Controlling systematics in ground-based CMB surveys with partial boresight rotation
MMNRAS , 1–11 (2019) Preprint 5 May 2020 Compiled using MNRAS L A TEX style file v3.0
Controlling systematics in ground-based CMB surveyswith partial boresight rotation
Daniel B. Thomas, (cid:63) Nialh McCallum, and Michael L. Brown Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy, The University of Manchester, Manchester M13 9PL, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Future CMB experiments will require exquisite control of systematics in order to con-strain the B -mode polarisation power spectrum. One class of systematics that requirescareful study is instrumental systematics. The potential impact of such systematics ismost readily understood by considering analysis pipelines based on pair differencing.In this case, any differential gain, pointing or beam ellipticity between the two detec-tors in a pair can result in intensity leakage into the B -mode spectrum, which needsto be controlled to a high precision due to the much greater magnitude of the totalintensity signal as compared to the B -mode signal. One well known way to suppresssuch systematics is through careful design of the scan-strategy, in particular makinguse of any capability to rotate the instrument about its pointing (boresight) direction.Here, we show that the combination of specific choices of such partial boresight ro-tation angles with redundancies present in the scan strategy is a powerful approachfor suppressing systematic effects. This mitigation can be performed in analysis inadvance of map-making and, in contrast to other approaches (e.g. deprojection or fil-tering), results in no signal loss. We demonstrate our approach explicitly with timeordered data simulations relevant to next-generation ground-based CMB experiments,using deep and wide scan strategies appropriate for experiments based in Chile. Thesesimulations show a reduction of multiple orders of magnitude in the spurious B -modesignal arising from differential gain and differential pointing systematics. Key words: (cosmology:) cosmic background radiation – cosmology: observations –methods: observational
One of the key observations for constraining the standardcosmological model is the Cosmic Microwave Background(CMB, see e.g. Durrer 2015; Staggs et al. 2018, for recent re-views). Over the last two decades, a host of experiments, in-cluding ground-based, balloon-based and satellite telescopes,have continuously pushed back the limits of CMB observa-tions. The three main CMB satellite missions,
COBE (Ben-nett et al. 1996),
WMAP (Bennett et al. 2013) and, mostrecently,
Planck (Planck Collaboration et al. 2018a) haveeach played pivotal roles in this progress.However, some of the most important scientific andtechnological breakthroughs have also been achieved us-ing ground-based and balloon-borne experiments. For exam-ple, the Boomerang balloon-based experiment (de Bernardiset al. 2000) provided the first definitive measurement ofthe position of the first acoustic peak in the CMB inten-sity power spectrum, while the first detection of CMB po- (cid:63)
E-mail: [email protected] larization was made using the ground-based DASI experi-ment (Kovac et al. 2002). In recent times, sub-orbital exper-iments such as the Atacama Cosmology Telescope (ACT,Louis et al. 2017) and the South Pole Telescope (SPT, Hen-ning et al. 2018), have focused on precisely characterisingthe CMB intensity signal on smaller angular scales, in ad-dition to more precise measurements of the polarizationanisotropies on all scales.Further characterisation of the polarisation signal is ofparticular interest for cosmology. In this respect, a key tar-get for forthcoming surveys is the “ B -mode” component ofthe CMB polarisation fluctuations, so-called because it isthe parity odd component of the spin-two polarisation field.The contribution to the B -mode power spectrum from gravi-tational lensing (Lewis & Challinor 2006) arises on relativelysmall angular scales and has now been detected by the BI-CEP2/Keck (BICEP2 Collaboration et al. 2018), SPTpol(Keisler et al. 2015) and PolarBear (POLARBEAR Collab-oration et al. 2017) experiments. However, there are only up-per limits on the amplitude of B -mode fluctuations on largeangular scales (BICEP2 Collaboration et al. 2018). Such a © a r X i v : . [ a s t r o - ph . C O ] M a y large-scale signal is expected to be present if there are signif-icant (primordial) gravitational waves in the early Universe(Kamionkowski et al. 1997; Seljak & Zaldarriaga 1997). Thisis a hugely important science goal for cosmology – a detec-tion of such a signal would provide a unique window into theperiod of inflation that is believed to have occurred in theearly universe (Guth 1981, see e.g. Baumann 2009 for a re-view). It is thus a key scientific driver for ongoing (Gualtieriet al. 2018; Appel et al. 2019; Ahmed et al. 2014; Suzukiet al. 2016) and forthcoming experiments (The LSPE col-laboration et al. 2012; Mennella et al. 2018).There is also keen scientific interest in high-precisionmeasurements of the lensing B -mode signal on smaller angu-lar scales, with several experiments now operating upgradedreceivers well suited to probing this signal (Benson et al.2014; Henderson et al. 2016). In particular, such measure-ments can be used to place strong constraints on the sumof the neutrino masses in a complementary way to ground-based particle physics experiments (e.g. Abazajian et al.2016; Ade et al. 2019).Beyond the current generation of experiments, proposedfuture experiments are targeting orders-of-magnitude in-creases in sensitivity across all angular scales. These fu-ture experiments include both satellites (LiteBIRD, Hazumiet al. 2019) and ground-based telescopes (CMB-S4, Abaza-jian et al. 2016) and will be sensitive enough to either detectthe inflationary B -mode signal or to rule out broad classes ofinflation models from its non-detection. In combination withfuture large-scale structure surveys they will also facilitatemajor advances in our understanding of neutrino physics,light relics, dark matter and dark energy (Abazajian et al.2016; Ade et al. 2019).In terms of the ground-based landscape beyond the cur-rently operating telescopes, there are two already-fundedexperiments that are likely to be key in driving the devel-opment of the field in advance of CMB-S4. These are theSimons Observatory (SO, Ade et al. 2019), due to beginoperations in the early 2020s from the Atacama Desert inChile, and the BICEP Array (Hui et al. 2018), an evolutionof the BICEP/Keck series of experiments and due to beginoperations on a similar timescale, from the South Pole.The B -mode spectrum is significantly smaller than boththe total intensity and the E -mode polarisation signals, mak-ing it a challenging observational target from a sensitivityperspective alone. However, this difficulty is compoundedby systematic errors. There are multiple types of system-atic that can contaminate a B -mode experiment, includingboth theoretical and instrumental systematic errors, as wellas biases introduced by astrophysical foregrounds. One in-strumental effect that is of particular concern is leakage ofthe intensity and E -mode signals into the B -mode signal(e.g. Hu et al. 2003; O’Dea et al. 2007; Shimon et al. 2008).The much larger size of these signals renders this leakagea significant problem. Here we focus on a specific subset ofinstrumental systematics, which are particularly relevant forpair-differencing analysis pipelines. In such approaches, thepolarisation signal is extracted by differencing the signalsfrom two detectors with orthogonal polarisation sensitivitydirections. Therefore any slight differences between the re-sponse of the detectors (such as differential gain, pointingor beam ellipticity) can result in significant temperature topolarisation leakage. Systematic errors can be dealt with in different ways.They can be measured directly through checks of the data(e.g. “jack-knife” tests) or they can be simulated, and thenremoved (see e.g. Bicep2 Collaboration et al. 2015a). In ad-dition, experimental design and hardware can be chosen toreduce them in the first place. One way to achieve this isthrough careful design of the scan strategy. The conventionalwisdom in the field is that it is important that each pointon the sky is scanned with multiple different “crossing” or“parallactic” angles (we will use these two terms interchange-ably), i.e. that each sky pixel is