Average fractional polarization of extragalactic sources at Planck frequencies
T. Trombetti, C. Burigana, G. De Zotti, V. Galluzzi, M. Massardi
AAstronomy & Astrophysics manuscript no. TTrombetti_etal_AvgFracPol_Src_AtPlanckFreqs_final c (cid:13)
ESO 2018October 3, 2018
Average fractional polarization of extragalactic sourcesat
Planck frequencies
T. Trombetti , , (cid:63) , C. Burigana , , (cid:63)(cid:63) , G. De Zotti (cid:63)(cid:63)(cid:63) , V. Galluzzi (cid:63)(cid:63)(cid:63)(cid:63) , and M. Massardi † INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, I-40129 Bologna, Italy Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, Via Giuseppe Saragat 1, I-44122 Ferrara, Italy INFN, Sezione di Ferrara, Via Giuseppe Saragat 1, I-44122 Ferrara, Italy INFN, Sezione di Bologna, Via Irnerio 46, I-40127 Bologna, Italy INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy INAF, Osservatorio Astronomico di Trieste, Via Giambattista Tiepolo 11, I-34143 Trieste, ItalyReceived ...; accepted ...
ABSTRACT
Recent detailed simulations have shown that an insu ffi ciently accurate characterization of the contamination of unresolved polarizedextragalactic sources can seriously bias measurements of the primordial cosmic microwave background (CMB) power spectrum ifthe tensor-to-scalar ratio r ∼ . , as predicted by models currently of special interest (e.g., Starobinsky’s R and Higgs inflation).This has motivated a reanalysis of the median polarization fraction of extragalactic sources (radio-loud AGNs and dusty galaxies)using data from the Planck polarization maps. Our approach, exploiting the intensity distribution analysis, mitigates or overcomesthe most delicate aspects of earlier analyses based on stacking techniques. By means of simulations, we have shown that the residualnoise bias on the median polarization fraction, Π median , of extragalactic sources is generally ∼ < . Π median (cid:39) . Π dusty ∼ < .
2% at 353 GHz and of ∼ < .
9% at 217 GHz. The contamination of CMBpolarization maps by unresolved point sources is discussed.
Key words. polarization – radio continuum: galaxies – cosmic background radiation
1. Introduction
The detection of the primordial B-mode polarization of the cos-mic microwave background (CMB) is currently the most press-ing question of particle physics and cosmological research be-cause the signal carries a clean signature of primordial inflation.It is very faint, however, because it is generated by tensor per-turbations, which are so weak as to defy detection so far. Thetightest upper limit on the ratio, r , of tensor-to-scalar fluctua-tions at the pivot scale of 0 .
05 Mpc − is r < .
07 at the 95%confidence level (BICEP2 Collaboration & Keck Array Collab-oration 2016).Measurements of Galactic (Choi & Page 2015; Planck Col-laboration Int. XXX 2016; Planck Collaboration XXV 2016;Krachmalnico ff et al. 2016; Planck Collaboration Int. L 2017)and estimates of extragalactic (Tucci et al. 2004; Tucci & To ff o-latti 2012; Bonavera et al. 2017a,b) polarized foreground emis-sion show that even in the cleanest 70% of the sky, this emis-sion dominates both primordial B-modes and instrumental noiseof the most sensitive forthcoming or proposed CMB polariza-tion experiments, such as the Lite (Light) satellite for the studiesof B-mode polarization and Inflation from cosmic background (cid:63) e-mail:[email protected] (cid:63)(cid:63) e-mail:[email protected] (cid:63)(cid:63)(cid:63) e-mail: [email protected] (cid:63)(cid:63)(cid:63)(cid:63) e-mail: [email protected] † e-mail: [email protected] Radiation Detection (LiteBIRD), the Cosmic ORigins Explorer(CORE) (Delabrouille et al. 2018), the Probe of Inflation andCosmic Origin (PICO), one of the eight Probe-Scale space mis-sions in the range of $ 400M – $ 1000 M that are being funded byNASA, the Primordial InflaXIon (Inflation) Explorer (PIXIE)(Kogut et al. 2016) and the CMB-Stage IV experiment (CMB-S4) (Abazajian et al. 2016).Thus, the capability of future CMB observations to measureor constrain CMB B-mode power spectra will be limited not bysensitivity of the instrument, but by the ability of removing fore-ground contamination with extreme accuracy. According to De-labrouille et al. (2018), foreground cleaning must reach at leastthe 99.9% level at (cid:96) (cid:39)
10, at least the 99% level at (cid:96) (cid:39) (cid:96) (cid:39) / or cosmic variance in multipole bins ∆ (cid:96)/(cid:96) = . (cid:96) ∼ >
10 if r ∼ < − due to gravitationallensing (cf. Fig. 1 of Delabrouille et al. 2018). Hence the detec-tion of primordial B modes requires a very accurate control oflensing e ff ects, which in turn are contaminated by fluctuationsof unresolved extragalactic sources.There is no definitive prediction for the magnitude of thetensor-to-scalar ratio, r . For inflationary models driven by a fun- http://litebird.jp/ https://zzz.physics.umn.edu/groups/ipsig/cmbprobe2016proposal Article number, page 1 of 10 a r X i v : . [ a s t r o - ph . C O ] J un & A proofs: manuscript no. TTrombetti_etal_AvgFracPol_Src_AtPlanckFreqs_final damental scalar field, the value of r is related to the total fieldexcursion (i.e., to the inflation field range in Planck units; Lyth1997; Baumann & McAllister 2007). Models currently of spe-cial interest (e.g., Starobinsky’s R and Higgs inflation) predict r ∼ .
