Sky rotation in ground-based cosmic microwave background experiments
VVersion February 5, 2021
Preprint typeset using L A TEX style openjournal v. 09/06/15
SKY ROTATION IN GROUND-BASED COSMIC MICROWAVE BACKGROUND EXPERIMENTS
Daniel B. Thomas
School of Physics and Astronomy, Queen Mary University of London, London, E1 4NS, UK
Nialh McCallum and Michael L. Brown
Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy, The University of Manchester, Manchester M13 9PL, UK (Dated: February 5, 2021)
Version February 5, 2021
ABSTRACTGround-based Cosmic Microwave Background (CMB) experiments (except those located at theSouth Pole) take advantage of sky rotation to achieve multiple crossing angles (i.e. scans across eachpixel in multiple directions). This crossing-angle coverage is important for controlling the magnitudeof some systematics, and for the removal of 1 /f noise during the map-making process. Typically,these experiments scan the sky using constant elevation scans, which puts strong constraints on thecrossing-angle coverage that is achievable, and therefore on the advantage gained from sky rotation.In this work we elucidate the relationship between scanning elevation and crossing angle, and derivesome general “rules of thumb” for scheduling constant elevation scans to maximise the crossing-anglecoverage. We derive some bounds on the quality of crossing-angle coverage that can be achievedindependently of detailed scheduling choices, which are an important consideration for map-makingalgorithms if upcoming CMB surveys are to achieve their aims, and we discuss how these results mightrelate to field selection. These bounds can be used to forecast the effect of scan-coupled systematicsfor upcoming surveys. We also show quantitatively how some simple choices of boresight rotationimprove the possible systematics mitigation from purely sky rotation. Our results are relevant forother surveys that perform constant elevation scans and may have scan-coupled systematics, such asintensity mapping surveys. INTRODUCTION
Ground-based Cosmic Microwave Background (CMB)surveys are a crucial part of ongoing (e.g. Ade et al.(2015a); Adachi et al. (2020); Sayre et al. (2020); Choiet al. (2020)) and future (e.g. Ade et al. (2019); Abaza-jian et al. (2019)) attempts to map the CMB temper-ate and polarisation in exquisite detail. These instru-ments are more limited than satellite-based experimentsin terms of which parts of the sky they can scan, andhow they can scan it.It is usual for ground-based CMB surveys to scan thesky with Constant Elevation Scans (CESs). This involveschoosing an observing elevation, and sweeping the tele-scope in azimuth. Due to the rotation of the Earth, thetargeted field on the sky will appear to move up (“ris-ing”) or down (“setting”) through the telescope’s field ofview. Once the field that is being observed is no longervisible, the elevation is increased or decreased (fields aretypically observed both when they are rising and set-ting ), and the telescope continues scanning. CESs areused because they are good for instrument stability, andbecause varying the elevation during a scan affects theamount of atmosphere that is being seen, resulting inmore complicated noise and systematic properties.However, CESs put strong constraints on the proper-ties of the survey, in particular the crossing-angle cover-age that can be achieved. The crossing-angle coverage [email protected] For the purposes of this work we will define azimuth as increas-ing clockwise from North, such that a “rising” field has an azimuth( θ ) of 0 ◦ < θ < ◦ . refers to the set of different angles at which the instru-ment scans across any given pixel, and is important be-cause it modulates the magnitude of systematics thatcouple to the scan strategy (see e.g. McCallum et al.(2020) and references therein). In addition, the crossing-angle coverage for CESs is closely related to the cross-linking. This refers to the different paths that the instru-ment takes across the sky as it scans, which can be usedto remove 1 /f noise during the map-making process (seee.g. Sutton et al. (2009); Poletti et al. (2017)).Both systematics and 1 /f noise could limit the resultsof future CMB surveys, particularly with the increas-ing sensitivity and ambitious science goals of upcomingground-based CMB surveys, notably attempts to mea-sure the B-mode polarisation signal. There are multi-ple methods for handling either 1 /f noise or systematicswithout using sky rotation (as is required for experimentslocated at the South Pole). These include using rotationof the instrument about its boresight (Ade et al. 2015b),rotating half wave plates that modulate the incomingphoton polarisation (see e.g. Maino et al. (2002); Brownet al. (2009); Johnson et al. (2007); MacTavish et al.