scanned multiple times, eachwith a different orientation of the instrument with respectto the sky. This can be achieved in multiple ways, makinguse of sky rotation, scan strategy design and any boresightrotation capability of the instrument.Specifically for pair-differencing experiments, it hasbeen shown through several formalisms (see e.g. Hu et al.2003; O’Dea et al. 2007; Shimon et al. 2008; Miller et al.2009; Ade et al. 2015; Wallis et al. 2017) that differen-tial systematics can be classified according to their spin-transformation properties. Then, for an ideal scan strategy ,it can be shown that differential gain effects (an incorrectrelative calibration of the two detectors within a pair) anddifferental pointing systematics (where the two detectors’pointing centres are slightly offset) vanish. Of course, sucha scan will never be achieved in practice, but these symme-tries can be used in other ways, for example the templatedeprojection technique used in the BICEP2 analyses (Bi-cep2 Collaboration et al. 2015a) or, as suggested in Milleret al. (2009), selectively choosing observations according tocertain criteria, such that the systematics should vanish.In Wallis et al. (2017), a new formalism for assessingthe impact of scan strategy choices on the achievable sup-pression of differential systematics in the B -mode spectrumwas developed. In this formalism, two detector pairs (an“instrument- Q ” pair and an “instrument- U pair) are consid-ered, which are oriented at 45 ◦ with respect to each other onthe focal plane. This matches how many focal planes are de-signed, e.g. the Planck (Maffei et al. 2010), SPTpol (Georgeet al. 2012) and ACTpol (Thornton et al. 2016) receivers.The focal planes for SO will also be at least partly com-posed of such combinations of detector orientations (Galitzkiet al. 2018). Though the formalism is specific to such focalplane layouts, it results in an arguably simpler approach foranalysing the impact of differential systematics.For experiments where the optical chain contains a con-tinuously rotating half wave plate (e.g. Kusaka et al. 2014;Kusaka et al. 2018), detector differencing is not required tocompute the I , Q and U Stokes parameters. In this case, allthree Stokes parameters can be measured using a single de-tector and an analytic formalism based on pair differencingwill not precisely describe the effect of systematics. Never-theless, pair differencing is often assumed in order to modelthe effect of systematics in such experiments, e.g. as wasdone recently for SO (Crowley et al. 2018), where the de-sign for the SO small aperture telescopes includes a HWP. Inaddition, multiple crossing angles within each pixel will still An ideal scan is one where all sky pixels are observed with aninfinite number of crossing angles uniformly distributed between and π . MNRAS , 1–11 (2019) artial boresight rotation in future CMB surveys be important in cases where pair differencing is not used, forexample in relation to the requirements of maximum likeli-hood map-making schemes – see the discussion in Stevenset al. (2018) and references therein.In this paper, we make use of the Wallis et al. (2017)formalism to identify specific choices in the scan strategydesign of a CMB polarisation experiment that will be op-timally suited for mitigating certain types of instrumentalsystematic effects. Specifically, we make recommendationsfor the boresight rotation schedule for experiments that havesuch a capability. We note that the analytic framework ofWallis et al. (2017) was developed under the assumption offull-sky coverage. Such coverage will not be achievable, evenfor satellite experiments (due to the need to excise regionsof the sky that are dominated by polarised Galactic fore-grounds). Therefore, in addition to identifying boresight ro-tation schedules based on the Wallis et al. (2017) formalism,we also demonstrate the mitigation of systematics using theidentified schedules with full time-ordered data simulationsof a generic ground-based experiment.The paper is organised as follows. In Section 2, we sum-marise the Wallis et al. (2017) formalism and describe how itcan be used to identifty recommendations for boresight ro-tation schedules. Here, we also discuss the adaptation of the(full-sky) Wallis et al. (2017) formalism to generic ground-based surveys. Section 3 describes the time-ordered data(TOD) simulations and scan strategies that we have usedto demonstrate our approach. In Section 4, we describe ouranalysis of the simulated data and present our main results.We conclude in Section 5. B -MODES We follow closely the formalism of Wallis et al. (2017) towhich we refer the reader for further details. In differencingexperiments, the detectors are arranged in pairs with orthog-onal polarisation sensitivity and their signals are differencedin order to nominally remove the temperature signal suchthat the signal for detector pair i is given by d i = (cid:16) d Ai − d Bi (cid:17) . (1)The differential gain comes from a miscalibration of thetwo detectors and the resulting intensity leakage into polar-isation is given by δ d gi = (cid:16) T B ( Ω ) − ( − δ g i ) T B ( Ω ) (cid:17) = δ g i T B ( Ω ) , (2)where δ g i is the differential gain in detector pair i and T B ( Ω ) is the beam-convolved intensity field on the sky. red Wealso note that, in the presence of frequency-dependent fore-grounds, a bandpass mismatch between the two detectorswill generate an effective differential gain, which is not pro-portional to the CMB temperature. The response I X of de-tector X to the foreground with frequency-dependent inten-sity I ( ν ) is (ignoring the beam convolution for simplicity) I X = ∫ d ν I ( ν ) f X ( ν ) , (3)where f X ( ν ) is the frequency dependent response of detector X . When the signals from the two detectors are differenced,if f X ( ν ) is different for the two detectors then I A (cid:44) I B , whichadds an additive signal with no crossing angle dependenceto the expected polarisation signal, analogous to the differ-ential gain case. We will not explicitly mention this further,but we note that the mitigation strategy for differential gaindeveloped here will apply to this bandpass mismatch and any other systematic that results in a crossing angle inde-pendent additive signal when the signals from the two de-tectors are differenced. Another advantage of the approachdeveloped later is that it is insensitive to the cause of thisadditive systematic, and thus doesn’t require the systematicto be known (and accurately modelled or having a knowntemplate) in order to remove the effect.The differential pointing refers to an incorrect alignmentof the beams of the two detectors by an angular separation ρ i , in a direction on the sky that is at an angle χ i with re-spect to the scan direction of the instrument, which in turnis at an angle ψ with respect to North. The resulting tem-perature leakage into the polarisation signal is approximatedby (assuming a flat sky with co-ordinates { x , y } ) δ d pi = (cid:20)(cid:18) ∂ T B ∂ y − i ∂ T B ∂ x (cid:19) ρ i e i ( ψ + χ i ) + c . c . (cid:21) . (4)Given these leakage expressions for a given “hit” of thedetector, the influence of the systematics on the polarisationsignal P = Q + iU reconstructed from two detector pairs (aninstrument- Q pair and an instrument U pair, oriented π / radians apart ) can be calculated as (Wallis et al. 2017) ∆ P g =
12 ˜ h ( δ g + i δ g ) T B ∆ P p =
14 ˜ h ∇ T B (cid:16) ρ e i χ + ρ e i ( χ − π / ) (cid:17) +
14 ˜ h ∇ ∗ T B (cid:16) ρ e − i χ + ρ e − i ( χ + π / ) (cid:17) , (5)where ∆ P is the spurious polarization signal resulting fromthe systematic and ∇ ≡ ( ∂ / ∂ y − i ∂ / ∂ x ) . The ˜ h n are the spin- n components of the crossing angles in a given pixel, i.e. theFourier series components of the real space field h ( ψ ) = π N hits (cid:213) j δ ( ψ − ψ j ) , (6)where ψ j is the crossing angle (orientation of the telescope)for the j th crossing of the pixel in question, and N hits isthe total number of crossings of that pixel. The ˜ h n quanti-ties thus characterise the relevant features of a given scanstrategy. Explicitly, in a single pixel, | ˜ h n | = (cid:169)(cid:173)(cid:171) N hits (cid:213) j cos ( n ψ j ) (cid:170)(cid:174)(cid:172) + (cid:169)(cid:173)(cid:171) N hits (cid:213) j sin ( n ψ j ) (cid:170)(cid:174)(cid:172) . (7) This assumption of both “ Q ” and “ U ” detector pairs is a cru-cial difference between the Wallis et al. (2017) analysis and otherformalisms (e.g. O’Dea et al. 2007; Shimon et al. 2008). It meansthat, in the absence of systematics, we only need a single ori-entation on the sky to reconstruct the full polarisation signal.Conversely, when this assumption is not made, multiple cross-ing angles are needed, even in the absence of systematics. Thisin turn results in our formalism being simpler but less generallyapplicable.MNRAS000
