The comptonization parameter from simulations of single-frequency, single-dish, dual-beam, cm-wave observations of galaxy clusters and mitigating CMB confusion using the Planck sky survey
aa r X i v : . [ a s t r o - ph . I M ] N ov The comptonization parameter from simulations of single-frequency, single-dish, dual-beam, cm-waveobservations of galaxy clusters and mitigating CMB confusion using the Planck sky survey
Bartosz Lew ∗ and Boudewijn F. Roukema Toru´n Centre for Astronomy, Faculty of Physics, Astronomy and Informatics,Grudziadzka 5, Nicolaus Copernicus University, ul. Gagarina 11, 87-100 Toru´n, Poland (Dated: Oct 10, 2016)Systematic effects in dual-beam, differential, radio observations of extended objects are discussed in thecontext of the One Centimeter Receiver Array (OCRA). We use simulated samples of Sunyaev–Zel’dovich (SZ)galaxy clusters at low ( z < . ) and intermediate ( . < z < . ) redshifts to study the implications ofoperating at a single frequency (30 GHz) on the accuracy of extracting SZ flux densities and of reconstructingcomptonization parameters with OCRA. We analyze dependences on cluster mass, redshift, observation strategy,and telescope pointing accuracy. Using Planck data to make primary cosmic microwave background (CMB)templates, we test the feasibility of mitigating CMB confusion effects in observations of SZ profiles at angularscales larger than the separation of the receiver beams.
Keywords: Sunyaev-Zeldovich effect – cosmological simulations – galaxy clusters – radio surveys – methods: observational
I. INTRODUCTION
Over the past several years many dedicated experimentshave been used to detect the Sunyaev–Zel’dovich (SZ) ef-fect (Sunyaev & Zeldovich 1970) from galaxy clusters at ra-dio wavelengths [e.g., Berkeley-Illinois-Maryland Associa-tion (BIMA) (Dawson et al. 2006); Combined Array for Re-search in Millimeter-wave Astronomy (CARMA) (Mantz et al.2014; Muchovej et al. 2012); the South Pole Telescope (SPT)(Reichardt et al. 2013); the Néel IRAM KIDs Array (NIKA)(Adam et al. 2014); the Atacama Pathfinder EXperimentSunyaev–Zel’dovich Instrument (APEX-SZ) (Bender et al.2016; Dobbs et al. 2006); the Arcminute Microkelvin Imager(AMI) (Rumsey et al. 2016; Zwart et al. 2008);
Planck
Sur-veyor (Planck Collaboration et al. 2011); the Atacama Cosmol-ogy Telescope (ACT) (Hasselfield et al. 2013); Array for Mi-crowave Background Anisotropy (AMiBA) (Lin et al. 2016)].Within the next few years, new observational facilities will be-come operational and will search for galaxy clusters, comple-menting the galaxy cluster census across the Universe [e.g.,New IRAM KID Array 2 (NIKA2) on the Institut de Radio As-tronomie Millimetrique 30 m telescope (Calvo et al. 2016)].The One Centimeter Receiver Array (OCRA) (Browne et al.2000; Peel et al. 2011) is one of the experiments capable of de-tecting the SZ effect at 30 GHz using beam-switching radiome-ters installed on a 32-meter radio telescope (Lancaster et al.2011, 2007). OCRA will be mostly sensitive to SZ clusterswith virial size > ′ and hence to clusters at redshifts in therange . < z < . and with masses M vir > × M ⊙ /h (Lew et al. 2015). However, a single frequency, beam-switchingsystem may suffer from confusion with the primordial cosmicmicrowave background (CMB) or suffer from systematic errorwhen observing extended sources.Confusion effects due to the CMB were investigated in detailby Melin et al. (2006) for AMI, SPT and Planck
Surveyor. Itwas found that for single frequency instruments, such as AMI ∗ [email protected] (a 15 GHz interferometer), the photometric accuracy that con-tributes to the accuracy of the reconstructed comptonization pa-rameter is strongly limited due to primary CMB confusion.In Lew et al. (2015) the impact of CMB flux density con-fusion at 30 GHz was investigated, in particular for theOCRA/RT32 (32 m Radio Telescope in Toru´n, Poland) exper-iment. It was found that the σ thermal SZ (tSZ) flux densityuncertainty due to CMB confusion should be of the order of for the range of clusters detectable with OCRA. However,in that work, the impact on the reconstructed comptonizationparameter in the presence of the CMB and radio sources wasnot calculated directly for the case of dual-beam differential ob-servations.The ≈ ′ separation of OCRA beams is very effective inCMB removal, but large correcting factors are required to com-pensate for the missing SZ signal (after accounting for pointsources) (Lancaster et al. 2007). Thus, there is a trade-off be-tween compromising photometry by the primary CMB signalversus losing flux due to the differential beam pattern. In be-tween these extremes, there should exist an optimal separationof differential beams that would need to be defined by criteriathat aim to maximize CMB removal and minimize SZ flux den-sity removal.In this paper, we reconsider the issue of systematic ef-fects on the reconstructed comptonization parameter from sin-gle frequency, beam-switched observations performed with acm wavelength radiometer. We consider a particular instrumen-tal setting for the OCRA/RT32 experiment and an extension tothe standard observation scheme that previously involved onlythe angular scales defined by the receiver feeds. The extensionadds additional beam pointings that map cluster peripheries, fur-ther from the central core than the initial pointings.The kinetic SZ (kSZ) may significantly modify the brightnessof the cluster peripheries that are integrated with the referencebeam. The significance of this effect depends on a combinationof the peculiar velocities of the intra-cluster medium (ICM) andinternal gas clumps, but at cm wavelengths, the kSZ only weaklymodifies the central brightness.With dual-beam observations, the reference beam back-ground coverage improves while integrating along arcs aroundthe cluster center as the field of view (FOV) rotates. However,due to the small angular size of the arcs, the chance of zeroingthe average background may be low, depending on the align-ment with the CMB pattern. We investigate the significance ofthis effect depending on observational strategy.For experiments limited by the size of the focal plane arraythe integration time required to generate a radio map and toprobe the outer regions of a galaxy cluster is significant andcan make the observation prohibitive. Therefore, previously,the method of reconstructing comptonization parameters fromOCRA observations of cluster central regions required inclu-sion of X-ray luminosity data in order to find the best fitting β -model for each cluster, and correction for the SZ power lostdue to the close beam separation. However, this approach re-lies on the cluster model assumptions and makes the radio SZmeasurements dependent on X-ray measurements of the cluster.Another possible approach is to observe SZ clusters out to largerangular distances but retain averaging over a range of parallacticangles. This is done at the cost of incurring extra noise due toweak tSZ in cluster peripheries and stronger systematic effectsdue to CMB.An OCRA-SZ observational program is presently underway.In support of this and similar efforts, we also investigate thepossibility of mitigating CMB confusion by using the available Planck data. Finally, we calculate the astrometric pointing andtracking accuracy requirements needed to attain a given accu-racy in flux density reconstruction.In Section II we review the current observing strategy and dis-cuss its possible extensions. In Section III we briefly outline ournumerical simulation setting. Section III B describes the con-struction of CMB templates from the currently available
Planck data. Section III C describes simulated samples of galaxy clus-ters used for the flux-density analyses. The main results are inSec. IV. Final remarks and conclusions are in Sections V andVI respectively.
