky, E - B mode separation would be cleaner inthe Fourier domain for interferometric data than in real-space; and (5) with the use of redundant baselines, sys-tematic errors can be averaged out. In addition, they of-fer a straightforward way to determine the angular powerspectrum since the output of an interferometer is the vis-ibility , that is, the Fourier transform of the sky intensityweighting by the response of the antennas.Interferometers have proved to be powerful toolsfor studying the CMB temperature and polarizationpower spectra. In fact, DASI (Kovac et al. 2002)was the first instrument to detect the faint CMB po-larization anisotropies. Pioneering attempts to mea-sure the CMB temperature anisotropy with interfer-ometers were made in the 1980s (Martin et al. 1980;Fomalont et al. 1984; Knoke et al. 1984; Partridge et al.1988; Timbie & Wilkinson 1988). Several groups havesuccessfully detected the CMB anisotropies. The CATtelescope was the first interferometer to actually de-tect structures in the CMB (O’Sullivan et al. 1995;Scott et al. 1996; Baker et al. 1999). CBI (Pearson et al.2003) and VSA (Dickinson et al. 2004; Grainge et al.2003) have detected the CMB temperature and po-larization angular power spectra down to sub-degreescales. In the next few years, the QUBIC instru-ment (Qubic Collaboration et al. 2011) based on thenovel concept of bolometric interferometry is expectedto constrain the tensor-to-scalar ratio to 0 .
01 at the 90%confidence level, with 1-year of observing.On the theory side, the formalism for analyz-ing interferometric CMB data has been well-developed (White et al. 1999; Hobson & Maisinger2002; Park et al. 2003; Myers et al. 2003, 2006;Hobson & Magueijo 1996). A pioneering study ofsystematic effects for interferometers based on ananalytic approach has been performed by Bunn (2007).However, this approach is of course only a first-orderapproximation for assessing systematics, since manyimportant effects, such as the configuration of the array,instrumental noise, and the sampling variance due tofinite sky coverage and incomplete uv -coverage, are nottaken into account. Any actual experiment thereforewill naturally require a complete simulation to assessexactly how systematic effects bias the power spectrumrecovery. In this concern, Zhang et al. (2012) havepresented a simulation pipeline to assess the systematicerrors, mainly focusing on pointing errors. With a fullmaximum likelihood (ML) analysis of mock data, thesimulation agrees with the analytical estimates andfinds that, for QUBIC-like experiments, the Gaussian-distributed pointing errors have to be controlled to thesub-degree level to avoid contaminating the primordial B -modes with r ≤ . B -modedetection and set allowable tolerance levels for those er-rors, we perform simulations for a specific interferomet-ric observation with an antenna configuration similar tothe QUBIC instrument. We also extend the analyticalexpressions (Bunn 2007) for characterizing systematic ef-fects on the full CMB power spectra. For verifying the power spectrum analysis, we employtwo completely independent codes based on the Gibbssampling algorithm and the maximum likelihood tech-nique. The use of Gibbs-sampling based Bayesian in-ference to interferometric CMB observations has beensuccessfully demonstrated by Sutter et al. (2012) andKarakci et al. (2013). It allows extraction of the under-lying CMB power spectra and reconstruction of the pureCMB signals simultaneously, with a much lower compu-tational complexity in contrast to the traditional maxi-mum likelihood technique (Hobson & Maisinger 2002).In this paper, for given input CMB angular powerspectra, we simulate the observed Stokes visibilities inthe flat-sky approximation. We believe that the flat-skysimulations are sufficiently accurate for the study of sys-tematic errors. First, in our simulation, we assume singlepointing observations with 5 ◦ beam width, correspond-ing to a sky coverage fraction of f sky = 0 . SYSTEMATICS
Instrument Errors and Beam Errors
In a polarimetric experiment, the Stokes parameters
I, Q, U and V can be obtained by using either linear orcircular polarizers. For a given baseline u jk = x k − x j , x k being the position vector of the k th antenna, the vis-ibilities can be written as a 2 × V jk (Bunn2007); V jk = Z d ˆ r A k (ˆ r ) R · S · R − A † j (ˆ r ) e − πi u jk · ˆ r , (1)where the 2 × A k (ˆ r ) is the antenna pattern and S = (cid:18) I + Q U + iVU − iV I − Q (cid:19) . (2)For a linear experiment , R is the identity matrix and fora circular experiment , R ( circ ) = 1 √ (cid:18) i − i (cid:19) . (3)Various systematic errors can be modeled in the defi-nition of the antenna pattern as follows (Bunn 2007) A k (ˆ r ) = J k · R · A ks (ˆ r ) · R − (4)where the Jones matrix J k represents the instrumentalerrors, such as gain errors and antenna couplings. Thematrix A ks is the antenna pattern that models the beamerrors, such as pointing errors, beam shape errors andcross-polarization. In an ideal experiment J k = I , where I is the identity matrix, and the antenna pattern is givenas A ks (ˆ r ) = A (ˆ r ) I , where A (ˆ r ) a circular Gaussian func-tion.In this paper we will consider only two types of in-strumental errors; antenna gain, parametrized by g k and g k , and couplings, parametrized by ǫ k and ǫ k . The cou-pling errors are caused by mixing of the two orthogo-nally polarized signals in the system. To account for thephase delays, the parameters g and ǫ are given as com-plex numbers. The Jones matrix for the k th antenna canbe written as (Bunn 2007) J k = (cid:18) g k ǫ k ǫ k g k (cid:19) . (5)For the beam errors, we will consider that each an-tenna has a slightly different beam width, ellipticity(beam shape errors) and beam center (pointing errors),as well as a cross-polar antenna response described byoff-diagonal entries in the antenna pattern matrix (Bunn2007); A ks = A k ( ρ, φ ) µ k ρ σ cos 2 φ µ k ρ σ sin 2 φ µ k ρ σ sin 2 φ − µ k ρ σ cos 2 φ ! . (6)where A k ( ρ, φ ) is an elliptical Gaussian function writ-ten in polar coordinates ( ρ, φ ), σ is the width of theideal beam and µ k is the cross-polarization parameterof the k th antenna. This particular form of the cross-polarization occurs, with µ k = σ /
2, when the curvedsky patch is projected onto a plane.