14 ˜ h ∇ ∗ T B (cid:16) ρ e − i χ + ρ e − i ( χ + π / ) (cid:17) , (5)where ∆ P is the spurious polarization signal resulting fromthe systematic and ∇ ≡ ( ∂ / ∂ y − i ∂ / ∂ x ) . The ˜ h n are the spin- n components of the crossing angles in a given pixel, i.e. theFourier series components of the real space field h ( ψ ) = π N hits (cid:213) j δ ( ψ − ψ j ) , (6)where ψ j is the crossing angle (orientation of the telescope)for the j th crossing of the pixel in question, and N hits isthe total number of crossings of that pixel. The ˜ h n quanti-ties thus characterise the relevant features of a given scanstrategy. Explicitly, in a single pixel, | ˜ h n | = (cid:169)(cid:173)(cid:171) N hits (cid:213) j cos ( n ψ j ) (cid:170)(cid:174)(cid:172) + (cid:169)(cid:173)(cid:171) N hits (cid:213) j sin ( n ψ j ) (cid:170)(cid:174)(cid:172) . (7) This assumption of both “ Q ” and “ U ” detector pairs is a cru-cial difference between the Wallis et al. (2017) analysis and otherformalisms (e.g. O’Dea et al. 2007; Shimon et al. 2008). It meansthat, in the absence of systematics, we only need a single ori-entation on the sky to reconstruct the full polarisation signal.Conversely, when this assumption is not made, multiple cross-ing angles are needed, even in the absence of systematics. Thisin turn results in our formalism being simpler but less generallyapplicable.MNRAS000 , 1–11 (2019) Examining equation (7), we note that for a crossing angle ψ , | ˜ h | is unchanged under the transformation ψ → ψ + π , and | ˜ h | is unchanged under ψ → ψ + π . More interestingly,the contribution of ψ to the sums has opposite sign when ψ → ψ + π / (for ˜ h ) or ψ → ψ + π (for ˜ h and ˜ h ). Thiswas noted before (in e.g. O’Dea et al. 2007; Shimon et al.2008; Miller et al. 2009; Ade et al. 2015) as being usefulwhen analysing the data from a specific instrument.Here, we further note that the requirements on a scanstrategy to ensure that such symmetries are present in thedata for all sky pixels are much less stringent than whatwould be required in order to approximate an “ideal” scan.Such near-ideal scans would also result in the ˜ h n quantitiesbeing driven to zero, but are not practically realisable.In addition to being used in post-hoc analyses of thedata, these systematic-mitigating symmetries can be incor-porated into the design of scan strategies. In principle, thescan strategy for any telescope with boresight rotation ca-pability can be designed to massively reduce the effect ofthese systematics long before the data is analysed by regu-larly repeating scan patterns (e.g. a set of constant elevationscans ) identically, but with a rotated instrument .In this respect, we note that the the Large ApertureTelescope (LAT) of the SO has “partial boresight rotation”capability, which is equivalent to the ability to rotate theentire optical assembly through π radians, thus allowing itto effectively mitigate the differential pointing systematic. The optical assemblies of SO’s three Small Aperture Tele-scopes (SATs) can be rotated through π / radians (in ad-dition to a number of other possibilities), thus allowing theeffective mitigation of differential gain systematics. The Wallis et al. (2017) analysis assumed a full sky sur-vey, which is unrealistic in practice, even for satellite exper-iments, due to masking of the galaxy and point sources. Wehave therefore adapted the formalism to deal with maskedskies. We will present this work, including simulations vali-dating our treatment, in a forthcoming publication (McCal-lum et al., in prep) and just give a brief summary here.To account for a mask, the expressions for the spuri-ous B -mode spectra induced by systematic effects (equations By set of constant elevations scans, we mean the complete spec-ification of a period of observing at a fixed elevation. I.e. a speci-fication of elevation, range of azimuth variation during each backand forth motion of the telescope, and starting and ending localsidereal times. This argument that the ˜ h n quantities can be cancelled using thescan strategy also applies in the alternate analyses of systematiceffects where only a single detector pair is considered (O’Dea et al.2007; Shimon et al. 2008) – the equivalent quantities to the ˜ h n salso cancel in these treatments. Thus, the utility of combiningscan strategies and boresight rotation is not restricted to focalplanes with both “ Q ” and “ U ” detector pairs. In addition, the LAT receiver has the capability to be rotatedby +/-45 degrees. Although this is not entirely equivalent to therotation of the whole telescope (since it doesn’t rotate the fulloptical response), it would likely be effective in mitigating gainsystematics as considered in this paper. E -mode spectraexpected on the full sky. Once these are specified, then thestandard pseudo- C (cid:96) approach (e.g. Brown et al. 2005) can beapplied to compute the total power in the pseudo- C (cid:96) B -moderesulting from a given systematic. We note that equations(26)–(28) in Wallis et al. (2017) are derived under the as-sumption that the a (cid:96) m s describing the signal caused by thesystematic are equal for the E - and B -modes (i.e. that thetemperature leaks equally into the parity even and parityodd polarisation modes). We also assume such a scenario inthis work, and consequently we model the full-sky E -modesystematic power as C EE , sys (cid:96) ≈ C BB , sys (cid:96) , (8)where the approximation indicates that we have neglectedcross-terms that can appear due to correlations between thesignal induced by the systematic and the cosmological signal.We have confirmed that these cross-terms do not contributesignificantly in our simulations, as can be seen by how wellour analytic forms describe our simulation results. Howeverit is possible that there are geometries and situations wherethese terms would contribute more significantly. A detailedinvestigation of these issues will be presented in our forth-coming work (McCallum et al., in prep). With analytic ap-proximations for the full-sky C BB (cid:96) and C EE (cid:96) in hand, thesecan be combined to predict the resulting pseudo- C (cid:96) B -modepower spectrum using the standard pseudo- C (cid:96) approach, (cid:101) C BB (cid:96) = (cid:213) (cid:96) (cid:48) (cid:16) M BB , BB (cid:96)(cid:96) (cid:48) C BB (cid:96) (cid:48) + M BB , EE (cid:96)(cid:96) (cid:48) C EE (cid:96) (cid:48) (cid:17) , (9)where M BB , BB (cid:96)(cid:96) (cid:48) and M BB , EE (cid:96)(cid:96) (cid:48) are the components of the usualpseudo- C (cid:96) coupling matrix for polarisation (Brown et al.2005). To complement the analytic-based arguments of the previ-ous section and to investigate the effect of partial boresightrotation in detail, we use TOD simulations as in Wallis et al.(2017). For simplicity, we consider two detector pairs, ori-ented at ◦ with respect to each other in order to simul-taneously measure the “ Q ” and “ U ” Stokes parameters. Inaddition, we choose not to add noise so that the impactof the systematic effect, and its mitigation through partialboresight rotation, can be clearly identified. Since one doesnot expect any correlation between random noise and thesystematic effects considered here, our conclusions regard-ing the ability of boresight rotation to mitigate systematiceffects should be insensitive to this choice.The input to our TOD simulation code consists of mapsof the CMB I , Q and U fields created using the SYNFAST routine of the
HEALPIX package (G´orski et al. 2005). Thecosmology used to generate the input CMB power spectrawas the best-fitting 6-parameter Λ CDM model to the 2015