II. OBSERVATIONAL STRATEGY
The common position-switching mode of observing(Lancaster et al. 2011) is that in which the reference beamnon-uniformly (due to varying FOV rotation speed) integratesthe background along arcs ≈ ′ from the source. In Fig. 1 ( top-right panel), circles denote OCRA beamwidths for a typicalobservational scheme (Birkinshaw & Lancaster 2005). First,the beam pair “A–B” measures the difference signal betweenthe cluster center and periphery, respectively. The beam pairis then “switched”, i.e. translated to configuration “C–A” witha swap of the roles of the primary and reference beams, sothat beams “C” and “A” now trace the cluster periphery andcenter, respectively. The position switching cycle is closed byreturning to the initial configuration “A–B” and the cycle isrepeated.As the Earth rotates, the reference beams sweep arcs aroundthe cluster center and probe different off-center background re-gions (beam B becomes B’ and C becomes C’). The beamposition-switching reduces fluctuations due to atmospheric tur-bulence on time scales of a few tens of seconds. At shorter time scales fluctuations due to receiver-gain instability and atmo-spheric absorption are reduced by switching and differentiatingsignals in receiver arms by means of electronically-controlledphase switches (Lancaster et al. 2011; Peel 2010). An extension to this pattern can be realized by adding an ex-tra beam pointing “D”. In this case an observation cycle wouldbe extended so that a beam pair would observe differences be-tween “A–B”, then “B–D” and followed by “A–B”, again prob-ing the cluster peripheries for varying parallactic angles. Thisscheme can be extended to both sides of the cluster and to largerdistances from the cluster center. In Fig. 1 ( top-left panel) theprojected beamwidths of the OCRA-f receiver focal plane areshown (black circles) overlaid on a nearby galaxy cluster seenthrough the SZ effect. The extra beam pointing “D” for thewhole array is shown with blue circles. In the following sectionswe investigate the implications of such an extended observationscheme using numerical simulations.
III. SIMULATIONSA. LSS and SZ effect
For the main results in this work we use the simulation ap-proach described in Lew et al. (2015), with a few modifications.In particular, for the same field of view ( ≈ . ◦ ) we use anincreased map resolution of ≈ . ′′ . In observational practise,often only the central comptonization parameter value is quoted,so we include the kSZ signal calculated as a contribution to themeasured Compton y -parameter for the appropriate frequency.In this analysis we neglect the large-scale foreground galacticsynchrotron, free–free and dust emissions. We assume these tobe smooth enough to be removed in differential observations.We assume that atmospheric effects and receiver noise aremitigated by sufficiently long integrations (see Sect. II andBirkinshaw & Lancaster (2005)). Systematic errors in flux den-sity estimation that we neglect include feed and elevation de-pendent beam response, feed and elevation dependent sidelobes,and elevation dependent antenna gain. We defer treatment ofthese effects to a separate analysis of end-to-end full OCRAfocal plane simulations. Although the simulated maps includesome of the effects of halo–halo LOS projections, the signifi-cance of the projection effects on the y -parameter photometryare not the main focus of the present study. These effects shouldbe small in this study, since we only consider the most massivesystems at the FOV generation stage.A cluster is not necessarily observed at the position that max-imises the SZ signal. For an X-ray selected galaxy cluster, SZobservations can be centered at the maximum of the X-ray sig-nal, but this may have a small offset with respect to the maxi-mum of the SZ signal, although the difference should be rather Beam switching is realized at the rate of 277 Hz which improves the /f knee of the resulting difference signal power spectrum roughly by an order ofmagnitude; typically down to frequencies . < f knee < . -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50∆λ [deg]-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50 ∆ b [ d e g ] A B DOCRA-f OCRA-p 0.00.40.81.21.62.02.4 × y -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50∆λ [deg]-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50 ∆ b [ d e g ] A BC B'C' D −0.16−0.12−0.08−0.040.000.040.080.120.16 ∆ T [ m K ] -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50∆λ [deg]-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50 ∆ b [ d e g ] −0.16−0.12−0.08−0.040.000.040.080.120.16 ∆ T [ m K ] -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50∆λ [deg]-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50 ∆ b [ d e g ] −0.16−0.12−0.08−0.040.000.040.080.120.16 ∆ T [ m K ] Figure 1: (
Top-left ) Projected map of the the line-of-sight (LOS) integrated comptonization parameter for a selected halo, and ( top-right ) simulatedCMB temperature fluctuations ∆ T including both primary CMB as well as tSZ and kSZ effects induced by the halo. ( Bottom-left ) 217 GHz
Planck resolution simulation – the Gaussian CMB ∆ T signal smoothed with a half-power beam width (HPBW) of ≈ ′ and including a realistic Planck noise realization (see Sec. III B). This map is used to make a high resolution CMB template by means of a smooth-particle interpolation. (
Bottom-right ) Residual map: primary CMB including tSZ minus interpolated
