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Ade, P. A. R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday,A. J.; Barreiro, R. B.; Bartolo, N.; Battaner, E.; Benabed, K.; Benoit-Levy, A.; Bernard, J-P.;Bersanelli, M.; Bielewicz, P.; Bonaldi, A.; Bonavera, L.; Bond, J. R.; Borrill, J.; Bouchet, F. R.;Boulanger, F.; Bracco, A.; Burigana, C.; Cardoso, J-F.; Catalano, A.; Chamballu, A.; Chary,R-R.; Chiang, H. C.; Christensen, P. R.; Colombo, L. P. L.; Combet, C.; Cri, B. P.; Curto, A.;Cuttaia, F.; Danese, L.; Davies, R. D.; Davis, R. J.; De Bernardis, P.; de Rosa, A.; de Zotti,G.; Delabrouille, J.; Delouis, J-M.; Dickinson, C.; Diego, J. M.; Dole, H.; Donzelli, S.; Dore, O.;Douspis, M.; Leon-Tavares, J.; Savelainen, M.Planck intermediate results
Published in:Astronomy and Astrophysics
DOI:10.1051/0004-6361/201526506
Published: 01/02/2016
Document VersionPublisher's PDF, also known as Version of record
Please cite the original version:Ade, P. A. R., Aghanim, N., Arnaud, M., Ashdown, M., Aumont, J., Baccigalupi, C., Banday, A. J., Barreiro, R.B., Bartolo, N., Battaner, E., Benabed, K., Benoit-Levy, A., Bernard, J-P., Bersanelli, M., Bielewicz, P., Bonaldi,A., Bonavera, L., Bond, J. R., Borrill, J. (2016). Planck intermediate results: XXXVIII. E- and B-modes of dustpolarization from the magnetized filamentary structure of the interstellar medium. Astronomy and Astrophysics,586, [141]. https://doi.org/10.1051/0004-6361/201526506
A&A 586, A141 (2016)DOI: 10.1051/0004-6361/201526506c© ESO 2016
Astronomy&
Astrophysics
Planck intermediate resultsXXXVIII. E- and B-modes of dust polarization from the magnetized filamentary
structure of the interstellar mediumPlanck Collaboration: P. A. R. Ade81, N. Aghanim55, M. Arnaud69, M. Ashdown66,5, J. Aumont55, C. Baccigalupi80,
A. J. Banday89,9, R. B. Barreiro61, N. Bartolo28,62, E. Battaner91,92, K. Benabed56,88, A. Benoit-Lévy22,56,88,J.-P. Bernard89,9, M. Bersanelli31,47, P. Bielewicz77,9,80, A. Bonaldi64, L. Bonavera61, J. R. Bond8, J. Borrill12,85,
F. R. Bouchet56,83, F. Boulanger55, A. Bracco55, C. Burigana46,29,48, E. Calabrese87, J.-F. Cardoso70,1,56,A. Catalano71,68, A. Chamballu69,14,55, R.-R. Chary53, H. C. Chiang25,6, P. R. Christensen78,33, L. P. L. Colombo21,63,C. Combet71, B. P. Crill63,10, A. Curto5,61, F. Cuttaia46, L. Danese80, R. D. Davies64, R. J. Davis64, P. de Bernardis30,
A. de Rosa46, G. de Zotti43,80, J. Delabrouille1, J.-M. Delouis56,88, C. Dickinson64, J. M. Diego61, H. Dole55,54,S. Donzelli47, O. Doré63,10, M. Douspis55, J. Dunkley87, X. Dupac35, G. Efstathiou57, F. Elsner22,56,88, T. A. Enßlin75,
H. K. Eriksen58, E. Falgarone68, K. Ferrière89,9, F. Finelli46,48, O. Forni89,9, M. Frailis45, A. A. Fraisse25,E. Franceschi46, A. Frolov82, S. Galeotta45, S. Galli65, K. Ganga1, T. Ghosh55,?, M. Giard89,9, E. Gjerløw58,J. González-Nuevo18,61, K. M. Górski63,93, A. Gruppuso46, V. Guillet55, F. K. Hansen58, D. L. Harrison57,66,
G. Helou10, C. Hernández-Monteagudo11,75, D. Herranz61, S. R. Hildebrandt63,10, E. Hivon56,88, A. Hornstrup15,W. Hovest75, Z. Huang8, K. M. Huffenberger23, G. Hurier55, T. R. Jaffe89,9, W. C. Jones25, M. Juvela24, E. Keihänen24,
R. Keskitalo12, T. S. Kisner73, R. Kneissl34,7, J. Knoche75, M. Kunz16,55,3, H. Kurki-Suonio24,41, J.-M. Lamarre68,A. Lasenby5,66, M. Lattanzi29, C. R. Lawrence63, R. Leonardi35, J. León-Tavares59,38,2, F. Levrier68, M. Liguori28,62,
P. B. Lilje58, M. Linden-Vørnle15, M. López-Caniego35,61, P. M. Lubin26, J. F. Macías-Pérez71, B. Maffei64,D. Maino31,47, N. Mandolesi46,29, M. Maris45, P. G. Martin8, E. Martínez-González61, S. Masi30, S. Matarrese28,62,39,P. McGehee53, A. Melchiorri30,49, A. Mennella31,47, M. Migliaccio57,66, M.-A. Miville-Deschênes55,8, A. Moneti56,
L. Montier89,9, G. Morgante46, D. Mortlock52, D. Munshi81, J. A. Murphy76, P. Naselsky78,33, F. Nati25, P. Natoli29,4,46,D. Novikov74, I. Novikov78,74, N. Oppermann8, C. A. Oxborrow15, L. Pagano30,49, F. Pajot55, D. Paoletti46,48,