Planck results (Planck Collaboration et al. 2015), specifiedby the following cosmological parameter values: H = . , Ω b = . , Ω cdm = . , τ = . , n s = . , A s = . × − .We do not include tensor modes ( r = . ) but our inputmaps do include lensing-induced B -modes (approximated as MNRAS , 1–11 (2019) artial boresight rotation in future CMB surveys Gaussian). The maximum multipole included when creatingour input maps is (cid:96) max = and the maps are generatedwith a HEALPIX resolution parameter N side = , corre-sponding to a pixel size of 1.7 arcmin. The maps are alsosmoothed with a Gaussian beam with a Full Width at HalfMaximum (FWHM) of 7 arcmin.Simulated TOD samples are then generated from themaps using a “synthetic” scan strategy as detailed below,with a fixed number of hits simulated for each pixel in thesky. For each of these hits, the code computes values for eachof the four detectors, interpolated from the input sky mapsat the appropriate location, and for the particular parallacticangle associated with that hit, as d Ai = I ( Ω ) + Q ( Ω ) cos ( ψ ) + U ( Ω ) sin ( ψ ) , (10)where the ψ values are offset by ◦ for the two detectorswithin a pair (labelled by the superscript A ), and by ◦ between the two pairs of detectors (labelled by the subscript i ). A differential gain systematic is included for each detec-tor pair by increasing the signal by some factor ( − δ g i ) inthe second detector d Bi , where we choose | δ g | = | δ g | = . for the simulations in this work. A differential pointing sys-tematic is included by offsetting the point seen on the sky bythe second detector of each pair, d Bi . The pointing offset isapplied to the HEALPIX latitude and longitude coordinates( θ, φ ) (see Wallis et al. 2017 for the coordinates conventionwe have used) as δθ = ρ i cos ( ψ + χ i ) δφ = ρ i sin θ sin ( ψ + χ i ) , (11)where we have set the pointing systematic level to ρ = ρ = . arcmin and χ = χ = . radians. These levels of system-atic are indicative of differential systematics seen in recentCMB ground-based surveys (e.g. Bicep2 Collaboration et al.2015b; Ade et al. 2014). We use two different scan strategies in our TOD simula-tions, which we refer to as “deep” and “wide” respectivelyand which are roughly modeled on the deep and wide surveysplanned for the SO (Ade et al. 2019; Stevens et al. 2018).The SO deep survey will be conducted using the SATs, andwill consist of a multi-frequency low-resolution survey (30arcmin FWHM resolution at 93 GHz) covering ∼
10% of thesky. The scientific aim of this survey is to constrain the am-plitude of the primordial B -mode signal which is expected topeak on relatively large angular scales of a few degrees, cor-responding to multipoles (cid:96) ∼ . The SO wide survey willbe conducted using the LAT, and will consist of a multi-frequency high resolution survey (2.2 arcmin FWHM reso-lution at 93 GHz) covering ∼
40% of the sky. While the SOwide survey will incorporate several distinct observationalprobes, facilitating a wide range of cosmological studies, theprobe which we focus on in this work is the measurement ofthe lensing B -mode power spectrum. This signal peaks onangular scales of a few arcmin, corresponding to multipoles (cid:96) ∼ .From equation (5), we can see that the gain system-atic couples directly to the total intensity field T B , while the pointing systematic couples to the gradient of the totalintensity ∇ T B . There will thus be an additional scaling with (cid:96) for the pointing systematic, such that we expect it to bemost prelevant on smaller angular scales. In addition, themuch higher angular resolution of the SO LAT means thatpointing errors are likely to be more relevant for the SO widesurvey. Thus, we will use a “deep” scan strategy for our TODsimulations investigating differential gain, and a “wide” scanstrategy for our TOD simulations investigating differentialpointing. We note again that the partial boresight rotationcapabilities of the SO SATs and LAT are appropriate forthose systematics that are most likely to present challengeson large scales (i.e. gain errors) and small scales (i.e. pointingerrors) respectively.We have developed a framework for creating “synthetic”scans which substantially decreases the computational timeand resources required by the TOD code, but is also stillable to capture the salient features of possible scans. In thisframework, we use two parameters per sky pixel to spec-ify the scan, and these can then be used to rapidly con-struct full-sky simulated maps for different choices of thescan strategy. A mask can then be applied to the simulatedmaps as required. The two parameters are the number ofdistinct crossing angles in each pixel, N φ , and the range ofcrossing angles, R . For the scans used in this paper, the pa-rameters are set to the same value in each pixel for simplicity.Allowing the two parameters to vary with sky location wouldfacilitate a more realistic distribution of these values for thesky pixels but, for the scans considered in this paper, thisincrease in complexity makes little quantitative difference tothe results.For each pixel in the sky, the TOD code selects N φ ran-dom crossing angles from the uniform range R . These cross-ing angles correspond to the “hits” of the detector in thispixel. Note that the degree to which a scan can mitigate asystematic effect is independent of the mean angle on whichthe range R is centred on, and indeed this central value canchange from pixel to pixel with no impact on the results.This is because the h n values of equation (7) are indepen-dent of the mean angle with which a pixel is observed. TheRA and Dec values associated to each hit are selected atrandom from the values inside the pixel. Thus, each hit iscompletely specified as it would be using a “real” scan strat-egy, but without any laborious creation or reading/writingof large files to/from disk.We leave a full exploration of the synthetic scans, theiruses, justification and relation to real scan strategies to aforthcoming work (McCallum et al., in prep). However, tojustify their use specifically for this work, we have testedour synthetic scan framework against prototype SO scansprovided by the SO collaboration for both the “deep” and“wide” surveys as follows.We characterised the SO prototype scan strategies bycalculating the mean range, and mean number of distinctcrossing angles per pixel, and then used these as the param-eter choices for our synthetic scans. The measured meanvalues were N φ = and R = . radians for the SO wide-type survey and N φ = and R = . radians for the SOdeep-type survey. We also verified that the true RA andDec values for the hits in the prototype scan strategies arenot preferentially located in any part of the sky pixels. Wehave also directly confirmed using a restricted number of full MNRAS000
40% of the sky. While the SOwide survey will incorporate several distinct observationalprobes, facilitating a wide range of cosmological studies, theprobe which we focus on in this work is the measurement ofthe lensing B -mode power spectrum. This signal peaks onangular scales of a few arcmin, corresponding to multipoles (cid:96) ∼ .From equation (5), we can see that the gain system-atic couples directly to the total intensity field T B , while the pointing systematic couples to the gradient of the totalintensity ∇ T B . There will thus be an additional scaling with (cid:96) for the pointing systematic, such that we expect it to bemost prelevant on smaller angular scales. In addition, themuch higher angular resolution of the SO LAT means thatpointing errors are likely to be more relevant for the SO widesurvey. Thus, we will use a “deep” scan strategy for our TODsimulations investigating differential gain, and a “wide” scanstrategy for our TOD simulations investigating differentialpointing. We note again that the partial boresight rotationcapabilities of the SO SATs and LAT are appropriate forthose systematics that are most likely to present challengeson large scales (i.e. gain errors) and small scales (i.e. pointingerrors) respectively.We have developed a framework for creating “synthetic”scans which substantially decreases the computational timeand resources required by the TOD code, but is also stillable to capture the salient features of possible scans. In thisframework, we use two parameters per sky pixel to spec-ify the scan, and these can then be used to rapidly con-struct full-sky simulated maps for different choices of thescan strategy. A mask can then be applied to the simulatedmaps as required. The two parameters are the number ofdistinct crossing angles in each pixel, N φ , and the range ofcrossing angles, R . For the scans used in this paper, the pa-rameters are set to the same value in each pixel for simplicity.Allowing the two parameters to vary with sky location wouldfacilitate a more realistic distribution of these values for thesky pixels but, for the scans considered in this paper, thisincrease in complexity makes little quantitative