Planck simulation (Sec. III B). The variance of the residual map (without SZeffects) is about one order of magnitude smaller than that measured using maps that include CMB. In the top-right panel the circles denote OCRAbeamwidths traversing over the background CMB for a typical observational scheme (see Sec. II for details). In the top-left panel black circlesrepresent the full focal plane of OCRA receivers. small. This will effectively translate into pointing errors that wemodel in this work.The OCRA beam separation is s OCRA = 0 . ± . ◦ (Fig. 1), which we adopt in this work. Within the measurementaccuracy this does not depend on sky direction. B. CMB templates
The
Planck mission has provided full sky maps of theCMB temperature fluctuations at several frequencies, including217 GHz, where the tSZ signal is minimal. This opens up thepossibility to correct the differential SZ observations at other frequencies by removing a customized CMB template and thusreduce the CMB confusion in single frequency observations,provided that: (i) the small scale noise level of the template islow, (ii) the angular resolution of the data is sufficient and (iii)diffuse Galactic foregrounds can be neglected. In practise, re-quirement (iii) may exclude all or most of the sky regions cov-ered by
Planck frequency maps on either side of the GalacticPlane.In order to test the usefulness of
Planck data in minimiz-ing the CMB-induced variance in the measured flux densitieswith a single-frequency instrument, we generate CMB simu-lations for
Planck ’s 217 GHz frequency band using simulatedCMB power spectrum (Lewis et al. 2000) and assuming cos-mological parameters as in Lew et al. (2015). We assume aGaussian beam transfer function defined by a full-width halfmaximum FWHM = . ′ (Planck Collaboration et al. 2016b)and use realistic and publicly available 217 GHz Planck re-ceiver noise simulations at Healpix (Górski et al. 2005) reso-lution n s = 2048 .We simulate the primary CMB up to ℓ max = 3500 , i.e.the OCRA beam separation. In order to create the final CMBtemplate with ≈ . ′′ resolution in small FOVs, we use asmooth-particle interpolation of the projected field (as discussedin Lew et al. 2015).Real observations at 217 GHz will contain kSZ contributions,which we ignore for generation of simulated templates. For eachrealization of the simulated template, we store maps with CMBsignal, and maps with the CMB smoothed with the instrumen-tal beam and contributed by a Planck noise realization (Fig. 1).The former is used for simulating astrophysical signals (SZ) inFOVs, while the latter is used for removing a
Planck -compatibleversion of CMB contamination, for the same FOVs. Each FOVsimulation is made at fixed galactic latitude b = 40 ◦ , but at adifferent galactic longitudes to account for variations due to di-rection dependent properties of Planck noise.The single frequency maps from
Planck mission are contami-nated by foregrounds other than the cluster tSZ and kSZ signals,especially at low galactic latitudes. Since in the current workwe do not investigate galactic foregrounds we additionally ana-lyze another set of simulations based on the foreground-reducedmap generated with the needlets-based internal linear combina-tion (NILC) algorithm (Planck Collaboration et al. 2016a). Thealgorithm (Delabrouille et al. 2009) provides a very clean pri-mary CMB map outside of masked regions which mask GalacticPlane and bright point sources (together ≈ . of the full sky).We simulate the NILC map using the published beam transferfunction, and use the full mission NILC rendition of ringhalf-1and ringhalf-2 half difference maps to generate a Planck
NILC-compatible noise realization. We verified that the resultingNILC simulations are compatible with the
Planck
NILC map interms of their angular pseudo-power spectra. Slight differencesin the high- ℓ regime of up to a few percent are present due to ourchoice of cosmological parameter values that is consistent withthe WMAP9 results.The NILC map resolution simulations are similar to those ofthe 217 GHz map, so in Fig. 1 we only show the map for the217 GHz case. C. Simulated galaxy cluster samples
For the analyses presented in Sect. IV, we construct twogalaxy cluster samples. The first one, hereafter referred as “tar-geted” (Fig. 2 thick solid lines) is constructed by selecting theheaviest halos [ M vir > × M ⊙ /h (see Table I)], from each http://pla.esac.esa.int/pla/ The variance lost due to neglecting even higher multipoles is negligibly small( . . ) and well below the cosmic variance uncertainty. −5 −4 −3log Y(θ<0.75 ′ ) [arcmin ]0.00.20.40.60.81.0 n o r m e d c o un t s [10 M ⊙ /h ] 0.2 0.6 1.0 1.4redshift targetedblindSPT Figure 2: Distribution of the solid-angle–integrated comptonizationparameter Y integrated within the angular radius θ < . ′ from thecluster center ( left ); distribution of cluster total mass measured within avolume with mean mass density times higher than the critical den-sity of the Universe at the cluster’s redshift ( middle ); and distributionof redshifts ( right ) for the simulated cluster samples (solid lines) andfor the SPT cluster sample (Bleem et al. 2015) (dashed lines). For eachsimulated sample only halos with M vir > M vir , min and z > z min arechosen (see Table I). The vertical dashed line shows the division intohigh- z and low- z sub-samples that is used later in the analysis.Table I: Selection criteria used for constructing galaxy cluster samples.Parameter Sample/Value “targeted” “blind” z min M vir , min [10 M ⊙ /h ] . ◦ × . ◦ FOVhalo count a
475 361Sub-sampleslow- z ≤ .
426 214high- z > .