003 (Martin et al. 2014; Abazajian et al. 2016; Finelliet al. 2018). Large-field inflation with super-Planckian field ex-cursions implies even lower values, r ∼ < . r (cid:39) − , the overall uncertainty on r is dominated by fore-ground residuals and that unresolved polarized point sources canbe the dominant foreground contaminant over a broad range ofangular scales ( (cid:96) ∼ > ff olatti 2012; Galluzzi & Massardi2016).Direct WMAP detections in polarization are limited to smallnumbers of sources (Wright et al. 2009; López-Caniego et al.2009). Planck , thanks to its higher sensitivity and angular reso-lution, has yielded more detections. The Second
Planck
Catalogof Compact Sources lists over 120 objects with polarized emis-sion significant at a > .
99% level, not considering the PCCS2Esub-catalog, whose reliability is unknown (Table 14 of PlanckCollaboration XXVI 2016). These sources, however, mostly lieat low Galactic latitudes so that only a minor fraction of themare expected to be extragalactic. The number of detections in theextragalactic zone ( | b | > ◦ ) ranges from 28 at 30 GHz to ∼ Planck frequencies (545 and 857 GHz)were not polarization sensitive. All detected sources are radio-loud AGNs.Follow-up polarization measurements at 8.4, 22, and 43 GHzof a complete sample of 199 extragalactic sources stronger than1 Jy in the 5 yr WMAP catalog were carried out by Jackson et al.(2010). Polarimetric observations of 211 radio-loud active galac-tic nuclei (AGNs) at 86 and 229 GHz were performed by Agudoet al. (2014). Their ≥ σ detection rate was 88% at 86 GHz and13% at 229 GHz. The sample selection was designed to be fluxlimited at 1 Jy at 86 GHz; however, 51% of the sources werefound to have S < S < . σ P (cid:39) . σ detection rate of 90%.Individual sources showed a broad variety of spectral shapes(flat, steep, upturning, peaked, inverted, downturning) both intotal intensity and in polarization, but with substantial variationswith frequency of the polarization fraction from one object to an-other. Because of this complexity, extrapolations to frequenciesof CMB experiments to mitigate the point source contaminationof CMB maps are unreliable.In the case of star-forming galaxies, the polarized emissionabove 100 GHz is dominated by dust. At lower frequencies, the synchrotron emission takes over, but at these frequencies, theextragalactic sky is far from being dominated by radio sources.Polarization properties of dusty galaxies as a whole at (sub-)millimeter wavelengths are almost completely unexplored. Theonly published measurement (Greaves & Holland 2002) yieldeda polarization fraction Π = .
4% for the prototype starburstgalaxy M 82. Integrating the
Planck dust polarization maps, DeZotti et al. (2018) found an average value of the Stokes Q param-eter of about 2.7%. If this is typical for late-type galaxies seenedge-on and the polarization fraction scales as cos( θ ), θ being theinclination angle, the mean polarization fraction, averaged overall possible inclinations, should be (cid:39) . Planck data (Planck Collaboration X 2016) have shown that thebrightness temperature spectra of di ff use polarized foregroundsdisplay a broad minimum (cf. Fig. 1 of Remazeilles et al. 2018).Although the number of sources detected in polarization by Planck in the extragalactic zone is quite limited, estimates of themean polarization fraction of fainter sources can be obtained us-ing stacking techniques, that is, by co-adding the polarized signalfrom many objects detected in total intensity but not in polariza-tion, to increase the signal-to-noise ratio (S / N). A first attemptin this direction was carried out by Bonavera et al. (2017a) forradio sources and by Bonavera et al. (2017b) for dusty galaxies.Bonavera et al. (2017a) applied the stacking techniques tothe 1560 30 GHz sources in the Second
Planck
Catalog of Com-pact Sources (PCCS 2; Planck Collaboration XXVI 2016), span-ning about a factor of 20 in flux density, and followed them inall
Planck maps in polarization (at 30, 44, 70, 100, 143, 217,and 353 GHz). The subsample outside the
Planck
GAL60 mask(covering about 40% of the sky, around the Galactic plane) andoutside the Magellanic Cloud regions, contains 881, likely extra-galactic, radio sources. The remaining 679 are probably mostlyGalactic.The application of stacking to high-resolution data is rela-tively straightforward, since only two ingredients need to be in-cluded: the (faint) signal of sources, and the noise. By coaddingmeasurements at the positions of n sources, the signals add upwhile Gaussian noise decreases as n − / . In the case of the low-resolution Planck data, the situation is much more complicatedbecause each resolution element containing a source also con-tains other polarized signals: the CMB itself, and di ff use syn-chrotron and dust emissions from the Galaxy. The Galactic con-tamination is particularly di ffi cult to include because of its highlynon-Gaussian statistics and its significant variations on the sky.Furthermore, the mean polarized flux