difference tothe results.For each pixel in the sky, the TOD code selects N φ ran-dom crossing angles from the uniform range R . These cross-ing angles correspond to the “hits” of the detector in thispixel. Note that the degree to which a scan can mitigate asystematic effect is independent of the mean angle on whichthe range R is centred on, and indeed this central value canchange from pixel to pixel with no impact on the results.This is because the h n values of equation (7) are indepen-dent of the mean angle with which a pixel is observed. TheRA and Dec values associated to each hit are selected atrandom from the values inside the pixel. Thus, each hit iscompletely specified as it would be using a “real” scan strat-egy, but without any laborious creation or reading/writingof large files to/from disk.We leave a full exploration of the synthetic scans, theiruses, justification and relation to real scan strategies to aforthcoming work (McCallum et al., in prep). However, tojustify their use specifically for this work, we have testedour synthetic scan framework against prototype SO scansprovided by the SO collaboration for both the “deep” and“wide” surveys as follows.We characterised the SO prototype scan strategies bycalculating the mean range, and mean number of distinctcrossing angles per pixel, and then used these as the param-eter choices for our synthetic scans. The measured meanvalues were N φ = and R = . radians for the SO wide-type survey and N φ = and R = . radians for the SOdeep-type survey. We also verified that the true RA andDec values for the hits in the prototype scan strategies arenot preferentially located in any part of the sky pixels. Wehave also directly confirmed using a restricted number of full MNRAS000 , 1–11 (2019)
Figure 1.
The current proposed SO wide survey area plannedfor the SO LAT targeting high (cid:96) science goals, as in Stevens et al.(2018), superimposed over the Planck thermal dust map (PlanckCollaboration et al. 2018b).
TOD simulations that using the SO prototype scan strate-gies in place of the synthetic scans in the following analysisdoes not significantly alter any of our conclusions.We have chosen to present the results using our syn-thetic scans because it allows us to run the simulationsquickly (for example when varying the precision of the bore-sight rotation of the telescope) and because it allows us tochoose a sensible (i.e. contiguous) mask, rather than use thesky coverage that results from the SO prototype scan strate-gies. These latter sky coverages have many unobserved pixelswithin the survey footprint for a single focal-plane elementsimulation – the SO prototype scans have not been designedsuch that every focal-plane element scans every sky pixel inthe survey area. The large number of resulting “holes” thatappear in the simulated maps when using the “real” scanstrategies makes the interpretation of our results less clear.The synthetic scans will also allow us to characterise theparameter space of possible ground-based scans (in a man-ner similar to what was done for satellite scans in Walliset al. 2017), and thus investigate the important propertiesof ground-based scans for mitigating systematics. We planto investigate this in detail in future work.Finally, in addition to specifying N φ and R for each pixelon the sky, in order to model a real ground-based survey, oneneeds to apply a survey mask to the simulated maps. Themasks used in this work to model the sky coverage for our“deep” and “wide” surveys are specified as in Stevens et al.(2018), and are representative of the likely SO survey areas.The scan areas used in this work are shown in figures 1(wide) and 2 (deep), plotted over the Planck dust intensitymap.We note that our simulations do not include the effectof a half wave plate as is included in the design for the SOSATs. In this work we are not interested in simulating SOSATs in detail; rather these are just a further example ofa scan strategy and systematic combination where partialboresight rotation might be useful. Figure 2.
The current proposed SO deep survey area planned forthe SO SATs targeting primordial B-modes, as in Stevens et al.(2018), superimposed over the Planck thermal dust map (PlanckCollaboration et al. 2018b).
There are multiple ways that the partial boresight rotationcan be implemented in to a given scan strategy, with associ-ated advantages and trade-offs. The specific combination ofscan strategy and boresight rotation considered here, whichwill result in the required cancellation of the ˜ h n quantities,can be performed as follows. Firstly, for a real observation,one could perform a set of Constant Elevation Scans (CES’s)as normal for a ground-based instrument. Once this is com-plete, the boresight could be rotated by the required angle,and then the same set of CES’s is repeated. Provided thisschedule of rotated scanning is implemented on a timescalethat is much shorter than the timescale on which the rele-vant systematic is varying, then such a scheme should resultin near-perfect cancellation of the systematic in question.We expect that “drift” of the systematic will be larger is-sue for gain than pointing, but here we only consider a timeindependent systematic.To incorporate partial boresight rotation into the sim-ulations, we add an additional hit for each observation ofeach sky pixel, where the parallactic angle of this additionalhit is increased by either ◦ (for differential gain) or ◦ (for differential pointing), depending on which systematic isbeing investigated. This represents the second repetition ofeach set of CES’s after the telescope has been rotated. Thesignal that would be observed by each of the four detectorsis then calculated as normal using this updated parallacticangle. The precise location (i.e. RA and DEC) of the obser-vation within the pixel is preserved for the boresight-rotatedhit. We create maps of the Q and U Stokes parameters fromour simulated TOD through simple averaging of the differ-enced signal within each detector pair, taking into accountthe parallactic angle of each observation. Note that for thesynthetic scan framework described above, this map-makingstep can be implemented (independently for each sky pixel)immediately after the creation of the full simulated TOD
MNRAS , 1–11 (2019) artial boresight rotation in future CMB surveys dataset for each sky pixel. Since our simulations do not in-clude noise this “na¨ıve” map-making algorithm performs aswell as would be achieved with a more optimal map-makingscheme. C (cid:96) B -mode power spectrum We want to understand the magnitude of the systematic sig-nal relative to the cosmological B -mode signal, so we presentour results in terms of residual B -mode power spectra con-structed as follows. The TOD simulations are run for thesame sky realisation with and without the relevant system-atic, and the HEALPIX routine
ANAFAST is used to estimatethe pseudo- C (cid:96) B -mode power spectrum ( (cid:101) C BB (cid:96) ) from the mapreconstructed from each simulation. Note that these spec-tra are “beam-smoothed” spectra – the map used as the in-put to the simulations was smoothed and we choose not todeconvolve this beam smoothing from the estimated powerspectra.We then subtract the (cid:101) C BB (cid:96) measured in the no-systematic simulation from the (cid:101) C BB (cid:96) recovered from the sim-ulation with the systematic present in order to isolate thespurious signal arising from the systematic effect. This sub-traction has several effects. Firstly it acts to reduce the in-herent scatter caused by only having a single realisation ofthe sky. Secondly it removes the contribution to the (cid:101) C BB (cid:96) spectrum, arising from both the cosmological B -mode signaland from the cosmological E -mode signal that leaks into the B -mode due to the mask. The resulting residuals thus showthe additional contribution to the pseudo- C (cid:96) B -mode powerspectrum arising from the systematic effect, incorporatingboth the B -mode directly induced by the systematic, as wellas the contribution to (cid:101) C BB (cid:96) from the systematic-induced E -modes leaking into the B -mode channel due to the mask.When analysing the simulations with the partial bore-sight rotation included, the no-systematic simulation is sub-tracted in the same way as for the simulations includingthe systematic and no partial boresight rotation. Note thatsome of the simulation points can be negative due to thissubtraction. Since we use