Confirming ALMA Calibration using Planck and ACT Observations
Gerrit S. Farren, Bruce Partridge, Rüdiger Kneissl, Simone Aiola, Rahul Datta, Megan Gralla, Yaqiong Li
DDraft version February 11, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Confirming the Calibration of ALMA Using Planck Observations
Gerrit S. Farren,
1, 2
Bruce Partridge, R¨udiger Kneissl,
3, 4
Simone Aiola,
5, 6
Rahul Datta, Megan Gralla, and Yaqiong Li Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridge CB3 0WA, UK Department of Physics and AstronomyHaverford College370 Lancaster AveHaverford, PA 19041, USA Atacama Large Millimetre/submillimetre ArrayALMA Santiago Central OfficesAlonso de Cordova 3107Vitacura, Casilla, 7630355, Santiago, Chile European Southern ObservatoryESO VitacuraAlonso de Cordova 3107Vitacura, Casilla, 19001, Santiago, Chile Joseph Henry Laboratories of PhysicsJadwin Hall, Princeton UniversityPrinceton, NJ 08544, USA Center for Computational AstrophysicsFlatiron Institute162 5th AvenueNew York, NY 10010, USA Department of Physics and AstronomyBloomberg Center for Physics and Astronomy, Johns Hopkins University3400 N. Charles StreetBaltimore, MD 21218, USA Department of Astronomy/Steward ObservatoryUniversity of Arizona933 N Cherry AveTucson, AZ 85721, USA Kavli Institute and Department of PhysicsCornell University245 East AveIthaca, NY 14850, USA
ABSTRACTWe test the accuracy of ALMA flux density calibration by comparing ALMA flux density measure-ments of extragalactic sources to measurements made by the Planck mission; Planck is absolutelycalibrated to sub-percent precision using the dipole signal induced by the satellite’s orbit around thesolar system barycenter. Planck observations ended before ALMA began systematic observations,however, and many of the sources are variable, so we employ measurements by the Atacama Cos-mology Telescope (ACT) to bridge the two epochs. We find the ALMA flux density scale (based onobservations of Uranus) is consistent with Planck; for instance ALMA flux densities in Band 3 ( ∼ Corresponding author: Gerrit S. [email protected] a r X i v : . [ a s t r o - ph . I M ] F e b Farren et al.
GHz) average 0 . ± .
02 times those measured by Planck. We also test the absolute calibration ofboth ACT and the South Pole Telescope (SPT), again with reference to Planck, as well as the internalconsistency of Planck, ACT and SPT measurements of compact sources.
Keywords:
ALMA, Planck, ACT, Compact Radio Sources, Calibration INTRODUCTIONAccurate and consistent calibration of flux densityscales is obviously important to radio and sub-millimeterastronomy, especially when observations made at differ-ent frequencies, or by different instruments, are to becompared. Ideally, such calibration would be absoluteas well as accurate. As shown by Partridge et al. (2016),refined and absolute calibration of ground-based radiotelescopes can be achieved by transferring the absolutecalibration of cosmic microwave background (CMB) ex-periments to these other instruments.The Planck satellite CMB mission (Planck Collabo-ration et al. 2016a), like the earlier WMAP (Wilkin-son Microwave Anisotropy Probe) mission, is absolutelycalibrated using measurements of the small (amplitude ∼ . ± . ± onfirming ALMA Calibration using Planck and ACT Observations CALIBRATION STANDARDSIn this section, we summarize the methods that havebeen used to calibrate the three instruments treated indepth in this paper. Briefly, as noted above, Planckis absolutely calibrated on the orbital dipole; ACT cali-bration is based on comparing its CMB maps and powerspectra to Planck CMB results; and ALMA is calibratedusing observations of planets, particularly Uranus.2.1.
Planck calibration
The calibration of the HFI instrument on Planckis extensively discussed in Planck Collaboration et al.(2020a) and earlier papers cited therein; see also PlanckCollaboration et al. (2016a). The calibration of the 100,143, 217 and 353 GHz instruments is ultimately deter-mined by the amplitude of the dipole signal induced inthe CMB by the satellite’s annual orbital motion aroundthe solar system barycenter. The amplitude of the in-duced dipole, to first order ∆ T = ( v/c ) T , depends onlyon the satellite’s velocity (known to better than 0.01%precision) and the temperature of the CMB, which wetake to be T = 2 . ± . S ∝ Ω);hence we will not add this uncertainty in quadrature toother observational errors. From Table 1 (column 3),however, we see that the Planck beam uncertainty liesbelow 0.15% for all HFI frequencies, and is ∼ which makes use of the instrument calibration used forthe 2015 Planck release. The calibration has been veryslightly updated for the 2018 release (Planck Collabora-tion et al. 2020a,b); the very small corrections are givenin Table 1 (column 6), and were applied to the PCCS2flux densities. PCCS2 flux densities are averaged overthe entire 2.5 years HFI operated. In contrast, the ear-lier PCCS1 catalog (Planck Collaboration et al. 2014a)provides flux densities averaged only over the period Au-gust 2009 to November 2010. As explained in sections4.2 and 5.1 below, we employ the earlier PCCS1 catalogwhen we compare Planck to ACT flux densities for thatlimited period. The PCCS1 flux densities were basedon a preliminary calibration, and hence need somewhatlarger 1-3% (and 7.7% at 353 GHz) corrections to matchthe 2015 calibration standard. The correction factorsare given in column 5 in Table 1; these include smalladjustments to the calculated beam solid angles madebetween 2013 and 2015.2.2. ACT calibration
The initial calibration of ACT data is based on obser-vations of Uranus (Choi et al. 2020). The final calibra- Note that we do not use any data from the lower quality “ex-tended” catalog, PCCS2-E (Planck Collaboration et al. 2016b).
Farren et al.
Table 1.
Observation frequencies, beam solid angles, calibration precision and calibrationshifts for Planck, ACT and SPT. The center frequency for the ACT measurements at ∼ % 2013 → → ± − ± < − ± < − ± ± ∼ ∼ ∼ ∼ ∼ ∼ ∼ tion, however, is ultimately tied to the CMB by directlycomparing ACT maps or angular power spectra of theCMB to those made by WMAP (Hajian et al. 2011)and later by Planck (Louis et al. 2014; Choi et al. 2020)over a range of angular scales. Louis et al. (2014) dis-cusses the process in detail, and also provides estimatesof the accuracy of ACT calibration. It also examinesPlanck-ACT calibration based on some preliminary ob-servations of compact sources at 148 and 218 GHz; wecompare those results with those determined here in Sec-tion 6. Since the release of Louis et al. (2014), the ACTflux density scales at 90 and 150 GHz have been slightlyaltered (Choi et al. 2020). The updated estimates of thecalibration precision for each band are given in Table1, as are the center frequencies appropriate for obser-vations of synchrotron sources. We include compara-ble quantities for the South Pole Telescope (Story et al.2013; Everett et al. 2020).2.3. ALMA calibration
ALMA’s flux density calibration employs Uranus asthe reference solar system object. Uranus was includedin a set of dedicated observation made in 2016 Novem-ber described below. Based on the CASA planetarymodel (Butler 2012) we obtain total flux densities forthis epoch of 8 .
26, 9 .
96 and 70 .
55 Jy for the primaryALMA frequencies employed here, 91 .
5, 103 . . COMPACT SOURCE FLUX DENSITIESIn this section, we describe briefly how flux densities ofcompact sources are determined from observations madeby the instruments treated here.3.1.
