Improved large scales interstellar dust foreground model and CMB solar dipole measurement
AAstronomy & Astrophysics manuscript no. solar˙dipole © ESO 2021February 22, 2021
Improved large scales interstellar dust foreground model and CMBsolar dipole measurement
J.-M. Delouis ∗ , J.-L. Puget , , and L. Vibert Laboratoire d’Oc´eanographie Physique et Spatiale (LOPS), Univ. Brest, CNRS, Ifremer, IRD, Brest, France Institut d’Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, Bˆat. 121, 91405 Orsay cedex, France Ecole Normale Sup´erieure, Sorbonne Universit´e, Observatoire de Paris, Universit´e PSL, ´Ecole normale sup´erieure, CNRS, Paris,FranceFebruary 22, 2021
ABSTRACT
The Cosmic Microwave Background anisotropies are di ffi cult to measure at large angular scales. In this paper, we present a newanalysis of the Planck
High Frequency Instrument data that brings the cosmological part and its major foreground signal close to thedetector noise. The solar dipole signal, induced by the motion of the solar system with respect to the CMB, is a very e ffi cient tool tocalibrate a detector or a set of detectors with high accuracy. In this work, the solar dipole signal is used to extract corrections of thefrequency maps o ff sets reducing significantly uncertainties. The solar dipole parameters are refined together with the improvement ofthe high frequency foregrounds, and of the CMB large scales cosmological anisotropies. The stability of the solar dipole parametersis a powerful way to control the galactic foregrounds removal in the component separation process. It is used to build a model forSpectral Energy Distribution spatial variations of the interstellar dust emission. The knowledge of these variations will help futureCMB analyses in intensity, and also in polarization to measure faint signal related to the optical reionization depth and the tensor-to-scalar ratio of the primordial anisotropies. The results of this work are: improved solar dipole parameters, a new interstellar dustmodel, and a large scale cosmological anisotropies map. Key words.
Surveys - Cosmology: cosmic microwave background - Cosmology: di ff use radiation - Methods: data analysis - ISM:dust
1. Introduction
Measuring accurately the di ff use extragalactic emissions at lowfrequencies on the nearly full sky is di ffi cult, especially at largeangular scales. It indeed implies being able to separate the atmo-spheric and the telescope emissions from the astronomical onesand, among those, to separate the extragalactic emissions fromthe galactic di ff use background ones. The Cosmic MicrowaveBackground (CMB) is the truly di ff use background bringing keycosmological information. At the the peak emission of the CMBfrequencies, it imposes to go to space, build cryogenic telescopesand instruments, and find compromises between constraints ofthe size of telescopes giving a limited angular resolution, andthe necessity to cover the whole sky.In the last 25 years, there has been three generations of CMBspace missions COBE, WMAP and Planck , the next one notproviding results before a decade. The Planck High FrequencyInstrument (HFI) observed the sky (at 100, 143, 217, 353, 545,and 857 GHz) with 50 bolometers cooled at 100 mK. The presentand future observations from the ground use large detector ar-rays, but strongly a ff ected by the atmospheric emission above200 GHz. Future space missions will allow to map the CMB andthe galactic foregrounds at higher frequencies. Progress of ex-periments on the ground will certainly outperform Planck capa-bilities on the CMB itself up to 150 GHz but not on the highfrequency galactic foregrounds, dominating signal at higher fre-quency.The large scale polarized B modes is a major goal for cos-mology which requires very high sensitivity at CMB frequen-cies (through more detectors and better control on polarized sys-tematic e ff ects). It also requires better measurements of the lowfrequency foregrounds with large ground-based surveys, and of ∗ Corresponding author: J.-M. Delouis,[email protected] high frequency foregrounds hard to achieve from the ground. Itis thus a key objective to squeeze out of the HFI data the infor-mation on the large scale high frequencies foregrounds. This isthe goal of this work.Although HFI achieved the remarkable result of being cos-mic background photon noise limited at the CMB peak fre-quency at the intermediate and small angular scales (a few de-grees to 5 (cid:48) ), at very large angular scales ( (cid:96) < ff erential detector requires taking advantage of thesolar system kinetic dipole. The solar kinetic dipole, common toall detectors and frequencies, is the strongest measurable di ff er-ential signal at large scales. Improving its determination is thusa key issue throughout this paper. Importantly, the solar kineticdipole is extracted from various splits of the HFI data (frequen-cies, galactic masks, component separation models), which con-strains an high frequency galactic foreground model. This pro-vides a tool for intercalibration of di ff erent instruments and mis-sions as well as constraining a better model of the high frequencyforeground systematics at large scales. These improvements, notachieved in the Planck
Legacy release, are introduced in thiswork where we develop a new interstellar dust model (the dom-inant foregrounds for the CMB above 100 GHz), self-consistentwith a better CMB anisotropies and the associated better solardipole measurement.The paper is organized as follow: Sect. 2 describes the con-text and goals of the paper; Sect. 3 describes the algorithm toextract coherently the parameters and templates maps of the a r X i v : . [ a s t r o - ph . C O ] F e b .-M. Delouis: CMB solar dipole and dust foreground model foreground, the map zero levels, and presents the resulting dustmodel; Sect. 4 build a self consistent CMB anisotropies map andsolves for solar dipole parameters.
2. Context and definitions
The relative motion between the
Planck spacecraft and the CMBrest frame generates, a dipolar modulation of the CMB intensityfield. The spacecraft motion with respect to the cosmologicalbackground is decomposed in two components.The first component, referred to as the “orbital dipole”,is caused by the relative velocity of the spacecraft at the L2Lagrange point following the earth motion with respect to thesolar system rest frame. It is known with a high accuracy at anytime. This orbital dipole of order 30 km s − does not project onthe sky maps. It is used to calibrate the absolute photometric re-sponse of each detector with a high signal to detector noise of or-der 10 − at the 100 and 143 GHz CMB dominated channels, and2 × − at 353 GHz (see Planck Collaboration III 2020, hereafterHFI-Legacy).The second component, referred to as the “solar dipole”, isthe consequence of the relative velocity of the solar system withrespect to the cosmological rest frame. Although it is an order ofmagnitude larger than the orbital dipole, it cannot be determinedfrom extragalactic astronomy, and thus cannot help to calibratethe absolute response of the detectors. The dipole signal associ-ated with this second component projects onto the sky maps withthe same Spectral Energy Distribution (SED) as the CMB cos-mological anisotropies. As such, it is not a cosmological e ff ect,but is a systematic e ff ect in the maps extracted by componentseparation methods on the basis of their SED. It is, nevertheless,not separable from the intrinsic dipolar term of the CMB primor-dial anisotropies. Using a Λ CDM model, the CMB anisotropiesmeasurement leads to an expected dipole amplitude of the or-der of 30 µ K. This induces an uncertainty of the order of onepercent on the kinetic solar system velocity. This dipolar termis thus included in the definition of the solar dipole, imposing anull dipolar term for the total CMB anisotropies maps.The solar system velocity measurement is independent of theCMB monopole temperature, known with an accuracy of 3 × − (Fixsen 2009). This introduces additional uncertainty when ex-pressing the intensity amplitude of the solar dipole in δ T CMB anisotropies. Nevertheless, for convenience, all intensity con-tributions to the dipole amplitudes are expressed in CMB tem-perature. As such, the conversion between velocity and µ K CMB depends directly on the CMB temperature, obtained by fittinga Planck function in the COBE-FIRAS data, which has beenrefined using a combination of the solar dipole direction fromWMAP and the FIRAS full spectrum measurements (Fixsen2009). In this regard, using the solar dipole to estimate the CMBmonopole temperature makes to conversion from velocity to µ K CMB slightly dependent of the dipole. However, the di ff erenceinduced by a new dipole determination leads to a negligible cor-rection.The amplitude of the solar dipole signal is proportional tothe response of each detector to a CMB intensity signal, andthus, very useful to test CMB relative