A numerical study of observational systematic errors in lensing analysis of CMB polarization
AA numerical study of observational systematic errors in lensinganalysis of CMB polarization
Ryo Nagata and Toshiya Namikawa Japan Aerospace Exploration Agency (JAXA), Institute of Space and AstronauticalScience (ISAS), Sagamihara, Kanagawa 252-5210, Japan Department of Applied Mathematics and Theoretical Physics, University ofCambridge, Cambridge CB3 0WA, United KingdomFebruary 2, 2021
Abstract
Impacts of observational systematic errors on the lensing analysis of the cosmic mi-crowave background (CMB) polarization are investigated by numerical simulations. Wemodel errors of gain, angle, and pointing in observation of the CMB polarization and simu-late polarization fields modulated by the errors. We discuss about response of systematicsinduced B-modes to amplitude and spatial scale of the imposed errors and show that re-sults of the lensing reconstruction and delensing analysis behave accordingly to it. It isobserved that error levels expected in the near future lead to no significant degradationin delensing efficiency.
Detection of the primordial gravitational waves (GWs) which originate from cosmic inflationis expected to provide us an opportunity to understand a very early stage of the universe. Anattempt to extract a signal of the primordial GWs from B -modes of the cosmic microwavebackground (CMB) polarization is one of the most promising ways to it [1, 2]. The tightestbound to date on amplitude of the primordial GWs characterized by the tensor-to-scalar ratio, r , is r < .
06 (95% C.L.) obtained from BICEP2/Keck Array [3] or r < .
044 (95% C.L.)from BICEP2/Keck Array and Planck [4].In recent observations of the CMB polarization, employment of integrated detector arraysis rapidly increasing measurement sensitivity. Observations in the near future will provide lownoise polarization maps of a few µ K-arcmin [5, 6, 7, 8]. It is expected that a major contaminantin observed polarization maps is disturbance due to gravitational lensing effects [9, 10]. Thegravitational lensing converts a small part of E -modes into B -modes [11]. The lensing B -modes have a spectrum like white noise in large scales and its amplitude is comparable to the B -mode spectrum of the primordial GWs origin with r = 0 .
01 at the scale of the recombinationbump. Delensing by which we remove the contamination from the lensing effects is consideredto be an indispensable analysis method in next generation high sensitivity experiments [12,13, 14, 15]. 1 a r X i v : . [ a s t r o - ph . C O ] J a n o achieve delensing, we need to identify lensing mass distribution which is characterizedby the quantity so-called lensing potential. CMB lensing mass maps have been reconstructedby multiple CMB observations such as ACTPol [16, 17], BICEP2/Keck Array [18], Planck[19, 20, 21], POLARBEAR [22, 23] and SPTpol [24, 25]. Ongoing, near term, and nextgeneration CMB observations such as Advanced ACT [26], Simons Observatory [5], SPT-3G[27], and CMB Stage-IV [15] will significantly increase sensitivity to the lensing potential.Those reconstructed mass maps can be used to form a lensing B -mode template which is thensubtracted from observed B -modes [10] or is incorporated into a likelihood analysis as a new“frequency” map [28]. In addition to the lensing maps measured internally with CMB data,we can also use external mass tracers which correlate with the CMB lensing signal efficiently,such as the cosmic infrared background (CIB) [29, 30, 31], radio and optical galaxies [32, 33],galaxy weak lensing [34], and intensity maps of high-redshift line emissions [35, 36]. In the lastfew years, several works have demonstrated delensing for small-scale lensed temperature andpolarization anisotropies using real data [37, 38, 39, 40, 41]. Recently, Ref. [28] demonstratedfor the first time delensing of large-scale B -modes using CIB as a mass tracer and obtain animproved constraint on r compared with that without delensing.Multiple works have explored possible systematics in delensing for future CMB experi-ments. For example, delensed CMB anisotropies in small scales have large biases arising fromhigher order correlations in the CMB anisotropies [42] and correlation between the lensingtemplate and the CMB anisotropies to be delensed [43]. This bias can be mitigated by severalproposed methods [44, 42, 40]. To constrain the primordial GWs, we need only large-scale B -modes and can simply mitigate this bias by using only small-scale CMB anisotropies toconstruct the lensing B -mode template [10, 45, 46]. Also, the lensing B -modes are known tobe a non-Gaussian field and non-Gaussian covariance of the lensing B -mode power spectrumdegrades cosmological parameter constraints [47]. However, non-Gaussian covariance of thedelensed B -mode power spectrum, which potentially degrades a constraint on r , is negligiblein large angular scales [48]. A power spectrum of B -modes delensed by using external masstracers could be biased by uncertainties of the tracers themselves. However, the delensed B -modes would not have significant contributions from non-linear growth of the large-scalestructure [49] and residual Galactic foreground emissions [50]. Further, the uncertainties ofthe mass tracers are, in general, constrained by power spectra