(2008); Buzzelli et al. (2018)), and strongly filtering thedata.Here, we focus on quantifying the crossing-angle cover-age that can be achieved using CESs, as a measure of theinherent advantage that can be gained from sky rotationalone towards solving these problems, as this essentiallycomes for “free” for instruments located away from thePoles (although we do consider some boresight rotationas a reference point for the ability of sky rotation aloneto control systematics). We do this using some simple a r X i v : . [ a s t r o - ph . I M ] F e b Thomas, McCallum, Brown“first principles” simulations.The majority of scan strategy investigations focus onscheduling (e.g. De Bernardis for the ACT collaborationet al. (2016); Goeckner-Wald et al. (2018)), i.e. choosingwhich elevation and range of azimuth to scan at a par-ticular time, in order to maximise the usable observingtime, equalise the depth of the field, and avoid prob-lems such as the Sun and the Moon. As we will see,for crossing-angle coverage, we can derive bounds for theperformance of CESs without considering scheduling indetail at all, which is the unique strength (and weakness)of our approach. As such our approach establishes whatcan be achieved in a best-case scenario, and thereforeacts as a target for detailed scheduling to aim for.These bounds can be compared to the required level ofcross-linking in map-making procedures that use cross-linking to remove 1 /f noise, and have consequences forfield selection and choice of observing elevation. Thesebounds can also be used to inform discussions of whetherthe benefits of sky rotation from CESs are sufficient, orif there is merit in more ambitious strategies (not lim-ited to CESs) that trade-off the advantages of CESs withthe improved crossing angle coverage and cross-linkingthat can be obtained. In combination with approachessuch as McCallum et al. (2020), our method can also beused to show the extent to which a careful choice of scanstrategy can reduce scan-coupled systematics, and fore-cast the impact of different scan-coupled systematics ina way that is agnostic of detailed scheduling considera-tions. Many of these results could be particularly usefulin the early stages of survey design.We describe our setup, code implementation and met-rics for crossing-angle coverage in section 2, and presentthe consequences of CESs, and resulting “rules of thumb”and optimal strategy in section 3. In section 4 we use thisoptimal strategy to examine the crossing-angle coverageachievable for surveys based in the Atacama Desert. Thisincludes an examination of the dependence on Declina-tion, and bounds on the degree to which scan-coupledsystematics can be modulated. We conclude in section5. CODE IMPLEMENTATION AND QUANTIFYINGCROSSING-ANGLE COVERAGE
We make extensive use of the python pyEphem(Rhodes 2011) and Healpy (Zonca et al. 2019; G´orskiet al. 2005) packages to derive our results. We define theobservatory using an ephem.Observer() object, where thekey property of the observatory for the results is the Lat-itude of the observatory. We take the Right Ascension(RA), and Declination (Dec), of the centre of a healpixpixel as an ephem.FixedBody() object. The FixedBody Compute method can then be used to compute the eleva-tion and azimuth required to see the healpix pixel fromthe given observatory at a particular time. This is thebasic building block of the code.We then want to calculate the trajectory of the instru-ment on the sky. Since we are restricting to CESs, weknow that the next observation will occur at the same el-evation, at a slightly increased azimuth value δAz . TheRA and Dec corresponding to the new azimuth can then http://healpix.sourceforge.net Or slightly decreased azimuth value if the telescope is currentlyscanning West to East, rather than East to West. However, in the be calculated using the radec of method in pyEphem.Given the RA and Dec of the pixel, and the RA and Decof the next observation, the direction of travel of the fo-cal plane with respect to the sky can be calculated usingthe “two-argument arctangent” as δ Dec = Dec − Dec (1) δ RA = RA − RA (2) ψ = arctan 2 (cid:18) δ Dec δ RA cos(Dec ) (cid:19) . (3)Appropriate choices of δAz depend on the scan speed andsampling frequency of the telescope. For this work weuse δAz = 0 .