density, P , is computedfrom the Stokes parameters Q and U as P = ( Q + U ) / . Anyquadratic sum of this kind is liable to the so-called noise bias,however (e. g., Wardle & Kronberg 1974): the errors on Q and U add a contribution to P . The standard methods for correcting forthis bias cannot be applied to stacking because sources are notindividually detected.Again, because sources are not detected, the desired result,that is, the mean polarization fraction, (cid:104) Π (cid:105) = (cid:104) P / S (cid:105) , where S is the total flux density, cannot be computed directly but is ap-proximated by (cid:104) P (cid:105) / (cid:104) S (cid:105) . Recent studies (at lower frequencies) didnot find evidence of systematic variations of the mean polariza-tion fraction with S (Hales et al. 2014; Galluzzi et al. 2017),but on the other hand, the spectra of polarized emission of indi-vidual sources are generally substantially di ff erent from those intotal intensity (Galluzzi et al. 2017, 2018). In addition, for thefaintest sources the polarized signal is dominated by other com-ponents or by noise, so that it is unrelated to S . The combination Article number, page 2 of 10. Trombetti et al.: Average fractional polarization of extragalactic sources at
Planck frequencies of these e ff ects may be a quite delicate point, in particular forflux densities di ff ering by more than one order of magnitude. Asdescribed in next section, our approach does not rely at all on the (cid:104) Π (cid:105) = (cid:104) P / S (cid:105) (cid:39) (cid:104) P (cid:105) / (cid:104) S (cid:105) approximation.When applied in this context, stacking techniques typicallyrequire simulations to include the noise bias. For example, thecorresponding correction found in Bonavera et al. (2017a) islower than (cid:39)
20% when applied to (cid:112) (cid:104) Π (cid:105) , but reaches a factorof approximately 3 to 6, depending on frequency, when appliedto (cid:104) Π (cid:105) .Bonavera et al. (2017b) applied the same approach to a sam-ple of 4697 dusty galaxies drawn from the PCCS2 857 GHz cat-alog, deriving average corrected polarization fractions and cor-responding median values at 217 and 353 GHz, with a tentativedetection at 143 GHz.Given the real, substantial di ffi culty of the problem and theimportance of reaching an assessment as solid as possible ofthe mean polarization properties of extragalactic sources in the Planck frequency range, we decided to carry out a new investiga-tion adopting an independent approach. In this paper we presenta simpler analytical approach and describe how we control thecritical aspects.The layout of the paper is the following. In Sect. 2 we outlineour method. In Sect. 3 we validate it by applying it to sourceswhose polarization has been measured by Jackson et al. (2010)at 43 GHz and by Agudo et al. (2014) at 86 and 229 GHz. InSect. 4 we report our results, which are summarized in Sect. 5.
2. Outline of the method
Our approach uses the intensity distribution analysis (IDA; DeZotti et al. 1989; Barcons et al. 1995). Briefly, this method con-sists of measurements of signals in a map at the positions ofa given source catalog. The distribution of signals is comparedwith that for the blank sky, measured at random positions, awayfrom sources (control fields). If some statistical test detects asignificant di ff erence, in the sense that the source distribution isshifted toward higher values than that of control fields, a signalis detected. For this purpose, we use the one-sided Kolmogorov-Smirnov (KS) statistics.The polarized flux density of sources is then estimated as P = (cid:16) P − P , median (cid:17) / , (1)where P s is the polarized signal in Planck maps at the sourceposition and P , median is the median value of P in the controlfields. The subtraction removes, in a statistical sense, the contri-butions to P of the noise, of the CMB, and of polarized Galacticemissions. It thus largely corrects for the noise bias, as verifiedthrough a comparison with direct polarimetric measurements(Sect. 3) and via simulations (Sect. 4.3). When P < P , median ,we set P = We have considered radio sources listed in the Second
Planck
Catalog of Compact Sources (PCCS2; Planck Collab-oration XXVI 2016), selecting those detected at 143 GHz andlocated at high Galactic latitude ( | b | ≥ ◦ ), excluding the ar-eas inside the mask adopted for the Planck polarization anal-ysis at 100 GHz at high resolution (COM_Mask_Likelihood-polarization-100_2048_R2.00.fits). We further excluded sourcesflagged as extended in the PCCS2. We repeated the analysis witha Galactic cut at | b | = ◦ , obtaining consistent results but with Since we computed median values, which requires positive signalsfor more than 50% of the sources, this choice does not a ff ect our results. a poorer statistics. Only the results for the cut at | b | = ◦ arereported.At high Galactic latitudes, objects above the the PCCS2 de-tection limit at 143 GHz are almost exclusively radio sources(cf. Negrello et al. 2013; Planck Collaboration Int. VII 2013;Mocanu et al. 2013). The contaminating fraction of dusty galax-ies or Galactic sources is negligibly small at low frequencies,but increases with increasing frequency because dust emissionsteeply rises with frequency at millimeter / submillimeter wave-lengths. An analysis of the brightest flux density bins at 353 GHz( S > Planck light maps in temperature ( T ) and in polariza-tion (Stokes parameters Q and U ) with N side = N side = / Planck