of the mass tracers as well ascross-spectra between the mass tracers and CMB lensing maps [30]. In actual observations,presence of survey boundaries, inhomogeneous scans and atmospheric noise could degradedelensing efficiency if we simply adopt sub-optimal filtering to observed CMB data [37]. How-ever, efficient delensing would be realized by modifying the filtering appropriately.In this paper, we investigate impacts of observational systematic errors, which are unavoid-able in practical situations, on reconstruction of the lensing potential and evaluate degradationof delensing efficiency caused by propagation of the systematics through mis-estimation of thelensing potential. Despite the multiple efforts on exploring feasibility of delensing describedabove, impacts of observational systematics on delensing have not yet thoroughly been inves-tigated for high precision CMB polarization experiments. This paper, therefore, provides afirst insight into response of the delensed B -mode spectrum to several major observationalsystematics.This paper is organized as follows: The procedure of our simulations is described in Sec. 2.In Sec. 3 and Sec. 4, we show results of the lensing reconstruction and subsequent delensinganalysis, respectively. Finally, some discussion and our conclusion are in Sec. 5.2 Simulation method
In early works of this research area, systematic errors are analytically modeled in terms oferror fields which characterize how CMB signals deteriorate at each sky position and areclassified according to their mathematical properties [51, 52]. A couple of works followedin order to apply the models to the lensing reconstruction analysis [53, 54]. Our study isin the line of such works. On the other hand, simulation based assessments for evaluatingpropagation of observational systematics in the lensing reconstruction analysis have beenperformed by several groups [23, 55]. They adopted detailed setups of actual experiments(such as patch geometries, scan patterns, detector distributions, e.t.c.) and carried out manysets of numerical simulations. Instead of performing simulations based on a specific experimentdesign, we take the former approach to obtain a correspondence between systematics anddelensing efficiency in a robust and concrete form.We investigate three types of major systematic errors common in CMB polarization ob-servations, i.e. gain error, angle error, and pointing error, the corrections of which still requireconventional calibration procedures. Intensity (temperature) to polarization leakage, whichwas one of the most significant systematic errors, is avoided in principle by measurementwith a polarization modulator [56] though other types of systematic errors associated withthe apparatus itself may arise instead [57, 58, 59]. Apart from them, also beam relatedmis-measurements such as elliptic deformation or sidelobe elongation are major sources ofsystematic errors. Since it is still developing to model beam systematics in terms of errorfields, we would like to leave them as future works.We focus on internal delensing analysis of the CMB polarization field in which we utilizestatistics of the polarization field to reconstruct the lensing mass distribution. We adopt thewell-established quadratic estimator method which is widely used in many works [60].In the simulations discussed in this paper, we assume that polarization fields are noiselessand free from image blurring due to finite beam width for the purpose of clarifying how signalmis-measurements themselves affect the lensing analysis. In the subsequent sections 3 and 4,we present results of our simulations in which a realization of the CMB polarization field isbasically fixed and the systematic errors have various configurations in contrast to it.Throughout this paper, we adopt the fiducial ΛCDM parameter set based on the Planck2018 baseline likelihood analysis [61]. The values of the parameters are h = 0 . b h =0 . c h = 0 .
12, Ω ν h = 0 . n s = 0 . A s ( k pivot = 0 . − ) = 2 . × − ,and τ = 0 . A linear polarization state at a sky position ˆ n is described in terms of Stokes parametersas [ Q ± iU ]( ˆ n ). In this subsection, we describe how this expression is modified by imposingmodulations due to the systematic errors on it.Uncertainty hosted in gain, which is correspondence of incident radiation to readout elec-trical signals, is a typical systematic error common among CMB polarization observations.Gain fluctuation caused by various instrumental and environmental factors during a scanobservation results in residual calibration uncertainty which causes modulation of the polar-3zation field. The polarization field modulated by the gain error is described as(1 + g ( ˆ n )) [ Q ± iU ]( ˆ n ) . (1)The quantity g ( ˆ n ) is a gain bias at a sky position ˆ n . Note that a homogeneous bias givesjust a scaling of the CMB field which is usually corrected by comparison with temperature or E -mode data provided by other observations.Mis-estimation of polarization orientation angles makes false B -modes leaked from E -modes. The angle error is sourced by thermal deformation and mechanical vibration of in-struments, errors in telescope-attitude determination, and those in optical system calibration.Systematic effects of this kind are also caused by cosmic birefringence (e.g. [62, 63]), which