Control Levels
The effect of errors on the power spectra can be de-scribed by the root-mean-square difference between theactual spectrum, C XYactual , which is recovered from thedata of an experiment with systematic errors, and theideal spectrum, C XYideal , which would have been recoveredfrom the data of an experiment with no systematic er-rors; ∆ C XY = D(cid:0) C XYactual − C XYideal (cid:1) E / (7)where X, Y = { T, E, B } .The strength of the effect of systematics can bequantified by a tolerance parameter α XY definedby (O’Dea et al. 2007; Miller et al. 2009; Zhang et al.2012) α XY = ∆ C XY σ XYstat (8)where σ XYstat is the statistical 1- σ error in XY -spectrumof the ideal experiment with no systematic errors.The main interest in a B -mode experiment isthe tensor-to-scalar ratio r which can be estimatedas (O’Dea et al. 2007) r = P b ∂ r C BBb ( C BBb − C BBb,lens ) / ( σ BBb,stat ) P b ( ∂ r C BBb /σ BBb,stat ) (9)where b denotes the power band, C BBb,lens is the B -modespectrum due to weak gravitational lensing and C BBb de-pends linearly on r through the amplitude of the primor-dial B -modes. The tolerance parameter of r is given by α r = ∆ r/σ r (O’Dea et al. 2007);∆ r = P b α BBb ( ∂ r C BBb /σ BBb,stat ) P b ( ∂ r C BBb /σ BBb,stat ) , (10a) σ r = X b ( ∂ r C BBb /σ BBb,stat ) ! − / . (10b)For good control of systematics, the value of α r is re-quired to stay below a determined tolerance limit. Analytical Estimations
Analytical estimations of the effect of systematic errorson the polarization power spectra are extensively exam-ined in Bunn (2007). Defining a vector of visibilities v = ( V I , V Q , V U ) corresponding to a single baseline u pointing in the x direction, for an ideal experiment, wecan write (cid:10) | V I | (cid:11) = C T Tℓ =2 πu , (11a) (cid:10) | V Q | (cid:11) = C EEℓ =2 πu c + C BBℓ =2 πu s , (11b) (cid:10) | V U | (cid:11) = C EEℓ =2 πu s + C BBℓ =2 πu c , (11c) h V Q V ∗ U i = C EBℓ =2 πu ( c − s ) , (11d) (cid:10) V I V ∗ Q (cid:11) = C T Eℓ =2 πu c, (11e) h V I V ∗ U i = C T Bℓ =2 πu c. (11f)where c , s and c are averages of cos (2 φ ), sin (2 φ ) andcos(2 φ ) over the beam patterns: s = R | f A ( k − π u ) | sin (2 φ ) d k R | f A ( k ) | d k = 1 − c , (12)where f A is the Fourier transform of the ideal beampattern squared. The unbiased estimator for C XY = D ˆ C XY E is obtained asˆ C XY = v † · N XY · v (13)where N XY is a 3 × s and c (see Ap-pendix). For a baseline pointing in an arbitrary directionthe analysis is done in a rotated coordinate system: v rot = θ sin 2 θ − sin 2 θ cos 2 θ ! v , (14) θ being the angle between u and the x -axis.The effect of errors on visibilities can be described, tofirst order, by v actual = v ideal + δ v . Combining v ideal and δ v into a 6-dimensional vector w = ( v , δ v ), we canwrite the first order approximation as (Bunn 2007)(∆ ˆ C XYrms ) = T r [( N XY ·M w ) ]+( T r [ N XY ·M w ]) , (15)where M w = (cid:10) w · w † (cid:11) is the covariance matrix of w and N XY = (cid:18) N XY N XY (cid:19) . (16)The error on a particular band power is, then, given asan expansion in terms of ideal power spectra:(∆ ˆ C XYrms,b ) = p rms X I,J κ XY,I,J C Ib C Jb (17)where p is the parameter that characterizes the error,such as gain, g , coupling, ǫ , or cross-polarization, µ , and I, J = { T T, T E, EE, BB } . This expression is valid for asingle baseline. For a system with n b baselines in band b ,∆ ˆ C XYrms,b must be normalized by 1 / √ n b , assuming there isno correlation between error parameters of different base-lines. Analytical estimations of the coefficients κ XY,I,J for various systematic errors are presented in the Ap-pendix. SIMULATIONS
The input
I, Q and U maps are constructed over30-degree square patches with 64 pixels per side asdescribed in Karakci et al. (2013) with the cosmolog-ical parameters consistent with the 7-year results ofWMAP (Larson et al. 2011; Komatsu et al. 2011). Thetensor-to-scalar ratio is taken to be r = 0 .