01 ˙3 ◦ , corresponding to for example a scanspeed of 0 . Combining the above, we have a routine that computesthe crossing angle ψ as a function of the observatoryLatitude, the time, and the RA and Dec of the targetedpixel. We use this in several ways to generate the resultsin the rest of the paper. In particular, we use this tosample the possible crossing angles of a given pixel overa fixed period of time (from a single day to 5 years) givena range of possible observing elevations. We also use it tofix the elevation, and then generate all possible crossingangles within each pixel that can be achieved from thatelevation.By setting up the problem as above, we have made twoimplicit assumptions. Firstly that every hit in a scan willbe at the centre of the corresponding pixel, and secondlythat the telescope will take no measurements during theacceleration/decceleration that accompanies the switchfrom an “East to West” scan to a “West to East” scan.The former introduces a small error, but for the pixelsizes typical of current surveys, the Declination variationwithin a pixel is small enough for this to be negligible.Regarding the latter, the time spent accelerating/decel-erating is not a substantial fraction of the scan time, andthe data taken during these times is typically problem-atic in other ways, so we think that neglecting this partof the scan is reasonable. Quantifying crossing angle coverage
We need a way to quantify the quality of the cross-ing angle coverage from the set of crossing angles withina given pixel. Following Wallis et al. (2016), we charac-terise the set of crossing angles ψ j using the set of metrics h k = 1 N hits (cid:88) j e ikψ j = 1 N hits (cid:88) j (cos( kψ j ) + i sin( kψ j )) ,(4)where k takes integer values. The value of | h k | rangesfrom 0 to 1, where 0 represents an ideal scan and 1 rep-resents the worst possible scan (every crossing angle in two situations, the orientation of the focal plane with respect to thesky doesn’t change, because the opposite edge of the focal planewill now be the leading edge with respect to the sky, and the twoeffects cancel. For simplicity in our code, we always treat δaz aspositive. This means that the crossing angle results derived here shouldapply for scan strategies with varying azimuth scan speed as longas it is within this range.
ESs and crossing angles 3the pixel is the same). These quantities measure thevariation within a set of crossing angles (note that onecan add a constant value to every angle in a set withoutchanging the value of | h k | for that set). The h and h quantities appear directly in simple binning polarisationmap-making, and h is the quantity most directly relatedto cross-linking (see e.g. the cross-linking metric used inChoi et al. (2020), which is similar to h ).Other h k are relevant for systematics, with h − themost relevant for intensity to polarisation leakage (seee.g. Wallis et al. (2016)), however there are systemat-ics that can couple to h k with k > h k to use to evaluate thecrossing-angle coverage depends on the question underconsideration. Here we will focus on k = 1 − (cid:112) | h | (in the absence of boresightrotation; see section 4.1.1). Using this metric, theACT team considered several cutoffs to determine thepixels with sufficient crossing angle coverage for theirscience analysis. Here, we do not consider a specificquantitative cutoff, however we note that map-makingmethods that are designed to remove 1 /f noise have animplicit requirement that this quantity is less than 1(Sutton et al. 2009; Poletti et al. 2017); this requirementshould be quantified where possible and compared toour results, to establish whether a particular observatoryand elevation range can achieve the requirement overthe whole targeted field. CES CONSEQUENCES AND “RULES OF THUMB”
In this section we use the codes described earlier to ex-amine what the assumption of CESs means for ground-based surveys. We then use this information to createa hypothetical optimal “scan strategy”, according to our h k metrics. This provides an upper bound on the crossingangle coverage, which corresponds to a lower bound onthe h k values, that can possibly be achieved with CESs.This bound is independent of any detailed schedulingchoice. We use the optimal strategy to present somegeneral “rules of thumb” that can be applied when con-sidering scheduling in detail. CES implications
Using the “first principles” tools developed above, weexplore the parameter space that could affect the crossingangle of a given observation. The parameters we variedwere RA; Dec; observing elevation; time, year and sea-son of observation; rising vs setting (azimuth less thanor greater than 180 ◦ ); and telescope latitude. We sum-marise our key findings here.1. During a 24 hour period, for a given Dec and ob-serving elevation, the point on the sky in questioncan be observed at two times, each with a differentazimuth; one rising and one setting.2. Each of these gives a single distinct crossing angle,so for N distinct observing elevations, there is amaximum of 2 N distinct crossing angles per pixel. 3. RA only makes for a very small correction to theabove, and causes a crossing angle variation of lessthan 0 . . −
1, within ∼
1% for the majority of pixels,and up to ∼
2% for the very extremes of the visiblefield.6. The largest and smallest possible observing eleva-tions bound the range of possible crossing angles,and correspond to the largest and smallest (rising)crossing angles.7. The latitude of the telescope matters for determin-ing which fields can be observed with good cross-ing angle coverage: fields with Dec ∼ Latitude, thatpass close to overhead, have the worst distributionof crossing angles.We illustrate these results in figure 1, where we showthe range of crossing angles available as a function of dec-lination, for a telescope located at Latitude -22:56.396.This Latitude corresponds to CMB experiments in theAtacama Desert, such as the Simons Observatory (SO)site. The blue lines correspond to the parallactic anglesobtained from nominal minimum and maximum observ-ing elevations of 30 ◦ and 65 ◦ respectively. The rangeof crossing angles possible for each Declination corre-sponds to the space between the two lines at each point.The narrowing of the range of crossing angles for fieldswith Dec ∼ Latitude is clearly visible. The entire shadedregion corresponds to the approximate range of Decli-nations that will be targeted by the SO Wide survey(Goeckner-Wald et al. 2018), and the two darker shadedareas correspond to the approximate ranges of Declina-tions that will be targeted by the SO North and SouthDeep surveys (Goeckner-Wald et al. 2018). We note thatthe deep scans avoid the range of Declinations where thecrossing angle coverage is worst. At the same time, it isclear from the figure that their placement in Declinationis not optimal from the point of view of maximising thecrossing angle coverage.