photometric calibrationof di ff erent CMB detectors. Its amplitude of order 3 000 µ K CMB is very large with respect to the sensitivity of the 143 GHz map(amplitude of the noise dipolar < − µ K). This 3 × ratiohas been used to measure the relative CMB photometric cali-bration on single detector maps. The consistency of the relativeCMB photometric calibration on single detector maps within a frequency has a noise limited accuracy of 10 − . The inter-calibration between HFI frequency maps is at the 4 × − levelfrom 100 to 353 GHz (HFI-Legacy). This work uses these sensi-tivities, and exploit the fact that all detectors measure the sameCMB solar dipole (called hereafter “the consistency argument”)to control the residuals from foregrounds on the largest angularscales.The CMB map is obtained from the frequency maps usingits properties: Planck function SED and Gaussian statistics. Theextraction of solar dipole amplitudes measured at various fre-quencies and on di ff erent sky fraction is the most powerful toolto detect the foreground residuals. This was used for example forthe HFI data showing residuals behaving with frequency like agalactic dust component as can be seen in Tables 7 and 8 of theHFI-Legacy are obviously due to the dust foreground removal,and showed that introducing large scale dipolar and quadrupolarterms to account for the large scale spectral energy distribution(SED) variation reduces very much these drifts for the highestfrequencies and sky fraction used to fit the solar dipole.Here, the solar dipole is extracted with an improved version SRoll2 (Delouis et al. 2019) of the
SRoll algorithm used toproduce the
Planck
Legacy HFI maps. This upgraded algorithm,named
SRoll2.2 , introduces improvements: – Use of 143, 217, 353, 545, and 857 GHz non-polarizedSpider Web bolometers (SWB) data, in addition to PolarizedSpider bolometers (PSB) data. This allows for reduced sys-tematic e ff ects for intensity only maps, while avoiding beinglimited by signal to noise ratios; – Removal of polarized parts of the signals due to the smallparasitic sensitivity to polarization of SWBs detectors; – Improved correction of the Analog-to-Digital Convertersnon-linearities, bringing the associated residuals below de-tector noise levels, even at the largest angular scales; – Computation of single detector maps removing polarizationpart in the signal; – Use of the full resolution for the bandpass leakage correctionwhere the signal is very strong, keeping the low resolutionof the foreground templates on the rest of the maps to avoidintroducing extra noise; – Removal the Cosmic Infrared Background (CIB) monopole(Lagache et al. 1999) and use CMB maps with zeromonopole; – Setting the initial zero level of the intensity maps to the zerolevel of the 21-cm emission of the H i galactic gas fore-ground. SRoll2.2 builds two types of single detectors intensitymaps where: i) the correction of all foreground signal band-pass mismatch between detectors is applied, ii) no correction ofthe foreground signal bandpass mismatch is applied. Those latestmaps, smoothed at a common 1 ◦ , are used in this work.
3. Foreground sky model and zero level offsets
The solar dipole amplitudes of Planck Collaboration Int. XLVI(2016) Table 2, are drifting away from the amplitude measured at100 and 143 GHz both for higher frequencies (217 to 545 GHz),and when increasing the sky fraction f sky used to fit the dipole.The increase of the amplitude errors with frequency follows atypical galactic dust SED. Such variations were expected fromprevious work on the COBE-DIRBE data (Lagache et al. 1998).HFI-Legacy shows that this is mostly cured by introducing, in Those maps are available at http: // sroll20.ias.u-psud.fr2.-M. Delouis: CMB solar dipole and dust foreground model the model, large scale variations of the dust SED. This result isobtained using the stability of the solar dipole direction, takingthe 100 GHz one as reference. The dipole direction (especiallythe longitude) drifts by 0.1 ◦ but the parameters for the 545 GHzstill exhibits large shift in amplitude and direction.These results indicate that the residuals from the dust emis-sion removal are a major concern when extracting the solardipole above 100 GHz in two ways: – first, indirectly by the dust residuals along the galactic ridge(central parts versus anticentre of the galactic disc). Thisforces the introduction of a galactic ridge mask M g where thethe CMB anisotropies cannot be removed accurately enough.This induces an unavoidable error when the dipolar term ofthe cosmological anisotropies is set to zero on the full sky(see Sect. 4.2); – second, the large scale galactic emission in the part of thesky where the solar dipole is fitted outside the galactic ridgemask a ff ects the residuals of the component separation pro-cess. We thus introduce, in our foreground dust emissionsky model, terms improving the modeling of the large scalesSED variations.The CMB anisotropies component of the maps, as well as thenoise and systematic e ff ects, are defined with a null monopoleterm. The HFI frequency maps do not have an absolute zerosetting calibration. Any error in the zero level changes the con-trast between the high and low galactic latitude foregrounds, thuschanges the foreground maps, especially where the intensity isvery low at high latitude. The large scale residuals of dust emis-sion depends strongly on the accuracy of these zero levels, andwe introduce consistency constraints from the solar dipole to re-fine the accuracy of the foreground model.To start with, the initial setting of the zero level of the Planck
Collaboration HFI frequency maps use the method in which theextrapolated zero intensity emission of the galactic foregroundsgas correspond to the extrapolated zero column density of gas(Planck Collaboration VIII 2014). The first step is performed byregressing the 857 GHz, fully dominated by the dust emissionintensity, to the neutral hydrogen interstellar gas column den-sity. Then the other frequencies are regressed to the 857 GHzemission brightness. This sets the initial zero levels of the fre-quency maps good enough accuracy as long as one does not re-quire highly accurate intensity on very large scales.To improve on these results, and especially on the dust tem-plate map zero level, a more accurate method is developed here . We decompose the intensity sky model as: sky model ∼ CMB tot + FreeFree + Synchrotron + CO + dust (1)In Eq. 1, there is only one extragalactic component, the CMBanisotropies. The CIB large scales ( (cid:96) < CMB tot is the total CMB anisotropies map contain-ing the solar dipole and the primary cosmological anisotropies.The
CMB tot intensity is described by a single map of δ T CMB atall frequencies.The three next terms of Eq. 1 are the low frequency fore-grounds: free-free, synchrotron emission, and CO emission The full schematics of the algorithm is available athttp: // sroll20.ias.u-psud.fr lines. In intensity, the free-free and synchrotron emissions arerespectively one, and two orders of magnitude smaller than theCMB anisotropies at 100 GHz. The free-free emission, which isonly one order magnitude lower than the dust at 100 GHz, ben-efits from a stable and accurately known SED. Furthermore, thesynchrotron emission is lower than other emissions by orders ofmagnitude at higher frequencies. Thus free-free and synchroronemissions are removed in an open loop, using Planck
Legacyresults (Planck Collaboration X 2016). In the HFI data, the COlines emission are derived using only the large di ff erences ofresponse between detectors within a frequency band containinga CO line and the Planck
Legacy CO template maps are used.Nevertheless, their amplitude are updated, consistently with thenew dust model.The last term of Eq. 1, the interstellar dust emission, is thedominant foreground at frequencies above 100 GHz. It is de-scribed by a new model: dust = f ( ν ) Dust + ∂ f ( ν ) ∂β ∆ Dust + ∂ f ( ν ) ∂β ∆ Dust , (2)where f ( ν ) = Planck function ( ν, T ) × (cid:16) νν (cid:17) β , ν beeing the refer-ence frequency of the Dust map.The temperature T and the index β are very correlated whenfitted in the range of frequencies used in this work, thus thetemperature is taken as constant, and the SED variations areonly described by the variations of the β parameter. Given theavailable number of degrees of freedom (5 frequencies), this hasbeen shown acceptable in the HFI frequency range from previ-ous work using a broader frequency range from the absolute in-tensity maps from COBE. The single temperature adopted 18 Khas been shown to give good SED fits for the H i dominant phaseof the interstellar medium but also for the subdominant H + , thatshow very similar SED (Lagache et al. 1998).The novelty of this dust model is the use of an analyticaldescription of the averaged SED, and the introduction of twoadditive correction modeling the e ff ects of the SED spatial vari-ations.The first term of Eq. 2, Dust , is the initial dust map (see fur-ther), propagated from the frequency ν to each single detectormap with a frequency ν . Furthermore, the frequency is taken atthe e ff ective frequency that takes into account the di ff erence ofthe response of the detector to a dust SED with the response tothe CMB orbital dipole signal used to calibrate the detector withhigh accuracy in K CMB .The two other terms of Eq. 2 describe the spatially variablepart of the SED, absorbing all di ff erences to the spatially con-stant SED first term. The analytical description of the SED f ( ν )allows a description of these variations as an expansion in termof its first and second derivatives with respect to β . The modelis not strictly described by the analytical expansion of the spec-tral energy distribution function f ( ν ) because the second spa-tial template ∆ Dust is independent from the first one, ∆ Dust ,which describes the first order expansion term. These two inde-pendent full sky maps with a null average have a resolution oforder 1 ◦ ( HEALPix (G´orski et al. 2005) N side = Dust is built from a map at high enoughfrequency ( ν =