Collaboration (PlanckCollaboration V 2014; Planck Collaboration IX 2014). We havechecked that we obtain good agreement with the PCCS2 fluxdensities in this way.The procedure was repeated for a large number of controlfields. The centers of these fields were required to be separatedby at least 2 FWHM from each other and from the sources. Thiswas e ffi ciently implemented using the properties of the hierar-chical equal area isolatitude pixelation (HEALPix; Górski et al.2005). It was enough to place the control field centers at thecenters of the N side =
64 and of the N side =
128 pixels for thefrequencies of the Low Frequency Instrument (LFI; 30, 44 and70 GHz) and of the High Frequency Instrument (HFI; 100, 143,217 and 353 GHz), respectively. We selected 26,231 and 111,442control fields at the LFI and HFI frequencies, respectively.Unlike the stacking technique, the IDA considers each sourceindividually, therefore it allows us to compute the mean and themedian P / S ratios directly. We preferred the median to the meanvalues because the latter are less stable, being very sensitive tothe presence of a few objects with exceptionally high polarizedflux densities. Analogously to the polarized flux density, the totalflux density of sources is estimated by subtracting, in a statisticalsense, the other contributions (mainly due to Galactic emissions)as S = S s − S CF , median , (2)where S s is the signal in Planck maps at the source position and S CF , median is the median value of S in control fields. Thus, thesource polarization fraction is estimated by Π = P / S , (3)with P and S given by Eqs. (1) and (2), respectively.In order to investigate the possible flux-density dependenceof the median polarization degree on flux density, we subdividedthe samples selected at each frequency into flux density binscontaining 30 sources each, starting from the 90% completeness http://wiki.cosmos.esa.int/planckpla2015/index.php/Unit_conversion_and_Color_correction http://healpix.sourceforge.net Article number, page 3 of 10 & A proofs: manuscript no. TTrombetti_etal_AvgFracPol_Src_AtPlanckFreqs_final
Table 1.
Comparison of the median polarization degrees, Π IDA = ( P / S ) median , at 44 GHz yielded by the IDA method with those measuredby Jackson et al. (2010) at 43 GHz, within the flux density range S range .The errors on Π IDA correspond to the 16th and 84th percentiles of thepolarization degree distribution divided by N / , N bin being the numberof sources in the bin, N PCCS2 is the number of them with polarizationmeasurements in the PCCS2, D is the KS statistics. The last columngives the probability of the null hypothesis (the distributions of signalsof sources and control fields are drawn from the same parent distribu-tion) given by the one-sided KS test. The bottom line refers to the fullsample. S range (mJy) N bin N PCCS2 Π Jackson Π IDA D Probability695–958 30 0 0.031 0.043 ( + . × − + . × − + . × − + . × − + . × − Table 2.
Comparison of the median polarization degrees, Π , at 100 GHzyielded by the IDA method with those measured by Agudo et al. (2014)at 86 GHz. The columns have the same meaning as in Table 1. Thebottom line refers to the full sample. S range (mJy) N bin N PCCS2 Π Agudo Π IDA D Probability300–550 30 0 0.035 0.040 ( + . × − + . × − + . × − + . × − + . × − + . × − + . × − limit given by the PCCS2 paper. This left fewer than 30 objectsin the brightest bin, which has the strongest signal, which is alsodetectable with fewer sources. Since the dusty sources were re-moved after the bins were defined, the final number of sources insome bins is also slightly smaller than 30. For the source binningwe used the DETFLUX photometry, as recommended for pointsources up to 217 GHz (Planck Collaboration XXVI 2016). Wechecked that using APERFLUX, the recommended photometryabove 217 GHz, the results at 353 GHz do not change signifi-cantly.For each bin we compared the distribution of polarized sig-nals with that of control fields using the one-sided KS statistics.Generally, no signal was detected for the faintest bins. When asignal was detected, the mean polarized flux density was com-puted using Eq. (1). This shows another advantage of the analy-sis by flux density bins over the stacking approach applied to thewhole sample: it allows us to include for the final estimates onlythe flux density bins that include signals; fainter objects whosepolarized emission is completely swamped by noise and fluctu-ations of other components can be excluded from the analysis.While the DETFLUX photomery was used for the binning,for estimating the total flux densities used to compute Π IDA = ( P / S ) median we adopted for uniformity the same approach as forthe polarization signals: we summed signals in pixels within aradius of FWHM /