Source extraction and flux density measurements
Planck, ACT and SPT compact sources are identifiedin sky maps using a spatial filter to minimize the ef-fect of CMB and other large-scale fluctuations as wellas noise. In the case of Planck, for instance, a Mexi-can Hat Wavelet filter is used to reduce both large-scalestructure and small-scale noise (Planck Collaborationet al. 2016b). ACT employs a similar approach usinga matched filter (see Marsden et al. 2014; Gralla et al.2020); for SPT, see Mocanu et al. (2013); Everett et al.(2020). Flux densities for significant detections are thendetermined; this step requires detailed knowledge of theinstrument’s beam solid angle and effective center fre-quency. Given the broad bands employed by all threeinstruments, both the beam solid angle and the effective onfirming ALMA Calibration using Planck and ACT Observations > Color corrections
The instruments treated here have substantial band-widths. Derived flux densities can thus depend onthe source spectral indices α (we use the convention S ∝ ν α ).In the case of Planck HFI, for instance, the cata-logued flux densities implicitly assume that all sourceshave α = −
1, and that the effective band centers areexactly 100, 143, 217 and 353 GHz (Planck Collabora-tion et al. 2014a). Small color corrections are needed,since virtually all of the sources we consider have flat-ter(closer to zero) synchrotron spectra instead. Thisissue is treated further in section 3.3 below. At ALMA,bandpass correction is an integral part of the calibrationprocedure. For the other two ground-based instruments,the required, spectrum-dependent, “color corrections”are smaller and are made by an appropriate adjustmentof the effective center frequency in each band, as wellas small, additional corrections depending on the sourcespectrum (for ACT, see Datta et al. 2019; Choi et al.2020). The center frequencies for ACT and SPT listedin Table 1 are those appropriate for compact sourceswith synchrotron spectra.3.3.
Planck flux densities
All of the raw (uncorrected) Planck flux density mea-surements considered here come either from the PCCS1 (Planck Collaboration et al. 2014a), which includes mea-surements made from August 2009 to November 2010, orthe PCCS2 (Planck Collaboration et al. 2016b), whichincludes measurements of flux density averaged over theentire mission (August 2009 to January 2012 for theHFI instrument). In the case of the PCCS2, unless oth-erwise noted, we employ only sources that appear in themain, high reliability catalog (see Planck Collaborationet al. (2016b) for details on the criteria employed aswell as source extraction). Flux densities in these cata-logs are averages over all observations of a given source.Some sources are observed at a given frequency for afew days every six months, that is once every survey.Others, near the ecliptic poles, are observed much morefrequently, given Planck’s scan strategy. Since the twocatalogs cover different (though partially overlapping)periods, we can use a comparison between flux densi-ties for a given source in the two catalogs to flag somestrongly variable sources.As noted above, cataloged Planck HFI flux densitiesneed to be color-corrected unless the source in questionhas a ν − spectrum. The color-corrections are a smoothfunction of the source spectral index, and are generallysmaller than a few percent (Planck Collaboration et al.2014a). We use Planck data to determine a preliminaryspectral index at each frequency for each source, andthen use these spectral indices to color correct the rawflux densities. These initial spectral indices for Planck’s143 GHz data, for instance, are determined by compar-ing flux densities at the neighboring frequencies of 100and 217 GHz. The color-corrected flux densities are thenused to re-compute corrected and final spectral indicesfor each source at each frequency. These corrected spec-tral indices are used when we make extrapolations orinterpolations from Planck’s center frequencies of 70.4,100, 143, 217 and 353 GHz to the frequencies used inground-based observations. While we list both raw andcolor-corrected flux densities for Planck in Table 3, weplot only the latter, and use only the color-correctedflux densities to compare with results from other instru-ments. 3.4. ACT flux densities
The ACT maps treated here are calibrated in twosteps. An initial calibration in temperature units isbased on observations of Uranus, array by array and sea-son by season. Note that at ACT frequencies, Uranusis a Rayleigh-Jeans emitter. These preliminary cali-brations are then adjusted by a small amount so thatthe ACT power spectra match Planck’s CMB powerspectrum in the overlapping range of spatial frequency(see Aiola et al. 2020; Choi et al. 2020, and references
Farren et al. therein). In all cases (except for ∼
145 GHz measure-ments by one array in 2016; see Section 4.3), these ad-justments are at the level of a few percent, and are mea-sured to ∼
1% accuracy. Thus the calibration of ACTmaps is ultimately tied to the absolute calibration ofPlanck. A matched filter is then applied to these cali-brated maps, and the amplitude of each compact sourcein the filtered map is measured. These values are thenconverted to flux density in Jy, taking account of thebeam solid angle and the frequency response of each ar-ray in each season. As discussed in greater detail inGralla et al. (2014), multiple, band-dependent sourcesof uncertainty contribute to the overall uncertainty onthe calibration of the ACT point source flux densities.These include uncertainty in the beam solid angle, un-certainty introduced by the mapmaking procedure, anduncertainty in the initial temperature calibration ob-tained from comparison to Planck (Choi et al. 2020).Combining these yields the overall uncertainties cited inTable 1. 3.5.
ALMA flux densities
The flux densities measured by ALMA and consideredhere derive from three different sources. ALMA, as partof routine observatory calibration, has a “grid monitor-ing” program of approximately two week cadence for theflux density of 40 bright quasars, which are used in PIscience observations as secondary flux calibrators (Fo-malont et al. 2014; Remijan et al. 2019). When theseare present in the ACT search area, monitoring mea-surements of these sources can be compared to ACTmeasurements made at roughly the same epoch; we willrefer to such measurements as “grid” observations.In addition, some weaker quasars and other radiosources are monitored irregularly in the vicinity of sci-ence targets depending on the scheduling of individ-ual PI projects in order to check their suitability asphase calibrators (“cone-searches”), especially for highfrequency and long baseline observations. These aremore heterogeneous in both frequency and quality. Theyspan the interval from 2012 to 2017, and are thus sub-ject to source variability (Guzm´an et al. 2019). Forthese reasons, we treat them separately when we com-pare ALMA results to those from ACT (section 5.2).We use only late 2015 and early 2016 ALMA observa-tions of this kind when comparing to 2016 ACT data. Inaddition, we accepted only those sets of measurementsthat included at least two ALMA frequency bands, sothat we could establish meaningful spectral indices fromthese data. When used, we refer to these lower-qualitymeasurements as “secondary” observations. Finally, we rely mainly on measurements of a set ofbright, extragalactic sources made in a dedicated pro-gram of ALMA observations in Bands 3 and 7 as partof the calibrator flux density update campaign; thesewe call “dedicated” observations. Specifically, we ob-served selected compact sources that lay in the ACTfields in a special, cone-search-like, calibrator run on ∼
50 ALMA phase calibrators. These targeted obser-vations were carried out between 8 and 10 November,2016, during ACT’s 2016 observing season, and resultedin 37 sources available for comparison to ACT. Thesededicated observations, centered at 91.5, 103.5 and 343.5GHz, used mainly the 12-m Array in configuration C40-6, with the 64-input correlator, and included Uranus asthe reference calibration object.Since we had simultaneous observations at widely dif-ferent frequencies, we employed spectral indices derivedfrom these dedicated ALMA observations to interpo-late (or extrapolate) to ACT’s central frequencies of 93and ∼
145 GHz, and to Planck’s central frequencies of100, 143, 217 and 353 GHz. We tested the results byemploying spectral indices derived instead from ACT 93and ∼
145 GHz observations (see Section 5.1.5). Typicaluncertainties in these dedicated ALMA measurementswere 10-20 mJy, and up to ∼