545 GHz) that it only needs to be cleaned fromthe CMB, the CO, and the free-free emissions in an open loop.The CMB large scales cosmological anisotropies map, takenfrom the
Planck
Legacy release, once propagated to ν , are thenthree orders of magnitude smaller than the dust emission. The ef-fects of the di ff erence between the initial CMB used at this stage with the CMB map obtained ultimately in this work, is thus neg-ligible.From Eq. 2, we obtain the dust map model D ( ν b ) for a singledetector (bolometer b ) map: D ( ν b ) = f ( ν b ) f ( ν ) Dust ν + ln( ν b ν ) ∆ Dust + (cid:32) ln( ν b ν ) (cid:33) ∆ Dust . (3)This equation is valid for maps expressed in spectral energy den-sity. The 545 and 857 GHz maps are calibrated on planets andare dealt with in spectral energy density expressed in MJy sr − .Nevertheless, 100-353 GHz detectors, calibrated on the orbitaldipole of the CMB in δ T CMB , are expressed in µ K CMB , and mustbe converted to spectral energy density. The lower frequenciesCMB maps, o ff sets and solar dipole amplitudes are finally beconverted back to µ K CMB for the convenience of comparisonwith similar maps in the literature.We know that SED variations exist at intermediate to largescales due to the gradients of i) stellar radiation field inten-sity changing the dust temperature, ii) the relative fraction oneach line of sight of the molecular, atomic and ionized gas frac-tion which are known to have slightly di ff erent dust properties.The two SED variation components are thus defined by mapssmoothed at 1 ◦ . The
SRoll2.2 single detector intensity maps are built after re-moving the polarized signal and the bandpass mismatch coe ffi -cients have been computed using template dust maps di ff erentfrom the ones build in the present work. We thus use the set ofmaps in which the color correction has not been corrected to beable to extract band pass mismatch coe ffi cients coherent with thedust model using an iterative process.We name I b the single detector intensity map built from thebolometer b data. The e ff ective frequency ν b , computed from thebandpass ground measurements, is used in the first iteration ofthe processing, and is corrected at each iteration when regressingthe single detector map against the dust initial template Dust ν .The dust emission is the only broad band foreground com-ponent not fully described by its SED, but modeled empiricallyfor the frequencies ν ≥
100 GHz as a map with an average SED,completed with two additive correction maps describing the spa-tially variable SED. These three maps are extracted in this work,using other constraints than previous component separation onesas discussed above (e.g. consistency of the solar dipole param-eters). The CO and dust foregrounds being partially correlated,the CO foreground map is built using a CO template map fromthe
Planck
Legacy results (Planck Collaboration X 2016) butwhose coe ffi cients are extracted coherently with our new dustmodel.Finally, the data contains noise and monopole o ff sets, not di-rectly measurable by the HFI di ff erential detectors, and whichare to be solved for, using consistency arguments and redundan-cies.Equation 4 defines the residual R b which only contains the CMB tot and the foreground components to be modeled. It is com-puted for each single detector map I b from which are removed,in an open loop, the low frequency foreground maps taken fromthe Planck legacy maps as described above. R b = I b − FreeFree b − Synchrotron b − O HI . (4) The initial o ff set O HI applied to the single detector map setsthe zero level of the single detector intensity maps I b , followingPlanck Collaboration VIII (2014) that sets this zero level to theextrapolation to zero column density of HI 21-cm emission of in-terstellar gas at high latitudes. This is a first approximation of theabsolute zero level of the dust foregrounds. The addition of theionized component traced by H α emission changes the zero levelby 3% within the uncertainties, and the molecular component isnegligible at high galactic latitudes. The o ff sets are further re-fined in this work with smaller uncertainties, and the final resultsshould be consistent with the initial o ff sets. The o ff set values fordetectors, either within the same frequency band or with di ff er-ent frequencies, change the contrast of the large scales at highlatitude but are constrained by the solar dipole signal which is acommon component to all bolometers.For the 100 to 353 GHz detectors, the single detector mapo ff set O b is adjusted in three steps:1. The first step uses all pair di ff erences that gives o ff sets withrespect to a zero average o ff set of all these bolometers.2. The second step adjusts the average o ff set by using the factthat the solar dipole vector seen by all the detectors is thesame.3. The third step is the determination of the 545 GHz frequencymap o ff set, performed by minimizing its projected e ff ect bythe dust emission SED from 545 GHz to the lower frequen-cies o ff sets. The initial dust template is extracted from the545 GHz frequency map, which is corrected at the end ofthis step. This provides a new dust template Dust ν , used atthe next iteration.The residual R b also contains the not-recoverable noise andsystematic e ff ects residuals N b (including the very small residu-als following the low frequency foregrounds removal).In order to improve the solar dipole measurement, an im-proved model of the two main partially correlated high fre-quency foregrounds components is built extracting simultane-ously the CO lines emission intensity coe ffi cients γ b and the ef-fective frequencies ν b and templates maps ∆ Dust and ∆ Dust describing the new thermal emission dust model. The constraintused is, for each sky pixel seen by all pairs of bolometers b b
2, to minimize the di ff erence ∆ R b , b = R b − R b , expressed in µ K CMB . The CMB tot term cancels and: ∆ R b , b = D ( ν b ) −D ( ν b ) + ( γ b − γ b ) (cid:103) CO + O b − O b + N b − N b (5)where D ( ν b ) is the dust emission (Eq. 3). The initial dust map D ( ν ) is computed from the SRoll2.2
545 GHz map, after sub-tracting in an open loop CO, free-free and CMB. For bolome-ter b , the specific projection coe ffi cient f ( ν b ) / f ( ν ) from thetemplate dust map computed from 545 GHz bandpass map, ac-count for the detector response to the dust SED, through thee ff ective frequency ν b . These e ff ective frequencies, initially setto the values computed from the ground measurements of thebandpass (Planck Collaboration IX 2014), are adjusted simulta-neously with the γ b coe ffi cients.The CO Planck