isconsidered to be induced by interaction between CMB photons and some hypothetical field,or other mechanisms. The polarization field with the angle error is described as e ± iα ( ˆ n ) [ Q ± iU ]( ˆ n ) . (2)The quantity α ( ˆ n ) is an angle bias at a sky position ˆ n . Here, we don’t restrict the angle errorto have spatial homogeneity which is often assumed in literatures.Mismatch between an actual pointing direction and its corresponding estimated directiondegrades spatial position accuracy in a polarization map. Pattern distortion due to the point-ing error modulates the polarization field and converts the associated parity modes from E to B (or vice versa) in a similar manner to the gravitational lensing effects. The pointing errorshares many of its physical origin with the angle error. The modulated polarization field isdescribed as [ Q ± iU ]( ˆ n + d ( ˆ n )) . (3)The vector d ( ˆ n ) stands for discrepancy between an actually observed sky position and acorresponding mapped position ˆ n due to the pointing error. It is generally decomposed intotwo components [64, 65, 66] as follows : d = ∇ ψ + ( (cid:63) ∇ ) (cid:36). (4) ψ and (cid:36) are potential functions of gradient (even parity) and curl (odd parity) modes, respec-tively. The (cid:63) operator denotes counterclockwise rotation by 90 degrees. The curl componentin this case is often non-negligible unlike in the case of the gravitational lensing effects. Infact, a scan orbit of a telescope boresight is often something like a periodic curve and a staticpointing bias tends to make the error vectors swirly distributed on the sky. Also, random er-rors don’t prefer one of the two components. It has been reported that fluctuation of samplingpositions due to pointing errors causes smearing of small-scale inhomogeneity of polarizationfields as an additional effect [55, 67]. That effect is not described by this error model. Inthose works, it is modeled as an additional suppression factor applied to multipole momentsof the CMB polarization. The suppression factor is parameterized by an effective beam widthwhich stands for a length scale of the smearing. We just comment that overall amplitude ofa lensing potential reconstructed in presence of the suppression factor is tend to be reducedand consequently delensing efficiency is degraded though the response is not very sensitive tothe amplitude reduction. We would like to report on a further study on the issue elsewherein future. 4 ( ˆ n ), α ( ˆ n ), and d ( ˆ n ) stand for net errors of gain, angle, and pointing at a sky position ˆ n ,respectively. A net error at a sky position is an average of instantaneous errors each of which isassociated with a single data sampled at the position during scan observation. Since each skyposition is visited many times during an entire observation period, the net error is averagedout in a statistical manner. For example, if fluctuation time scale of an error is shorter thanrevisiting intervals, net amplitude of the error is suppressed by a factor of 1/ √ N , where N isa number of visiting times. In most cases of contemporary experiments, net amplitude of anerror is smaller by one or more orders than instantaneous amplitude of the error.In a single simulation case, the field of the systematic error ( g ( ˆ n ), α ( ˆ n ), or d ( ˆ n )) is set tohave a non-vanishing value of multipole moment only at a single multipole ( (cid:96) = (cid:96) in ) so thatwe can clearly see scale dependence of analysis results. Then, given the satistical isotropy ofthe CMB polarization field, we set its directional eigenvalue fixed to be zero ( m = 0). Wenormalize overall amplitude of the systematic error by quantifying a spatial root-mean-square(RMS) value of the error field.We utilize routines of the Healpix library to manipulate numerical data of the polariza-tion and error fields and Lenspix to make lensed CMB polarization maps to be imposed thesystematic errors on. To reconstruct the lensing potential, we adopt the EB quadratic estimator [60], which hasdecisive reconstruction performance in the situation concerned here, given as[ (cid:98) φ EBLM ] ∗ = A EBL (cid:88) (cid:96)(cid:96) (cid:48) (cid:88) mm (cid:48) (cid:18) (cid:96) (cid:96) (cid:48) Lm m (cid:48) M (cid:19) g EB(cid:96)(cid:96) (cid:48) L (cid:98) E (cid:96)m (cid:98) B (cid:96) (cid:48) m (cid:48) . (5) (cid:98) E (cid:96)m and (cid:98) B (cid:96) (cid:48) m (cid:48) are observed harmonic coefficients of E - and B -modes, respectively. Hereafterin this subsection , we denote quantities to be affected by the systematic errors as characterswith (cid:98) while quantities theoretically predetermined are without such symbols. We define thequantities g EB(cid:96)(cid:96) (cid:48) L and A EBL as follows : g EB(cid:96)(cid:96) (cid:48) L = [ f EB(cid:96)(cid:96) (cid:48) L ] ∗ C EE(cid:96) C BB(cid:96) (cid:48) ,A EBL = (cid:40) L + 1 (cid:88) (cid:96)(cid:96) (cid:48) f EB(cid:96)(cid:96) (cid:48) L g EB(cid:96)(cid:96) (cid:48) L (cid:41) − . (6) C EE(cid:96) and C BB(cid:96) are the angular power spectra of the lensed E - and B -modes, respectively.The weight function f EB(cid:96)(cid:96) (cid:48) L , which is defined as f EB(cid:96)(cid:96) (cid:48) L = iC EE(cid:96) S ( − ) (cid:96) (cid:48) (cid:96)L , (7) https://healpix.jpl.nasa.gov/. http://cosmologist.info/lenspix/.