49 147 a The actual number of halos used in statistical analyses are slightly differentas they are further screened for halos that lie well within the projected FOV,which is required for simulating dual-beam observations at all possible paral-lactic angles and beam separations in a consistent way. independent simulation volume and using each recorded simula-tion snapshot. We impose a low redshift cut-off z > z min to re-move very extended clusters. The choice of redshifts for whichsimulation snapshots are taken is made such that the simulationvolume continuously fills comoving space out to the maximalredshift (see Fig.1 of Lew et al. (2015)). For each simulationvolume we apply random periodic coordinate shifts of the par-ticles within, and we apply random coordinate switches. This(i) improves redshift space coverage and (ii) yields cluster SZsurface brightness profiles in different projections, at the cost ofgenerating a partially correlated sample.The second sample, hereafter referred to as “blind” (Fig. 2thin solid lines), is generated using a blind survey approach (asin Lew et al. (2015)). We generate 37 FOV realizations each ≈
27 deg together covering a sky area of ≈ . Fromeach realization we select halos with virial masses M vir , c > × M ⊙ /h .The solid angle integrated comptonization for any given halo( Y = R y (ˆ n ) d Ω ) depends on a combination of halo redshift andmass. The “blind” sample is dominated by lighter halos thanthose found in the SPT sample (Fig. 2), although redshift spacedistributions of the two are similar. Hence, the bulk of the “blind” sample halos yields lower Y ( θ < . ′ ) values thanthose in the SPT sample (Fig. 2). Although increasing the low-est mass limit for the halos of the “blind” sample tends to makeits mass and redshift distributions more consistent with thoseof the SPT sample, it reduces the numbers of halos, thus in-creasing Poisson noise. For the statistical analysis in this work,larger simulations and more FOV realizations than are currentlyavailable would be required to reach consistency. Therefore, weuse this sample for tSZ analyses of simulated dual beam obser-vations, bearing in mind that in this limit of weak SZ effects,CMB confusion is expected to be the most significant. On theother hand, the “targeted” sample is expected to be less affectedby CMB confusion.In order to investigate the differences between compact andextended SZ clusters we further split our cluster samples byredshift at z = 0 . (Table I). This split roughly corresponds tohalf of the radial comoving distance to z = 1 . , beyond whichwe do not observe any heavy (Fig. 2) halos in our simulations.We find that the low- z and high- z samples mainly differ due tothe strength of the SZ effects, and due to the presence of sub-structures, being respectively stronger and more abundant in thelow- z subset. Examples for halos from low- z and high- z sam-ples are shown below in Fig. 7. IV. ANALYSIS AND RESULTSA. Systematic effects from beam separation
Dual beam difference observations capture only a fraction ofthe intrinsic flux density, depending on the physical extent of thesource, its redshift and the angular separation of the beams. Wedefine this fraction as F ( θ b , s, q max ) = (cid:28) S x0 ( θ b , s ) − S x r ( θ b , s, ˆn i ) S tSZ0 ( θ b ) (cid:29) i , (1)where S x0 is the measured central flux density per beam inducedby effect “x”, e.g. x = tSZ, S tSZ0 is the true central flux densityper beam due to tSZ (neglecting CMB, point sources, and othereffects), S xr is the flux density per beam due to effect “x” inthe reference beam direction ( ˆn i ), θ b is the instrumental half-power beam width (HPBW), and s is the angular separation ofthe beams. N r = 500 reference beam directions ( ˆn i ) are chosenrandomly from a uniform distribution of parallactic angles ( q ∈ [0 , q max ] ), where the upper limit q max is a free parameter.For each halo, we measure these fractions F by integratingspecific intensity directly from high resolution maps, and usingthe mean over the N r values of q . If there is no CMB contami-nation, i.e. setting x = tSZ, so that S x0 = S tSZ0 and S xr = S tSZr ,then F ≤ and − F represents the fraction of the signal lostonly due to the closeness of the beam angular separation.The impact of beam separation on the dual-beam observationsis shown in Figs. 3 and 4 for the “blind” and “targeted” sam-ples respectively, for an idealistic case of exact pointing—i.e.,no pointing inaccuracies are allowed ( ǫ p = 0 ). In these figures,the median F (from all halos matching the selection criteria) isplotted along with a 68% confidence region. Clearly, dual-beam observations at larger beam separations are less biased than ob-servations at smaller beam separations, and the 68% confidencerange generally shrinks as s increases.The significance of the primary CMB fluctuations for thedual-beam observations is estimated by setting x = tSZ + CMB,i.e., S x0 = S tSZ+CMB0 , S xr = S tSZ+CMBr . While the median F does not differ significantly from the pure tSZ case, the 68%confidence region significantly increases with beam separationsdue to primary CMB confusion. For example, since a primor-dial CMB fluctuation has a good chance of being of the samesign as the SZ signal at the cluster center but of the oppo-site sign in a distant reference beam, F can easily be greaterthan unity, as is clear in Fig. 3. As expected, the increase isstronger in the “blind” sample/high- z sub-sample than in the “targeted” sample/low- z sub-sample, due to differences in am-plitudes of SZ effects compared to the level of CMB fluctua-tions.Comparing Figs. 3 and 4 it is clear that the main differenceis the relative significance of the CMB as a source of confusionand the amount of residual biasing. However, for any individualhigh- z and/or low-mass cluster observation, the measured fluxdensity can be biased substantially. This can be inferred fromthe size of the σ tSZ+CMB confidence region. Even observa-tions of the most massive clusters, which are the least affectedby the presence of the CMB, can be biased substantially depend-ing on the angular scales being measured ( s ) (Fig. 4 left panels).In the figure, the trade-off between CMB confusion due to ob-servations at larger angular scales and the level of biasing ( F ) inthe limit of small s is clearly seen.For clusters that are small relative to the beam size, measure-ments far away from the cluster center are not really needed asthe F values approach unity relatively fast (e.g. “blind” / high- z sample in Fig. 3). At the OCRA beam separation (the verti-cal line in the figures) the primary CMB does not strongly con-tribute to the scatter in flux density measurements. This is evenmore so in the case of the “targeted” sample of heavy and low- z clusters. On the other hand, the most massive halos (Fig. 4) re-quire significant ( > ) flux density corrections even at largebeam separations (although these may partially be generated byprojection effects discussed in Sec. V).It is clear that in the two cluster samples, kSZ only slightlyincreases the scatter in F at the OCRA beam separation, as ex-pected at 30 GHz.The impact of Planck based CMB template removal isshown in green. The calculation is done by setting x =tSZ + CMB − template in Eq. 1, i.e., S x0 = S tSZ+CMB − template0 , S xr = S tSZ+CMB − templater . From Figs. 3 and 4 it is clear thatat the OCRA beam separation, and for the full range of paral-lactic angles, the Planck template does not significantly help, ordoes not help at all, in reducing the confusion due to primaryCMB. However, in observations that probe