3. Validation with external data
To validate our approach, we have exploited the ground-basedpolarization measurements by Jackson et al. (2010) and byAgudo et al. (2014) at frequencies close to those of a
Planck channel. At 43 GHz Jackson et al. (2010) detected 167 sources. Wedivided those detected by
Planck in total flux density at 44 GHz(Planck Collaboration XXVI 2016) within the area specifiedabove (135 sources) into bins containing 30 sources each, exceptfor the brightest bin, which contains 15 sources. The KS test de-tected signal in
Planck polarization maps for all bins, except forthe 1740–3510 mJy bin. All detections are highly significant, asshown by the last column of Table 1. In this table (and in thefollowing tables), D is the KS statistics, that is, the largest dis-crepancy between the cumulative distributions of sources in a binand of control fields. The probability of the null hypothesis (nodi ff erence between the two) was computed by approximating thedistribution of X = D mnm + n (4)with the chi-square distribution with two degrees of freedom(Siegel & Castellan 1988). Here m and n are the numbers ofsources and of control fields, respectively.For the full sample, the IDA method yielded estimates of themedian polarization degrees consistent with those measured byJackson et al. (2010), within the errors that correspond to the16th and 84th percentiles of the polarization degree distribution,divided by the square root of the number of sources in the bin(cf. Table 1). Agudo et al. (2014) reported ≥ σ linear polarization de-tections of 183 sources at 86 GHz (88% of the sample detectedin total flux density at this frequency) and of 23 sources at229 GHz (13% of those detected in total flux density); 156 ofthe sources detected in polarization at 86 GHz are listed in thePCCS2 100 GHz catalog within the area considered here. Thesesources were again divided into flux density bins containing30 sources each, except for the brightest bin, which contains 6sources. Highly significant polarization signals are detected forall bins (cf. Table 2). The method yields values of the medianpolarization degrees in close agreement with the ground-basedmeasurements in this case as well.Of the 23 sources detected in polarization at 229 GHz byAgudo et al. (2014), 18 are within the region we considered,but one of them was not detected by Planck at 217 GHz. Forthe remaining 17 sources, the KS test gives a detection proba-bility of 92% (i.e., a probability of the null hypothesis, sourcesextracted from the same distribution as control fields, of 8%).The median polarization degree measured on
Planck maps is4 . + . , − . . + . , − . Planck maps, which are averages over fivescans of the sky spanning 2.5 years, is somewhat lower.
4. Results
We detected significant polarization signals of PCCS2 radiosources in the region specified in Sect. 2 for more than two fluxdensity bins at 30, 44, 100, and 143 GHz. The median polar-ization degrees yielded by the IDA method for each bin are re-ported in Tables 3–6. Π IDA was computed using the median of Π , Note that di ff erences larger than statistical errors can be expected formeasurements made at di ff erent epochs, as a consequence of variability.Article number, page 4 of 10. Trombetti et al.: Average fractional polarization of extragalactic sources at Planck frequencies
Fig. 1.
Distribution of the polarized signals of PCCS2 radio sources and dusty galaxies compared with those of control fields (signals within aradius of FWHM / S lim , dusty =
300 mJy (see text).
Table 3.
Median polarization degrees, Π IDA = ( P / S ) median , at 30 GHzof PCCS2 radio sources in the region specified in Sect. 2. The columnshave the same meaning as in Table 1. The first bin is the faintest forwhich the test detected a significant signal. S range (mJy) N bin N PCCS2 Π IDA D Probability596–621 29 0 0.054 ( + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − Eq. (3), over the considered sample. The errors correspond to the16th and 84th percentiles of the polarization degree distribution,divided by the square root of the number of sources in the bin.
Table 4.
Same as in Table 3, but at 44 GHz. S range (mJy) N bin N PCCS2 Π IDA D Probability873–945 30 0 0.035 ( + . × − + . × − + . × − + . × − + . × − + . × − + . × − Table 5.
Same as in Table 3, but at 100 GHz. S range (mJy) N bin N PCCS2 Π IDA D Probability663–728 30 0 0.037 ( + . × − + . × − + . × − + . × − + . × − + . × − + . × − The first bin in each table is the faintest for which we de-tected a significant positive signal. The significance of the detec-
Article number, page 5 of 10 & A proofs: manuscript no. TTrombetti_etal_AvgFracPol_Src_AtPlanckFreqs_final
Table 6.
Same as in Table 3, but at 143 GHz. S range (mJy) N bin N PCCS2 Π IDA D Probability525–580 30 0 0.005 ( + . × − + . × − + . × − + . × − + . × − + . × − + . × − + . × − Table 7.