5s the combination of the lensed E -mode spectrum [68, 69] and the mode coupling function( S ( − ) (cid:96) (cid:48) (cid:96)L ), which is given by S ( − ) (cid:96)(cid:96) (cid:48) L = 1 − ( − (cid:96) + (cid:96) (cid:48) + L (cid:114) (2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)(2 L + 1)16 π × [ − (cid:96) ( (cid:96) + 1) + (cid:96) (cid:48) ( (cid:96) (cid:48) + 1) + L ( L + 1)] (cid:18) (cid:96) (cid:96) (cid:48) L − (cid:19) . (8)By use of the reconstructed lensing potential, we make a template of the lensing B -modesfor delensing by convolving the E -modes with the reconstructed lensing potential as (cid:98) B lens (cid:96)m = − i (cid:88) (cid:96) (cid:48) m (cid:48) (cid:88) LM (cid:18) (cid:96) (cid:96) (cid:48) Lm m (cid:48) M (cid:19) S ( − ) (cid:96)(cid:96) (cid:48) L W E(cid:96) (cid:48) W φL (cid:98) E ∗ (cid:96) (cid:48) m (cid:48) (cid:98) φ ∗ LM , (9)where the Wiener filters are defined as W φL = C φφL C φφL + A EBL ,W E(cid:96) = 1 . (10)We don’t apply filtering to the E -modes because noise is absent in our case. Residual B -modesafter delensing are evaluated as (cid:98) B res (cid:96)m = B (cid:96)m − (cid:98) B lens (cid:96)m . (11)To estimate delensing efficiency itself, we set the first term of the right-hand-side of theequation above to be a multipole moment of the lensing B -modes which is initially preparedand does not include biases from the systematic errors, except for the case of the gradient-type pointing error. (See Sec. 3.3 and 4.3 for details.). The lensing template described abovedoes not include higher-order terms of the lensing potential, but the power spectrum of thedelensed B -modes does not have significant contribution from O ( φ ) terms because of partialcancellation as shown in Ref. [70].It is reported that overlap of multipole ranges used for reconstruction and delensing causesundesirable bias in the residual lensing B -modes defined above [45, 46]. We incorporate onlymultipoles between (cid:96) = 300 and 2048 into the reconstruction analysis to avoid such delensingbias and evaluate the residual lensing B-modes only between (cid:96) = 2 and 299.We compute the theoretical angular power spectra of the CMB polarization and lensingpotential with CAMB . The library used for the lensing analysis described above is made publicat GitHub . Contributions from the systematic errors to polarization fields are not in a state of pure evenparity which is kept by the primary CMB polarization of the density fluctuation origin. In https://camb.info/ https://github.com/toshiyan/. -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓℓ in = 600 RMS (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓℓ in = 600 RMS (b)
Figure 1: Dependence on RMS amplitude of the gain error.
Left : angular power spectra of B -modes induced by the gain error. The thick black curve is the theoretical spectrum of thelensing B -modes. Right : angular power spectra of lensing potentials reconstructed in presenceof the gain error. The thick black curve is the theoretical spectrum of the lensing potential.fact, E - and B -modes induced by the systematic errors have similar amplitude. Consideringthe fact that the lensing B -modes are smaller than the CMB E -modes by a few orders, wecontrast the systematics induced B -modes to the lensing B -modes to illustrate impacts on the EB quadratic estimator. In this section, we compare angular power spectra of the induced B -modes with reconstructed lensing potentials and discuss about response of the reconstructedpotentials to the systematic errors imposed to the CMB polarization field.We evaluate angular power spectra of reconstructed lensing potentials as [71] (cid:98) C φφL = 12 L + 1 L (cid:88) M = − L | (cid:98) φ LM | − N (0) L , (12)where the Gaussian bias term is given by N (0) L = A L in the fullsky idealistic case [60]. Wesimply use the normalization factor to evaluate the Gaussian bias though the realization-dependent method [72] is more effective to mitigate systematics in N (0) L which are oftenpresent in non-idealistic cases [55]. Although we can evaluate N (1) bias contribution to thespectrum by means of an analytic calculation or a Monte Carlo simulation [19], we just omitthe correction. These treatments do not affect our discussion in this paper. The figure 1a shows angular power spectra of B -modes induced by the gain error with (cid:96) in =600. The RMS values of the error fields are 0 .
01, 0 .
03, and 0 .
10. The figure 1b shows theangular power spectra of the reconstructed lensing potentials in the respective cases. Atfirst, let us mention the overall appearance of the figures. While the shapes of the B -modespectra are apparently similar to that of the lensing B -modes, the spectra of the reconstructedlensing potentials have spiky features at the multipoles of the imposed error field and its7 -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓ RMS ℓ in = 1003006001000 (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ RMS ℓ in = 1003006001000 (b) Figure 2: Dependence on modulation scale ( (cid:96) in ) of the gain error. Left : angular power spectraof B -modes induced by the gain error. The thick black curve is the theoretical spectrum ofthe lensing B -modes. Right : angular power spectra of lensing potentials reconstructed inpresence of the gain error. The thick black curve is the theoretical spectrum of the lensingpotential.overtones, which is clearly seen in the case of RMS 0 .
10. This gives us a simple intuitionthat multipole moments of reconstructed lensing potentials exhibit such sharp response topower of error fields at corresponding multipoles. Also, it is found that a trend of deviationfrom the theoretical spectrum of the lensing potential is observed both in the scale rangesof several dozen degrees and sub-degree. The bias in the latter range possibly contaminatesthe delensing analysis, which is discussed in detail in Sec. 3.4 and 4. Next, we discuss aboutdependence of reconstructed lensing potentials on RMS values of fields of the gain error. Inthe case of RMS 0 .
01, the amplitude of the induced B -mode spectrum is about 1% of that ofthe lensing B -mode spectrum and the reconstructed lensing potential is almost identical tothe one ideally reconstructed in absence of any systematic errors. When we increase the RMSvalue from 0 .
01, the spectrum of the reconstructed lensing potential begins to perceptiblydeviate from the ideally reconstructed spectrum around RMS 0 .
03 in the case of which theamplitude of the induced B -mode spectrum is about 10% of that of the lensing B -modespectrum.The figure 2a shows angular power spectra of B -modes induced by the gain error thefields of which have their RMS values fixed to be 0 .