Optimal “scan strategy”
From the intuition developed in the previous section,and restricting ourselves to a single pixel, we can con-sider the effect on the h k metrics of viewing that pixelwith multiple elevations. This requires setting minimumand maximum rising crossing angles, corresponding tothe minimum and maximum observation elevations, andthen sampling and averaging cos ( kψ ) and sin ( kψ ) overthat range of angles. We can also choose to include thesetting crossing angles, corresponding to the same rangeas the rising angles but with opposite signs (the distribu-tion of setting crossing angles is the distribution of risingcrossing angles, mirrored in ψ = 0).For the overwhelming majority of possible pixels andelevation bounds, the h k metrics are optimised (min-imised) through the simple choice of observing only at the Thomas, McCallum, Brown Fig. 1.—
The crossing angles as a function of Declination, for SOlocation and elevation bounds 30 − ◦ , are shown by the two bluelines. The range of crossing angles possible for each Declinationcorresponds to the space between the two lines at each point.The larger shaded region corresponds to the approximate range ofDeclinations that will be targeted by the SO Wide survey, and thetwo darker shaded areas correspond to the approximate ranges ofDeclinations that will be targeted by the SO Deep surveys. two extremal elevations, and observing the pixel when itis both rising and setting at these two elevations. Most ofthe exceptions to this are unrealistic combinations. Werefer to this approach as the Two Elevation Rising andSetting (2ERS) strategy. For a given pixel, telescope lati-tude and minimum and maximum elevation bounds, thisstrategy sets the optimal h k values that can be achievedin that pixel, for any detailed choice of scheduling; usingany intermediate elevations will not improve the crossing-angle coverage. We use this “strategy” in the rest of thiswork to create bounds on what is theoretically possibleassuming CESs.There is a caveat to the optimality of the 2ERS strat-egy. It is possible to choose specific patches in Declina-tion such that the h k values are lower than for a 2ERSstrategy when observing with one of the elevation boundsas the only observing elevation. We present this case inappendix A. Here we just note that even allowing forstrategies with a single elevation, 2ERS is optimal whenconsidering the whole visible field, in the sense that itminimises both the mean and the mode of the distribu-tion of | h k | values over the field. Moreover, it is alwaysclose to the true lower bound, even when restricting toonly part of the entire visible area and allowing for singleelevation scans. General “rules of thumb” for scheduling and fieldselection
The results from the previous two subsections can becombined into a set of general “rules of thumb” that canbe used to guide detailed scheduling choices and fieldselection. • The crossing-angle bounds from 2ERS comprise abest case scenario (except for the caveat exploredin appendix A), and can act as a target for detailedscheduling • Choose two elevations that are as far apart as pos- sible given the other constraints, and observe bothof them when rising and setting • As far as possible, equalise the number of hits ineach pixel at each elevation, and for each of risingand setting • Intermediate elevations do not improve the cross-ing angle coverage (but they do allow pixels to beobserved at times when the telescope would other-wise be idle) • Declinations that are similar to the Latitude of thetelescope (i.e. that pass overhead or nearly over-head) will always have the worst crossing-angle cov-erage and are bad fields to target (all other thingsequal) • Since the visible area changes with the maximumallowed observing elevation, in practice reachingthe edges of a given field might require a differentmaximum observing elevation to the central areasof the fields. CROSSING ANGLE COVERAGE FROM SKY ROTATION
We now illustrate the power of our simple approach tosurveys using CESs, by using the 2ERS strategy to inves-tigate the best achievable crossing-angle coverage for dif-ferent elevation bounds. We quantify this using | h k | , for k = 1 −
4. We work within the elevation range 30 − ◦ ,and we set the Latitude to −
22 : 56 .