Legacy maps are used as a template to ex-tract a single e ff ective set of coe ffi cients the CO lines response γ b at 100, 217, 353 GHz. The fact that there is no CO line in the143 GHz frequency band is used to regularize the solution, re-quiring that the detectors have a negligible response correlatedwith the (cid:103) CO template. Weaker interstellar molecular lines nev-ertheless exist in the 143 GHz frequency band, but this a ff ectsthe CO coe ffi cients by an error of 1.1%, inducing an error on the solar dipole amplitude of 0.06 µ K and 10 − degree in direction,thus negligible.The o ff sets O b are strongly degenerate with the dust largescale SED variation, thus are also solved for simultaneously us-ing the same minimization of the di ff erences ∆ R b , b used tocompute the dust large scale SED variation. The quantity to min-imize, based on the internal stability of the frequency maps, is,for each pixel p : χ h f = (cid:88) ( b , b (cid:88) p (cid:16) ∆ R b , b , p − D ( ν b ) + D ( ν b ) − ( γ b − γ b ) (cid:103) CO p − O b + O b (cid:17) . (6)Conditions on the mean parameter values between all single de-tector are needed to minimize Eq. 6 that is based on map di ff er-ences. This is possible thanks to several closure conditions. – For O b , ∆ Dust , and ∆ Dust , we impose the mean values tobe null. – For γ b , we force the mean values to be null at 143 GHz. – The level of residual dust emission in the mean map over alldetectors after removing the dust and the CO model, and aninitial CMB anisotropies map, is related to a miss estimationof the mean f ( ν b ). Setting this level to 0, provides a closurecondition for all ν b . At the first iteration, we use the CMB SMICA map.From this minimization, we obtain: – the correction to the response to a dust SED for bolome-ters calibrated on the CMB orbital dipole as an e ff ective fre-quency ν b , – the additive SED spatial variation described by the ∆ Dust and ∆ Dust maps, – the CO e ff ective coe ffi cients γ b , – the relative o ff sets O b of the single detector maps.Furthermore, a global 100-353 GHz o ff set ¯ O must be intro-duced to adjust the o ff sets of the 100-353 GHz maps with respectto the unknown 545 GHz map o ff set. Moreover, considering thelarge uncertainty on the 545 GHz map o ff set, a correction hasalso to be introduced. This will correct the large scales initialdust map template taken from the 545 GHz frequency map. Thisis performed by adding a constraint on the solar dipoles built foreach detector (Sect. 3.3). The dust SED spatial variations are strongly degenerate with theaverage zero levels of the frequency maps. Equation 6 mini-mization only constrains the relative adjustments of these o ff -sets, and do not a ff ect the initial band average zero levels of thefrequency maps which have rather large uncertainties with re-spect to the relative ones, especially for the 353 and 545 GHzfrequency band.A global (100 to 353 GHz) o ff set ¯ O is introduced in the algo-rithm. It is determined thanks to the fact that the spatial distor-tions of the dust foreground initial dust map uncertainties, biasesthe solar dipole vector directions. Thus, the minimization of thedispersion of the single detector CMB solar dipole vector overthe 43 (100 to 353 GHz) HFI bolometers allows to compute ¯ O .To remove the solar dipole, the cosmological CMBanisotropies map must be removed from the CMB tot map.Nevertheless this CMB anisotropies map, extracted by compo-nent separation methods, is not reliable in the narrow galactic ridge that needs to be masked and filled with a constrained real-ization within a galactic ridge mask M g before setting the dipoleterm to zero on the whole sky. The mask M g is defined as thesky contained between two symmetrical latitudes centered onthe galactic plane and characterized by the sky fraction f sky itcovers.The CMB anisotropies full sky map noted CMB M g is the CMB tot map from a component separation masked with M g inwhich the dipole term has been set to zero. It depends on the sizeof the galactic mask M g , and on the procedure used to fill thismask with CMB anisotropies constrained realizations. Beforehaving a CMB tot anisotropies map consistent with the new fore-ground model, we use, at the first iteration, the
Planck
LegacyCMB
SMICA map, estimated reliable for cosmology outside a5% galactic mask. At the second and following iterations, aCMB large scales anisotropies map, coherent with Sect. 3, resultcan now be built. We remove from all single intensity bolometermaps R b from 100 to 353 GHz, the current dust model and COtemplate, and the adjustments of the o ff sets obtained in Sect. 3.The weighted average over all bolometers provides the new cur-rent CMB tot .A new three-components (CMB, dust and CO) separation al-gorithm, named
Bcsep , is developed using di ff erence betweensingle detector maps in nested iterative loops, also using theconsistency of the solar dipole parameters between frequencies.This requires to start with good enough approximations and theconvergence of the algorithm is the test of this condition. Thisnew component separation is performed by subtracting the dustor the CMB maps from the frequency residual map R b and im-proving iteratively CMB tot and the three maps of the dust modelper frequency:
CMB tot = (cid:88) b σ b − (cid:88) b σ b (cid:16) R b −D ( ν b ) − γ b (cid:103) CO − O b − ¯ O (cid:17) . (7)where σ b is the variance of the cleaned maps for f sky = R b asthe residual map ℵ b , M g after removal of the high frequency fore-grounds model, the current CMB anisotropies CMB M g , and therelative o ff sets extracted previously. ℵ b , M g is thus the map con-taining only the CMB solar dipole, together with the noise andresiduals from the foregrounds removal: ℵ b , M g = R b − CMB M g − D b − γ b (cid:103) CO − O b (8) = A b , M g , ¯ O (cid:88) (cid:96) = , m = , ± A b , M g Y + ¯ O + N b , M g , where – A b , M g , ¯ O is the solar dipole amplitude extracted from CMB M g , – A b , M g , ¯ O are the three a (cid:96), m vector components ( a , , a , realand imaginary parts) of the solar dipole direction. – Y are the three spherical harmonics Y (cid:96) = , m component mapsof the solar dipole. – ¯ O is the global o ff set of the detector maps 100-353 GHz ini-tially set to zero.To constrain ¯ O , we impose that all detectors show a minimaldispersion of the solar dipole direction vector. Starting with ¯ O =
0, we loop on Eqs. 8, 9, and 10 and solve for the ¯ O value. Thesolar dipole parameters for bolometer b are fitted within a setof 10 masks M taken in the sky region left outside the galacticridge M g to smaller and smaller areas identified by their f sky . The minimization Eq. 9 extracts, for a given detector, a run-ning ¯ O and mask M g , the dipole amplitude A b , M g , ¯ O and directionvector A b , M p , ¯ O by minimizing the di ff erence taken over for amask M between the residual map ℵ b , p , M g map and the modelsolar dipole map plus the current ¯ O : χ b , M , M g = (cid:88) p M p ℵ b , p , M g − A b , M g , ¯ O (cid:88) (cid:96) = m = , ± A b , M p , ¯ O Y p − ¯ O . (9)Then ¯ O in carried back into Eq. 8 if we change M g or intoEq. 9 if we change only M after the optimization of M g , untilconvergence.This should not be a ff ected at the first iteration by the choiceof the CMB cosmological anisotropies SMICA map as it con-verges to a stable ¯ O value. χ b , M g = N b (cid:88) b = N M (cid:88) M = A b , M p , ¯ O − N b (cid:88) b = N M (cid:88) M = A b , M p , ¯ O N M N b . (10)At this stage, the 545 GHz map initial o ff set uncertainty stilla ff ects all lower frequencies zero levels indirectly through thedistortion of the initial dust template and the minimization Eq. 6to Eq. 10, and directly by the propagation of the 545 GHz o ff setresidual to lower frequencies through the SED f ( ν ).We define the 100 to 353 GHz o ff sets O ν as the average of( O b + ¯ O ) over the frequency band ν . The correction of the initialo ff set at 545 GHz, O , which also a ff ects all the O ν (followingthe f ( ν ) / f ( ν ) emission law), is the one which minimizes Eq. 11: χ = (cid:88) ν = σ HI (cid:32) O ν − (cid:32) f ( ν ) f ( ν ) (cid:33) O (cid:33) , (11)where σ HI is the rms of the initial zero level determination un-certainties at frequency ν . Thus this correction will a ff ect mainlythe 353 GHz.The next section (Sect. 3.4) discusses the coherence of theo ff set determination. The absolute zero levels of the maps are critical for the knowl-edge of the foregrounds that need to be modeled with the rightzero level. In Sects. 3.2 and 3.3, we develop a method to con-strain these o ff sets using the solar dipole strong signal thatshould be seen in all single detector maps with the same direc-tion vector.Four map o ff sets are determined in sequence:1. The initial setting of the zero levels in the SRoll2.2 maps O HI are computed following Planck Collaboration VIII(2014). Their uncertainties, although small in the CMB fre-quencies, a ff ect the dust emission large scales at high lati-tudes at 353 and 545 GHz.2. Equation 6 describes the computation of the relative o ff setsof the 100 to 353 GHz single bolometer maps, O b . The con-straint is the consistency between all pairs of single bolome-ter maps. To allow the minimization based on di ff erencesof maps, the mean of all these relative o ff sets per singlebolometer is set to 0. 3. Section 3.3 deals with the zero level mean correction overall 100 to 353 GHz bolometers, ¯ O . It is obtained by mini-mizing, over all single detectors and masks, the variance ofthe solar dipole directions using Eq. 9. The single bolometersolar dipoles are extracted using Eq. 10. Finally, we mesure¯ O = µ K. The relative corrections are bigger on the low-est frequencies.4. Finally, Equation 11 adjusts the 545 GHz o ff set, O , byminimizing the lower frequencies o ff set corrections a ff ectedby the projection of O following the dust SED f ( ν ). Thiso ff set is applied to the initial dust map Dust ν , and thewhole process is iterated over.The results after convergence are reported in Table 1. The data Table 1.