10. The adopted multipoles of the errorfields are 100, 300, 600, and 1000. The figure 2b shows the angular power spectra of thereconstructed lensing potentials in the respective cases. When we increase (cid:96) in , the induced B -modes substantially grow especially in small scales. The multipoles of the spikes in each ofthe reconstructed lensing potentials are consistent with the adopted multipole of the corre-sponding case. The coherent deviation from the theoretical spectrum of the lensing potentialin the scale ranges of several dozen degrees and sub-degree shows response to the amplitudeof the induced B -modes in small scales. 8 -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓℓ in = 600 RMS (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓℓ in = 600 RMS (b)
Figure 3: Dependence on RMS amplitude of the angle error.
Left : angular power spectraof B -modes induced by the angle error. The thick black curve is the theoretical spectrumof the lensing B -modes. Right : angular power spectra of lensing potentials reconstructed inpresence of the angle error. The thick black curve is the theoretical spectrum of the lensingpotential.
The figure 3a shows angular power spectra of B -modes induced by the angle error with (cid:96) in = 600. The RMS values of the error fields are 20 arcmin, 60 arcmin, and 180 arcmin.The figure 3b shows the angular power spectra of the reconstructed lensing potentials inthe respective cases. The induced B -modes and the reconstructed lensing potentials exhibitsimilar behavior to those in the case of the gain error, which is shown in the figure 1, exceptfor the spike which is observed only at the overtone scale of the adopted multipole (see alsoFig. 9b).The figure 4a shows angular power spectra of B -modes induced by the angle error thefields of which have their RMS values fixed to be 180 arcmin. The adopted multipoles of theerror fields are 100, 300, 600, and 1000. The figure 4b shows the angular power spectra of thereconstructed lensing potentials in the respective cases. The induced B -mode spectra appearto be overlapping each other. Also the spectra of the reconstructed lensing potentials aresimilar to each other. It is well known that a spectral shape of false B -modes leaked from E -modes due to a spatially homogeneous angular bias is identical to that of the E -modes.Actually, when we further decrease (cid:96) in from 100, it is found that the spectral shape of theinduced B -modes approaches that of the CMB E -modes and the induced B -mode spectrumexhibits prominent oscillatory structure which comes from the acoustic peaks seen in theCMB E -mode spectrum. In that case, overall amplitude of the induced B -mode spectrumdoes not significantly change and the reconstructed lensing potential keeps its spectrum almostidentical to that in the case of (cid:96) in = 100. 9 -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓ RMS ℓ in = 1003006001000 (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ RMS ℓ in = 1003006001000 (b) Figure 4: Dependence on modulation scale ( (cid:96) in ) of the angle error. Left : angular power spectraof B -modes induced by the angle error. The thick black curve is the theoretical spectrumof the lensing B -modes. Right : angular power spectra of lensing potentials reconstructed inpresence of the angle error. The thick black curve is the theoretical spectrum of the lensingpotential. -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓℓ in = 600 RMS (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓℓ in = 600 RMS (b)
Figure 5: Dependence on RMS amplitude of the gradient-type pointing error.
Left : angularpower spectra of B -modes induced by the gradient-type pointing error. The thick black curveis the theoretical spectrum of the lensing B -modes. Right : angular power spectra of lensingpotentials reconstructed in presence of the gradient-type pointing error. The thick black curveis the theoretical spectrum of the lensing potential.10 -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓ RMS ℓ in = 1003006001000 (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ RMS ℓ in = 1003006001000 (b) Figure 6: Dependence on modulation scale ( (cid:96) in ) of the gradient-type pointing error. Left :angular power spectra of B -modes induced by the gradient-type pointing error. The thickblack curve is the theoretical spectrum of the lensing B -modes. Right : angular power spectraof lensing potentials reconstructed in presence of the gradient-type pointing error. The thickblack curve is the theoretical spectrum of the lensing potential. -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓℓ in = 600 RMS (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓℓ in = 600 RMS (b)
Figure 7: Dependence on RMS amplitude of the curl-type pointing error.
Left : angularpower spectra of B -modes induced by the curl-type pointing error. The thick black curveis the theoretical spectrum of the lensing B -modes. Right : angular power spectra of lensingpotentials reconstructed in presence of the curl-type pointing error. The thick black curve isthe theoretical spectrum of the lensing potential.11 -6 -5 -4 -3 -2 -1
10 100 1000 ℓ ( ℓ + ) C ℓ BB / π [ µ K ] ℓ RMS ℓ in = 1003006001000 (a) -9 -8 -7 -6
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ RMS ℓ in = 1003006001000 (b) Figure 8: Dependence on modulation scale ( (cid:96) in ) of the curl-type pointing error. The thickblack curve is the theoretical spectrum of the lensing B -modes. Left : angular power spectraof B -modes induced by the curl-type pointing error. Right : angular power spectra of lensingpotentials reconstructed in presence of the curl-type pointing error. The thick black curve isthe theoretical spectrum of the lensing potential.