01. The angu-lar resolution of the signal maps is 28 arcminutes, cor-responding to a maximum available multipole of ℓ max =384. The ideal primary beam pattern, A (ˆ r ), is modeledas a Gaussian with peak value of unity and standard de-viation of σ = 5 ◦ , which drops to the value of 10 − atthe edges of the patch, reducing the edge-effects causedby the periodic boundary conditions of the fast Fouriertransformations. Although the patch size is too largefor the flat-sky approximation, the width of the primarybeam is small enough to employ the approximation.The interferometer configuration is a close-packedsquare array of 400 antennas with diameters of 7.89 λ .The observation frequency is 150 GHz with a 10-GHzbandwidth. This configuration is similar to the QUBICdesign (Qubic Collaboration et al. 2011). With this fre-quency and antenna radius, the minimum available mul-tipole is ℓ min = 28. The baselines are uniformly rotatedin the uv -plane over a period of 12 hours while observingthe same sky patch. The resulting interferometer patternis shown in Figure 1.The noise at each pixel for the temperature data is ob-tained from the total observation time that all baselines Interferometer Pattern, 150 GHz-40 -20 0 20 40 60u [ λ ]-40-20 0 20 40 60 v [ λ ] Figure 1.
Interferometer pattern created over an observation pe-riod of 12 hours by 20 ×
20 close-packed array of antennas of radius7 . λ . spend in the pixel. The noise covariance for the baseline u kj is given as (White et al. 1999) C kjN = (cid:18) λ T sys η A A D (cid:19) (cid:18) ν t a ¯ n (cid:19) δ kj (18)where T sys is the system temperature, λ is the observa-tion wavelength, η A is the aperture efficiency, ∆ ν is thebandwidth, ¯ n is the number of baselines with the samebaseline vector, and t a is the integration time. The noisevalue is normalized by a constant to have an rms noiselevel of 0 . µK per visibility, yielding an average overallsignal-to-noise ratio of about 5 for the Q and U maps.The systematic errors are introduced by calculatingthe visibilities in each pixel according to Eq. 1. Eachantenna has random error parameters for gain, cou-pling, pointing, beam shape, and cross-polarization er-rors drawn from Gaussian distributions with rms valuesof | g rms | = 0 . | ǫ rms | = 5 × − , δ rms = 0 . σ ≈ . ◦ , ζ rms = 0 . σ ≈ . ◦ , and µ rms = 5 × − , respectively.Here δ is the offset of the beam centers of the anten-nas and ζ is the deviation in the beam width along theprincipal axes of the elliptical beams. As the baseline ro-tates, the beam patterns of the corresponding antennasget rotated as well. Whenever a baseline crosses a newpixel, the visibility within the pixel, given by Eq. 1, iscalculated again with the rotated beam patterns. Thedata in a given pixel is taken as the average of all thevisibilities calculated in that pixel.In a circular experiment, the Stokes variables Q and U can be simultaneously obtained for the same baseline.Thus, for a circular experiment, p Qcirc = p Ucirc . However,for a linear experiment, direct measurement of Q requiresperfect cancellation of the much larger I contribution inEq. 2. Practically, a linear experiment only measures U .Since U → Q under a 45 ◦ clockwise rotation, Q can bemeasured by measuring U with 45 ◦ -rotated linear polar-izers. Since Q and U are not measured simultaneously bythe same baseline, in general, the error parameters p Qlin and p Ulin are treated as the distinct parameters in a linearexperiment, i.e., p Qlin = p Ulin . To simulate this, we calcu-late V U with a set of error parameters, p Ulin . Then Q and U in Eq. 2 are replaced by − U and Q , respectively, and V U is calculated again with a different set of parameters, p Qlin , to obtain V Q . The simulation requires 4.5 CPU-hours for the circular experiment and 13.5 CPU-hoursfor the linear experiment. ANALYSIS METHODS
Maximum Likelihood Analysis
The scheme for the maximum likelihood (ML) analy-sis of CMB power spectra from interferometric visibil-ity measurements is presented in Hobson & Maisinger(2002); Park et al. (2003); Zhang et al. (2012), whichwe briefly summarize here. The ML estimator of thepower spectrum has many desirable features (Bond et al.1998; Stuart & Ord 1987) and has been widely applied inCMB cosmology (Bond et al. 1998; Bunn & White 1997;Hobson & Maisinger 2002).In practice, we divide the total ℓ -range into N b spectral bands, each of bin-width ∆ ℓ . The powerspectrum C ℓ thus can be parametrized as flat band-powers C b ( b = 1 , . . . , N b ) over ∆ ℓ to evaluate the like-lihood function (Bunn & White 1997; Bond et al. 1998;Gorski et al. 1996; White et al. 1999). In each of theband-powers, we assume ℓ ( ℓ + 