100 mJy for the bright-est sources, i.e. very roughly 1/2 the Planck uncertain-ties. As expected, flux density uncertainties for the ”sec-ondary” observations were often larger. See section 5.1for further details.The absolute calibration of all three data sets is basedon Uranus, as noted in section 2.3 above. We notethat the brightness temperature of Uranus has been con-firmed by absolutely-calibrated measurements made byWMAP (Weiland et al. 2011). THE INTERNAL CONSISTENCY OF ACT ANDPLANCK FLUX DENSITY MEASUREMENTSThe main goal of this paper is to compare flux densi-ties measured by one instrument (or at one time) withflux densities measured by another instrument (or ata different time). We thus begin in section 4.1 with abrief description of how we made these flux/flux com-parisons, then turn to the issue of internal consistencyin the Planck and ACT data.The ACT measurements were made over several differ-ent observing seasons (a season at ACT typically spansApril through the following December or early January),and employed several different receiver arrays as the Specifically, for each source, we calculated spectral indices from91.5 to 343.5 GHz, and from 103.5 to 343.5 GHz, then took theinverse-variance weighted average of the two. onfirming ALMA Calibration using Planck and ACT Observations
General features of all flux density comparisons
As noted, the instruments considered here operate atmany different frequencies. We therefore need to inter-polate or extrapolate flux densities to obtain a propermatch. To do so, we use the best available spectralindex for each source, as described in detail for eachcomparison. In most cases, these are based on the multi-frequency Planck results (see section 3.2); in some cases,we employ ACT-based or ALMA-based values (section3.5). We also record for each comparison the frequencyat which the comparison is made. Except for ALMAdata in some bands, these extrapolations involve adjust-ments to the flux density of a few percent or less.The instruments also have different detection thresh-olds: Planck in particular is much less sensitive tocompact sources than the larger ground-based instru-ments. As noted in section 3.1, this raises the possi-bility of Eddington bias or other biases when we com-pare Planck results with ground-based results. As inPartridge et al. (2016), we mitigate the effect of bias byforcing the fit in flux vs flux plots to pass through (0,0). Unless otherwise specified all the figures and resultscited below are subject to this constraint. Thus, theslope of the resulting linear fit provides a direct com-parison of the flux density scales of the two data setsbeing compared; a slope of unity is expected for fluxdensity scales in perfect agreement.As noted in section 1 above, most ofthe Planck sources are AGN, and therefore likely tobe variable. Since flux densities can either increase ordecrease, variability should not on average introduce abias in our comparisons, but it will increase the scat-ter. The scatter in most of the comparisons shown in figures in this paper is dominated by source variability.We attempted to reduce the effect of variability-inducedscatter on flux/flux plots by first fitting a line to all thedata, then dropping outliers and re-computing the slopeof the fit. Specifically, we first fit all the data usingorthogonal distance regression (ODR). This model as-sumes that given any observational data point (ˆ x i , ˆ y i ),where both ˆ x i and ˆ y i are subject to error, there ex-ist true values ( x i , y i ) such that y i = f ( x i ; β ). f issome fitting function dependent on parameters β (Inour case the fitting function is simply f ( x ; m ) = mx ).The observed values are related to the “true” valuesas x i = ˆ x i + δ i and y i = ˆ y i + (cid:15) i where δ i and (cid:15) i areunknown.The best fit line is determined by minimizing the lossfunction L ( δ , (cid:15) ) = (cid:88) i δ i + (cid:15) i (1)over all δ i and (cid:15) i . Since we assume that y i = f ( x i ; β ) wecan express this in terms of our observations, the fittingfunction and the parameters β as L ( δ , β ) = (cid:88) i δ i + [ f (ˆ x i + δ i ; β ) − ˆ y i ] . (2)In order to account for unequal errors in the twodata sets and between different data points we inverse-variance weight this loss function L w ( δ , β ) = (cid:88) i δ i σ x i + 1 σ y i [ f (ˆ x i + δ i ; β ) − ˆ y i ] . (3)The fitting is performed using the ODR routines avail-able in SciPy (Virtanen et al. 2020) which are based onthe Fortran package ODRPACK (Boggs et al. 1989).To perform the iterative fitting we compute theweighted root-mean-square (rms) distance of all ob-served data points from their respective “true” values d rms = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i (cid:32) δ i σ x i + (cid:15) i σ y i (cid:33) . (4)Here N is the total number of data points fit in each it-eration. We then iteratively exclude all data points witha weighted distance d i = (cid:114) δ i σ xi + (cid:15) i σ yi of more than threetimes d rms until convergence is reached. In the figures,the sources dropped in this iterative process (“outliers”)are shown in grey. In most cases only a small numberof iterations (0-5) were required before the slope of thefit converged to a final value. In all cases, the numberof sources dropped by this iterative process, and thusnot included in the final fits, was a small fraction of thetotal. Farren et al.
We also excluded a small number of bright, extendedor known variable sources at the outset from all flux/fluxcomparisons and all plots shown here. Specifically, weexcluded 33 bright (and often extended) sources suchas Cen A, PKS 0521-36, the Orion Nebula, 3C279, Pic-tor A and 3C454.3 from all the comparisons made here.The number of sources excluded from each comparisonvaries, however, since not all of these sources are locatedin sky regions accessible to ACT, SPT or ALMA.4.2.
Internal Consistency of Planck data
In the case of Planck, we do not have year-by-year orsurvey-by-survey flux density measurements available.Instead, as noted in section 3.1, we have flux densitiesaveraged over the first ∼
15 months of Planck observa-tions, as well as flux densities averaged over the entire( ∼
30 month) lifetime of the HFI instrument (Aug. 2009to Jan. 2012). These are listed in the Planck Catalogof Compact Sources (PCCS1; see Planck Collaborationet al. 2011) and the Second Planck Catalog of Com-pact Sources (PCCS2; see Planck Collaboration et al.2016b), respectively. We do not expect flux densitiesfor a given source drawn from the two catalogs to agreeexactly, both because of instrument noise and becauseof possible variability in the sources. We can, however,ask whether, on average, flux densities of a large sampleof sources were the same in the two catalogs. In theprocess of making this consistency check, we can see ifthe Planck measurements themselves reveal sources thatvaried strongly during the HFI observations; these ap-pear as outliers in the plots of PCCS1 vs PCCS2 fluxdensities such as Fig. 1.We performed this test by matching sources in PCCS2with sources in the older PCCS1 at each frequency, us-ing a search radius of 6 (cid:48) . From the outset, 33 bright,extended sources were excluded. The number of remain-ing matches ranged from ∼
700 to ∼ S(PCCS2)@143GHz [mJy] S ( P CC S , c o rr ec t e d ) @ GH z [ m J y ] best fitoutliers Figure 1.
Corrected flux densities of ∼ . ± . S ), even though the fits were made to lineardata. combined shift in the flux density scale at 143 GHz isthus +2.40%; see Table 1 for corresponding shifts forother Planck bands. These changes in calibration wereapplied to the older PCCS1 flux density values beforethey were compared to either the PCCS2 values (as inFig. 1) or to ACT and ALMA measurements. Therewere also much smaller shifts in calibration between the2015 and the final 2018 release of Planck data (col. 6 ofTable 1); we take the 2018 calibration as our standard,and have corrected both PCCS1 and PCCS2 values ac-cordingly.As Fig. 1 demonstrates, the slope of the fit for 143GHz is clearly close to unity, but there are a number ofoutliers (shown in grey in the figure). We assume theseare variable sources. The slope of the fit to all the datais 1 . ± . σ criterion of section 4.1,the slope settles to 0 . ± . . ± . . ± .
33 % below the newer PCCS2 ones.We repeated this comparison for the four other Planckbands employed here, 70.4, 100, 217 and 353 GH (Table2). Again, flux densities from PCCS1 were corrected onfirming ALMA Calibration using Planck and ACT Observations Table 2.
Flux densities for the two Planck cat-alogs are compared, after correction for knownchanges in calibration and beam solid angle. Wetake PCCS2 and the 2018 Planck calibration asthe standard, and hence have corrected the earlierPCCS1 data by the small factors given in Table 1before making the comparison.Frequency, GHz Ratio S (PCCS1) /S (PCCS2)70.4 1.0037 ± ± ± ± ± by the small changes in overall calibration (columns 5and 6 of Table 1) and the errors include uncertainty inthe beam solid angle. With the exception of 217 and 353GHz measurements, we see that the two Planck catalogsof compact source flux densities are closely compatible,once small, known changes in overall calibration between2013 and 2018 are accounted for. We have also measuredwith sub percent precision the small residual differences(e.g., 0 . ± .
33 % at 143 GHz). These small, additionalcorrection factors are included when we compare Planckflux densities from PCCS1 to those from ground-basedinstruments in section 5.4.2.1.