The figure 5a shows angular power spectra of B -modes induced by the gradient-type pointingerror with (cid:96) in = 600. The RMS values of the error fields are 4 arcsec, 12 arcsec, and 36 arcsec.The figure 5b shows the angular power spectra of the reconstructed lensing potentials in therespective cases. The mechanism of E to B leakage due to the gradient-type pointing error ismathematically equivalent to that of the gravitational lensing effects and actually the shapesof the induced B -mode spectra quite resemble that of the lensing B -mode spectrum. In thecase of RMS 36arcsec, the induced B -mode spectrum is almost degenerated with the lensingB-mode spectrum. While 36arcsec is substantially smaller than 2arcmin which is the RMSamplitude of deflection angles due to the gravitaional lensing effects, power of the pointingerror is localized at (cid:96) in = 600 in this case (see also the next paragraph). The spectra of thereconstructed lensing potentials exhibit biases similar to those in the case of the gain error,i.e. spikes at multipoles relevant to imposed error fields and coherent deviation from thetheoretical spectrum in the scale ranges of several degrees and sub-degree. These biases arenot mere contamination in the lensing reconstruction analysis. Fields of the pointing errorare simultaneously reconstructed and the biases in reconstructed lensing potentials can beutilized for removing the pointing error from polarization fields. In Sec. 4.3, we discuss aconsequence of this property in the delensing analysis.The figure 6a shows angular power spectra of B -modes induced by the gradient-typepointing error the fields of which have their RMS values fixed to be 36 arcsec. The adoptedmultipoles of the error fields are 100, 300, 600, and 1000. The figure 6b shows the angularpower spectra of the reconstructed lensing potentials in the respective cases. Error fields withsmall (cid:96) in cause spatially coherent displacement of local structure in the CMB polarization field.When we further decrease (cid:96) in from 100, the acoustic peaks in the CMB E -mode spectrum12re gradually appearing in the induced B -mode spectrum. On the other hand, the inducedB-modes grow as we increase (cid:96) in and the high- (cid:96) bias in the reconstructed lensing potentialsincreases accordingly. In addition, we can clearly see the correspondence between (cid:96) in and themultipoles of the spikes in Fig 6b.The figure 7a shows angular power spectra of B -modes induced by the curl-type pointingerror with (cid:96) in = 600. The RMS values of the error fields are 2 arcsec, 6 arcsec, and 18 arcsec.The figure 7b shows the angular power spectra of the reconstructed lensing potentials inthe respective cases. Unlike other error species, induced B -mode power of this error kind islocalized in small scales because of geometrical reason [71]. An induced B -mode spectrumfor a given set of error parameters (i.e. a RMS value and (cid:96) in ) has higher amplitude around (cid:96) = 1000 than that in the case of the gradient-type pointing error for the same parameterset, which results in more impacts on the subsequent lensing analysis. The spectra of thereconstructed lensing potentials are responding to the induced B -mode amplitude around (cid:96) = 1000 similarly to the case of the gain error though the spikes in the spectra are seen onlyat the overtone scale as in the case of the angle error (see also Fig. 9d).The figure 8a shows angular power spectra of B -modes induced by the curl-type pointingerror the fields of which have their RMS values fixed to be 18 arcsec. The adopted multipolesof the error fields are 100, 300, 600, and 1000. The figure 6b shows the angular power spectraof the reconstructed lensing potentials in the respective cases. The dependence on (cid:96) in is almostthe same as that in the case of the gradient-type pointing error except for the low- (cid:96) behaviorof the induced B -modes. It is known that spatial modulations applied to the CMB polarization field make biases inreconstructed lensing potentials which are called as mean-fields. In such a case, the estimatordefined in Eq. 5 is a biased estimator and the associated angular power spectrum is biasedby contribution from a mean-field. Some of the behavior observed in the spectra of thereconstructed lensing potentials shown in the previous subsections can be understood in termsof the mean-field bias.Formulation of the mean-field theory is found in literature, e.g. [72]. Unlike the simula-tions in the previous subsections, in the case of each error species, we repeat the simulationprocedure 200 times to obtain two sets of 100 Monte Carlo samples in which realizations ofthe CMB polarization are independent and parameters for the systematic error are fixed. Themean field of the reconstructed lensing potential is evaluated as the average of the 200 samplesof the estimator. Also, we obtain the associated Gaussian bias by cross-correlating E -modesin the one sample set with B -modes in the other sample set or vice versa. The figures 9a to 9dshow spectra of mean-fields from the systematic errors discussed in the previous subsections.It is clearly found that the biases in the large scales and the spikes at the multipolesrelevant to the imposed error fields are manifestations of the mean-fields. On the other hand,we can see that the mean-fields do not cause the coherent deviations in the small scales whichhave major impacts on the delensing analysis. Instead, they are understood to be due tothe biases in high- (cid:96) power in the respective B -mode spectra which increase reconstructionnoise relevant to the Gaussian biases. More specifically, they do not come from biases in thereconstructed lensing potentials but come from increase in variance around the true lensingpotential. In addition, by performing mean-field subtraction from the reconstructed lensing13 -11 -10 -9 -8 -7 -6 -5 -4
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ Gain error :
RMS ℓ in = 600 (a) -11 -10 -9 -8 -7 -6 -5 -4
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ Angle error :
RMS ℓ in = 600 (b) -11 -10 -9 -8 -7 -6 -5 -4
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ Gradient-type pointing error :
RMS ℓ in = 600 (c) -11 -10 -9 -8 -7 -6 -5 -4
10 100 1000 ℓ ( ℓ + ) C ℓ φφ / π ℓ Curl-type pointing error :
RMS ℓ in = 600 (d) Figure 9: Spectra of mean-fields from the systematic errors (dot-dashed coral) comparedwith those of their respective reconstructed lensing potentials (dotted blue). The associatedGaussian biases are not subtracted from the spectra of the reconstructed lensing potentials.The thick black curve is the theoretical spectrum of the lensing potential. The upper andlower dashed gray curves are the Gaussian biases and spectra of Monte Carlo errors in theevaluated mean-fields, respectively. 14 C ℓ BB , r e s / C ℓ BB ℓ 〈 ℓ in = 600 RMS (a) C ℓ BB , r e s / C ℓ BB ℓ 〈 RMS ℓ in = 1003006001000 (b) Figure 10: Residual fractions of the lensing B -modes in the case of the gain error. Thethick black line is that in the case without any systematic errors. Left : dependence on RMSamplitude of the gain error.