396 correspondingto the Atacama Desert, a common site for CMB exper-iments. However, note that due to the symmetry of thesituation, Northern Latitudes (+ φ ◦ ) behave the same asSouthern Latitudes ( − φ ◦ ).We show maps of | h | in figure 2 for three differentpairs of elevations. There are several main features ofthese maps. The first is that the range of pixels seenby both elevations reduces as the upper elevation is in-creased as expected. The second is that there is a clearpattern to the h maps as a function of Dec, most notablya band of declinations within the visible area where thecrossing angle coverage is worst. These features persistfor all pairs of elevation bounds in the range 30 − ◦ . Fi-nally, the worst (peak) | h | values are lowest (i.e. best)for the map with the widest range of observing elevations.A key observation from these maps is that the achiev-able crossing-angle coverage with CESs is unlikely to bereasonably homogeneous over a larger field. Moreover,since the 2ERS approach is an upper bound on the cross-ing angle coverage that CESs can achieve, then to makethe field homogeneous would in practice simply amountto making the crossing-angle coverage worse over most ofthe field. Map-makers seeking to take advantage of skyrotation in surveys using CESs need to be able to achievetheir goals in the presence of these limitations.This is further explored in figure 3, showing the vari-ation of | h | with Declination as the lower elevationbound is changed (top plot) and upper elevation bound ischanged (bottom plot). The decrease in the peak (worst)values with increasing difference between the elevationbounds is clear in both plots. This peak decrease isnot large, however it may be important for map-makingmethods seeking to remove 1 /f noise, as they are morelikely to fail as | h | approaches unity. The lower plotESs and crossing angles 5 Fig. 2.— | h | maps for a telescope at Latitude -22:56.396 for the2ERS strategy with elevation bounds (from top to bottom) (30 , , , also shows the decrease in visible area, and slight shift ofthe peak location, as the upper elevation increases.Going beyond | h | , figure 4 shows | h | , | h | and | h | as a function of Declination, for the same Latitude.Again, the 2ERS strategy is used, with a lower elevationbound of 30 ◦ and varying upper elevation bound. Theshape of these functions is different to | h | , however, asbefore the peaks (worst values) are reduced when thereis a greater difference between the elevation bounds. Inaddition, as for | h | , there is a small shift in the locationof the peak and the shape as the upper elevation boundis changed. Fig. 3.— | h | as a function of Declination as the elevationbounds are changed for the 2ERS strategy and Latitude -22:56.396.Top: the upper elevation is fixed to 65 ◦ and the lower bound is var-ied as 30 ◦ (blue, solid), 40 ◦ (purple, dashed), 50 ◦ (red, dot-dashed),60 ◦ (oranged, dotted). Bottom: the lower elevation is fixed to 30 ◦ and the upper bound is varied as 40 ◦ (blue, solid), 50 ◦ (purple,dashed), 60 ◦ (red, dot-dashed), 65 ◦ (orange, dotted). In both cases,increasing the difference between the elevations reduces the peak.The lower plot also shows the decrease of visible area, and move-ment of the peak, as the upper elevation is increased. Note thatchoosing different RA values makes a negligible difference to thisplot. h k tables for systematics Although the values of | h | for the worst Declinationsare likely to be the important factor for map-making con-siderations, as discussed above, the same is not true forsystematics. In this case, the average value of | h k | acrossthe map (for the relevant k values for each systematic) isa good approximation to the effect of the scanning strat-egy on the contaminating systematic (Wallis et al. 2016).In tables 1-4 we show how the | h k | values (for k = 1 − | h | this trend largely van-ishes and the values vary little as the elevation boundsvary. To further test the efficacy of the 2ERS strategy Thomas, McCallum, Brown Fig. 4.— | h | (top), | h | (middle) and | h | (bottom) as afunction of Declination as the upper elevation bound is changed forthe 2ERS strategy and Latitude -22:56.396. The lower elevationis fixed to 30 ◦ and the upper bound is varied as 40 ◦ (blue, solid),50 ◦ (purple, dashed), 60 ◦ (red, dot-dashed), 65 ◦ (orange, dotted).As with | h | , a greater difference between the elevation boundslowers the peaks, and increasing the upper elevation decreases thevisible area and moves the position of the peak. TABLE 1 h values for different elevation bounds using 2ERSstrategy.
30 35 40 45 50 55 6030 0 .
23 0 .
21 0 .
20 0 .
19 0 .
18 0 .
17 0 . .