1) C ℓ to be a constantvalue to characterize the averaged C ℓ over ∆ ℓ andhas C b ≡ π | u b | S ( | u b | ) as the flat-sky approxima-tion (White et al. 1999).In our case, the CMB signals and the instrumentalnoise are assumed to be Gaussian random fields. There-fore, for a given set of CMB band-power parameters {C T Tb , C EEb , C BBb , C T Eb , C T Bb , C EBb } , the signal covariancematrices can be written as C ijZZ ′ = N b X b =1 X X,Y C XYb Z | u b || u b | π dww × W i,jZZ ′ XY ( w ) , (19)where we introduced the so-called window functions W ijZZ ′ XY given by W ijZZ ′ XY ( | w | ) = Z π dφ w ω ZX ω Z ′ Y ˜ A ( u i − w ) ˜ A ∗ ( u j − w ) , (20)where Z, Z ′ = { I, Q, U } and X, Y = { T, E, B } with ω IT = 1, ω UE = sin 2 φ w , ω UB = cos 2 φ w , ω QE =cos 2 φ w , ω QB = − sin 2 φ w and otherwise zero.Due to the fact that the window functions W ijZZ ′ XY ( | w | ) are independent of C b , the integralsof the window functions over w in Eq. 19 only haveto be calculated once before evaluating the covariancematrices. Additionally, if the primary beam pattern A ( x ) is Gaussian, the window functions can be expressedanalytically (see details in Hobson & Maisinger (2002);Park et al. (2003); Zhang et al. (2012)).We evaluate the likelihood function by varying theCMB band-powers using the above parametrization. Fol-lowing Hobson & Maisinger (2002); Park et al. (2003);Myers et al. (2003); Zhang et al. (2012), the logarithm ofthe likelihood function for interferometric observations is given byln L ( {C b } ) = n log π − log | C V + C N |− d † V ( C V + C N ) − d V , (21)where C V is the predicted signal covariance matrix and C N is the instrumental noise covariance matrix, d V isthe observed visibility data vector constructed by d V ≡ ( · · · ; V I ( u i ) , V Q ( u i ) , V U ( u i ); · · · )( i = 1 , . . . , n ) where i denotes the visibility data contributed from the pureCMB signals and the instrument noise at the i -th pixelin the uv -plane and we have a total of n data points.As mentioned by Hobson & Maisinger (2002) and ref-erences therein, the combination of the sparse matrixconjugate-gradient technique and Powell’s directional-set method give a sophisticated and optimized numerical al-gorithm for maximizing the likelihood function to findthe “best-fitted” CMB power spectrum. With an appro-priate initial guess to start iteration, independent line-maximization is performed for each band-power param-eter in turn, while fixing the others. Typically, this pro-cess requires a few iterations, of order N b , to achieve themaximum-likelihood solution. For about 4000 visibilitiesin a QUBIC-like observation, the maximum-likelihoodsolution of 6 × ∼ Gibbs Sampling Method
As discussed in Karakci et al. (2013), the method ofGibbs sampling has been applied to interferometric ob-servations of the polarized CMB signal in order to recoverboth the input signal and the power spectra.The CMB signal is described as a 3 n p dimensional vec-tor, s , of the Fourier transform of the pixelated signalmaps of n p pixels; s = ( ..., ˜ T i , ˜ E i , ˜ B i , ... ); i = 0 , ..., n p − s , and the signal covariance, S = (cid:10) s s † (cid:11) , from thejoint distribution P ( S , s , d V ) by successively samplingfrom the conditional distributions in an iterative fashion(Larson et al. 2007; Karakci et al. 2013): s a +1 ← P ( s | S a , d V ) (22a) S a +1 ← P ( S | s a +1 ) . (22b)After a “burn-in” phase, the stationary distribution ofthe Markov chain is reached and the samples approxi-mate to being samples from the joint distribution.To determine that the stationary distribution of theMarkov chain has been reached, the Gelman-Rubin(GR) statistic is employed (Gelman & Rubin 1992;Sutter et al. 2012; Karakci et al. 2013). For multiple in-stances of chains, when the ratio of the variance withineach chain to the variance among chains drops to a valuebelow a given tolerance, the convergence is said to be at-tained. The convergence of the Gibbs sampling is reachedroughly in 30 CPU-hours. RESULTS
Power Spectra
The mean posterior power spectra, together with theassociated uncertainties at each ℓ -bin, obtained by themethods of Gibbs Sampling (GS) and Maximum Likeli-hood (ML) for the ideal linear experiment, are shown inFigure 2. The input power spectra, which are used toconstruct the signal realization, and the spectra of thesignal realization are also shown in Figure 2. Almost allof our estimates fall within 2 σ of the expected value. Effect of Errors
In order to estimate α we ran 30 realizations of eachsystematic error simulation for both linear and circularexperiments. To keep the value of α r less than 10%tolerance limit at r = 0 .