logarithmic axes to best displayour results, we have plotted the absolute magnitude of the (cid:101) C BB (cid:96) values.We can use the formalism of Section 2 to predict whatwe expect to see in the residual power spectra derived fromthe simulations, based on the expressions in equation (5).These expressions can be used to derive the full-sky E - and B -mode power spectra resulting from the systematic effect(see Wallis et al. 2017, as well as equation 8 and the pre-ceding discussion). The resulting predictions for the full-skysystematic E - and B -mode power spectra can then be con-verted to a prediction for the systematic pseudo- C (cid:96) B -modepower spectrum as discussed earlier. Since the cosmologicalsignal is removed from the simulated power spectra by theprocess described above, the remaining signal seen in thesimulations should be primarily due to the intensity to po-larisation leakage caused by the systematic effect. We expectthis to be well approximated by these predictions.In order to assess the relevance of the systematic ef-fects, and their mitigation through the boresight rotationtechnique, we compare them to the cosmological signals thatthey will contaminate. To do this we calculate the cosmo- logical contribution to the (cid:101) C BB (cid:96) spectrum from the B -modespectrum only, i.e. essentially setting the cosmological E -mode signal to zero. Of course, in a real experiment, it isnot so simple to just subtract this part, and any error indoing so would result in an additional source of error, whichshould be compared to the power spectra that we presentin our figures. An investigation of this effect is beyond thescope of this paper. However we expect it to be smaller thanthe effects of the systematics that we consider here (see e.g.Ade et al. 2016).Finally, we recall that the cosmological signal of primaryinterest is dependent on the angular scale being probed. Wetherefore consider two different contributions to the cosmo-logical B -mode spectrum. Firstly, for assessing the impactof the gain systematic in the deep survey, on large angularscales, we subtract the lensing B -mode signal and comparethe systematic-induced signal with the primordial B -modesignal for a range of values of the tensor-to-scalar ratio, r .Secondly, in order to assess the impact of the pointing sys-tematic in the wide survey, on small angular scales, we set r = and compare the simulation results against the lensing B -mode signal for a range of values of the lensing amplitudeparameter A lens (Calabrese et al. 2008). The main results of this paper are shown in Figs. 3 and 4.The spectra shown are calculated as described above, forthe wide survey simulations including pointing systematics(Fig. 3) and for the deep survey simulations including gainsystematics (Fig. 4). Both of these figures show a strongreduction in the size of the spurious B -mode power spectrumwhen the partial boresight rotation is used. We will nowexamine these results in more detail.In Fig. 3, we show that the pointing systematic at thelevel considered in this work is similar to a cosmologicalsignal with A lens = . . Using partial boresight rotation withthe scan strategy results in a reduction of this by three ordersof magnitude, to a level equivalent to a cosmological signalwith A lens = . . This residual is scattered around zero,suggesting that the cancellation is working perfectly (or veryclose to it), as predicted by the approximate analytic form.The shape and magnitude of the systematic signal when theboresight rotation is not used is also well described by theapproximate analytic form.In Fig. 4, we see that a gain systematic at the levelconsidered in this work results in a spurious B -mode sig-nal that is greater than the primordial B -mode signal fortensor-to-scalar ratios r = . – . (depending on the exact (cid:96) range being considered). Using partial boresight rotationwith the scan strategy the spurious signal is reduced by morethan two orders of magnitude on the largest scales, and bybetween one and two orders of magnitude on small scales.Depending on the (cid:96) value being considered, the residual sys-tematic signal is of the same order as a primordial B -modesignal corresponding to r < . on the largest scales, and aprimordial signal corresponing to r > . on smaller scales.However, we note that, in contrast to the case of the pointingsystematic, this residual signal is not centred on zero. Thatis there is a small but positive residual that is not predictedby the exact cancellation in the approximate analytic forms.We will discuss this below. The shape and magnitude of the MNRAS000
ANAFAST is used to estimatethe pseudo- C (cid:96) B -mode power spectrum ( (cid:101) C BB (cid:96) ) from the mapreconstructed from each simulation. Note that these spec-tra are “beam-smoothed” spectra – the map used as the in-put to the simulations was smoothed and we choose not todeconvolve this beam smoothing from the estimated powerspectra.We then subtract the (cid:101) C BB (cid:96) measured in the no-systematic simulation from the (cid:101) C BB (cid:96) recovered from the sim-ulation with the systematic present in order to isolate thespurious signal arising from the systematic effect. This sub-traction has several effects. Firstly it acts to reduce the in-herent scatter caused by only having a single realisation ofthe sky. Secondly it removes the contribution to the (cid:101) C BB (cid:96) spectrum, arising from both the cosmological B -mode signaland from the cosmological E -mode signal that leaks into the B -mode due to the mask. The resulting residuals thus showthe additional contribution to the pseudo- C (cid:96) B -mode powerspectrum arising from the systematic effect, incorporatingboth the B -mode directly induced by the systematic, as wellas the contribution to (cid:101) C BB (cid:96) from the systematic-induced E -modes leaking into the B -mode channel due to the mask.When analysing the simulations with the partial bore-sight rotation included, the no-systematic simulation is sub-tracted in the same way as for the simulations includingthe systematic and no partial boresight rotation. Note thatsome of the simulation points can be negative due to thissubtraction. Since we use logarithmic axes to best displayour results, we have plotted the absolute magnitude of the (cid:101) C BB (cid:96) values.We can use the formalism of Section 2 to predict whatwe expect to see in the residual power spectra derived fromthe simulations, based on the expressions in equation (5).These expressions can be used to derive the full-sky E - and B -mode power spectra resulting from the systematic effect(see Wallis et al. 2017, as well as equation 8 and the pre-ceding discussion). The resulting predictions for the full-skysystematic E - and B -mode power spectra can then be con-verted to a prediction for the systematic pseudo- C (cid:96) B -modepower spectrum as discussed earlier. Since the cosmologicalsignal is removed from the simulated power spectra by theprocess described above, the remaining signal seen in thesimulations should be primarily due to the intensity to po-larisation leakage caused by the systematic effect. We expectthis to be well approximated by these predictions.In order to assess the relevance of the systematic ef-fects, and their mitigation through the boresight rotationtechnique, we compare them to the cosmological signals thatthey will contaminate. To do this we calculate the cosmo- logical contribution to the (cid:101) C BB (cid:96) spectrum from the B -modespectrum only, i.e. essentially setting the cosmological E -mode signal to zero. Of course, in a real experiment, it isnot so simple to just subtract this part, and any error indoing so would result in an additional source of error, whichshould be compared to the power spectra that we presentin our figures. An investigation of this effect is beyond thescope of this paper. However we expect it to be smaller thanthe effects of the systematics that we consider here (see e.g.Ade et al. 2016).Finally, we recall that the cosmological signal of primaryinterest is dependent on the angular scale being probed. Wetherefore consider two different contributions to the cosmo-logical B -mode spectrum. Firstly, for assessing the impactof the gain systematic in the deep survey, on large angularscales, we subtract the lensing B -mode signal and comparethe systematic-induced signal with the primordial B -modesignal for a range of values of the tensor-to-scalar