24 0 .
22 0 .
21 0 .
20 0 .
18 0 . .
25 0 .
23 0 .
22 0 .
20 0 . .
26 0 .
24 0 .
22 0 . .
28 0 .
25 0 . .
29 0 . . TABLE 2 h values for different elevation bounds using 2ERSstrategy.
30 35 40 45 50 55 6030 0 .
54 0 .
53 0 .
52 0 .
51 0 .
50 0 .
49 0 . .
54 0 .
52 0 .
51 0 .
50 0 .
50 0 . .
52 0 .
51 0 .
50 0 .
50 0 . .
51 0 .
50 0 .
49 0 . .
50 0 .
49 0 . .
49 0 . . approach, we repeated the simulations for these tableswhere a third observing elevation was added, optimallychosen for each pixel to give a crossing angle midway be-tween the two rising crossing angles from the elevationbounds. In all cases the average | h k | values across thefields were the same or worse than for the 2ERS case,although they typically do not degrade by much (up to0.06). The diagonal entries of each table show that re-stricting to a single elevation is never better (averagedover the whole visible area) than a 2ERS strategy (ex-cept for | h | which, as noted, varies little across thewhole table).We note that there are some complications here re-garding sky coverage. Firstly, we have not applied anymasks; a mask that systematically affects some declina-tions more than others could change the results in thesetables. Secondly, the values in the tables are computedfrom the whole visible area, which changes along eachrow (but not down each column). Judging from the ear-lier figures showing | h k | as a function of Declination,we expect the results to not change when restricting el-ements along a row to only the common area.The trends highlighted in this section apply for anysensible pair of elevation bounds. Since the chosen Lat-itude is the Atacama Desert, the bounds in this sec-tion represent the best possible (smallest) | h k | (averagedover the visible sky area) that can be achieved from skyrotation by experiments in the Atacama Desert (such asACT or Simons Observatory) using CESs. In the samefashion, bounds can be constructed for any ground-basedobservatory using our approach.The bounds in tables 1-4, and the maps in figure 2ESs and crossing angles 7 TABLE 3 h values for different elevation bounds using 2ERSstrategy.
30 35 40 45 50 55 6030 0 .
50 0 .
48 0 .
46 0 .
44 0 .
41 0 .
39 0 . .
49 0 .
48 0 .
46 0 .
43 0 .
40 0 . .
49 0 .
48 0 .
46 0 .
43 0 . .
49 0 .
48 0 .
46 0 . .
49 0 .
48 0 . .
49 0 . . TABLE 4 h values for different elevation bounds using 2ERSstrategy.
30 35 40 45 50 55 6030 0 .
49 0 .
47 0 .
43 0 .
39 0 .
34 0 .
30 0 . .
49 0 .
47 0 .
44 0 .
39 0 .
34 0 . .
50 0 .
48 0 .
43 0 .
38 0 . .
50 0 .
47 0 .
43 0 . .
50 0 .
47 0 . .
49 0 . . (and the equivalent maps for h , h and h ; with a maskapplied if required) can be used in combination with theexpressions in Wallis et al. (2016); McCallum et al. (2020)to predict the magnitude of the distortion due to scan-coupled systematics on both CMB power spectra andCMB maps. This can be done without detailed evalua-tion of the scheduling, allowing it to be used early in theforecasting and design process to get approximate valuesand best case scenarios, that can be used to inform fieldselection and the other design aspects of the surveys.For example, a differential pointing systematic causes atemperature to polarisation leakage modulated by h and h : from the tables given here, the values 0 . − .