01, we set the rms values ofthe parameters for gain errors to | g rms | = 0 .
1, for cou-pling errors to | ǫ rms | = 5 × − , for pointing errors to δ rms ≈ . ◦ , for beam shape errors to ζ rms ≈ . ◦ , andfor cross-polarization errors to µ rms = 5 × − .Figure 3 shows the mean values of α XY for beam er-rors, averaged over 30 realizations. The results from MLand GS methods are in good agreement for both linearand circular experiments. In all three cases α BB ∼ . ℓ , as expected. Although the cross polarization hasa much smaller error parameter, its effect on the powerspectra is comparable to the pointing and shape errors.The reason for this is the leakage from T T power into BB power that is caused by the off-diagonal elements ofthe beam pattern, whereas the source of α BB for point-ing and shape errors is the EE → BB leakage (Bunn2007).The mean values of α XY for instrumental errors areshown in Figure 4. For gain and coupling errors, α XY is roughly at the 10% level. The main contribution forthe α BB comes from the leakage from EE power into BB power for gain errors. As in the case of cross-polarizationerrors, despite having a much smaller parameter thangain, α BB ∼ . ℓ for antenna coupling errorsbecause of T T → BB leakage.We simulated the systematics by turning on one errorat a time. However, in a realistic experiment, all system-atic errors act together simultaneously, causing a largereffect on the spectra. In order to see this combined effectwe ran 30 realizations with all the systematic errors dis-cussed in previous sections turned on at once. The resultsare also shown in Figure 4. As expected, the combinedeffect is almost twice as large as the individual cases. Comparison to Analytical Estimations
Analytical estimations for α XY are obtained from thequadrature difference of Eq. 15, normalized by the num-ber of baselines. In general α XY has a polynomial de-pendence on s . For our interferometer configuration s is roughly s ∼ . /ℓ . The explicit forms of the un-normalized estimations are given in the Appendix.In general, our simulated results are larger than theestimations in all ℓ -bins. This is expected because our analytical estimations are only first order approximationswhere it is also assumed that the errors associated withbaselines are uncorrelated, making them lower boundsfor the estimations. In reality, there is a correlation be-tween errors associated with baselines having commonantennas, a fact that is captured by our simulations.Upper bounds for the estimations can be found by un-realistically assuming full correlation of errors betweenbaselines, where each baseline has the same error. Forour interferometer design, this corresponds to roughly 65times larger values. We expect our results to fall betweenuncorrelated and fully correlated estimations. In order tocompare our results with the analytical ones, we considerthe rms values of α XY averaging over the ℓ -bins. Figure5 shows the ratios of α XYrms obtained by ML and GS meth-ods to the estimated α XYrms . In most cases, both methodsare in agreement with the analytical results within a fac-tor of 6.
Biases in Tensor-to-Scalar Ratio
The major goal of QUBIC-like experiments is to detectthe signals of the primordial B-modes, the magnitudeof which is characterized by the tensor-to-scalar ratio r . In this context, it is necessary to propagate the ef-fects of systematic errors through to r to assess properlythe systematic-induced biases in the primordial B-modemeasurements.The shape of the primordial BB power spectrum C BBℓ,prim is insensitive to r but the amplitude is directlyproportional to r . We can straightforwardly convert theamplitude of the systematic-induced false BB into thebias in r by writing C BBℓ,prim = r C BBℓ,r =1 in Eqs. 9 and 10where C BBℓ,r =1 is the CAMB (Lewis et al. 2000) calculatedprimordial BB power spectrum at r = 1. The tensor-to-scalar ratios obtained from ideal linear experiment byGibbs sampling and maximum likelihood methods arefound as r GS = 0 . ± .
012 and r ML = 0 . ± . r canbe obtained without subtracting the lensed spectrum inEq. 9 and by taking only the first bin where the ef-fect of lensing is the least; r lensedGS = 0 . ± .
014 and r lensedML = 0 . ± . r are illus-trated in Figure 6, evaluated by both the GS and MLmethods based on the simulations performed in the lin-ear and circular bases. Both methods demonstrate goodagreement, within a factor of 2.5. Although the mockvisibility data are simulated based on only one realiza-tion of CMB anisotropy fields, drawn from the powerspectra with input BB for r = 0 .
01, the resulting false BB band-powers for the different systematic errors areexpected to be a good approximation for other r valuessince the leading-order false B -modes are contaminatedonly by the leakage of T T , T E and EE power spectra,which are independent of r .The simulations show that, due to the leakage of T T C l TT l ( l + ) / π [ µ K ] lTT Angular Power SpectrumGSMLInputRealization -60-30 0 30 60 90 120 150 50 100 150 200 250 300 350 C l T E l ( l + ) / π [ µ K ] lTE Angular Power SpectrumGSMLInputRealization 0 5 10 15 20 25 50 100 150 200 250 300 350 C l EE l ( l + ) / π [ µ K ] lEE Angular Power SpectrumGSMLInputRealization 0 0.01 0.02 0.03 0.04 0.05 50 100 150 200 250 300 350 C l BB l ( l + ) / π [ µ K ] lBB Angular Power SpectrumGSMLInputRealization-4-3-2-1 0 1 2 3 50 100 150 200 250 300 350 C l T B l ( l + ) / π [ µ K ] lTB Angular Power SpectrumGSMLInputRealization -0.3-0.2-0.1 0 0.1 0.2 50 100 150 200 250 300 350 C l EB l ( l + ) / π [ µ K ] lEB Angular Power SpectrumGSMLInputRealization Figure 2.