larger angular sepa-rations, the CMB template removal can substantially reduce the σ contours. The template removal may also be useful for ob-servations of high- z massive clusters for which mapping largerangular distances away from the central directions still appearsto be well motivated. Both in the high- z and low- z sub-samplesof the “targeted” sample the template reduces the tSZ+CMBscatter nearly down to the level limited by the intrinsic tSZ scat- F TSZTSZ+KSZTSZ+CMBTSZ+CMB-PLANCK template sample: blind / LOW-ZCMB template: 217GHz median TSZmedian TSZ+CMBmedian TSZ+CMB-PLANCK template sample: blind / HIGH-ZCMB template: 217GHz F sample: blind / LOW-ZCMB template: NILC sample: blind / HIGH-ZCMB template: NILC Figure 3: Simulated fractions ( F ) of SZ effect flux density at 30 GHz recovered from difference, dual beam observations of clusters from “blind” sample as a function of beam angular separation s and redshift range. The shaded/hatched regions map the 68% confidence regions(CR) in the distribution of F . The scatter in F calculated from maps containing only tSZ signal is shown in gray. The backslash-hatched regionshows the effects of primary CMB on biasing the tSZ flux density measurements. The forward-slash–hatched region shows the intrinsic scatterdue to kSZ when converted and embedded into the 30 GHz thermal SZ effect maps. The green region shows the improvements in decreasingthe intrinsic scatter in F as a result of subtracting the Planck CMB template from CMB+tSZ simulated maps prior to flux density calculations.The median F values are shown as lines. The 68% confidence regions about the medians become asymmetric as the beam separation increases(simulation sample error also becomes obvious in the TSZ+CMB case by comparing upper to lower plots; the TSZ, TSZ+KSZ, and TSZ+CMBcases are statistically equivalent between the upper and lower panels). The vertical dashed line marks the actual separation of OCRA beams fixedby the telescope optics. It is assumed that the reference beam covers an annulus around a galaxy cluster within parallactic angle range [0 ◦ , ◦ ] on either side of the central direction, and that pointing error ǫ p = 0 (see Sec. IV F). ter for the full range of s studied here (Fig. 4). B. Parallactic angle dependence
In practise, the OCRA observations exploit beam and positionswitching (Sect. II) but it is unrealistic to cover the full parallac-tic angle range: i.e. q ∈ [0 ◦ , q max ] where q max = 360 ◦ .It was already known that position switching(Birkinshaw & Lancaster 2005) significantly mitigates at-mospheric instabilities over the time scale of tens of seconds by(i) subtracting linear drifts caused by large-scale precipitablewater vapor (PWV) fluctuations (Lew & Uscka-Kowalkowska2016), (ii) accounting for beam response asymmetries, and(iii) maximizing the probability of avoiding (masking out)intervening radio sources that can significantly bias the SZmeasurement. In this section, we show that position switchingis also efficient in mitigating the confusion due to primordialCMB, even with a very modest coverage of parallactic angles.There should not be any statistical correlation between pri- mordial CMB fluctuations and the locations of heavy halos.Moreover, galaxy clusters have small angular sizes comparedthose representing most of the CMB power. Thus, clustersshould mostly lie on slopes rather than peaks or troughs in theCMB map. Hence, sampling SZ flux density differences at op-posite sides of a galaxy cluster core should help average out theprimordial CMB in comparison to one-sided observations. Weconfirm that this is indeed the case and find that this improve-ment is reached at even moderate values of q max .We calculate F ( s ) [Eq. (1)] for maps containing tSZ andCMB using mean flux density estimates either according to theposition switching observation scheme or without it. As before,each measurement is an average of dual-beam pointings atdifferent q but drawn randomly from within the range [0 ◦ , q max ] where q max ∈ { ◦ , ◦ , ◦ , . ◦ } .The result is shown in Fig. 5 for q max = 22 . ◦ . By compar-ing the left panel of this figure with the top-left panel of Fig. 3 itis clear that even strongly incomplete coverage of the parallacticangles does not cause significant broadening of the 68% confi-dence level (CL) contours. However, when position switching F TSZTSZ+KSZTSZ+CMBTSZ+CMB-PLANCK template sample: targeted / LOW-ZCMB template: 217GHz median TSZmedian TSZ+CMBmedian TSZ+CMB-PLANCK template sample: targeted / HIGH-ZCMB template: 217GHz F sample: targeted / LOW-ZCMB template: NILC sample: targeted / HIGH-ZCMB template: NILC Figure 4: As in Fig. 3 but for the “targeted” sample. F TSZTSZ+CMB sample: blind / LOW-ZPosition switching: ON sample: blind / LOW-ZPosition switching: OFF F TSZTSZ+CMB sample: targeted / LOW-ZPosition switching: ON sample: targeted / LOW-ZPosition switching: OFF
Figure 5: As in Figs 3 and 4, but for observations where the referencebeam covers an annulus around a galaxy cluster within parallactic anglerange [0 ◦ , . ◦ ] on either side of the cluster direction (left), or onlyon one side of the cluster direction (right) for the “blind” (top) and “targeted” (bottom) samples. is not used (right panels in Fig. 5), the confusion due to primor- dial CMB is stronger. As before, the “targeted” sample of theheaviest halos is less affected by the presence of the CMB, butthe effect of not using position switching is still visible, even at s = s OCRA (vertical line in Fig. 5, bottom panels).
C. Systematic effects in redshift space
The F factor depends on a cluster’s angular size, which inturn depends on the cluster’s redshift. We model the dependenceof F ( θ b , s, p ) (Eq. 1) on redshift by defining: F m ( θ b , s, β, θ c ) = 1 − R b ( ˆn , θ b , s ) I SZ ( ˆn , β, θ c ) dΩ R b ( ˆn , θ b , s = 0) I SZ ( ˆn , β, θ c ) dΩ , (2)where b ( ˆn , θ b , s ) is a Gaussian beam profile with beam width θ b , offset by angular distance s from the cluster center direc-tion ˆn . We choose s = s OCRA to simulate the position ofthe OCRA reference beam when the primary beam points at thecluster center. I SZ ( ˆn , β, θ c ) is a normalized, LOS integrated β profile that represents the SZ effect surface brightness: I SZ ( ˆn , β, θ c ) ∝ (cid:18) θ θ c (cid:19) − β , (3)where θ c = 2 r c /d A ( z ) is the angular diameter of the observedgalaxy cluster defined in terms of its core size r c , θ is the an-gle from ˆn to ˆn , and d A ( z ) is the angular diameter distance. F m depends on the choice of cosmological parameters and on Table II: Parameters of halos selected from Fig. 6. The parameter y isthe maximal value of the LOS integrated comptonization parameter.ID z F M vir y × comment a [10 M ⊙ /h ] “targeted” sample (rect. “1” selection)1 0.552 0.39 4.6 1.38 P2 0.520 0.43 4.7 3.40 P3 0.885 0.49 4.4 0.71 P, D4 0.429 0.47 4.6 2.10 P, E5 0.366 0.44 5.1 1.75 P6 0.370 0.53 4.9 2.10 P7 0.349 0.50 5.1 3.10 P8 0.337 0.34 5.4 3.70 P9 0.388 0.50 4.4 2.2 P “targeted” sample ( M vir > . × M ⊙ /h )10 0.103 0.41 13.8 11.2 D11 0.109 0.56 14.6 25.6 R, P12 0.193 0.49 12.8 5.6 E, D, S, P “blind” sample (rect. “1” selection)13 0.367 0.44 2.1 0.93 P “targeted” sample (rect. “2” selection)14 0.182 0.64 11.3 6.68 D15 0.268 0.77 12.6 9.50 D16 0.169 0.75 10.1 10.3 R, S a P - reference beam flux density contamination from another halo due to LOSprojection; D - disturbed morphology; E - elongated shape; R - regular morphol-ogy (virialized halo); S - sub-halo(s) present; the chosen cluster density profile. We calculate F m for Λ CDMcosmological parameters: h = 0 . , Ω m = 0 . , Ω Λ = 