Frequency-to-frequency consistency of Planckmeasurements of compact sources
We also confirmed the band-to-band consistency ofPlanck flux density measurements of compact sources.As shown in Planck Collaboration et al. (2020a), thereis agreement at the 0.5% level or better between CMBanisotropy measurements made at 100, 143 and 217GHz. As noted in section 1, however, the CMB sig-nal is typically larger than the beam size at these fre-quencies unlike the case for compact sources; does thegood inter-frequency agreement extend to Planck mea-surements of compact sources? That is examined in de-tail in Appendix A, where we compare measurementsfrom one frequency band to predictions based on inter-polation from the two neighboring frequency bands. Totest the consistency of 143 GHz measurements, for in-stance, we use the corrected flux densities of sources at100 and 217 GHz.For 143 GHz, we find a 1 . ± . . ± . . ± .
3% higher than the predictedvalues. We suggest in Appendix A that this discrepancycould be caused by slight curvature in the spectra of thesources used in the test.4.3.
Internal Consistency of ACT data
As noted above, different receiver arrays were em-ployed as ACT evolved. In 2008-10, an early receiverarray called MBAC (Swetz et al. 2011) was used. It wasreplaced by new receiver arrays in 2012 (se Thorntonet al. 2016) The ACT maps for 2013 to 2016, array byarray and season by season, form part of data releaseDR4; see Aiola et al. (2020). When flux densities de-rived from the various season-by-season maps were av-eraged, we used inverse variance weights to produce thesource lists such as the “ACTall” list defined in 4.3.1.In contrast, the flux densities derived from the earlierMBAC runs (Marsden et al. 2014; Gralla et al. 2020)are archived values drawn from LAMBDA .Given thsee changes in the detectors employed at ACT(and some changes in the analysis pipeline as well), weinvestigated whether ACT flux density measurementsof compact sources, once properly calibrated, were in-ternally consistent across arrays, sky area and time.For instance, observations during the 2013, 2014 and2015 seasons were made with two different, polarization-sensitive, receiver arrays (called PA1 and PA2), withvery slightly different operating center frequencies near144.5 GHz. A third array, PA3, including 93 GHz detec-tors, was added in 2015, and the earliest ACT measure-ments (in 2008-2010) were made with an entirely differ-ent instrument, MBAC (see Swetz et al. 2011) with aneffective center frequency of 147.6 GHz for synchrotronsources. We arbitrarily selected PA2 as our standardfor comparison of flux densities of compact sources fromdifferent arrays and used 3 (cid:48) as a matching radius. Thesechecks of internal consistency, array by array, are de-tailed in Appendix B. With one exception, measure-ments of flux densities in each ACT band agree to 1-2%.The exception is PA3 operating at 144.1 GHz in sea-son 2016, which recorded flux densities on average 7%higher than PA2 for the same season and band. Hencewe dropped PA3 season 2016 data at 144.1 GHz fromfurther consideration.4.3.1. Consistency of ACT data from season to season
Next, we turn to consistency of ACT measurementsmade in different seasons. It is vital to the use of ACTas a bridge to connect Planck and ALMA measurements lambda.gsfc.nasa.gov/product/act/act point sources get.cfm Farren et al. that we confirm the consistency of ACT flux densitiesacross time, and that we measure any residual calibra-tion changes accurately. Of particular importance is thecomparison between ACT observations made in 2008-2010, when Planck was active, to the later observations(2013-16) when ALMA was operating. It is this compar-ison that determines the quality of the bridge we buildbetween Planck and ALMA flux densities. Given thelarge span of time involved, we expect variability of thesources to play a prominent role. To mitigate to somedegree variability during each of the two periods 2008-10 and 2013-16, we compare weighted averages of fluxdensities made for each of the two periods. For the laterACTPol observations (2013-16) specifically, we use in-clusive catalogs of all 93 and ∼
145 GHz sources (“AC-Tall”); the flux density for each is the inverse-varianceweighted average of measurements across all arrays andall seasons 2013-16 (except PA3 in 2016 for reasons citedabove). The same is true for the earlier MBAC mea-surements made in 2008-10 (Marsden et al. 2014; Grallaet al. 2020). Changes in luminosity of sources betweenthe two epochs, of course, remain and are responsible formost of the scatter evident in Fig. 2. In principle, givena large enough sample, source variability should not af-fect the slope of the fitted line, but only the scatter. InAppendix C we reconsider the validity of this assump-tion in the case when the two catalogs to be comparedhave very different flux density thresholds, but that isnot the case here. To mitigate any remaining bias dueto variability, and to decrease the scatter it introduces,we took all the steps described in 4.1 above.While the scatter of the ∼
300 points is large, the aver-age slope is extremely close to unity at 1 . ± . ∼
145 GHz results,source by source, before the fit shown in Fig. 2.We will also need to compare ACT observations madesolely in 2016 (PA2 only; center frequency 144.7 GHz)with ACTall. In this case, a much smaller correction forthe difference in central frequencies is needed; for typicalsynchrotron spectra it is 0.1%. With this small correc-tion included, we find PA2 season 2016 = 1 . ± . S(ACT MBAC)@150GHz [mJy] S ( A C T a ll ) @ + GH z [ m J y ] best fitoutliers Figure 2.
Comparing ACT measurements made with theMBAC array in 2008-10 with later (2013-2016) ACTPol mea-surements made at ∼
145 GHz with polarized arrays (“AC-Tall”). We omit PA3 results from 2016 in the latter. Theslope of the best-fit line (shown) is very close to unity at1.0092. Recall that outlier sources (grey) are not included inthe fit and that we fit to linear data even though we showlog-log plots. and further details are provided in Appendix B. We takethese small differences into account when we use ACTobservations as a bridge — see section 5.1.4.3.2.
Consistency of 93 and ∼
145 GHz ACT data
Are ACT measurements at 93 and ∼
145 GHz consis-tent? Since we have only two bands, we cannot em-ploy the same test of consistency used for Planck. Asa rough sanity check, we plotted 93 GHz flux densi-ties measured in 2016 for ∼ . ± . S (93, extrapolated to 144.7 GHz) =0 . ± . S (144 . Internal Consistency of SPT data
As this paper was nearing completion, an extensivecatalog of compact sources was released by the SPT onfirming ALMA Calibration using Planck and ACT Observations ∼
95, 150 and 220 GHz are provided; for observationsof sources with synchrotron spectra, the effective cen-ter frequencies are closer to 97.43, 152.9 and 215.8 GHz(T. Crawford, private communication). Since three fre-quency bands are available, we may use the same pro-cedure as for Planck to check frequency-to-frequencyconsistency. For roughly 700 sources, we find that theflux densities measured at 152.9 GHz on average lie1 . ± .
25% above the values predicted from the 97.43and 215.8 GHz measurements. This small discrepancycould be present because the SPT flux densities at 215.8GHz are biased low, a point we return to below. PLANCK BASED CALIBRATION OF ALMA,USING ACT OBSERVATIONS AS A “BRIDGE”Since we have measured the small, time-dependentchanges in the calibration of both ACT and Planck, wecan include these small factors when comparing mea-surements of compact sources made by the two instru-ments (or by ALMA) no matter when they were made.We expect, and find, that source variability dominatesthe scatter seen in any such comparison, despite thesteps listed in 4.1 to mitigate it.5.1.
Linking ALMA to Planck
We outline here how we connect ALMA observationsmade in 2012 to 2017 with Planck observations made atan earlier epoch, 2009 August to 2012 January. We con-centrate on frequencies between 90 and 150 (and ALMABands 3 and 4) at first.5.1.1.