Right : dependence on modulation scale of the gain error.potentials, we confirmed that the spikes also contribute somewhat to degradation in delensingefficiency depending on their amplitude and multipoles. Note that the subtraction is notactually applied in the delensing analysis described in the next section (see the paragraphbelow).In the Monte Carlo simulations, we fixed the configurations of the error fields to extracttheir respective mean-fields. However, in practical situations, the systematic errors discussedhere are not expected to make such fixed modulations predictable in advance of observa-tion. Mitigation of mean-fields from potential systematic errors is a considerable task to beaddressed. The bias hardening [72, 73] is a possible method to do it.
In this section, we compare results of the delensing analysis in the cases of the systemtic errorsdescribed in the previous sections. Residual lensing B -modes after delensing are defined inEq. 11 and we make the multipole moments up into the associated angular power spectrum( (cid:98) C BB, res (cid:96) ). We define delensing efficiency as fractional difference of this power spectrum fromthe theoretical spectrum of the lensing B -modes. The figure 10a shows residual fractions of the lensing B -modes after delensing performed withlensing potentials reconstructed in presence of the gain error. They correspond to the casesshown in the figure 1. The figure 10b shows those which correspond to the cases in the figure2. The levels of overall amplitude of the residual lensing B -modes totally depend on thebias levels of the respective reconstructed lensing potentials. The spectrum of the residuallensing B -modes in the case of RMS 0 .
01 is indistinguishable from that in the case withoutany systematic errors. The amplitude of the spectrum is about 25% of that of the original15 C ℓ BB , r e s / C ℓ BB ℓ 〈 ℓ in = 600 RMS (a) C ℓ BB , r e s / C ℓ BB ℓ 〈 RMS ℓ in = 1003006001000 (b) Figure 11: Residual fractions of the lensing B -modes in the case of the angle error. Thethick black line is that in the case without any systematic errors. Left : dependence on RMSamplitude of the angle error.
Right : dependence on modulation scale of the angle error.lensing B -modes. With increase in the RMS value from 0 .
01, the delensing efficiency begins todegrade around RMS 0 .
03 in the case of which the amplitude of the induced B -mode spectrumis about 10% of that of the lensing B -modes.It is noted that the delensing efficiency in the case of 0 .
10 RMS and (cid:96) in = 600 is sub-stantially degraded but still about 50%. In that case, the residual lensing B -modes are muchsmaller than the induced B -modes which are comparable to the lensing B -modes (see Fig.1a). In such a case, importance of systematics removal is comparable to that of delensing.In addition to conventional error correction, there may be a possibility of reconstruction ofsystematic errors which is formulated in the same way as the lensing reconstruction analysis[74]. However, it is not a trivial problem to find an appropriate filter function such as theWiener filter in the case of the delensing analysis. An example of tackling the problem isfound in Ref. [75]. The figure 11a shows residual fractions of the lensing B -modes after delensing performed withlensing potentials reconstructed in presence of the angle error. They correspond to the casesshown in the figure 3. The figure 10b shows those which correspond to the cases in the figure4. The behavior of the residual lensing B -modes is similar to that in the case of the gain errorthough their dependence on (cid:96) in is in opposite sense. These are implied by the figures 1b, 2b,3b, and 4b.In the case of RMS 180arcmin and (cid:96) in = 600, the residual lensing B -modes are definitelysmaller than the induced B -modes (cf. Fig. 3a). There are examples of angle error recon-struction applied to real observation data, which illustrates effectiveness of such methods andpossibility of further removing false signals in B -modes [76, 63].16 C ℓ BB , r e s / C ℓ BB ℓ 〈 ℓ in = 600 RMS (a) C ℓ BB , r e s / C ℓ BB ℓ 〈 RMS ℓ in = 1003006001000 (b) Figure 12: Residual fractions of the lensing B -modes in the case of the gradient-type pointingerror. The thick black line is that in the case without any systematic errors. Left : dependenceon RMS amplitude of the gradient-type pointing error.
Right : dependence on modulation scaleof the gradient-type pointing error.