For the HFI frequency bands, O HI gives the initial mapo ff sets based on the zero HI column density extrapolation; O b gives the relative bolometer o ff sets averaged over the frequencyband; column + ¯ O gives the o ff sets after adding the global o ff -set ¯ O ; column Final gives the final o ff sets after removing the545 GHz o ff set. Last column gives the final uncertainty on thesezero levels. P lanck T his work Frequency O HI O b + ¯ O Final rms[GHz] [ µ K] [ µ K] [ µ K] [ µ K] [ µ K]100 . . . -3.5 ± ± ± ±
27 6.26 11.0 6.17 12.2[ MJy sr − ] [ MJy sr − ] [ MJy sr − ]545 . . . -0.884 ± ± zero level o ff set corrections, averaged per HFI frequency band,are given as cumulative o ff set values obtained after each correc-tion of the algorithm refining the initial o ff set O HI . The conver-gence of the iterative process shows the stability of the algorithmwhich leads to a minimal dispersion of the solar dipole vectordirection for the single detector CMB anisotropies maps in thefrequency bands 100-353 GHz.The final absolute zero level uncertainties are computed us-ing a set of ten simulations for the four lower frequencies. Thosesimulations have representative noise and systematic e ff ects butno attempt is done to quantify the uncertainties due to the dustmodel by simulations which have no reliable statistical model.These uncertainties at 100, 143, and 217 GHz are reflected inthe dispersion of the single bolometers at each frequency. Thus,these errors are probably dominant over the instrumental sys-tematic e ff ect dispersion. The uncertainty at 353 GHz is aboutone third of the dispersion between bolometers which indicatethat the zero level correction reflects the dust foreground ef-fects. The uncertainty at 545 GHz is computed using a MonteCarlo approach based on uncertainties at the lower frequencies.The O correction induces a marginally significant correctionat 353 GHz only. These corrections steps are weakly correlated,and the convergence confirms that the algorithm steps are e ffi -cient to improve the o ff sets.Figure 1 displays the map o ff set results, the initial ones from Planck in red, and the final ones as a function of the frequencybands. The o ff set corrections from this work, in blue, are consis-
100 143 217 353Frequency [GHz]30201001020 Z e r o l e v e l c o rr e c t i o n [ K C M B ] PlanckThis study 5450.010.000.01 Z e r o l e v e l c o rr e c t i o n [ M j y . s r ] Fig. 1.
Zero levels of the maps corrections, in red the initial one,and in blue the final cumulated o ff sets corrections taken as ref-erence, with their uncertainties.tent with the initial frequency map o ff sets but with much betteraccuracies by factors 2.5 to 5. Figure 2 presents these additive correction maps (as
Healpix maps at N side = ∆ Dust and ∆ Dust maps, respec-tively weighted by the frequency dependent ratios ∂ f ( ν ) /∂β and ∂ f ( ν ) /∂β . Those maps are compared with the HFI-Legacyones (left column), where each frequency SED spatial variationmap has been determined independently. The large scales aresimilar although obtained in di ff erent ways, and despite the factthat the initial dust maps were taken from di ff erent frequencies:857 GHz for HFI-Legacy and 545 GHz for this work.The SED spatial variations are described for by an expansionin ln( ν b /ν ) at the second order, but associated with two indepen-dent large scales maps providing more degrees of freedom. Thetwo bottom maps in Fig. 3 shows these additive corrections mapsfor the 143 GHz frequency band. The top map shows an estimateof an e ff ective β corresponding to the total correction of the twoterms.To be more quantitative, Fig. 4 compares the ratio of thepower spectra of the first and second order additive SED cor-rection maps at 143 GHz, with di ff erent f sky . There is a stronggalactic center anti-center dipole term ( (cid:96) = (cid:96) = (cid:96) >
15 shows the noise contribution. For f sky = < (cid:96) < K CMB K CMB K CMB K CMB
100 GHz -4 4
143 GHz -5 5
143 GHz -5 5
217 GHz -17 17
217 GHz -17 17
353 GHz -75 75
353 GHz -75 75
Fig. 2.
Right column shows the additive correction maps fromthis work. The equivalent correction maps from HFI-Legacy,which only contains dipole and quadrupole terms, are shown inthe left column. The 100 GHz map is not displayed as it was usedas the reference for the dipole direction.
4. CMB large scales results and solar dipoledetermination
The solar dipole is extracted from each single detector
CMB tot map from which we can subtract CMB anisotropies. The first it-eration uses the
Planck
Legacy CMB anisotropies
SMICA map,after masking the galactic ridge keeping 95% of the sky asrecommended in Planck Collaboration IV (2020). In later iter-ations, the Bcsep
CMB tot map from Equation 8 is used.
To illustrate the accuracy of the CMB cosmological anisotropiesmap built by the
Bcsep method, we use an end-to-end simu-lation which contains CMB, noise, systematic e ff ects residuals,and also a CO model and a dust model as per Eq. 2. Thus, thissimulation does not intend to evaluate the Bcsep ability to solvefor the dust model, but only for the noise and systematic e ff ectsresiduals.Figure 5 shows the di ff erence between the simulated inputCMB primary anisotropies map (without solar dipole) and theretrieved one. The narrow galactic ridge towards the inner galaxy K K
Effective f ( ) Dust at 143GHz -10 10 f ( ) Dust at 143GHz -1 1 Fig. 3.
The top map is the e ff ective dust parameter β variationmap. In the narrow galactic ridge, a single β correction per line ofsight is not meaningful considering the large variations of SED.The bottom maps show the two independent intensity correc-tions maps associated with the first and second β order deriva-tive. Multipole 10 D U S T () / D U S T () f sky = 0.18 f sky = 0.50 f sky = 0.82 f ( ) Dust f ( ) Dust Fig. 4.
For di ff erent high latitude masks, the continuous linesshow the power spectrum ratio of the ∆ Dust term to the dustemission. The dashed lines show the power spectrum ratio of the ∆ Dust term to the dust emission. These results are given for the143 GHz.is the only visible contributor to a significant dipole term, witha main vector direction toward b = l = ◦ . To avoid this bias,the solar dipole is extracted outside the galactic ridge mask M g .The error induced on the solar dipole amplitude is 0.103 µ K with f sky = µ K with f sky = ff ect on the CMB maps removal outside M g , within this range of f sky , is not a direct problem for the solardipole determination. The dust foreground residuals remains themain issue.Figure 6 displays the di ff erence maps between the CMB tot maps obtained by
Bcsep and the four
Planck
Collaboration methods
Commander , NILC , SEVEM , and
SMICA (Planck Collaboration IV 2020). The residuals are dominated bythe galactic foregrounds at low latitudes. At high galactic latitude( b > ◦ ), NILC , and
SMICA , which do not depend on a physicalmodel of the galactic components to extract the Gaussian CMB,are in very good agreement. The di ff erence map between SMICA K cmb K cmb Input CMB -300 300
Retrieved CMBDifference -3 3
Fig. 5.