The ACT-Planck Comparison
The first step is to compare Planck observations madein 2009-10 with ACT observations made in a roughlysimilar time-frame, 2008-2010. We thus compare PlanckPCCS1 measurements (corrected to the 2018 calibra-tion, as per section 4.2) to the ACT MBAC observa-tions at 147.6 GHz (no 93 GHz ACT data were avail-able until 2015) as described by Gralla et al. (2020) andMarsden et al. (2014); as noted, we take final flux den-sity values from LAMBDA. The Planck 143 GHz dataare color corrected and very slightly extrapolated to theMBAC center frequency using spectral indices derivedfrom Planck data (section 3.2). Specifically, we employthe weighted average of the spectral indices between 100and 143 GHz and between 143 and 217 GHz for eachsource. Since the 2008-2010 ACT observations coveredonly a small region of the sky, the number of matches islimited (only 34) and thus the uncertainty in this com-parison is the dominant contributor to the uncertaintyin the overall ALMA-Planck comparison. S(PCCS1, extrapolated) [mJy] S ( A C T M B A C ) @ GH z [ m J y ] best fitoutliers Figure 3.
Comparing ACT flux densities measured byMBAC in 2008-2010 to those measured by Planck HFI at143 GHz in 2009-10 (after color-correction and extrapola-tion to 147.6 GHz to match the center frequency of ACT’sMBAC array). In this case, only a single iteration was re-quired, with one source (shown in grey) excluded from thefit. Note that the fit (dashed line) is extremely close to unity.
For this comparison, like all the others described inthis paper, we take all the precautions described in 4.1to limit the effect of source variability. One further stepwe take from the beginning is to limit the comparisonto sources with flux densities S ≥
220 mJy in the ACTMBAC catalog. This threshold was chosen as the ap-parent flux density level that matches the 90% com-pleteness in the 143 GHz Planck catalog. Adoptingsuch a threshold ensures that there are not large num-bers of faint ACT sources without matches in the PlanckPCCS1 catalog (see Table 3).The two lists of sources being compared here have verydifferent threshold sensitivities. That introduces whatwe call “variability bias,” a topic we address in more de-tail in Appendix C. Briefly, a variable source that hap-pened to be bright when Planck observed it, but fadedduring the ACT observations, would still be detected byACT given its much greater sensitivity. But the oppo-site is not true. Indeed, we find three sources with ACTflux density above our chosen threshold of 220 mJy withno match in the Planck data, presumably because theyfell below Planck’s detection limit in 2009-2011. Note The official 90% completeness given in Planck Collaboration et al.(2014b) is 190 mJy, but we instead infer from figure 6 in that samepaper that 220 mJy is a more conservative value, and we adoptit. Cutting at 190 mJy instead changes the slope of the fit by ∼ . σ . Farren et al. that this number (3 of 34) is consistent with what is ex-pected given our use of the 90% completeness limit, sotheir absence from the Planck catalog is not necessarilydue to variability. If we simply ignore these “missing”matches, we might bias our results. Consequently, wereplace the Planck flux density of these three sources byeither 320 mJy, or 190 mJy, values chosen as reasonablelimits on the possible flux density of the three sourcesnot found in the Planck catalog (see Appendix C). Weassign an uncertainty of ±
40 mJy to these three mea-surements, a value typical of Planck flux density uncer-tainties. The resulting slopes for the two different limitswere 1 . ± . . ± . ∼ . σ ; and(b) replacing the three sources not found in Planck withan intermediate value S = 220 ±
40 mJy instead, chang-ing the slope to 1 . ± . ∼ . σ . Theseand other tests are described in more detail in AppendixC. For the remainder of this work, we adopt a value ly-ing between the two slopes set by plausible limits on theflux density of the 3 missing Planck sources, taking theinverse-variance weighted average, we find ACT MBACfluxes on average = (1 . ± . ± . ± . . (5)Since ACT observations at the lower frequency of93 GHz began only in 2015, we have no simultaneousPlanck data with which to compare them. Instead, wesimply compare ACT 93 GHz flux densities from 2016with those from the Planck 100 GHz PCCS2 catalog,after color-correcting and extrapolating the latter us- ing Planck-based spectral indices. Since the approxi-mate 90% completeness limit at 100 GHz is higher thanfor 143 GHz, we employ a threshold of 330 mJy in thecomparison, and replace any ACT sources not found inPlanck with a Planck flux density set to 400 ±
50 mJy(the uncertainty roughly matches typical Planck valuesat 100 GHz). At 93 GHz, we findACT 93 GHz = (1 . ± . . (6)We have added 0.13% Planck beam uncertainty to theerror listed. Including a small number of 93 GHz ob-servations made in 2015 changes the slope by ∼ . σ .We return to the evidence that ACT flux densities at93 GHz may be slightly overestimated in section 5.1.4.Recall that any small bias in ACT values will cancel outto a large degree, since we compare ACT to both Planckand ALMA. 5.1.2. Resolution differences
The solid angle of Planck beams is roughly an or-der of magnitude larger than those of ACT or SPT; weneed to check whether Planck measurements of compactsources could be biased high by background sources inthe Planck beam. While the DETFLUX method usedto determine Planck flux densities listed in PCCS2 av-erages out the contributions from a uniform backgroundof other sources, sources preferentially clustered arounda Planck source could bias Planck measurements high.Welikala et al. (2016) present evidence that Planck’slarge beam is picking up flux from star-forming galax-ies along the line of sight to strongly lensed dusty starforming galaxies. On the other hand, that same papershows no such effect for (unlensed) synchrotron sourceslike those examined here. In addition, the amplitude ofthe signal observed for the lensed sources is at most afew mJy, <
1% of the flux density of Planck sources weconsider.We also estimated the probability of finding a back-ground source brighter than either 22 or 4.4 mJy withinthe 6 (cid:48) search radius, assuming those sources are ran-domly placed on the sky. We selected these two limitssince they represent ∼
10% and ∼
2% of the flux densityof the faintest sources we use in the Planck-ACT com-parison. To make these estimates, we used the 150 GHzsource counts of Everett et al. (2020); ACT data, pub-lished by Datta et al. (2019) is in general agreement withthe SPT source counts, but the SPT counts are easierto use. The probability of finding a source with
S > (cid:48) is lessthan 4%, so we might expect to find one or two suchrandom background sources among the 34 Planck-ACT onfirming ALMA Calibration using Planck and ACT Observations Table 3.
Flux densities from Planck PCCS1 at 143 GHz, both before and after color correction and extrapolation to 147.6 GHz. We take PCCS2and the 2018 Planck calibration as the standard, and hence have corrected the PCCS1 data by the small factors given in Table 1 before enteringthem in col. 5. Values in col. 6 have been color corrected and extrapolated as well. These may be compared to ACT MBAC measurements forthe same sources (col.7). Two known MBAC sources are inexplicably missing from LAMBDA; these have no ACT name, and are entered as a , Corrected/Extrapolated MBAC Flux, mJymJy Planck Flux, mJyPCCS1 G115.22-64.77 ACT-S J003820-020738 9.58 − ±
48 433 ±
48 442.5 ± ±
35 187 ±
35 252.2 ± ±
48 343 ±
48 374.8 ± − ±
45 445 ±
44 339.0 ± − ±
35 222 ±
35 240.6 ± − ±
39 1565 ±
38 1718.3 ± ±
53 1213 ±
52 871.2 ± − ±
39 1239 ±
38 828.7 ± − ±
27 526 ±
27 434.4 ± ±
39 1200 ±
39 1438.0 ± − ±
48 309 ±
47 371.5 ± ±
38 487 ±
38 626.4 ± ±
41 277 ±
41 235.6 ± ±
41 558 ±
41 399.0 ± ±
47 222 ±
46 267.3 ± ±
36 517 ±
36 510.3 ± ±
39 729 ±
39 669.9 ± ±
51 483 ±
50 356.7 ± − ±
48 278 ±
47 246.1 ± ±
52 336 ±
52 266.6 ± − ±
43 1331 ±
43 1264.0 ± ±
39 1202 ±
38 1149.0 ± − ±
42 375 ±
42 372.1 ± − ±
32 272 ±
32 220.5 ± ±
37 316 ±
36 245.0 ± ±
31 267 ±
31 330.6 ± ±
35 265 ±
35 252.3 ± − ±
38 332 ±
38 347.7 ± ±
40 273 ±
40 234.2 ± − ±
41 265 ±
41 246.3 ± − ±
44 288 ±
43 242.6 ± ± − ± ± a Calibration corrected to 2018 level Farren et al. matches. The probability for finding a weaker (4.4 mJy)source is larger, ∼ (cid:48) of eachPlanck source. Eleven of the 34 sources plotted on Fig.5.1.1 do have one or more secondary sources nearby; inonly 3 cases, does the flux density of a secondary sourceexceed 22 mJy. Neither value is strongly inconsistentwith the expectations for randomly placed sourcesPlanck’s broad beam (with a full width at half max-imum of ∼ . (cid:48) at 150 GHz) will incorporate some ofthe flux from these secondary sources. How much eachsecondary source contributes depends on its distancefrom the primary source. If, for instance, the secondarysource lies 3 . (cid:48) away, only of its flux is included in thePlanck measurements. When this convolution with thePlanck beam is taken into account, we find that the ob-served secondary sources can contribute ∼
1% to ∼ ∼ σ/
6. The matched filterused in Planck source extraction would reduce this stillfurther; hence we continue to use the value shown in Eq.5.5.1.3.