Mean posterior power spectra obtained by Gibbs Sampling (GS) for each ℓ -bin are shown in black. The power spectraestimations obtained by Maximum Likelihood (ML) method are shown in blue. Dark and light grey indicate 1 σ and 2 σ uncertainties forGibbs sampling results, respectively. The binned power spectra of the signal realization are shown in pink. Red lines are the input CMBpower spectra obtained by CAMB for a tensor-to-scalar ratio of r = 0 . signals into BB , even though the cross-polarization andcoupling errors are very small, e.g., µ rms = 5 × − and | ǫ rms | = 5 × − , the resulting biases in r are compa-rable to those induced by relatively larger pointing, gainand shape errors. In addition, when increasing the cross-polarization and coupling errors by a factor of 10, thesimulations show that the resulting biases would roughlyincrease by the same factor. As expected, the systematicerrors are approximately linearly proportional to theirerror parameters. We also find that the combined sys-tematic effects (referred to as “c” in Figure 6) wouldincrease the biases and their values are consistent withthe quadrature sum of the individual errors within 10%.If we set up an allowable tolerance level of 10% on r ,where r is assumed to be r = 0 .
01, for QUBIC-like ex-periments the error parameters adopted as in Figure 6 satisfy this threshold when each systematic error occursalone during observations. But if all the systematic er-rors are present at the same time, on average, we requireroughly 2 times better systematic control on each errorparameter. Although the tolerance level for r is chosento be α r = 0 .
1, our results can directly apply to anyother desired threshold level as long as the linear depen-dence of systematic effects on error parameters is a goodapproximation for sufficiently small error parameters. DISCUSSIONS
In this work a complete pipeline of simulations is devel-oped to diagnose the effects of systematic errors on theCMB polarization power spectra obtained by an interfer-ometric observation. A realistic, QUBIC-like interferom-eter design with systematics that incorporate the effects δ=0.7 o Antenna Pointing Errors (TT & TE)50 100 150 200 250 300 350Multipole l ∆ C l / σ l δ=0.7 o Antenna Pointing Errors (EE & BB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l δ=0.7 o Antenna Pointing Errors (TB & EB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l ζ=0.7 o Beam Shape Errors (TT & TE)50 100 150 200 250 300 350Multipole l ∆ C l / σ l ζ=0.7 o Beam Shape Errors (EE & BB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l ζ=0.7 o Beam Shape Errors (TB & EB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l µ=5 x10 −4 Beam Cross Polarization (TT & TE)50 100 150 200 250 300 350Multipole l ∆ C l / σ l µ=5 x10 −4 Beam Cross Polarization (EE & BB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l µ=5 x10 −4 Beam Cross Polarization (TB & EB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l Figure 3.
Beam errors. The values of α XY , averaged over 30 simulations, obtained by both maximum likelihood (ML) method (triangles)and the method of Gibbs sampling (GS) (solid dots) are shown. The three rows indicate, from top to bottom, pointing errors with δ rms ≈ . ◦ , beam shape errors with ζ rms ≈ . ◦ , and beam cross-polarization with µ rms = 5 × − . Left panel shows α TT (red)and α TE (blue). Middle panel shows α EE (red) and α BB (blue). Right panel shows α TB (red) and α EB (blue). Linear and circularexperiments are shown by solid and dashed lines, respectively. of sky-rotation is simulated. The mock data sets are an-alyzed by both the maximum likelihood method and themethod of Gibbs sampling. The results from both meth-ods are found to be consistent with each other, as wellas with the analytical estimations within a factor of 6.In order to assess the level at which systematic effectsmust be controlled, a tolerance level of α r = 0 . B -signal at r = 0 .
01 level (O’Dea et al. 2007).We see that, for a QUBIC-like experiment, the contami-nation of the tensor-to-scalar ratio at r = 0 .
01 does notexceed the 10% tolerance level in the multipole range28 < ℓ <
384 when the Gaussian-distributed system-atic errors are controlled with precisions of | g rms | = 0 . | ǫ rms | = 5 × − for antenna cou-pling, δ rms ≈ . ◦ for pointing, ζ rms ≈ . ◦ for beamshape, and µ rms = 5 × − for beam cross-polarizationwhen each error acts individually. However, in a real-istic experiment all the systematic errors are simultane- ously present, in which case the tolerance parameter of r roughly reaches the 20% level, suggesting that bettercontrol of systematics would be needed.Apart from the systematics presented in the paper, wealso ran simulations to analyze the effects of uncertaintiesin the positions of the antennas. In order to have aneffect on the order of α BB = 0 .