Median polarization degrees of radio sources at the
Planck polarization-sensitive frequencies. All values were corrected for thesmall biases discussed in Sect. 4.3. ν (GHz) S min (mJy) N N
PCCS2 Π IDA D Probability30 596 438 35 0.033 ( + . × −
44 873 228 9 0.022 ( + . × −
70 3749 15 4 0.028 ( + . × −
100 663 224 14 0.019 ( + . × −
143 525 227 15 0.029 ( + . × −
217 1085 48 8 0.031 ( + . × −
353 784 41 1 0.030 ( + . × − tions (not to be confused with the S / N) is at least at the (cid:39) σ level and in most cases is much higher, as shown in the last col-umn of Tables 3–6. There is no significant dependence of Π IDA on flux density at any frequency.In Fig. 1 the distributions of polarized signals of radiosources is compared with those of control fields. The source dis-tributions refer to the total flux densities above the S lim listedin Table 7. The shift of the distributions toward higher valuesof the polarized flux density, compared to control fields, can beperceived by eye.Table 7 and Fig. 2 show the median polarization degrees,with their errors, for the full flux density ranges over which a sig-nal was detected, at all the Planck polarization-sensitive frequen-cies. The KS test always detects signals with high significance,and the estimates have an S / N > Π IDA is frequency independent, a minimum χ fitgives (cid:104) Π IDA (cid:105) = . + . , − . χ , χ r (cid:39) The median polarization degree does not show any statisti-cally significant frequency dependence, consistent with the re-sults of Battye et al. (2011) and Galluzzi et al. (2017, 2018),but not with the increase between 86 and 229 GHz by factors of ∼ . ∼ . (cid:104) Π (cid:105) = .
85% with first and third quartile valuesof 1.17% and 3.29%, respectively, to be compared with our me-dian values at 30 GHz of 3 . Π median = . . Π median = .
9% with first and thirdquartiles of 1.8% and 4.8%, respectively, to be compared withour median value at 100 GHz of 1 . Planck range and derived aweighted mean over all the channels of
Π = . Π = . , are slightly lower, and close to ours. When the bias correction was not applied, we obtained (cid:104) Π IDA (cid:105) = .
75% and a slightly larger χ r ( (cid:39) Fig. 2.
Median polarization degrees for radio sources at
Planck frequen-cies with their errors (blue data points), and upper limits (90% and 68%confidence) for dusty galaxies (red arrows). All values were correctedfor the small biases discussed in Sect. 4.3. The solid blue line shows theminimum χ value of the median polarization degree of radio sources,assumed to be frequency independent. The dashed and dot-dashed linesshow the 68% and 95% limits, respectively. The same approach was applied to investigate the polarizationproperties of dusty galaxies. Samples of these objects within thesame area considered for radio sources were extracted from thePCCS2 at 217 and 353 GHz. The selection of dusty galaxies wasmade taking into account only sources without a counterpart at143 GHz, where radio sources dominate, as mentioned above.We collected 616 and 678 dusty galaxies brighter than152 mJy at 217 GHz and than 304 mJy at 353 GHz. As in thecase of radio sources, these objects were subdivided into totalflux density bins containing 30 dusty galaxies each, except forthe brightest bins, which contain 16 and 18 objects, respectively.The KS test did not detect any significant positive shift of thedistribution of polarized flux densities compared to that of con-trol fields. Examples are shown in the bottom panels of Fig. 1.At 353 GHz, we adopted S lim =
784 mJy, the same as for radiosources. The 68% and 90% confidence upper limits to the polar-ization degree are Π , dusty ∼ < .
1% and ∼ < . S lim . At 217 GHz, there are no objects above 1085 mJy (the limitfor radio sources) and only 11 objects above 500 mJy. Thus, thehistogram refers to S lim , dusty =
300 mJy. Correcting for the smallbias, the 68% and 90% confidence upper limits are Π , dusty ∼ < .
0% and ∼ < . S lim . The number of sources rapidly decreases forhigher S lim , and the limits become unstable.We therefore do not confirm the conclusion by Bonaveraet al. (2017b), who derived mean fractional polarizations of(3 . ± . . ± . . ± . . ± . In particular, we found very stable 68% (90%) upper limits for S lim between 300 mJy (500 mJy) and 1200 mJy.Article number, page 6 of 10. Trombetti et al.: Average fractional polarization of extragalactic sources at Planck frequencies values are consistent with our 68% confidence upper limit at217 GHz and with our 90% confidence upper limit at 353 GHz.
Although the comparison with the polarization measurements byJackson et al. (2010) and by Agudo et al. (2014) has shown thatthe IDA approach accounts for and remarkably well corrects thenoise bias (cf. Sect. 3), Eq. (1) can still be a ff ected by bias residu-als, whose magnitude can be estimated by means of Monte Carlosimulations.In the ideal case, the polarized flux density of the i –th sourceis P i = Q i + U i and, for an ensemble of N s sources, the medianpolarization degree is Π median = P i S i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) median . (5)In our approach, Eq. (5) becomes Π = ( P , i − P , median ) / S obs , i − S CF , median (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) median , (6)where ( P or S ) CF , median is the median over all the N CF con-trol fields (with N CF >> N s ). The signal in the j –th controlfield depends on noise and “foregrounds” (here including theCMB as well), that is, P , j = ( Q n + Q f ) j + ( U n + U f ) j and S CF , j = ( S n + S f ) j , while for the i -th source, we have P , i = ( Q s + Q n + Q f ) i + ( U s + U n + U f ) i and S obs , i = ( S s + S n + S f ) i .The ( Q , U ) cross-product terms of source with noise and fore-grounds are not subtracted in the numerator of Eq. (6) and theircontribution is not fully negligible, in principle, even though theyare uncorrelated, hence vanishing on average, and expected to besignificantly suppressed in the median.To quantify these potential residuals, we performed MonteCarlo simulations consisting of 1000 realizations. Since di ff useforegrounds vary especially with Galactic latitude, we analyzedthe following cases: Galactic latitude | b | > ◦ , | b | ∈ (20 , ◦ , | b | ∈ (40 , ◦ , and | b | > ◦ . The simulated sources in the mockcatalogs (chosen in number as in column 3 of Table 7) were ran-domly located at the positions of some control fields, in orderto avoid regions where real sources are detected. We generatedsource flux densities in the same ranges as real sources at eachfrequency, in keeping with their di ff erential number counts.Mock radio sources and dusty galaxies were assigned Π RS = .