Using ACT 147.5 GHz observations as a bridge toALMA
The next step is to use additional ACT measurementsof extragalactic sources as a bridge. We begin with thecomparison of 2008-10 MBAC measurements to the all-season, all-array catalog of DR4 ACTPol measurementsmade in 2013-16 (“ACTall;” recall that this omits ∼ ∼ σ to eliminate many weaksources. The lower flux density cut allows many morematches between the two ACT data sets. The resultis that the error is smaller than for the earlier MBAC-Planck comparison. From section 4.3.1, we take ACTall= (1 . ± . . ± . . ± . , (7)where we have summed the uncertainties from all pre-vious steps in quadrature. Note that, for now, we stillretain “excess” significant figures to enable precise com-binations of the various results. ACT measurements ofcompact sources at ∼
145 GHz match Planck’s withinerrors, but may run a bit high. To test this finding, wecompare extrapolated and color-corrected PCCS2 mea-surements at 143 GHz directly with ACTall. Since thetwo sets of observations do not overlap in time, variabil-ity increases the scatter, but we have many more sourcesin common. We find ACTall = 1 . ± . ∼
145 GHz.ALMA = (0 . ± . , (8)as shown in Fig. 4, and consequently,ALMA = (0 . ± . , (9)thus linking ALMA calibration in band 3 to the absolutecalibration of the Planck satellite. Calibration of the twoinstruments agrees well within errors (which are domi-nated by the first step in the process just described, theMBAC vs. Planck comparison).We can eliminate one step in the bridge, namely theuse of the multi-season ACTall data, by comparing theMBAC measurements to both Planck and the laterseason 2016 PA2 ACT measurements, then comparingthe latter to ALMA. Omitting ACTall, we find ALMA= 0 . ± .
038 Planck. The problem here is the longgap between 2008-10 MBAC measurements and season2016. We may also drop the early MBAC observationsand link Planck to ACTall, then ACTall to the season2016 PA2 measurements and hence to ALMA: ALMA= 1 . ± .
028 Planck. Eliminating season 2016 PA2 in-stead, by linking ACTall directly to ALMA and throughMBAC to Planck, yields marginally consistent resultsbut larger error: ALMA = 0 . ± .
042 Planck. Wemay also drop both MBAC and season 2016 PA2, anduse ACTall as the sole link between Planck and ALMA,yielding ALMA = 0 . ± .
036 Planck. Finally, we cansimply compare the ACT PA2 measurements made in2016 with the corrected PCCS2 Planck data to move di-rectly to the equivalent of Eq. 7, after adding the 0 . onfirming ALMA Calibration using Planck and ACT Observations S(ACT S16PA2)@150GHz [mJy] S ( A L M A N ov16 ) [ m J y ] best fitoutliers Figure 4.
Dedicated Nov. 2016 ALMA measurements com-pared to ACT measurements in the same year to minimizethe effect of source variability (note the reduced scatter com-pared to Figs. 2 and 3). ALMA-based spectral indices wereused to interpolate the ALMA fluxes to 144.7 GHz to matchACT. Only one outlier source (shown in grey) was droppedfrom this fit; this source, J2236-1433, is known to vary overshort time scales. uncertainty in the Planck beam solid angle to the error:PA2 S16 = (1 . ± . , (10)in excellent agreement with the result found earlier.Combining this result with Eq. 8 yieldsALMA = (0 . ± . , (11)consistent with Eq. 9. All but one of the ALMA-Planckcomparisons are consistent with unity within the errorbars. With that same exception, they are internally con-sistent within the errors as well.5.1.4. Using ACT 93 GHz observations
We can repeat the last step using 2016 ACT data at93 GHz, comparing it to the extrapolated and color-corrected Planck 100 GHz data, as well as to ALMAdata. From section 5.1, we have ACT(93 GHz) =(1 . ± . . ± . , (12)again suggesting that ACT 93 GHz values may be a fewpercent high. Combining this result with Eq. 6, we find:ALMA = (1 . ± . . (13) Since both Planck and ALMA are compared to the 93GHz ACT data, small errors in the latter cancel to somedegree. Employing entirely different Planck and ACTcatalogs produces results in acceptable agreement withthose found in Eq. 9, though somewhat larger.5.1.5. Using ACT not ALMA spectral indices
A potential problem with the process just outlined isour reliance on spectral indices from ALMA observationsat widely separated frequencies ( ∼
100 and ∼
340 GHz).We can instead use spectral indices for each source de-rived from the 2016 ACT measurements at 93 and ∼ . ± . , (14)ALMA 103.5 GHz = (0 . ± . . (15)We simply take the inverse variance weighted averageof these to obtainALMA = (0 . ± . , (16)To reach this result, we have used both 93 and ∼ . ± . ∼ . ± . Including other ALMA observations
We now turn to ALMA observations from the largerlist of ALMA observations from cone searches and gridmonitoring, as described in section 3.5. As noted there,these are of mixed quality and cover a wider range of6
Farren et al. time. Source variability was therefore a more signifi-cant problem. To minimize it, we began by separatingthe list of observations by year, retaining only observa-tions made in late 2015 and 2016. These were comparedto ACT measurements made in the same 2016 season.To calculate ALMA spectral indices, we required sourceswith nearly simultaneous observations in a least two dif-ferent ALMA bands, one of them Band 3. As in 5.1,these were used to interpolate (or extrapolate) ALMAobservations at various frequencies to ACT’s central fre-quency of ∼
145 GHz.After these cuts, we found 57 sources with usefulALMA observations, including the dedicated Novemberobservations; we then compared these measurements tothose made by ACT in the 2016 season, using ALMA-based spectral indices. Then, as we did above (Eq. 7),using Planck-based spectral indices, we compare ACTto Planck. For this wider set of ALMA observations, wefind: ALMA = (1 . ± . , (18)fully consistent with the earlier result (Eq. 9). We mayalso omit various portions of the “bridge” linking Planckto ALMA, as we did in section 5.1.3. For instance, if weuse ACTall as the sole link from Planck to ALMA, wenow obtain 0.968 ± ± . σ , except for the case where we use ACTallas the sole bridge between Planck and ALMA (here,adding the grid observations raised the slope by ∼ σ ).Using ACT 93 GHz observations, as in section 5.1.4,as a “bridge” between Planck 100 GHz observationsand the wider set of ALMA observations yields 1.026 ± . ± . Other ALMA Bands
We cannot employ ACT observations as a bridge tolink Planck measurements at 353 GHz and ALMA ob-servations at 343 GHz (Band 7) as we did at lower fre-quencies since ACTPol data are available only at 93 and ∼
145 GHz. Nevertheless, we can set some rough con-straints on ALMA flux density calibration in this band.Since we employed ALMA observations in Band 7 in sec-tion 5.1 to calculate spectral indices, we can claim thatthere is no gross discrepancy between ALMA Band 7calibration and the absolute calibration of Planck. Forinstance, a 10% error in calibration at 343.5 GHz wouldproduce a ∼
3% mismatch in the ALMA-Planck calibra-tion, or a roughly 1 σ shift in our results.Finally, since the absolute calibration of ALMA isbased on observations of Uranus, our results also val-idate models of the microwave emission of Uranus (e.g.Butler 2012). For direct measurements of Uranus byWMAP and Planck see Weiland et al. (2011) and PlanckCollaboration et al. (2017) respectively. Unlike the casefor lower frequencies, the Planck 545 GHz channel wasalso calibrated on Uranus (and Neptune), not the an-nual dipole (Planck Collaboration et al. 2020a). In thatpaper, it is pointed out that the CMB dipole derivedfrom this calibration at 545 GHz agrees with the dipolemeasured at lower frequencies to 1% precision. This im-plies that, in principle, an accurate flux density scale isavailable in ALMA Bands 8 and 9 as well.5.4. SPT
SPT measurements of extragalactic sources were madein the interval 2008-11 (Everett et al. 2020), and henceroughly overlap with the Planck mission. We can thuscompare them directly to Planck observations. As wedid for ACT data, we apply all the measures from section4.1 to limit the effects of variability, color correct andextrapolate the Planck data, and set thresholds of
S >
330 mJy at 90 GHz and
S >
220 mJy for 150 GHz. Theagreement between 97.43 GHz flux densities from SPTand Planck 100 GHz measurements is good, as is theagreement between 152.9 and Planck 143:SPT (97.43 GHz) = 1 . ± .