1, we found that theuncertainty in the position of each antenna should be onthe order of 50% of the length of the uv -plane. Sincesuch an error is unrealistically large, we conclude thatthe effect of antenna position errors on power spectra isnegligible in an interferometric observation.We have shown that a QUBIC-like experiment hasfairly manageable systematics, which is essential for thedetection of primordial B -modes. Since our interferome-ter design has a large number of redundant baselines (ap-proximately 10 baselines per visibility), as a further im-provement, a self-calibration technique can be employedto significantly reduce the level of instrumental errors g=0.1 Antenna Gain Errors (TT & TE)50 100 150 200 250 300 350Multipole l ∆ C l / σ l g=0.1 Antenna Gain Errors (EE & BB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l g=0.1 Antenna Gain Errors (TB & EB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l ε=5 x10 −4 Antenna Coupling Errors (TT & TE)50 100 150 200 250 300 350Multipole l ∆ C l / σ l ε=5 x10 −4 Antenna Coupling Errors (EE & BB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l ε=5 x10 −4 Antenna Coupling Errors (TB & EB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l Combined Systematic Errors (TT & TE)50 100 150 200 250 300 350Multipole l ∆ C l / σ l Combined Systematic Errors (EE & BB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l Combined Systematic Errors (TB & EB)50 100 150 200 250 300 350Multipole l ∆ C l / σ l Figure 4.
Instrumental and combined systematic errors. The values of α XY , averaged over 30 simulations, obtained by both maximumlikelihood (ML) method (triangles) and the method of Gibbs sampling (GS) (solid dots) are shown. Top row: antenna gain with | g rms | = 0 . | ǫ rms | = 5 × − . Bottom row: combined effect of beam and instrumental systematic errors. Leftpanel shows α TT (red) and α TE (blue). Middle panel shows α EE (red) and α BB (blue). Right panel shows α TB (red) and α EB (blue).Linear and circular experiments are shown by solid and dashed lines, respectively. (Liu et al. 2010). ACKNOWLEDGMENTS
Computing resources were provided by the Univer-sity of Richmond under NSF Grant 0922748. Our im-plementation of the Gibbs sampling algorithm uses theopen-source PETSc library (Balay et al. 1997, 2010)and FFTW (Frigo & Johnson 2005). G. S. Tucker andA. Karakci acknowledge support from NSF Grant AST-0908844. P. M. Sutter and B. D. Wandelt acknowledgesupport from NSF Grant AST-0908902. B. D. Wandeltacknowledges funding from an ANR Chaire d’Excellence,the UPMC Chaire Internationale in Theoretical Cosmol-ogy, and NSF grants AST-0908902 and AST-0708849.L. Zhang and P. Timbie acknowledge support from NSFGrant AST-0908900. E. F. Bunn acknowledges sup-port from NSF Grant AST-0908900. We are grateful forthe generous hospitality of The Ohio State University’sCenter for Cosmology and Astro-Particle Physics, which hosted a workshop during which some of these resultswere obtained.
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Following Bunn (2007), we obtain first order approximations for the ∆ ˆ C XYrms given, for a single baseline, in Eq. 15.For a baseline lying on the x -axis, the matrices in Eq. 15 are given as N T T = ! , N EE = (cid:20)(cid:16) c (cid:17) − (cid:16) s (cid:17) (cid:21) − c
00 0 − s , N BB = (cid:20)(cid:16) c (cid:17) − (cid:16) s (cid:17) (cid:21) − − s
00 0 c N T E = 12 c ! , N T B = 12 c ! , N EB = 12( c − s ) ! . The covariance matrix can be written in block-matrix form as M w = (cid:18) M M M † M (cid:19) where M = C T T C T E c C