Retrieved primordial CMB anisotropies
Bcsep map com-pared to the input CMB (top maps). The bottom map shows thedi ff erence between input and output with a color scale 100 timessmaller.and Bcsep at b > ◦ does not show significant residuals either.All di ff erences containing either the Commander or the
SEVEM maps show zodiacal dust emission residuals along the eclipticequator.We note that the
CMB tot
Bcsep map is only aimed at a sig-nificant improvement at (cid:96) <
30 and outside the bright galacticridge M g where it has smaller large scale dust residuals than the Planck legacy ones.
Bcsep does not intend to build a CMB mapaimed at the analysis of cosmological parameters mostly basedon CMB power spectra at (cid:96) >
30, and is not characterized forthat purpose.
Lacking a model to build statistically representative and con-trolled dust foreground maps, the foreground residuals cannotbe fully evaluated by simulations. A semi-quantitative approachis nevertheless possible by comparing data and simulations. Thebehavior of the solar dipole parameters with frequency and skyfraction are sensitive probes to test both the CMB anisotropiesremoval method and the galactic foreground residuals.Large scale foreground residuals in the part of the sky wherethe dipole is fitted, induce a source of systematic e ff ects on thedipole parameter retrieval. These e ff ects are introduced in thesimulation by adding an empirical estimate of the foregroundresiduals. This residual is defined as the di ff erence map from twocomponent separation method : SMICA - Bcsep shown in Fig. 6.We now use the
CMB tot
Bcsep map built in Sect. 4.1. Whenextracting the solar dipole from the full sky
CMB tot , the so-lar dipole parameters are biased by the galactic ridge residu-als seen in Fig. 6. This leads to the use of the galactic ridgemask M g where the CMB tot is a ff ected by large galactic residu-als. This mask M g is filled with a simulated CMB constrainedat the boundaries (Wandelt et al. 2004; Thommesen et al. 2020).By definition, the all sky CMB cosmological anisotropies maphas no dipolar term, and shall be built by removing the dipolarterm from the CMB tot map. The solar dipole is thus fitted in amask M taken as the complement of M g . On the one hand, themasked sky replaced by a constrained CMB, induces an uncer-tainty on the dipole parameters. This uncertainty is evaluated by Bcsep-CommanderCommander-NILC Commander-SEVEM Commander-SMICABcsep-NILCNILC-SEVEM NILC-SMICA Bcsep-SEVEMSEVEM-SMICA Bcsep-SMICA -10 +10 K CMB
Fig. 6. Di ff erence between the CMB maps built by Bcsep and the four
Planck
Collaboration component separation methods.the dispersion of the retreived solar dipole parameters from 1000Monte Carlo realizations with various M g . On the other hand, thebias, induced by the dust residuals in the central galactic ridge,decreases when increasing M g . The combination of these two ef-fects presents a minimum, which defines the optimal. The largescale dust residuals in the CMB tot map also a ff ect slightly thedipole measurement (Sect. 4.1). Any detectable dust residual ef-fect will then appear as a frequency and / or an f sky dependenceof the dipole parameters when changing M g .We want to compare these two e ff ects: the large scale SEDspatial variations of the dust emission, and the CMB tot fillingof M g . These behaviors, observed with the data, are simulated.Figure 7 shows the computed solar dipole parameters for the100-353 GHz frequencies, when varying the galactic ridge mask M g in which the CMB tot
Bcsep map is filled with constrainedCMB realizations. The dipole parameters are plotted as a func-tion of the sky fraction left outside M g used to fit the dipole.At first, the systematic e ff ects a ff ecting the retrieved dipoleparameters are tested both on simulations and data to ensure thatthe simulations are a good representation of the data. The twofirst rows if Fig. 7 use the processing algorithm without SEDvariation correction. The input solar dipole parameters for thesimulated data are taken from the final determination from thiswork, and given by the black lines with grey bands. In these twofirst rows, we see the remaining e ff ects of the dust SED varia-tions e ff ects (included in the simulations and not corrected for)well simulated when compared to the results obtained on thedata. The magnitude and sign of the residuals are qualitativelydemonstrating the quality of the simulation even though the dustbehavior is not quantitatively exact as expected, lacking a reli-able physical model of the dust SED variations. In the third andfourth rows of Fig. 7, the algorithm now includes the dust SED spatial variation correction. Both show a spectacular reduction ofthe previous trends. This demonstrates that the simulation takesinto account the SED variations with the right order of magni-tude, and that also the algorithm captures them well.The dipole term has to be removed from the CMB tot map,well known only outside a galactic ridge mask M g where thereare significant galactic dust residuals. This masking induces anerror due to leakages between the very large scale anisotropiesand the dipole term. Figure 8 shows the results on the recoveredparameters of the solar dipole as a function of the sky fractionleft outside of the varying M g where the dipole is fitted. Thedispersion of the solar dipole parameters is evaluated by sim-ulation with 1000 realizations of the CMB filling of M g . Theplain gray wedges show the dipole term dispersion when noth-ing is done to compensate this leakage. The dashed (resp. doted)gray wedges show the dispersion of the dipole term after filling M g with a smoothed 1 ◦ (respectively 6 ◦ ) CMB realization of thelarge scales only (up to (cid:96) =
5) constrained by the cosmologi-cal parameters HFI best-fit (Planck Collaboration V 2020), andcontinuity constraints at the boundaries. Once the dipole term isremoved from the
CMB tot map, as per the adopted definition, itis used to remove the CMB anisotropies from the full sky map.Then, the solar dipole is fitted in M . This removal and fillingprocedure introduces a non-recoverable uncertainty, increasingwith M g estimated using results shown in Fig. 8.The simulations with the qualitatively representative dustcomponent residuals defined above, show a bias as expectedin the amplitudes and in direction (longitude bias as expectedlarger than the latitude one) of the solar dipole parameters. Thebiases are maximal when the galactic ridge mask M g is null, anddecreases with decreasing f sky following the predicted behav-ior. For no masking of the galactic ridge, the direction is shifted f sky SIMULATION:Dust SED NOT corrected
Amplitude [ K CMB ] f sky Latitude [°] f sky Longitude [°] f sky DATA:Dust SED NOT corrected f sky f sky f sky SIMULATION:Dust SED corrected f sky f sky f sky DATA:DUST SED corrected f sky f sky Fig. 7.
Reconstructed solar dipole parameters along sky fraction used to remove the galactic ridge from the retrieved CMB.Frequencies are color coded from 100 to 353 GHz. Black lines reflect the input dipole parameters (taken from the final resultof this work). The grey band gives the uncertainties on these parameters.by about 2 (cid:48) of longitude. When f sky decreases to 0.95, the lon-gitude bias is reduced by a factor of 2, and crosses the longi-tude dispersion for 1 ◦ smoothing. For this sky fraction, the am-plitude parameter shows a bias of 0.25 µ K. The dispersion for6 ◦ smoothing crosses the bias for f sky = M g mask leav-ing f sky = ◦ smoothing. Figure 9 is a zoom-in of Fig.7 third row. The dipole parame-ters for all frequencies show the behavior described in Sect. 4.2with a bias increasing for increasing from f sky = ◦ smoothing), de-creasing rapidly with increasing f sky . The four frequencies showvery similar patterns. The longitude, the most sensitive parame-ter, shows a minimal error for the optimal M g size to f sky = f sky < .