The figure 12a shows residual fractions of the lensing B -modes after delensing performedwith lensing potentials reconstructed in presence of the gradient-type pointing error. Theycorrespond to the cases shown in the figure 5. The figure 12b shows those which correspondto the cases in the figure 6. While the residual lensing B -modes mostly show the similarbehavior to that in the case of the gain error, the dependence on (cid:96) in is slightly stronger. Thisis related to bias in spectral amplitude of reconstructed lensing potentials at multipoles ofseveral hundreds, which can be seen in Fig. 2b and 6b. Also the presence of the higherspikes is contributing to such scale dependence (see Fig. 9). Unlike other error species, thegradient-type pointing error has the same properties as the gravitational lensing effects. In thelensing reconstruction analysis, also the error fields are reconstructed simultaneously with thelensing potentials. Since the estimators contain the contributions from the pointing error inthe proper way, we subtract the template B -modes from the B -modes including the pointingerror contributions instead of the true lensing B -modes themselves. Without such treatment,we suffer over-subtraction which causes bias in the spectra of the residual lensing B -modes.This means that the usual lensing analysis which is not specialized for error correction hasthe power to reject this kind of pointing error (partially at least).The figure 13a shows residual fractions of the lensing B -modes after delensing performedwith lensing potentials reconstructed in presence of the curl-type pointing error. They corre-spond to the cases shown in the figure 7. The figure 13b shows those which correspond to thecases in the figure 8. Also in this case, the responses of the residual B -modes to the imposederror fields are similar to those in the case of the gain error. Although the induced B -modeslack their power in large scales as shown in Fig. 7a and 8a, we just mention the fact thatthe reconstruction analysis of curl-type gravitational lensing effects, which is formulated forthe purpose of detecting the primordial GWs or other objects of exotic origin [65, 66], can be17 C ℓ BB , r e s / C ℓ BB ℓ 〈 ℓ in = 600 RMS (a) C ℓ BB , r e s / C ℓ BB ℓ 〈 RMS ℓ in = 1003006001000 (b) Figure 13: Residual fractions of the lensing B -modes in the case of the curl-type pointing error.The thick black line is that in the case without any systematic errors. Left : dependence onRMS amplitude of the curl-type pointing error.
Right : dependence on modulation scale ofthe curl-type pointing error.applied for removing this systematic error.
We have illustrated the effects of the observational systematic errors on the lensing analysis ofthe CMB polarization. We considered three kinds of systematic errors, i.e. gain error, angleerror, and pointing error, which were modeled in terms of error fields. A suite of simula-tions, which consists of generating a modulated polarization field, lensing reconstruction, andsubsequent delensing, was performed for each error species and was repeated with differentvalues of error parameters which are RMS amplitude and a multipole of a single non-vanishingmultipole moment of the error field.The systematic errors cause biases in reconstructed lensing potentials. Spatial modula-tions due to the systematic errors make mean-fields. The mean-fields exhibit sharp responseto multipole moments of imposed error fields at multipoles relevant to the multipoles mo-ments. The mean-fields appear also as biases in reconstructed lensing potentials in largescales. Apart from mean-fields, additional variance in reconstructed lensing potentials insmall scales comes from extra B -mode power induced by the systematic errors and ends upbeing a major contaminant for the delensing analysis.Impacts on results of the lensing analysis depend on amplitude of the systematic errors.If amplitude of the induced B -modes is smaller than a few percent of that of the lensing B -modes in the power spectrum domain, they have little impact on the lensing analysis. Asdescribed in Sec. 2, the error fields discussed in this work stand for net errors at each skyposition which are averages of instantaneous errors. Given the fact, some of the RMS errorvalues adopted in Sec. 3 and 4 are excessively large. In actual observations, levels of gainand angle accuracy have already reached to sub-percent and a few dozen arcmin, respectively[77, 78, 79]. Also, planned future projects are targeting to achieve pointing accuracy of a18ew arcsec [6, 7]. However, in actual data analysis, we should be aware that there may beother possibilities of indirect influence such as derivation of errors of different species fromthe errors concerned.In practical situations, power of the error fields does not localize in a narrow multipolerange and their spectra are usually multi-modal. Also multipole moments to which the induced B -modes do not sensitively respond have finite power. For example, in the case of the EPICbaseline scan [80], a pointing error which is assumed to be static in the telescope frame makesan error field dominated by the curl-type pointing error and the spectrum consists of a whitenoise component, a scan synchronous component associated with its harmonics, and an inversepower-law component which is dominant in large scales. With normalization of RMS 18arcsecapplied to the error field, the spectrum of the induced B -modes has amplitude smaller thanthat of the lensing B -modes by about one order (cf. Fig. 7a and 8a).Finally, we comment on cases with observation noise. Inclusion of observation noise intoour simulations leads to increase in variance of reconstructed lensing potentials. Unlike thesystematics induced biases, statistical properties of the additional variance are theoreticallypredictable on the basis of experimental specifications. Due to the additional variance, frac-tional contributions from the systematic errors to whole reconstruction errors are reduced.Consequently, difference in delensing efficiency between cases with and without the systematicerrors decreases in presence of observation noise. Acknowledgment
We acknowledge the use of
CAMB [81],
Healpix [82], and
Lenspix [83]. This work is supportedby JSPS KAKENHI Grant Number JP17K05478.
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