T B cC T E c C EE c + C BB s C EB ( c − s ) C T B c C EB ( c − s ) C EE s + C BB c . For a baseline lying in an arbitrary direction, these matrices must be transformed as M → R − M R and N XY → R − N XY R , where R is the rotation matrix given in Eq. 14. The resulting expression will, then, be averaged over θ .For instrumental errors δ v = E · v , which gives M = M · E † and M = E · M · E † .GAIN ERRORS g = 12 ( g i + g i + g j ∗ + g j ∗ ) , g = 12 ( g i − g i + g j ∗ − g j ∗ ) Linear Basis γ = 12 ( g Q + g U ) , γ = 12 ( g Q − g U ) , γ = 12 ( g Q + g U ) , E gainlinear = γ γ γ + γ
00 0 γ − γ ! C T Trms ) = 8 Re { γ } ( C T T ) (∆ ˆ C T Erms ) = (6 Re { γ } + Re { γ } − Im { γ } )( C T E ) + (2 Re { γ } + | γ | ) C T T C EE (∆ ˆ C EErms ) = (8 Re { γ } + 4 Re { γ } )( C EE ) (∆ ˆ C BBrms ) = s | γ | ( C EE ) (∆ ˆ C T Brms ) = (3 Re { γ } − Im { γ } )( C T E ) + ( | γ | + 8 Re { γ } ) C T T C EE (∆ ˆ C EBrms ) = Re { γ } ( C EE ) + ( | γ | + 2 Re { γ } ) C EE C BB Circular Basis E gaincircular = g g ig − ig g ! (∆ ˆ C T Trms ) = 8 Re { g } ( C T T ) (∆ ˆ C T Erms ) = 6 Re { g } ( C T E ) + 2 Re { g } C T T C EE (∆ ˆ C EErms ) = 8 Re { g } ( C EE ) (∆ ˆ C BBrms ) = 2 | g | C EE ( C BB + s C EE )(∆ ˆ C T Brms ) = ( Im { g } − Re { g } )( C T E ) + | g | C T T C EE (∆ ˆ C EBrms ) = 2 Im { g } ( C EE ) COUPLING ERRORS e = 12 ( e i + e i + e j ∗ + e j ∗ ) , e = 12 ( e i − e i − e j ∗ + e j ∗ ) Linear Basis ǫ = 12 ( e Q + e U ) , ǫ = 12 ( e Q − e U ) , ǫ = 12 ( e Q + e U ) , ǫ = 12 ( e Q − e U ) , E couplinglinear = ǫ ǫ + ǫ − ǫ − ǫ ǫ − ǫ ǫ − ǫ ! (∆ ˆ C T Trms ) = (3 Re { ǫ } − Im { ǫ } )( C T E ) + | ǫ | C T T C EE (∆ ˆ C T Erms ) = 2( Re { ǫ } + Re { ǫ } )( C T T ) (∆ ˆ C EErms ) = 2( | ǫ | + | ǫ | ) C T T C EE (∆ ˆ C BBrms ) = 2( | ǫ | + | ǫ | ) C T T C BB + 2 s ( | ǫ | + | ǫ | ) C T T C EE (∆ ˆ C T Brms ) = 2( Re { ǫ } + Re { ǫ } )( C T T ) (∆ ˆ C EBrms ) = ( | ǫ | + | ǫ | ) C T T C EE + (3 Re { ǫ } + 3 Re { ǫ } − Im { ǫ } − Im { ǫ } )( C T E ) Circular Basis E couplingcircular = e ie e ie ! C T Trms ) = (3 Re { e } − Im { e } + 2 Im { e } )( C T E ) + ( | e | + | e | ) C T T C EE (∆ ˆ C T Erms ) = ( Re { e } + Im { e } )(( C T T ) + ( C T E ) + C T T C EE )(∆ ˆ C EErms ) = (3 Re { e } − Im { e } + 2 Im { e } )( C T E ) + ( | e | + | e | ) C T T C EE (∆ ˆ C BBrms ) = ( | e | + | e | ) C T T ( C BB + s C EE )(∆ ˆ C T Brms ) = ( Re { e } + Im { e } )( C T T ) (∆ ˆ C EBrms ) = ( | e | + | e | ) C T T C EE + (3 Re { e } + 3 Im { e } − Im { e } − Re { e } )( C T E ) POINTING ERRORSDefining δ ˆ r k as the deviation of the k th antenna’s pointing center, we can write, to the first order, A j (ˆ r ) A k (ˆ r ) = exp [ − (ˆ r − σ δ ) / σ ] , where δ = ( δ ˆ r j + δ ˆ r k ) / σ. δV Z = − iσ Z d k ˜ Z ( k )[ ˜ A ( k − π u )] ∗ [( k − π u ) · δ Z ] h V X δV ∗ Y i = 0 and h δV X δV ∗ Y i = ( δ X · δ Y ) h V X V ∗ Y i . Linear Basis δ = 12 ( δ Q + δ U ) , δ = 12 ( δ Q − δ U )(∆ ˆ C T Trms ) = | δ | ( C T T ) (∆ ˆ C T Erms ) = | δ | ( C T E ) + (4 | δ | + | δ | ) C T T C EE (∆ ˆ C EErms ) = ( | δ | + | δ | )( C EE ) (∆ ˆ C BBrms ) = | δ | C BB ( C BB + 2 s C EE ) + | δ | C EE ( C BB + s C EE )(∆ ˆ C T Brms ) = | δ | C T T ( C BB + s C EE ) + | δ | C T T C EE (∆ ˆ C EBrms ) = | δ | C EE ( C BB + s C EE ) + | δ | ( C EE ) Circular Basis δ = 0SHAPE ERRORSThe product of two elliptic Gaussian beams can be written as a single elliptic Gaussian: A j (ˆ r ) A k (ˆ r ) = exp (cid:20) − ( x cos β + y sin β ) σ + σ x ) − ( y cos β − x sin β ) σ + σ y ) (cid:21) , where β is the angle between the major axis of the resulting ellipse and the x -axis. δV Z = − σ Z d k ˜ Z ( k )[( ^ A ∆ Z )( k − π u )] ∗ where ∆ Z ( x, y ) = x ( ζ Zx cos β + ζ Zy sin β ) + y ( ζ Zy cos β + ζ Zx sin β ) + xy ( ζ Zx − ζ Zy ) sin 2 β , and ζ Zx,y = σ Zx,y /σ.