95, within the range predicted by the gray wedge. It isat present impossible to draw high quality statistically represen-tative galactic foreground maps, and thus fully evaluate the fore-ground residuals. The uncertainties are given by the dispersionbetween detectors in each frequency band, and increase with fre- quency. Calibration errors resulting from noise and systematice ff ects, both evaluated by simulations, are visible as a variationin amplitude between frequencies. We want now to test the e ff ect of the dust residuals on the stabil-ity of the solar dipole parameters, when increasing frequencies,and varying M keeping the optimal M g constant ( f sky = M spanning from f sky = M nor with frequency. The most noticeable discrepancies are inamplitude and nearly all within 1 σ . The final results of this pa-per (average over the three lower frequencies) is shown as blacklines and grey band uncertainties. We thus conclude that there isno dust foreground residuals detectable above the noise.Table 2 shows the final results of the solar dipole parame-ters, computed as the weighted averaged per frequency over M with f sky = ff ects.HFI-Legacy uses 100 full end-to-end simulations of the overall abso-lute calibration process based on the recovery of the solar dipoleinput. These simulations retrieve the rms uncertainties of the cal- A m p li t u d e [ K C M B ] rms dipole rms constrained [FWHM=1°]rms constrained [FWHM=6°]residual [FWHM=1°]residual [FWHM=6°] f sky L a t i t u d e [ ° ] L o n g i t u d e [ ° ] Fig. 8.
The retrieved solar dipole parameters as a function of thesky fraction outside the galactic ridge mask M g out of which thesolar dipole is extracted. M g is filled with a contrained realiza-tion of CMB anisotropies. The blue and red lines show the biasfor two smoothing (1 ◦ and 6 ◦ ). The gray wedges show the disper-sion of the solar dipole parameters.ibration measured by the solar dipole amplitude rms: 1 . × − at frequency from 100 to 217 GHz increasing to 4 × − at353 GHz. Our results in amplitude are coherent with the HFI-Legacy calibration mismatch estimates. Thus, the weighted av-erage (AVG) of the 100, 143, and 217 GHz results provides thebest dipole parameters estimate. The AVG amplitude uncertainty(0.36 µ K), computed as the dispersion between the three fre-quencies, is significantly larger than the intra-frequency uncer-tainties, but reflects the calibration error due the residual system-atic e ff ects.For direction, the uncertainties are the statistical errors.The HFI-Legacy results are obtained by alignment of the 143-353 GHz frequencies solar dipole directions on the 100 GHz so-lar dipole direction, assuming it was the best reference, becauseof its best combination of high CMB sensitivity and low fore- Mask [percent]3360.53361.03361.53362.03362.53363.03363.5 A m p li t u d e [ K C M B ] DATA f sky L a t i t u d e [ ° ] L o n g i t u d e [ ° ] Galactic ridge residualCMB large multipole leakages
Fig. 9.
Zoom-in of the Fig. 7 for simulated data. The purple linesshow the galactic ridge emission residual after the
Bcsep com-ponent separation. The gray lines show the dispersion of the ef-fect of M g filling. Table 2.
Solar dipole parameters averaged per frequency and perfitting mask M . AVG is the average over CMB dominated fre-quencies 100, 143, and 217 GHz. The 353 GHz is also given; ithas larger uncertainties because of the much larger dust emis-sion. Frequency
A l b [GHz] [ µ K] [deg] [deg]100 . . 3361 . ± .
04 263 . ± .
003 48 . ± . . ± .
04 263 . ± .
002 48 . ± . . ± .
11 263 . ± .
002 48 . ± . . . ± ± ± AVG . 3361 . ± .
36 263 . ± .
003 48 . ± . A m p li t u d e [ K ]
100 GHz 143 GHz 217 GHz 353 GHz
BCSEP Commander NILC SEVEM SMICA f sky L a t i t u d e [ ° ] L o n g i t u d e [ ° ] Fig. 10.
Variation of the solar dipole parameters (in rows) for di ff erent frequency bands (in columns) as a function of the galacticmask in which the dipole is fitted. The black lines and grey band uncertainties show the final results of this work, using the Bcsep
CMB anisotropies map. For reference, the pink lines and bands show the HFI-Legacy measurements. The four
Planck
Legacy CMBanisotropies maps are also tested, and are shown color coded.grounds. In the present work, this assumption is replaced by theconstraint of minimum dispersion of the dipole directions for all100 to 353 GHz detectors, a more refined dust model of the SEDspatial variation, and coherent large scales CMB anisotropies.We also introduce a better masking-filling procedure for the re-moval of the galactic ridge residuals in the CMB cosmologicalanisotropies. Figure 10 compares the present results with theHFI-Legacy results, shown as the pink lines and bands. The re-sults for the amplitude and latitude are well compatible withintheir error bars. Nevertheless, the longitude is not, although verystable when varying f sky , and consistent with SMICA , and
NILC .An empirical test of the systematic e ff ects related to the com-ponent separation is to use di ff erent CMB maps. This is donealso in Fig. 10, and the four component separation method usedfor the CMB anisotropies are shown with thin lines. Although using CMB from Commander , NILC , SEVEM , and
SMICA is notfully consistent with the new and better foreground dust modeldeveloped in this work, the comparison remains interesting.When other component separations are used to get the CMBanisotropies, the stability with varying f sky ( M ) parameter is notas good as the one achieved by the Bcsep one. It strengthensthe statement that the CMB removal, when done with an opti-mal galactic ridge mask, has little e ff ect on the solar dipole pa-rameters fitted in a mask M covering a very broad range of skyfraction, thus showing no sign of large scales dust residuals.Thommesen et al. (2020) and Planck Collaboration et al.(2020) increasing the galactic mask up to 80% of the sky, isfar from optimal. The dispersion of the amplitude in these pa-pers grows from ± . µ K for M g in the range f sky ± µ K for f sky = duced when setting the dipole term to zero in a larger and largergalactic mask. Conversely, the method described in the presentwork, optimizes this mask and reduce by large factors the uncer-tainties making full use of the CMB anisotropies accurately mea-sured up to 95% of the sky. We also note that the 4 σ calibrationdiscrepancy between 100 and 143 GHz mentioned in Table 10if Planck Collaboration et al. (2020) is of the same order as theabsolute calibration reflected by the shift of the amplitude of thesolar dipole by 4 µ K in their paper showing a photometric cali-bration problem.Figure 11 shows three combinations of the solar dipole pa-rameters, results of this work, and of the HFI-Legacy results(in red). The longitude is the only discrepant parameter, being L a t i t u d e [] HFI2018No CMB fillingCMB filling K cmb ]48.2048.2548.3048.3548.40 L a t i t u d e [] A m p li t u d e [ K c m b ] Fig. 11.
Three combinations of the solar dipole parameters asa cross together with uncertainty ellipses. Red color shows theHFI-Legacy results. The blue color is the present work but with-out filling the galactic ridge mask M g , dotted when ignoring thesystematic e ff ect on the CMB anisotropies dipolar term, full linewhen taking into account the systematic e ff ect. The black coloris for the results of the present work with galactic ridge M g filledwith a constrained CMB. shifted by − . (cid:48) (2 . (cid:48) on the sky) well outside the error ellipses.There is a key di ff erence between the extraction method usedin this work and the one used in HFI-Legacy. The latter fit thedipole parameters outside of a galactic ridge mask for which thezero dipolar term of the CMB anisotropies has been removedwithout filling the mask with constrained CMB realizations noncontaminated by dust residuals. This induces a bias on the dipo-lar component in the part of the sky outside the galactic ridgemask M g . This bias has been simulated on the present final maps,by removing the step of the gap filling, and fitting the dipole pa-rameters outside the optimal mask M g . The dipole direction isdisplayed as the blue cross and dotted error ellipse in the toppanel of Fig. 11. This point falls near the HFI-Legacy red point,as expected as both of them have not the optimal gap filling.The bias on the sky is 1 . (cid:48) when the shift observed is 2 . (cid:48) . Thissimulation shows the bias induced when M g is not filled with aconstrained CMB, and we added this bias to the uncertainties ofthe legacy shown by the full blue line ellipse.Averaging the three lowest Planck-HFI CMB frequencies(Table 2) with uncertainties including the systematic e ff ectsleads thus to the final solar dipole parameters: A = (cid:2) . ± .
04 (stat.) ± .