0 . , andfor an Einstein–de Sitter cosmological model. For our redshiftrange we find that the dependence on cosmological parametersis weak compared to the dependence on the halo density profile(Fig. 6). We also calculate F m for the case of a Gaussian halobut find that such profile is strongly disfavored by simulationsas F m approaches unity at fairly low redshifts.The simplest β -model does not allow for the steepening ofdensity profiles with increasing θ . However, X-ray observationssuggest that such steepening is real (e.g. Vikhlinin et al. 2006),and it is expected that at large distances from cluster cores (orhigher redshifts) the β -model yields lower F m ( z ) values thanthose predicted by simulations, as seen in Fig. 6.Clearly, the “targeted” sample has a large scatter in F val-ues at high redshifts. Some of that scatter is due to projectioneffects, which we discuss latter. Heavy clusters of the “tar-geted” sample appear more compatible with the β -model atlower β values than the lower-mass clusters of the “blind” sam-ple. At the OCRA beam separation, the low-mass clusters inboth samples show very weak effects of biasing ( F ≈ ) at thehighest redshifts. On the other hand heavy halos require largecorrections, some of which do not result from simple projectioneffects. In the next section, a selection of halos are investigatedindividually. D. Analysis of individual clusters
In Fig. 6 some of the halos are selected by rectangles in the z − F diagram. The properties of some of these halos are givenin Table. II. Fig. 6 shows that only the lightest halos in our sam-ples are found to be strong outliers, which is unsurprising.We visually inspected all the clusters listed in Table II andverified that each of the clusters from rectangle “1” (halo IDsfrom 1 to 9, and 13) lie at sky positions that are partially withinanother cluster’s atmosphere and also within the angular dis-tance of the reference beam. An example of such overlap isshown in Fig. 7 (top panels).Inspection of the three highest mass clusters in the “tar-geted” sample (halo IDs 10, 11, and 12; black dots in right panelof Fig. 6) show that two of them (IDs 11 and 12) are also af-fected to some degree by a LOS projection, but the morphologyof halo 10 shows no signs of another halo in the composite high-resolution map. Instead, the SZ signature has a disturbed mor-phology with angular extents larger than a single OCRA beamseparation even though all three are at redshift z > . . Thisresults in small F values, and motivates measurements at largerangular separations.In order to test whether high redshift clusters that significantlycontribute to the scatter in the F – z plane (Fig. 6) could also ben-efit from observations out to angular distances beyond s OCRA ,we investigate the three most massive clusters from rectangle“2” (Fig. 6, IDs: 14,15 and 16). Their corresponding F values(Tab. II) do not seem to result from projection effects. Instead,these clusters have extended atmospheres and/or strongly dis-turbed and asymmetric SZ profiles (e.g. cluster 14, Fig. 8).Some of the heavy clusters have surface brightness profiles(Fig. 8) that are strongly inconsistent with an axially-symmetric β -profile. This necessitates using more sophisticated two-dimensional profiles at the data analysis stage (Lancaster et al.2011; Mirakhor & Birkinshaw 2016). Clearly, heavy halos gen-erate low F values and require large flux density correctionswith an OCRA type standard observational strategy (Sec. II).These low F values may partially stem from spurious projectioneffects (e.g. halos 11 and 12) which arise at the FOV generationstage for halos from the “targeted” sample (see Sec. V).The outlying halos (rectangle “1”) are either mergers (closepairs of SZ-strong halos), or have elongated of disturbed mor-phology (e.g. halos 3 and 4), or have small scale sub-structures.However, in many cases these properties occur at spatial scalesthat will not be resolved in OCRA SZ observations and/or maybe relevant only as galaxy scale SZ effects that are too faint tobe detected. E. Practical aspects of using CMB templates
By subtracting the templated version of the CMB map(Sec. III B) from the pure CMB simulation, it is easy to esti-mate the upper limit of the residual CMB signal captured in theOCRA difference beam observations. The
Planck
217 GHz andNILC maps have enough pixels to create a template of resolu-tion of the order of an arcminute, and at least the former shouldcontain only a negligible tSZ signal. F ( z ) r C =0.1 Mpc β =1.0r C =0.1 Mpc β =0.85 r C =0.2 Mpc β =1.0r C =0.2 Mpc β =0.85 M v i r [ M ⊙ / h ] F ( z ) r C =0.1 Mpc β =1.0r C =0.1 Mpc β =2/3 r C =0.2 Mpc β =1.0r C =0.2 Mpc β =2/3 M v i r [ M ⊙ / h ] Figure 6: Simulated fractions ( F ) as a function of redshift and virial mass for clusters from “blind” (left) and “targeted” (right) samples andfor observations at effective beams angular separation s = 0 . ◦ (the separation of OCRA beams). The fractions were measured from mapscontaining tSZ signal only. The lines trace the dependence for a halo described by a β -model according to Eq. 2 with parameters given in theplot legend. The rectangles “1” and “2” mark strong outliers and some halos from the main group that are inspected individually (see text fordiscussion). Although subtracting CMB templates from the pure CMBmaps decreases the large-scale variance by an order of magni-tude, the residual variance in the map is carried by high fre-quency noise that will generate a small amount of dispersion indifference observations (Fig. 1 bottom-right panel). However,the residual small-scale noise should approximately average outunder rotation of the beams in the sky, and since the large-scalepower is effectively removed, increasing the effective separa-tion should not suffer from exponential variance growth due toprimordial CMB at arcminute angular scales.How reliable are the 217 GHz or NILC
Planck templates incorrecting single frequency SZ observations for confusion withthe primordial CMB? The 217 GHz map is foreground contam-inated and the NILC map, although foreground cleaned, stillmay contain residual tSZ signals at scales least optimized in theneedlet space.In order to quantify the foregrounds and residual tSZ con-taminaiton in each map, we calculate histograms of the tem-perature fluctuation distribution outside of a mask that re-moves the full sky except for the directions towards
Planck -detected galaxy clusters from the PCSS SZ union R.2.08 catalog(Planck Collaboration et al. 2016d). Each non-masked region isa circular patch of radius a = { . ′ , ′ , ′ , ′ } . Foregroundswill generate strong positive skewness in the temperature dis-tribution, while the presence of residual tSZ in the NILC mapshould manifest itself by either a positive or negative skew de-pending on the frequency weights in the internal linear combi-nation.While the results of the test for the Planck
NILC map (Fig. 9)do not give strong deviations from Gaussian simulations, the217 GHz map generally does. The data are inconsistent withGaussian simulations even at high galactic latitudes (Fig. 10),although the significance of the foregrounds seems to dependon the size of the circular patch. This implies that the 217 GHz frequency map cannot readily be used to mitigate the confusiondue to CMB in OCRA observations without further assumptionson the foregrounds’ frequency dependence. However, it shouldbe interesting to quantify the significance of the arcminute scaleGalactic foregrounds at 30 GHz at high and intermediate lati-tudes for OCRA difference observations with small beam sep-arations.