75% and Π dusty = Q i , U i ) were derivedfrom a uniform distribution of polarization angles. Applying theIDA method, we generally found for radio sources bias valuesof δ Π RS (cid:39) − (0 . − . Planck maps are particularly noisy) and at 353 GHz (where the statisticsis poor); at these frequencies we have δ Π RS , (cid:39) − .
12% and δ Π RS , (cid:39) − . δ Π dusty , (cid:39) − .
1% and a negligiblevalue at 353 GHz.These results refer to | b | > ◦ , but the bias amplitudes do notshow any significant dependence on Galactic latitudes, except at353 GHz, where they somewhat increase for | b | ∈ (40 , ◦ and | b | > ◦ because of the poorer statistics of control fields thatis due to the smaller sky fraction. In general, the bias values aremuch lower than the uncertainties on Π ; nevertheless, the me-dian polarization degrees in Table 7 and in Fig. 2 include thiscorrection. The errors on the retrieved median values are about afactor of 2 lower than those estimated for real sources. The rea-son probably is that the simulations do not allow for the intrinsicdispersion of the source fractional polarization. Figure 3 compares the polarization power spectra of radiosources (solid blue lines) and of dusty galaxies (dashed red lines,shown only for ν ≥
100 GHz) with the CMB power spectra. Thedot-dashed and dotted green lines show the E-mode and lens-ing B-mode power spectra for the
Planck best-fit cosmologicalparameters (Planck Collaboration XIII 2016). The dashed greenlines show the primordial B-mode power spectra for four valuesof the tensor to scalar ratio.The power spectra of sources are the sum of Poisson andclustering contributions. Poisson fluctuations have a white-noisepower spectrum, independent of the multipole number (cid:96) : C (cid:96), Poisson = (cid:90) S d dNdS S dS , (7)where dN ( S ) / dS are the di ff erential number counts per steradianof sources weaker than the detection limit S d .In the case of extragalactic radio sources, the contributionof clustering can be neglected (cf., e.g., Delabrouille et al.2013). To compute the Poisson power spectrum, counts in po-larized flux density were estimated from counts in total flux den-sity, S , adopting a polarization fraction of 2.83%, that is, setting S p = . S . For the counts in total flux density, we exploitedthe models by de Zotti et al. (2005) up to 70 GHz and by Tucciet al. (2011) at higher frequencies. In the relevant flux-densityrange, the slope of the dN ( S ) / dS of extragalactic radio sourcesis ∼ <
2, so that the largest contribution to C (cid:96), Poisson comes fromflux densities just below the detection limits in polarization. Fig-ure 3 shows two cases, considering the polarized flux densitydetection limits expected for next-generation CMB experiments(cf. Fig. 8 of De Zotti et al. 2018).Conversely, in the case of dusty galaxies, the clustering con-tribution dominates most of the multipole range of interest here.The power spectra in polarized flux density were derived fromthose in total flux density given by the Cai et al. (2013) modelthat accurately reproduce both the
Planck (Planck CollaborationXVIII 2011; Planck Collaboration XXX 2014) and the
Herschel (Viero et al. 2013) measurements. For these objects, the countsare very steep (slope > S d .Again, the power spectra in polarization were scaled from thosein total flux density for two choices of the polarization degree,3%, close to the mean value found by Bonavera et al. (2017b),and 1%, close to our 68% confidence upper limit at 353 GHz.For the sake of illustration, we applied this approach down to100 GHz. We note, however, that already at 143 GHz, the dustygalaxies in Planck maps are too faint to allow the derivation ofreasonably accurate constraints on their average polarization de-gree.Unresolved point sources contribute, on average, equally toE- and B-mode power spectra. Thus, for comparison with theCMB polarization modes (Fig. 3), the total power spectra dis-cussed above were divided by a factor of 2.The contamination of CMB polarization maps by extragalac-tic radio sources was previously discussed by Tucci et al. (2004),Tucci & To ff olatti (2012), and Curto et al. (2013). Our analysisagrees with their conclusion that these objects are not a strongcontaminant to the CMB E-mode polarization, but can constrainthe detection of cosmological B-modes if r ∼ < . Article number, page 7 of 10 & A proofs: manuscript no. TTrombetti_etal_AvgFracPol_Src_AtPlanckFreqs_final
Fig. 3.