36 (syst.) (cid:3) µ K; l = ◦ ± ◦
003 (stat.) ± ◦
017 (syst.); b = ◦ ± ◦
001 (stat.) ± ◦
007 (syst.) . (12)This result is consistent with HFI-Legacy for amplitude and lat-itude. The longitude of the dipole axis is shifted by − . (cid:48) on thesky. This is fully explained by the di ff erence of procedure, hav-ing introduced the filling of the galactic ridge mask with con-strained CMB anisotropies realizations before setting its dipolarterm to zero.
5. Conclusion
This work provides the best measurement of the Solar dipoleparameters as reminded in Table 3, showing an unprecedentedstability. This measurement, together with the large scale CMBcosmological anisotropies map can be used for intercalibrationof future CMB experiments, and testing the removal of fore-grounds as done in this work.The dust emission is the dominant large scale anisotropiescomponent above 100 GHz a ff ecting strongly the CMB at thelargest angular scales. Such non-Gaussian foreground criticallydepends on the zero level of the frequency maps which canonly be constrained for di ff erential experiments by the coher-ence of the dipoles. We have developed here a method whichdoes improve these zero levels which have much better accuracythan the extrapolation to zero column density of interstellar gas.Moreover, the component separation between the CMB and thedominant dust foreground is performed iteratively and allows tobuild intermediate and large scales maps of the dust SED varia-tions.The data products associated to this paper are thus: – improved zero levels of the HFI single detector maps, – refined CMB solar dipole vector parameters, – a new CMB anisotropies map at large scales, –
100 to 857 GHz dust intensity maps of our new dust modelincluding the SED spatial variation propertiesFigure 12 illustrates this dust model, which can be used tobetter understand the polarized dust emission, that should be af-fected by similar spatial variation.
Table 3.
Measurements of the solar dipole parameters for di ff erent Collaborations and data sets. Reference Collaboration data set Amplitude Longitude Latitude[ µ K] [deg] [deg]Fixsen et al. (1996) . . . . . . . . . . . . COBE FIRAS 3372 ± . ± .
30 48 . ± . ± . ± .
14 48 . ± . + LFI 3364 . ± . . ± .
03 48 . ± . . ± .
09 (stat.) 264 . ± .
003 (stat.) 48 . ± .
001 (stat.) ± .
45 (syst.) ± .
008 (syst.) ± .
004 (syst.) ± .
45 (cal.)This work . . . . . . . . . . . . . . . . . . Bware HFI 3361 . ± .
04 (stat.) 263 . ± .
002 (stat.) 48 . ± .
001 (stat.) ± .
36 (syst.) ± .
017 (syst.) ± .
007 (syst.)
K100GHz-2 2 K143GHz-4 4K217GHz-13.5 13.5 K353GHz-100 100MJy.sr Dust at 545GHz0 3MJy.sr Fig. 12.
Maps of the dust model. The top map shows the
Dust template extracted from the 545GHz map. The other maps showthe spatial variation of the dust SED.The
Planck
HFI data will stay, for at least a decade, the bestall sky data at the high frequencies. At frequencies higher than200 GHz, the sky signal is strongly a ff ected by the atmosphere.The present study products are thus unique and useful to com-bine with balloon borne and ground based CMB data waiting forthe next CMB space mission. Acknowledgements. This work is part of the Bware project supported by CNES.The authors acknowledge the heritage of the Planck-HFI consortium regardingdata, software, knowledge. The program was granted access to the HPC re-sources of CINES (http: // // References
Delouis, J. M., Pagano, L., Mottet, S., Puget, J. L., & Vibert, L., SRoll2: animproved mapmaking approach to reduce large-scale systematic e ff ects inthe Planck High Frequency Instrument legacy maps. 2019, A&A, 629, A38,arXiv:1901.11386Fixsen, D. J., The Temperature of the Cosmic Microwave Background. 2009,ApJ, 707, 916, arXiv:0911.1955Fixsen, D. J., Cheng, E. S., Gales, J. M., et al., The Cosmic MicrowaveBackground Spectrum from the Full COBE FIRAS Data Set. 1996, ApJ, 473,576, arXiv:astro-ph / / / / Planck
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Planck ff use component separation:Foreground maps. 2016, A&A, 594, A10, arXiv:1502.01588Planck Collaboration III, Planck
Planck ff use component separation.2020, A&A, 641, A4, arXiv:1807.06208Planck Collaboration V, Planck
Planck intermediate results. XLVI. Reductionof large-scale systematic e ff ects in HFI polarization maps and estimation ofthe reionization optical depth. 2016, A&A, 596, A107, arXiv:1605.02985Thommesen, H., Andersen, K. J., Aurlien, R., et al., A Monte Carlo compari-son between template-based and Wiener-filter CMB dipole estimators. 2020,A&A, A179Wandelt, B. D., Larson, D. L., & Lakshminarayanan, A., Global, exact cos-mic microwave background data analysis using Gibbs sampling. 2004,Phys. Rev. D, 70, 083511, arXiv:astro-ph / Appendix A: Consistency of the dust modelextended to 857 GHz
Figure 2 shows similarity between the SED correction mapsfrom the HFI-Legacy(based on the 857 GHz map) and the present dust model (based on the 545 GHz). This suggests thatthe dust model can be extrapolated to 857 GHz.The 857 GHz map is the sum of the dust emission, anunknown o ff set correction, noise and systematics. The CMBanisotropies and low frequency foregrounds are fully negligiblehere. Thus, the 857 GHz map is used as a dust template which ispropagated at 545 GHz using Eq. 3: D (cid:48) ( ν ) = f ( ν ) f ( ν ) × (A.1) (cid:34) Dust ν − ln (cid:32) ν ν (cid:33) ∆ Dust − ln (cid:32) ν ν (cid:33) ∆ Dust (cid:35) A 545 GHz dust model D (cid:48) ( ν ) is built from the 857 GHz mapby propagating the SED variation dust model. This model is re-moved from the 545 GHz single detector map, only leaving aCMB solar dipole map plus noise and systematics residuals. Toconstrain the 857 GHz map o ff set O , we compute, for eachsingle detector 545 GHz map, the solar dipole parameters fol-lowing the method described in Sect. 4. The 857 GHz o ff set isadjusted by minimizing the di ff erence between the 545 GHz so-lar dipole directions and the 100-217 GHz average (Table 2), andfound to be O = − .Figure A.1 shows the CMB dipole parameters extracted fromthe three single detector 545 GHz maps for both a null 857 GHzo ff set and for the solved one. The convergence and stability (am-plitude dispersion about 50 µ K) at such high frequencies demon-strate the quality of the dust model and the fact that the devia-tions do not come from the dust removal (like in the HFI-Legacy)but more from the photometric calibration errors associated withinstrument systematic e ff ects.Figure A.2 shows the consistency of the two dust maps at545 GHz (top row). The di ff erences are shown in the bottomrow for the full resolution, and at 5 ◦ smoothing. Even if thereare some di ff erences near the galactic ridge, both maps are veryconsistent at high latitudes, demonstrating that the dust modelstands at large scale and high latitude up to 857 GHz. A m p li t u d e [ K ] L o n g i t u d e [ ° ] f sky
46 47 48 49 50 L a t i t u d e [ ° ] O = 0 O = 0.124 Mjy . sr Fig. A.1.
Solar dipole parameters extracted from single detec-tor maps at 545 GHz to which a dust map from 857 GHz hasbeen removed. The parameters are shown for the three 545 GHzbolometers, and an o ff set correction null or optimized to mini-mize the di ff erence with solar dipole direction. K SED variation from model -1000 1000 K -1000 1000K
Residual -1000 1000 K
Residual [smoothed 5°] -200 200
Fig. A.2.
Top left map is the computed emission associated to theSED dust spatial variation applied to 545 GHz. Top right is the545 GHz map correction after subtracting linearly the 857 GHzmap. The bottom maps are the di ff erences between the two topmaps, smoothed at 1 and 5 ◦ . Maps are in µ K CMB to allow com-parisons with the dipole amplitude.to allow com-parisons with the dipole amplitude.