Planck -LFI data might also help reduce these fore-grounds, though we do not study this here.Since the foreground cleaned NILC map is statistically con-sistent with Gaussian simulations (Planck Collaboration et al.2016c,e) towards the
Planck -detected galaxy clusters (Fig. 9), itshould also be suitable for mitigating CMB confusion in OCRAobservations in directions outside of the mask where clustersundetected by
Planck lie.
F. Pointing requirements
In order to quantify the implications of telescope pointing er-rors on the reconstruction of Compton y -parameters, and to de-fine pointing requirements, we introduce a pointing precisionparameter ǫ p that defines the maximal angular distance that aprimary beam can have from the intended position, and then werepeat the analysis of Sec. IV A. The pointing error p is drawnfrom a uniform distribution on [0 , ǫ p ] , since the RT32 pointingand tracking are dominated by systematic errors, and we inves-tigate different values of ǫ p . Since galaxy cluster SZ profilesare typically steep functions of angular separation, any point-ing inaccuracy will lead to biasing measurements of the centralcomptonization parameter when taking averages from multipleobservational sequences.Figure 11 shows that for the “targeted” sample, i.e. typicallyheavy clusters, pointing error up to ǫ p ≈ HPBW / should notlead to strong ( > ) extra biases relative to the ǫ p = 0 case.0 -2.8 -1.4 -0.0 1.4 2.8x [Mpc]-2.8-1.40.01.42.8 y [ M p c ] (4) 0.0 5.0∆θ x [arcmin]0.05.0 ∆ θ y [ a r c m i n ] (4)-5.5 -3.7 -1.8 -0.0 1.8 3.6 5.4x [Mpc]-5.5-3.7-1.8-0.01.83.65.4 y [ M p c ] (10) 0.0 5.0∆θ x [arcmin]0.05.0 ∆ θ y [ a r c m i n ] (10)-3.6 -1.8 -0.0 1.8 3.6x [Mpc]-3.6-1.8-0.01.83.6 y [ M p c ] (11) 0.0 5.0∆θ x [arcmin]0.05.0 ∆ θ y [ a r c m i n ] (11)-5.1 -3.4 -1.7 -0.0 1.7 3.3 5.0x [Mpc]-5.1-3.4-1.7-0.01.73.35.0 y [ M p c ] (12) 0.0 5.0∆θ x [arcmin]0.05.0 ∆ θ y [ a r c m i n ] (12) Figure 7: Selection of simulated Compton y -parameter profiles (in ar-bitrary units) for halos from Table. II. The panels show profiles forindividual halos in physical coordinate space ( left ), and their coarse-grained version obtained from high resolution maps in angular spacewith contributions from other halos along the LOS ( right ). The posi-tion of halos in the left-hand side panels is defined by a box size thatcontains all FOF particles of the halo associated with a given cluster. Inthe right-hand side panels the SZ peak for the cluster is located in theplot center. For any given cluster the flux density calculation is done atthe sky position of the peak. For each cluster the black circles denoteOCRA FWHMs and their relative separation ( s OCRA ). Measurements of the “blind” sample, i.e. typically less massiveclusters, are more sensitive to pointing errors, but if the pointingaccuracy is better than ǫ p = 0 . ◦ ( θ OCRAb ≈ . ′ ) the ad-ditional systematic effects will be smaller than . However,larger pointing errors should be taken into account at the dataanalysis stage. An observational campaign is currently underway to improve RT32 pointing accuracy. -3.4 -1.7 -0.0 1.7 3.4x [Mpc]-5.1-3.4-1.7-0.01.73.45.16.8 y [ M p c ] (14) 0.0 5.0∆θ x [arcmin]0.05.0 ∆ θ y [ a r c m i n ] (14)-3.2 -1.6 0.0 1.6 3.2 4.7x [Mpc]-3.2-1.6-0.01.63.1 y [ M p c ] (15) 0.0 5.0∆θ x [arcmin]0.05.0 ∆ θ y [ a r c m i n ] (15)-3.6 -1.8 -0.0 1.8 3.6x [Mpc]-3.6-1.80.01.83.6 y [ M p c ] (16) 0.0 5.0∆θ x [arcmin]0.05.0 ∆ θ y [ a r c m i n ] (16) Figure 8: As in Fig. 7 but for the selection of halos from rectangle “2”.See Table. II for details. −20020 r e s i d u a l c o un t s r=2.5' −40040 r=5' −0.4 −0.2 0.0 0.2∆T [mK]−150−5050150 r e s i d u a l c o un t s r=10' −0.4 −0.2 0.0 0.2∆T [mK]−300−100100300 r=15' Figure 9: Residual histogram (i.e. observational pixel frequencies mi-nus median pixel frequencies estimated from an ensemble of GaussianNILC map simulations) of the CMB temperature fluctuations at andaround the
Planck
SZ galaxy clusters, measured in the
Planck
NILCinside circular apertures of radius r centered at the clusters’ positions(solid); and σ , σ and σ confidence contours of the pixel frequenciesin these simulations (dashed). −55 r e s i d u a l c o un t s r=2.5' −20020 r=5' −0.4 −0.2 0.0 0.2∆T [mK]−5050150 r e s i d u a l c o un t s r=10' −0.4 −0.2 0.0 0.2∆T [mK]−2000200 r=15' Figure 10: As in Fig. 9 but for the
Planck
217 GHz frequency map andonly for clusters at galactic latitude b > ◦ . V. DISCUSSION
The map-making procedure that has been tested for recov-ering the source intensity distribution from OCRA differencemeasurements ( s = s OCRA ) assumes a flat background. Thisis not a problem for reconstructions of comptonization param-eters from SZ observations of heavy clusters, as with the stan-dard OCRA beam-pair separation the corrections due to CMBbackground are small. However, reconstructing cluster SZ pro-files out to larger angular distances could benefit from correctingthe difference measurements according to the
Planck
CMB tem-plate. For example, Fig. 4 (left panel) shows that an observationat s = 2 s OCRA decreases bias by ∆ F ≈ . . At the same timeCMB confusion broadens the 68% CR by ∆ F ≈ . , but ap-plying a Planck