29 and0 . − . Combined sky rotation and boresight rotation
The 2ERS strategy also gives us a simple way to exam-ine the effect of combining boresight rotation with skyrotation on the quality of the crossing angle coveragefor the purpose of minimising systematics. Note that inthis section the crossing-angle metrics are no longer rep-resentative of the cross-linking, because boresight rota-tion changes the distribution of angles in a pixel withoutchanging the path of the instrument across the sky . A similar comment applies to crossing-angle coverage in thepresence of a half wave plate, although see Wallis et al. (2015) forhow to extend the h k metrics to half wave plates whilst separatelytracking crossing angles and half wave plate modulation. TABLE 5Effects on h k for different choices of boresight rotationangle with 2ERS strategy. ◦ ◦ ◦ h . − × worse 1 . − × worse reduced to zero h − × better reduced to zero no effect h − × better 2 × better reduced to zero h reduced to zero no effect no effect We consider the case where the instrument has twovalues for the boresight rotation, taken to be 0 ◦ and theboresight rotation value. In principle we could extendthis to many rotation angles, for example the set of eightangles separated by 45 ◦ as used by BICEP (Ade et al.2015a), however for this number of angles the effects ofsky rotation are largely negligible, and the minimisationof systematics is almost independent of the chosen scan-ning elevations and observatory location. Indeed, for thisset of eight angles, the h k metrics for k = 1 − ◦ rotation has no effect on h ,but reduces h to zero. We consider three angles: 180 ◦ ,90 ◦ and 45 ◦ , and present their effects on the differentmetrics in table 5. The key result here is that even theinclusion of a single boresight rotation angle can result ina strong improvement on the mitigation of scan-coupledsystematics compared to solely using sky rotation.The powerful effect of the 45 ◦ rotation on most of thesemetrics occurs for the same reasons found in similar con-texts. In particular, the choice to use focal planes withtwo detector pairs, oriented 45 ◦ apart (Couchot et al.1999), and the work done by BICEP on choosing deckangles to use at the South Pole. However, it is interest-ing that h (which couples to e.g. differential pointingsystematics) is not improved, and is in fact slightly wors-ened by this choice of boresight rotation. This is not astrong argument against this choice of boresight rota-tion, as the worsening is not substantial. Nonetheless,this shows that a single fixed boresight rotation anglecannot simultaneously improve every aspect of crossing-angle coverage that is relevant for systematics. It alsoshows that it is important to consider h k for all rele-vant k values when evaluating possible boresight rotationchoices. CONCLUSION
We have used simple first-principles simulations to ex-plore the possible crossing-angle coverage that can beachieved by surveys using sky rotation and Constant El-evation Scans, using the h k metrics from Wallis et al.(2016); McCallum et al. (2020) to evaluate the crossingangle coverage. This work is the first step towards quan-tifying in detail whether CESs and existing map-makersare sufficient to achieve the ambitious goals of upcomingsurveys. The key results and conclusions are1. For a given experiment, the crossing-angle coverage Thomas, McCallum, Browndepends on the Declination, and the range of cross-ing angles is poorest for fields with Dec ∼ Latitude.This could be used to inform field selection.2. Given upper and lower bounds on the observing el-evation, the 2ERS strategy (elucidated in section3.2) gives the best possible values of | h k | achiev-able in each pixel (with some caveats noted in ap-pendix A). This can be used as a target for detailedscheduling to aim for.3. We have illustrated this by showing how | h k | varies as a function of Declination for variouschoices of elevation bounds. The best case sce-nario in each pixel depends strongly on Declina-tion; if one is aiming to maximise the cross-linkingin the worst areas of the map, it is always bestto choose a larger difference between the elevationbounds. These bounds on crossing-angle coverageare an important consideration for map-making al-gorithms seeking to remove 1 /f noise, particularlyfor the minimum level of cross-linking that they canwork with. These bounds also show that the cross-linking is unlikely to be homogenous over largerfields.4. For systematics, it is more important to look at theaverage value of | h k | over the visible field. Tables1-4 show how these values change with the eleva-tion bounds. Except for | h | , where the elevationbounds make little difference to the average valueover the field, we get the same result that the bestchoice is to increase the difference between the el-evation bounds.5. It is known that including boresight rotation canimprove the crossing-angle coverage for the purposeof mitigating some systematics. We use our 2ERSstrategy to quantify the improvement that can be made for several choices of a single boresight rota-tion angle, when combined with sky rotation. Wefind that 45 ◦ boresight rotation has a strong impacton h k for k = 2 −
4, although we note that suchboresight rotation actually worsens h . Boresightrotation of 90 ◦ or 180 ◦ does not improve h k for allof k = 2 −