ratio, r .Secondly, in order to assess the impact of the pointing sys-tematic in the wide survey, on small angular scales, we set r = and compare the simulation results against the lensing B -mode signal for a range of values of the lensing amplitudeparameter A lens (Calabrese et al. 2008). The main results of this paper are shown in Figs. 3 and 4.The spectra shown are calculated as described above, forthe wide survey simulations including pointing systematics(Fig. 3) and for the deep survey simulations including gainsystematics (Fig. 4). Both of these figures show a strongreduction in the size of the spurious B -mode power spectrumwhen the partial boresight rotation is used. We will nowexamine these results in more detail.In Fig. 3, we show that the pointing systematic at thelevel considered in this work is similar to a cosmologicalsignal with A lens = . . Using partial boresight rotation withthe scan strategy results in a reduction of this by three ordersof magnitude, to a level equivalent to a cosmological signalwith A lens = . . This residual is scattered around zero,suggesting that the cancellation is working perfectly (or veryclose to it), as predicted by the approximate analytic form.The shape and magnitude of the systematic signal when theboresight rotation is not used is also well described by theapproximate analytic form.In Fig. 4, we see that a gain systematic at the levelconsidered in this work results in a spurious B -mode sig-nal that is greater than the primordial B -mode signal fortensor-to-scalar ratios r = . – . (depending on the exact (cid:96) range being considered). Using partial boresight rotationwith the scan strategy the spurious signal is reduced by morethan two orders of magnitude on the largest scales, and bybetween one and two orders of magnitude on small scales.Depending on the (cid:96) value being considered, the residual sys-tematic signal is of the same order as a primordial B -modesignal corresponding to r < . on the largest scales, and aprimordial signal corresponing to r > . on smaller scales.However, we note that, in contrast to the case of the pointingsystematic, this residual signal is not centred on zero. Thatis there is a small but positive residual that is not predictedby the exact cancellation in the approximate analytic forms.We will discuss this below. The shape and magnitude of the MNRAS000 , 1–11 (2019) systematic signal when the boresight rotation is not used iswell described by the approximate form as expected.The good agreement between the approximate analyticforms and the TOD simulations is evidence that the effectof the systematic on the maps, and their leakage into thepower spectra, is being modeled correctly, even for the caseof masked skies (which were not considered in Wallis et al.2017). Furthermore, it suggests that the complicated detailsand structure of the scan strategies are not driving the ef-fect of the systematic in the TOD simulations. We refer thereader to Wallis et al. (2017) for a more detailed discussionof these approximate forms and how they relate to the fullstructure of the scans.The perfect cancellation of the systematics that is pre-dicted by the analytic treatment of Section 2 does not man-ifest exactly for either the gain or the pointing systematic.Detailed investigations have shown that neither of theseresiduals is due to the breakdown of the approximate an-alytic forms of Wallis et al. (2017) at the map level, which isperhaps surprising as these approximations are expected tobreak down at some level due to the exact structure of thescan strategy being neglected in their derivation. Rather, wehave confirmed that, for the gain, the remaining residual isprimarily due to additional leakage from E -modes into B -modes caused by the instrumental systematic. We have alsoconfirmed that, for the pointing, the residuals are due to thecorrelation between the systematic a (cid:96) m s (which depend ontemperature) and the cosmological E -mode polarisation sig-nal. This correlation is related to the cross term mentionedin Section 2 and that is neglected in equation (8). This willbe examined in further detail in a forthcoming work (Mc-Callum et al., in prep). We have checked that if the simu-lations incorporating the partial boresight rotation are runwith zero input Q and U fields, then the residual systematiclevels are negligible as expected. We do not present these re-sults here as they are not realistic simulations of future CMBsurveys, but the results nevertheless demonstrate that themitigation of temperature to polarisation leakage predictedby the analytic approximations holds to a very high degreeof accuracy.We also investigate how well the partial boresight rota-tion works if the rotation angle is not precisely the desiredangle. The results of this study are shown in Fig. 5 (for thepointing) and Fig. 6 (for the gain). In these figures we showhow the residual effect of the systematic (the blue points inFigs. 3 and 4) varies if the boresight rotation angle is notprecisely ◦ or ◦ as required. In particular, we plot theresults for a boresight rotation angle that is ◦ from the idealangle ( ◦ and ◦ for gain and pointing respectively), anda rotation that is ◦ from the ideal angle ( ◦ and ◦ re-spectively). These plots reveal a number of notable features.Firstly, as illustrated by the ◦ and ◦ points, theeffect we are looking at here is a continuum: being able tomake a significant rotation to the telescope during the scanstrategy, even one that is not of the precise angle requiredto cancel a systematic, has a strong effect on the systematic, We also simulated a range of other angles, whose outputsshowed a smooth interpolation between the angles plotted here.We have only plotted a few of these angles in order to make theplots as clear as possible.
Figure 3.
Impact of the pointing systematic on the B -modepower spectrum recovered from TOD simulations of a high-resolution wide area survey. The spectra plotted here are theresidual (cid:101) C BB (cid:96) spectra, with the contributions of both the cos-mological B -mode and the cosmological E -mode subtracted – seetext for details. The orange points show the power spectrum re-covered from the maps reconstructed from the simulated datawhere no partial boresight rotation is used. The blue points showthe power spectrum recovered from the simulation when partialboresight rotation is implemented in the scan strategy. The black(dashed) curve shows the level of systematic predicted by theapproximate analytic expressions resulting from the analysis ofSection2. All three of these curves have the cosmological B -modesignal also subtracted, so they show purely the signal generatedby the systematic effect. The purple, red and green curves showthe expected theoretical B -mode spectra for r = and differentvalues of the A lens parameter (see text for details): 0.001, 0.05 and1.0 respectively. The approximate analytic form describes the re-sults from the simulation well. The magnitude of the signal dueto the systematic is similar to that due to A lens = . , and thisis reduced by three orders of magnitude by the partial boresightrotation. without requiring any data to be excluded during the analy-sis. This is expected from the definition of the ˜ h n quantities:adding extra crossing angles to each pixel, that are quitedifferent to the existing angles, will reduce the averages ofthe trigonometric quantities that make up the ˜ h n ’s. For boththe gain and the pointing, a rotation by an angle that is ◦ different from the ideal angle, still yields an order of magni-tude reduction in the signal due to the systematic. However,what is especially notable in these results, is that the situ-ations simulated here are far from being ideal scans: thereare only a few crossing angles per pixel. Despite this, therepetition of the scans in combination with the boresightrotation is making a significant difference, even when theboresight rotation angle is not close to the ideal angle for agiven systematic.Secondly, these results show that the majority of theimprovement from using the partial boresight rotation doesnot require the ideal angle to be achieved with a high levelof accuracy. For the deep survey simulations including gainsystematics, the partial boresight rotation efficacy is affectedvery little if the boresight rotation is incorrect by ◦ or less.At this point the residual saturates due to the E to B leakage MNRAS , 1–11 (2019) artial boresight rotation in future CMB surveys Figure 4.