CMB template reverses this effect almost downto the intrinsic tSZ+kSZ scatter.As discussed in Sec. III C the “blind” and “targeted” sam-ples represent quite opposite observational approaches. How-ever, since halos of the “targeted” sample were selected fromfull simulation volumes (rather than from light-cone sections),mock maps for this sample contain clusters with angular sizescalculated according to their redshifts and physical extents, asin the case of FOV simulations, but are placed in the map atrectilinearly projected locations. This contaminates the result-ing maps with halos that would not fall into the assumed FOVin the standard light-cone approach. These spurious halo–halooverlaps may somewhat enlarge the 68% CR contours of various F distributions (e.g. x=tSZ or x=tSZ+CMB). A possible mod-ification of the calculation scheme for the “targeted” samplewould be to consider each halo independently, thus completelyignoring the intrinsic projection effects that exist in the light-cone approach, or by extending the FOV to a hemisphere (whichwould probably require implementing adaptive resolution mapsto maintain the angular resolution of the present calculations).The cluster samples that we analyze were not screened to se-lect virialized clusters. Although we analyzed sub-samples se-lected using a virialization criterion (based on ratios of potential to kinetic energy of FOF halo particles) the results presentedhere are based on the full sample in order to retain a morpholog-ical variety of SZ galaxy cluster profiles (Figs. 7 and 8), and toexpose the complexity of SZ flux density reconstructions fromobservations that do not intend to create multi-pixel intensitymaps. VI. CONCLUSIONS
We quantify the significance of systematic effects arisingin dual-beam, differential observations of Sunyaev-Zel’dovich(SZ) effect in galaxy clusters. We primarily focus on effects rel-evant to the reconstruction of comptonization parameters fromsingle frequency flux-density observations performed with theOne Centimeter Receiver Array (OCRA) – a focal plane receiverwith arcminute scale beamwidths and arcminute scale beam sep-arations – installed on the 32 m radio telescope in Toru´n.Using numerical simulations of large scale structure forma-tion we generate mock cluster samples (i) from blind surveys insmall fields of view and (ii) from volume limited targeted ob-servations of the most massive clusters (Sec. III C). Using mockintensity maps of SZ effects we compare the true and recoveredSZ flux densities and quantify systematic effects caused by thesmall beam separation, by primary CMB confusion and by tele-scope pointing accuracy.We find that for massive clusters the primary CMB confusiondoes not significantly affect the recovered SZ effect flux den-sity with OCRA beam angular separation of ≈ ′ . However,these observations require large corrections due to the differ-ential observing strategy. On the other hand, measurements ofSZ-faint (or high redshift . < z < . ) clusters may havetheir SZ photometry erroneously estimated by or more dueto CMB confusion, which becomes stronger in observations thatmap larger angular distances from cluster centers.We investigate the possibility of mitigating the CMB confu-sion in SZ observations that map scales beyond ′ from clus-ter centers by using Planck
CMB 217 GHz and foreground re-duced NILC maps as primary CMB templates. Using simula-tions we find that these templates have sufficiently high angu-lar resolution/low noise to significantly mitigate CMB confu-sion in 30 GHz observations of high- z clusters, given that thetemplates do not contain residual foregrounds. Using a simpleone-point statistic, targeted towards directions of known clus-ters, we verify that at least the Planck
NILC map should also besufficiently free from foregrounds to serve as a primary CMBtemplate (Sec. IV E) that could improve OCRA-SZ or similarobservations extended to larger angular scales (Sec. II).Finally, we find that RT32 telescope pointing and tracking ac-curacy ǫ p < . ◦ should keep systematic errors in recoveredSZ flux densities (comptonization parameters) below ≈ (af-ter correcting for other systematic effects) even in observationsof SZ-weak clusters (Sec. IV F).2 F medianTSZ, 68% CL sample: blind / LOW-Zs=3.156' sample: blind / HIGH-Zs=3.156' p [HPBW]0.30.40.50.60.70.80.91.0 F sample: blind / LOW-Zs=6.312' p [HPBW] sample: blind / HIGH-Zs=6.312' F medianTSZ, 68% CL sample: targeted / LOW-Zs=3.156' sample: targeted / HIGH-Zs=3.156' p [HPBW]0.30.40.50.60.70.80.91.0 F sample: targeted / LOW-Zs=6.312' p [HPBW] sample: targeted / HIGH-Zs=6.312' Figure 11: Systematic effects in tSZ flux density reconstruction from dual-beam observations as a function of telescope pointing errors ( ǫ p ) andbeam separation s for the “blind” sample (left) and for the “targeted” sample (right). The reference beam is assumed to cover all possibleparallactic angles for any given galaxy cluster. Acknowledgments
Thank you to Mark Birkinshaw for discussion on OCRA ob-servational strategies, and to an anonymous referee for usefulcomments. This research has made use of a modified versionof the GPL-licensed ccSHT library. We also acknowledge use of the matplotlib plotting library (Hunter 2007). This workwas financially supported by the Polish National Science Cen-tre through grant DEC-2011/03/D/ST9/03373. A part of thisproject has made use of computations made under grant 197 ofthe Pozna´n Supercomputing and Networking Center (PSNC).
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