026 PCCS2 (100 GHz)(20)SPT (152.9 GHz) = 0 . ± .
030 PCCS2 (143 GHz)(21)At the higher frequency, the match is less good;SPT (215.8 GHz) = 0 . ± .
044 PCCS2 (217 GHz) . (22)We employed Planck-based spectral indices to makethe small adjustments for the differences between SPTand Planck frequencies. Since there are three SPT fre-quencies, we could have used SPT-based spectral indicesinstead: this would have changed the results by ∼ . σ or onfirming ALMA Calibration using Planck and ACT Observations ∼
150 GHz run ∼ σ belowPlanck; for ACT at ∼
145 GHz, we earlier found valuesin close agreement or ∼ σ above Planck. The SPT mea-surements at 215.8 GHz fall significantly below Planck217 GHz values; a miscalibration could explain the smallfrequency-to-frequency discrepancy noted in section 4.4.The optics of SPT are a better match to ACT thanPlanck. In addition, a portion of the ACT fields overlapsthe SPT survey area. So we compared SPT measure-ments at 90 and 150 GHz with season 2016 ACT values,in this case using a threshold of 20 mJy for both catalogs.After correcting for the different central frequencies ofthe two experiments, using ACT-based spectral indices,we find:ACT = 0 . ± .
016 SPT (97.43 GHz) . (23)ACT = 0 . ± .
018 SPT (152.9 GHz) . (24)Using SPT-based spectral indices to correct for the dif-ference in central frequencies produces results consistentwith the above within ∼ σ . It may appear surprisingthat we find some SPT flux densities lower than Planck,but above ACT while ACT matches Planck fairly well.This is possible because very different sets of sources areinvolved in these various comparisons (e.g., only brightsources when Planck is involved). Given the reasonablematch between SPT and Planck at 97.43 GHz, we triedusing SPT observations as a link in the bridge betweenPlanck and ALMA: we find ALMA = 0 . ± . . ± .
044 Planck, and do not consider this resultfurther. DISCUSSIONThe most important result in this paper is the con-firmation of the calibration accuracy of the ALMA fluxdensity scale. We employed a variety of means to linkALMA observations to the earlier, absolutely calibrated,Planck results, generally using ACT measurements as abridge. These results appear in Eqns. 9, 11, 13 and 18among others. For convenience, we assemble in Table4 the many results from the analysis in section 5. Thetests of ALMA calibration at 91.5 and 103.5 GHz arepresented in finer detail here than in the text of section5.1.5. Although these comparisons of ALMA and Planckmeasurements were obtained from different data sets inmany different combinations, the values are in generalinternally consistent, and in all but two cases they areconsistent with unity as well, thus confirming the abso-lute accuracy of ALMA in band 3. Readers may wish to weight or combine the results in different ways, but wesuggest a reasonable summary of all the tabulated re-sults is that ALMA Band 3 calibration runs only 1 ± . ± . . It should be noted that this does not rep-resent the minimum variance estimate, given that weexpect significant covariance between fits based on thededicated ALMA measurements and the corresponding8
Farren et al.
Table 4.
Comparing ALMA flux density measurements to those from Planck, for various different combinations of thedata. Most of the entries come from the text (as indicated in column 1). Entries in parentheses in column 2 are based onthe more extensive set of ALMA measurements described in section 5.2.See in text S ALMA /S Planck
Notes1 Eq. 9 0.985 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± measurements employing ALMAall. The error was com-puted under the assumption that those measurementsare fully degenerate, taking into account the resultingoff-diagonal elements of the covariance matrix. The lat-ter assumption is conservative, but still yields an er-ror estimate lower than 0.02. All these results suggestthat a ratio of 0 . ± .
02 is reasonable. All the valuesjust listed lie well within the ±
5% normally assumedfor ALMA calibration uncertainty based on planet ob-servations. Assessing the uncertainty in these ratios isnon-trivial since many of the entries in Table 4 use over-lapping data sets. We suggest that ±
2% is a reasonableestimate, and could be used as the standard figure forALMA flux density calibration uncertainty rather thanthe ±
5% now employed, at least in ALMA band 3.An earlier paper (Louis et al. 2014) compares PlanckPCCS1 flux densities with preliminary MBAC measure-ments at 148 and 218 GHz. At both frequencies, theearly ACT MBAC measurements ran 1-2% high com-pared to Planck PCCS1 values after removing a fewknown variable sources from the fits. When we correctthe Planck calibration to its final values using Tables 1and 2, the results reported in Louis et al. (2014) become: MBAC = 0 . ± .
021 Planck (143 GHz), (25)MBAC = 0 . ± .
031 Planck (217 GHz). (26)The first of these is in acceptable agreement with theresults of Eq. 5. As was the case for SPT (Eqn. 22),the ground-based ACT measurements at ∼
217 GHz arelower than Planck values (by ∼ σ for ACT).A comment on a related test of the accuracy of thecalibration of the Jansky Very Large Array. Partridgeet al. (2016) earlier found that flux density measure-ments by the VLA ran a statistically significant ∼ ± . onfirming ALMA Calibration using Planck and ACT Observations Python 3.7 andmany of its standard packages (Van Rossum & Drake1995, 2009). Furthermore, we use the scientific comput-ing packages
SciPy (Virtanen et al. 2020) and
NumPy (Oliphant 2006; Walt et al. 2011) as well as
Astropy (Astropy Collaboration et al. 2013; Price-Whelan et al.2018).APPENDIX A. FREQUENCY-TO-FREQUENCY CONSISTENCY OF PLANCK MEASUREMENTS OF COMPACTSOURCESWe provide here more detail on how we confirmed the channel-to-channel consistency of Planck flux density mea-surements on compact sources. To test the consistency at 100, 143 and 217 GHz, for instance, we constructed aband-matched catalog of Planck sources from the PCCS2 catalog, again using a search radius of 6 (cid:48) . We also correctedthe PCCS2 flux densities by the very small factors shown in column 6 of Table 1 to update them to the final Planckcalibration. We then color-corrected flux density measurements at 100 and 217 GHz for each source, and used thespectral index α derived from these corrected values to predict the flux density at the intermediate frequency of 143GHz: S (143 GHz) pred = S (100 GHz) (cid:18) (cid:19) α (A1)These predicted values were then compared to the measured, color-corrected values at 143 GHz. Note that thisprediction does not take account of any curvature in the spectra of sources; we return to this point below.Before making the comparison, we excluded as usual 33 bright or extended sources. We also employed a filter onthe spectral index to exclude sources with evidence of thermal emission in order to keep the color corrections small.0 Farren et al.