4. This underlines the need to consider h k for all relevant values of k when evaluating scanstrategy choices. These results also show that skyrotation by itself is of limited utility compared toboresight rotation for controlling systematics.Given an observatory location and choice of elevationbounds, the framework shown here can be used to rapidlydetermine whether CESs can in principle deliver suf-ficient crossing angle-coverage over the whole field formap-making algorithms attempting to remove 1 /f noiseto work. In addition, the bounds in tables 1-4, and themaps in figure 2 can be used in combination with the ex-pressions in Wallis et al. (2016); McCallum et al. (2020)to predict the magnitude of the distortion due to scan-coupled systematics on both CMB power spectra andCMB maps. This can be done without detailed evalua-tion of the scheduling, allowing it to be used early in theforecasting and design process to get approximate valuesand best case scenarios, that can be used to inform fieldselection and the other design aspects of the surveys.We expect this approach to be useful for guiding de-tailed scan strategy considerations for the next genera-tion of ground based CMB surveys, leading up to andincluding the “Stage IV” programme. In addition, theseconsiderations may be relevant for other cosmologicalsurveys, such as intensity mapping.DBT acknowledges support from Science and Tech-nology Facilities Council (STFC) grants ST/P000649/1,ST/T000414/1 and ST/T000341/1. NM is supported byan STFC studentship. MLB acknowledges support fromSTFC grant ST/T007222/1. The codes used to generatethese results are available from the authors on request. APPENDIX
SINGLE ELEVATION SCANS
As mentioned in the main text, whilst the optimality of the 2ERS strategy is quite general (in that intermediateelevations are never useful unless they give access to extra observing time, and 2ERS has the smallest peak and bestaverage over the whole range of visible declinations), there are specific cases where 2ERS isn’t the optimal strategy. Thisoccurs when the Latitude and targeted Declinations conspire to make observations using a single elevation optimal;in this case the optimal choice is one of the two limiting elevations, and adding any other elevations will degradethe quality of the crossing-angle coverage. Consequently, if using two observing elevations, choosing a wider pair ofelevations will no longer be a better choice.Limiting scans to a single elevation has other effects, such as limiting the amount of time that the target field canbe scanned. However, if scanning with a single elevation (and therefore having a maximum of 2 crossing angles withineach pixel) is acceptable, and one is observing around a specifically chosen Declination with a sufficiently relativelyrestricted range (∆Dec < ◦ ), then the bounds from the 2ERS approach will not quite be strict bounds, althoughthey will still be close to the optimum values. Note that as shown in tables 1-4, restricting to a single elevation isnever better than 2ERS when averaged over the whole visible area.To compare this case to the 2ERS approach in the main text, we consider scans performed with only a singleelevation, and set this elevation to be one of the 2ERS elevations. Typically the average value of h k does not changesignificantly between 2ERS and the optimal single elevation scan for the targeted Declination range, but the peakvalue over the targeted Declinations may change. This could be relevant for map-making algorithms seeking to remove1 /f noise. We illustrate this in figure 5. These plots show that 2ERS has the lowest peak over the whole visiblefield, is never the worst strategy over any range, and gives a competitive bound when averaged over the whole visiblearea. However, specific Declination ranges can be chosen such that a single elevation strategy is optimal for the entireESs and crossing angles 9 Fig. 5.— h as a function of Declination, comparing 2ERS to single elevation scans for the upper and lower elevations. In each case theblack (solid) line is the 2ERS strategy, the blue (dashed) line is the single elevation strategy for the lower bound and the red (dotted) lineis the single elevation strategy for the upper bound. The left plot has elevation bounds (30 ,
65) and the right plot has elevation bounds(40 , targeted Declination range. For these specific Declination ranges, 2ERS is not a strict lower bound on the h k metrics(but it is close), and the respective single elevation scan provides the true bound.In more detail, a single elevation scan using the lower elevation from a 2ERS scan typically cuts through the 2ERSpeak at a slightly lower value (but is worse for all other Declinations). Conversely, a single elevation scan using theupper elevation is typically better for the range of Declinations either side of the peak. For example, considering theleft hand panel of figure 5, if one is targeting a Declination range of ∼ − → − ◦ , then 2ERS would not be a strictlower bound for the average value over the targeted field, and perhaps not the peak, although it does approximatethe correct value. If greater precision is required, the single elevation scan at the upper limit of 65 ◦ does provide thisbound, and this can be used for simple scheduling-independent investigations, as described in the main text. Notethat as soon as the targeted declination range goes above ∼ − ◦ , then the 2ERS will again give the correct boundon the worst pixels (peak | h | values) in the field.As a simple example, we can consider what these caveats mean for the fields shown in figure 1. The Wide survey isfar too large for these considerations to apply; so a single elevation strategy will be a poor choice. For the Deep fields,as these cover the Declinations either side of the peak, a single elevation strategy (taking the upper bound of possibleobserving elevations) could have slightly better bounds than a 2ERS strategy. REFERENCESAde, P. A. R., Ahmed, Z., Aikin, R. W., et al. Bicep2/keck arrayv: Measurements of b-mode polarization at degree angularscales and 150 ghz by the keck array.
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