Polarization E / B-mode power spectra of radio sources and of dusty galaxies compared with the CMB E mode and B mode for four valuesof the tensor to scalar ratio r = . , . , . , . tion level of 1%, and by Bonavera et al. (2017b). The latter au-thors used log-normal distributions for the polarization degreesand took into account only the Poisson contributions. Extrapo-lating to lower frequencies the estimates or the upper limits ob-tained at 217 and 353 GHz, we find that the contamination bydusty galaxies may be comparable to that of radio sources at100–143 GHz; it becomes dominant at higher frequencies andrapidly fades away at lower frequencies.
5. Conclusions
We have revisited the estimates of the mean polarization frac-tion of extragalactic sources (radio-loud AGNs and dusty galax-ies) based on data from the
Planck polarization maps at 30, 44,70, 100, 143, 217, and 353 GHz. Although the earlier analysesby Bonavera et al. (2017a,b) based on stacking techniques werecarefully made, there are several tricky aspects and subtletiesthat call for an independent analysis. This is particularly impor-
Article number, page 8 of 10. Trombetti et al.: Average fractional polarization of extragalactic sources at
Planck frequencies tant in relation to the forthcoming or proposed CMB polarizationexperiments aimed at detecting primordial B modes.The importance of a careful control of foregrounds hasbeen demonstrated by detailed sky simulations (Remazeilleset al. 2018), which included Galactic and extragalactic polarizedemissions in addition to the CMB, based on state-of-the-art ob-servations. These simulations have shown that for r at the perthousand level (i.e., at the level predicted by models currentlyof special interest), or smaller (as in the case of large-field in-flation with super-Planckian field excursions), the overall uncer-tainty on this parameter can be dominated by the contaminationof unresolved polarized extragalactic sources. An insu ffi cientlyaccurate characterization of this component could lead to a biasin the reconstruction of the primordial CMB B-mode signal.Our independent reanalysis overcomes the two most delicateaspects of the application of stacking techniques: the approxi-mation of the average polarization fraction, (cid:104) Π (cid:105) = (cid:104) P / S (cid:105) , withthe ratio of the mean polarized flux density to the mean total fluxdensity, (cid:104) P (cid:105) / (cid:104) S (cid:105) , and the need of simulations to correct for thenoise bias.Our approach considered the objects one by one. This al-lowed us to identify the flux density range that contributes sig-nificantly to the polarization signal; thus we can exclude fainterobjects from the analysis that may a ff ect results because of theserious limitation by noise and background signals. For objectsabove the flux density threshold, we directly computed the me-dian P / S . We find that the method allows us to detect, on Planck maps, mean polarized flux densities at few tens of mJy levels.For comparison, the detection limits in total intensity are at thefew to several hundred mJy levels (cf. Table 13 of Planck Col-laboration XXVI 2016).In addition, the subtraction of the median of the polariza-tion signal of control fields largely corrects for the contributionsof the noise and of the other polarized components (CMB andGalactic emissions). By means of simulations, we have foundthat residual biases on the median polarization fractions are gen-erally below 0.1%, which is much smaller than the estimatederrors.For radio sources, we find a median polarization degree, av-eraged over frequencies, Π IDA , median (cid:39) . Π on eitherflux density or frequency, in agreement with earlier analyses atfrequencies up to 43 GHz, but not in agreement with the increaseof Π from 86 to 229 GHz claimed by Agudo et al. (2014, 2018).At variance with Bonavera et al. (2017b), we do not de-tect any polarization signal from dusty galaxies, although theirmedian values are consistent with our upper limits. For theseobjects we derive a 90% confidence upper limit at 353 GHz Π dusty ∼ < . ∼ < . ∼
100 GHz. The amplitude of their power spectra depends ontheir detection limit in polarization, S d . For the values of S d ex-pected for next-generation CMB experiments, we confirm that at (cid:39)
70 GHz, that is, in correspondence to the minimum Galacticemission, the point source (radio source) contamination is wellbelow primordial E modes, as found by previous analyses. Onthe other hand, it is close to the level of lensing B modes andof primordial B modes for r (cid:39) .
01. The contribution of dusty galaxies to the point source power spectra is still poorly con-strained, but may be substantial or even dominant at ∼ >
100 GHz.
Acknowledgements.
Thanks are due to Z.-Y. Cai and to M. Tucci for havingprovided the power spectra of dusty galaxies and the high-frequency counts ofradio sources, respectively, yielded by their models. We also thank the anony-mous referee for comments that helped improve the paper. We gratefully ac-knowledge financial support from ASI / INAF agreement n. 2014-024-R.1 for the
Planck
LFI Activity of Phase E2, from the ASI / Physics Department of the uni-versity of Roma–Tor Vergata agreement n. 2016-24-H.0 for study activities ofthe Italian cosmology community and from the Italian Ministero dell’Istruzione,Università e Ricerca through the grant ‘Progetti Premiali 2012-iALMA’ (CUPC52I13000140001). Some of the results in this paper have been derived usingthe HEALPix (Górski et al. 2005) package.
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