Table 5.
Array-to-array consistency of ACT ∼
145 GHzmeasurements of compact sources in a given season. Notethe outlier, PA3 in 2016; we exclude it from all analyseshere.Season Arrays Compared Comparison of OverallA B Calibration, S ( A ) /S ( B )2014 PA1 PA2 1.011 ± ± ± ± ± Specifically, we required the 100-143 GHz spectral indices to lie in the interval − ≤ α <
0. That left us with 461sources.If the 3 Planck bands we are testing are consistent, we expect the measured and predicted flux densities to match onaverage (unit slope in the fit), absent spectral curvature. In fact, we find 0 . ± . ∼
1% level. The small amount of scatter is due in part to measurement error,but some may also be due to source variability (a given source swept through Planck’s beams for different frequenciesat slightly different times). Ten outliers were excluded by the iterative procedure described earlier.To perform a similar check of the 217 GHz flux densities, we used color-corrected Planck data at 143 and 353GHz to predict flux densities at 217 GHz, and compared these predictions to the observed flux densities. There werefewer sources that met the spectral index selection criterion (148), and the observed slope was a bit above unity at1 . ± . . ± .
3% higher than the predicted values. Thisresult could be due to a slight curvature in the average SED of sources around 100 GHz, or possibly to contaminationby Galactic CO (1 →
0) line emission leaking into the Planck 100 GHz channel. We tested for the latter by excludingall sources with Galactic latitude below 20 degrees, on the assumption that Galactic CO contamination would bemore prevalent at low latitudes Planck Collaboration et al. (2020b). The result was largely unchanged: a difference of2 . ± . α = − .
13 at 100 GHz would explain the 2.7% excess in measured flux densities we detect.We conclude that the catalogued Planck flux densities at 143, 217 and 353 GHz are consistent to 2% or better, andthat there is only a mild inconsistency at 100 GHz. B. CONSISTENCY OF ACT DATA ACROSS ARRAYS AND EPOCHSAs noted in section 4.3, given the changes in the detectors employed at ACT (and some changes in the analysis pipelineas well), we sought to confirm that ACT flux density measurements of compact sources, once properly calibrated, wereinternally consistent across detector arrays, sky area and time. For instance, observations during the 2014 season weremade with two different, polarized receiver arrays (called PA1 and PA2), operating with center frequencies of 144.3and 144.7 GHz, respectively (for sources with spectral indices near − .
5; see Choi et al. 2020). For these comparisons,we arbitrarily selected PA2 as our standard for comparison.In 2014, two disjoint sky regions were observed; we combined source lists, and as usual dropped 33 bright or extendedsources. For all comparisons, we also excluded numerous weak sources by fixing the threshold to 20 mJy. That left323 sources in all for 2014: when we compared observations made by the two arrays, the slope was close to unityat 1 . ± . S >
20 mJy, or > σ ). onfirming ALMA Calibration using Planck and ACT Observations Table 6.
Season-to-season consistency of ACT measurements ofcompact sources at ∼
145 GHz (93 GHz on last line). Recall thatwe do not use PA3 results at 145 GHz from 2016. In 2015, therewere few and scattered 93 GHz observations, so we rely on the 2016measurements.Array(s)/Seasons(s) compared Comparison of OverallA B Calibration, S ( A ) /S ( B )MBAC (2008-10) ACTall (2013-16) 0.9909 ± ± ± ± ± ± ± ± We conclude that results from the two detector arrays used in 2014 are consistent at the ∼
1% level. In 2015, theagreement was not as close; we found PA1 = (1 . ± . ∼
145 GHz. We again restricted the comparison to sources with
S >
20 mJy. In 2015, where this threshold was ∼
10 times the noise, the fit was acceptable at 1 . ± . . ± . > ∼ σ discrepancy between 2013 and 2015 PA1 measurements can be ascribed to both the longer interval betweenobservations and the small number of sources in the 2013 observations. C. VARIABILITY BIASIn this appendix we consider the potential bias introduced by source variability in the comparison of flux densitiesfrom two catalogs with very different thresholds. That is the case for Planck and ACT; the noise levels differ by afactor of ∼
20, greater than typical variations in luminosity of AGN. Consequently, a variable source that happened tobe bright when Planck observed it, but faded for the later ACT observations, would still be detected by ACT given itsmuch greater sensitivity. But the opposite is not true. Hence a simple list of matching sources could be biased. Oneremedy, to use a third, “neutral” catalog to select sources, is not available at the frequencies we consider; the highestfrequency wide-area radio survey is at 20 GHz. We consider this bias in general terms, then describe the specific stepswe took to minimize in the present work.C.1.
Using cuts in flux density as a remedy.
Consider two catalogs, one made at epoch t , with noise level N , and a nominal 100% completeness at S , and thesecond catalog with t , N , and S . Also, let N be << N , as is the case for ACT and Planck. In principle, thebias we are considering could be removed by setting a threshold on the flux densities from catalog 2 so high thatevery remaining source is matched in the noisier catalog 1. In the context of this paper, however, that would meandropping nearly half of the 34 entries in Table 3.2 Farren et al.
An alternative is to fix a threshold based on the nominal completeness limit of catalog 1, dropping all sources fromcatalog 2 with
S < S . In the absence of variability, this would produce an unbiased set of matches. It is possible,however, for a source to have flux density S > S at epoch t , but a much lower flux density at epoch t , and henceto be absent in catalog 1. We call these “missing sources.” As already noted, simply ignoring these would bias thecomparison. Can we place any limits on the flux density of these missing sources at t ? The flux densities of themissing sources at t cannot in principle exceed S , providing an upper limit. On the other hand, in principle there isno lower limit, but we know it is unusual for sources to fade by more than a factor 10. If we knew the probability offractional variation P(V) for the class of sources we are considering, we could establish a reasonable lower limit asfollows. Say that 90% of variable sources have V < V t , then adopt (1- V t ) times the observed flux density in catalog2 as the probable lower limit on the flux of each source missing from catalog 1 (and S as the upper limit). If thereare many missing sources, lower limits could be assigned by drawing from P(V). A further refinement would be toconsider the completeness curve for catalog 1.In this paper, since the number of matched sources is relatively small, we adopt a simpler approach. To retainmore sources when we compare Planck and ACT flux densities, we apply a threshold to catalog 2 (the ACT cat-alog) corresponding not to the 100% completeness level of catalog 1 (Planck) but instead to the 90% completenesslevel. Hence we cut the ACT catalog at 220 mJy rather than 320 mJy. We find that only 3 Planck sources are missingin the sense just defined. Since only 3 of 34 are missing, we do not bother with a full treatment using P(V). Insteadwe fix one limit by assigning each of the 3 missing sources the maximum flux density S , in this case 320 mJy, andassign a typical Planck uncertainty of ±
40 mJy to it. By assigning the maximum flux density to each missing Plancksource, we establish a lower limit on the slope of ACT versus Planck comparisons, like those shown in Fig. 3. Theresulting slope is 0 . ± . ∼
25% of ACT sources at 144 GHzvary by more than 40% over a five year time span. We then set the Planck flux density for all 3 missing sources to(1-V) = 0.6 of the corresponding ACT flux density. For computational ease, we use the same value for each of thethree missing sources: 190 mJy. As expected, with lower values inserted for the flux of the missing Planck sources,the slope increases by ∼ . σ to 1 . ± . . ± . ∼ . σ lower than our standard value).Given the central role of this ACT-Planck comparison, we also made a number of other tests. First, as usual, wecut the ACT catalog at 220 mJy, but replaced the 3 missing source with an intermediate flux density of 220 ±