Impact of the gain systematic on the B -mode spectrumfrom simulations of a low-resolution deep survey. See the captionof Fig. 3 for a description of the quantities plotted for the case ofthe simulations and analytic approximation. The purple, red andgreen curves show the expected theoretical B -mode spectra fordifferent values of the tensor-to-scalar ratio r : 0.001, 0.01 and 0.1respectively, where the lensing contribution to the B -mode hasbeen subtracted. The approximate analytic form describes thesimulation output well. The magnitude of the signal due to thesystematic is between r = . − . (depending on the exact (cid:96) rangebeing considered). This is reduced by between one, and more thantwo, orders of magnitude by the partial boresight rotation, whichcorresponds to a cosmological signal ranging between r < . onthe largest scales, to r > . on smaller scales. discussed above. As noted, this saturated level is a significantreduction of the signal due to the systematic.For the wide survey simulations including pointing sys-tematics, there is also a highly significant reduction of thesystematic-induced signal for a telescope boresight rotationangle within ◦ of the ideal value. However, unlike the gainsystematic, this residual does not saturate at this point: asthe rotation is moved ever closer to ideality, the residual fallsuntil it is scattered around zero, showing an essentially per-fect cancellation of the signal due to the systematic. Thisis an important result, because it gives us a benchmark forthe required tolerance on the boresight rotation in upcomingand future CMB instruments. We have shown explicitly using TOD simulations that thereare specific combinations of instrument rotation and scanstrategy choices that would lead to natural suppressionof systematic errors for ground-based CMB experiments.Specifically, we recommend repeating the same constant el-evation scans, targeting the same sky areas, with either a ◦ (to reduce differential gain errors) or ◦ (to reducedifferential pointing errors) rotation of the instrument. Forthe surveys and telescope details considered here, which aresimilar to the specifications for the Simons Observatory, thelevel of the gain systematic is reduced by between one, andmore than two, orders of magnitude, but still leaves a smallresidual due to E to B leakage effects (Fig. 4). The level of Figure 5.
The different levels of residual for the wide survey, in-cluding pointing systemaitcs, as the boresight rotation angle usedin the boresight rotation technique is varied. The orange pointsare for the case with no partial boresight rotation and the bluepoints show the residual for the ideal boresight rotation, both asin Fig. 3. The red points show a boresight rotation that is ◦ lessthan ideal ( ◦ ), and the green points show a boresight rotationthat is ◦ less than ideal ( ◦ ). In all cases the residual signaldue to the systematic is significantly reduced, and this remainingresidual approaches zero as the ideal rotation is reached. Figure 6.
The different levels of residual for the deep surveysimulation, including gain systematics, as the boresight rotationangle used in the partial boresight rotation technique is varied.The orange points are for the case with no partial boresight rota-tion and the blue points show the residual for the ideal boresightrotation, both as in Fig. 4. The red points show a boresight rota-tion that is ◦ less than ideal ( ◦ ), and the green points show aboresight rotation that is ◦ less than ideal ( ◦ ). In all cases theresidual signal due to the systematic is significantly reduced, al-though this residual saturates around ◦ away from the ideal value(due to irreducible E to B leakage caused by the gain systematic)and the residual doesn’t improve further.MNRAS000
The different levels of residual for the deep surveysimulation, including gain systematics, as the boresight rotationangle used in the partial boresight rotation technique is varied.The orange points are for the case with no partial boresight rota-tion and the blue points show the residual for the ideal boresightrotation, both as in Fig. 4. The red points show a boresight rota-tion that is ◦ less than ideal ( ◦ ), and the green points show aboresight rotation that is ◦ less than ideal ( ◦ ). In all cases theresidual signal due to the systematic is significantly reduced, al-though this residual saturates around ◦ away from the ideal value(due to irreducible E to B leakage caused by the gain systematic)and the residual doesn’t improve further.MNRAS000 , 1–11 (2019) the pointing systematic is reduced by around three orders ofmagnitude, to a signal that is scattered around zero (Fig. 3).Importantly, we note that this suppression of the systematicresults in no filtering of the signal or signal suppression. Asdiscussed above, the residuals are due to other effects, ratherthan a breakdown of the analytic approximations used topredict the results.These results were derived using a new simulation ap-proach which we refer to as “synthetic” scans. Our techniqueretains the important features of typical scan strategies, butalso facilitates the rapid creation of large numbers of sim-ulated maps. In particular, the technique does not requirelarge scan files to be created and/or read from disk by aTOD simulation code. We will present the full details of thesynthetic scan framework in a future work (Mccallum et al.,in prep), where we will also present the details of our ex-tension of the analytic formalism of Wallis et al. (2017) toground-based surveys.We have also investigated the performance of the partialboresight rotation technique in the case that the telescopeboresight rotation is not exact. We showed that, even forscans that are far from ideal, the effect is a continuum, withexponential reductions in the signal due to the systematicas the ideal rotation angle is approached. For the the deepsurvey simulations including gain errors, the residual sat-urates at a rotation that is non-ideal by ◦ (Fig. 6). Forthe wide survey simulations including pointing errors, theresidual continues to reduce as the non-ideality of the ro-tation is reduced below ◦ , until the residual is scatteredaround zero for the ideal rotation. For both cases, for sen-sible tolerances on the precision of the boresight rotation,the residual B -mode contamination is orders of magnitudesmaller than the systematic B -mode signal that would ap-pear in an equivalent experiment where boresight rotation isnot used. Furthermore, these results suggest that other, lessspecific, ways of combining partial boresight rotation withthe scan strategy could still deliver significant reductions insystematic levels. We conclude that telescope boresight ro-tation could thus play a key role in reaching the sensitivityrequired for precision measurements of the B -mode polari-sation signal.Finally, we note that the assumption of pair differenc-ing is fundamental to this work. This assumption shouldresult in the predictions from our formalism being an upperbound on the level of systematic contamination that canbe expected in an analysis. We leave it to future work toinvestigate how these instrumental systematics manifest indifferent analysis pipelines, and to what extent the incorpo-ration of these specific instrument rotation capabilities intothe scan strategies in these particular ways also reduces thesesystematics in different analysis pipelines. In particular, halfwave plates are a common reason that the analysis pipelinemight be different. In principle it might be posible to recre-ate some of our results with no partial boresight rotation,but by instead stepping a half wave plate at specific anglesand combining this with the scan strategy in specific ways. ACKNOWLEDGEMENTS
We thank C Wallis for providing us with the TOD simulationcode and for useful discussions. We thank Julien Peloton and the SO collaboration for providing us with the LaFabriquecode for generating prototype SO scan strategies. We thankthe SO scan strategy working group for useful discussions.DBT acknowledges support from Science and TechnologyFacilities Council (STFC) grant ST/P000649/1. NM is sup-ported by a STFC studentship. This is not an official SimonsObservatory Collaboration paper.
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