F. Pasian45, O. Perdereau67, V. Pettorino40, F. Piacentini30, M. Piat1, E. Pierpaoli21, S. Plaszczynski67,E. Pointecouteau89,9, G. Polenta4,44, N. Ponthieu55,51, G. W. Pratt69, S. Prunet56,88, J.-L. Puget55, J. P. Rachen19,75,
W. T. Reach90, R. Rebolo60,13,17, M. Reinecke75, M. Remazeilles64,55,1, C. Renault71, A. Renzi32,50, I. Ristorcelli89,9,G. Rocha63,10, C. Rosset1, M. Rossetti31,47, G. Roudier1,68,63, J. A. Rubiño-Martín60,17, B. Rusholme53, M. Sandri46,D. Santos71, M. Savelainen24,41, G. Savini79, D. Scott20, P. Serra55, J. D. Soler55, V. Stolyarov5,66,86, R. Sudiwala81,
R. Sunyaev75,84, A.-S. Suur-Uski24,41, J.-F. Sygnet56, J. A. Tauber36, L. Terenzi37,46, L. Toffolatti18,61,46,M. Tomasi31,47, M. Tristram67, M. Tucci16, G. Umana42, L. Valenziano46, J. Valiviita24,41, B. Van Tent72, P. Vielva61,
F. Villa46, L. A. Wade63, B. D. Wandelt56,88,27, I. K. Wehus63, D. Yvon14, A. Zacchei45, and A. Zonca26
(Affiliations can be found after the references)
Received 9 May 2015 / Accepted 22 September 2015
ABSTRACT
The quest for a B-mode imprint from primordial gravity waves on the polarization of the cosmic microwave background (CMB) requires thecharacterization of foreground polarization from Galactic dust. We present a statistical study of the filamentary structure of the 353 GHz PlanckStokes maps at high Galactic latitude, relevant to the study of dust emission as a polarized foreground to the CMB. We filter the intensity andpolarization maps to isolate filaments in the range of angular scales where the power asymmetry between E-modes and B-modes is observed.Using the Smoothed Hessian Major Axis Filament Finder (SMAFF), we identify 259 filaments at high Galactic latitude, with lengths larger or equalto 2◦ (corresponding to 3.5 pc in length for a typical distance of 100 pc). These filaments show a preferred orientation parallel to the magnetic fieldprojected onto the plane of the sky, derived from their polarization angles. We present mean maps of the filaments in Stokes I, Q, U, E, and B,computed by stacking individual images rotated to align the orientations of the filaments. Combining the stacked images and the histogram ofrelative orientations, we estimate the mean polarization fraction of the filaments to be 11%. Furthermore, we show that the correlation between thefilaments and the magnetic field orientations may account for the E and B asymmetry and the CT E
` /CEE` ratio, reported in the power spectra analysis
of the Planck 353 GHz polarization maps. Future models of the dust foreground for CMB polarization studies will need to take into account theobserved correlation between the dust polarization and the structure of interstellar matter.
Key words. polarization – galaxies: ISM – submillimeter: ISM – ISM: general
? Corresponding author: T. Ghosh, e-mail: [email protected]
Article published by EDP Sciences A141, page 1 of 17
A&A 586, A141 (2016)
1. Introduction
Recently, Planck1 has reported an asymmetry in power betweenthe dust E- and B-modes in its 353 GHz observations (PlanckCollaboration Int. XXX 2016; Planck Collaboration X 2015).This power asymmetry has been observed outside masks cov-ering 20 to 70% of the sky, excluding the Galactic plane. Theratio of the dust B- to E-mode power amplitudes is about a halfover the multipole range 40 < ` < 600 (Planck CollaborationInt. XXX 2016). The source of this power asymmetry in the dustpolarization data is currently unknown. Models of the Galacticmagnetic field (GMF) used in the Planck sky model (PSM,Delabrouille et al. 2013) and the FGPol model (O’Dea et al.2012) produce an equal amount of power in E- and B-modesoutside the regions covered by the sky masks. These models,which were used to estimate the dust polarization foreground(BICEP2 Collaboration 2014; BICEP2/Keck Array and PlanckCollaborations 2015), include an analytical model of the large-scale GMF (BGMF) and a statistical description of the turbulentcomponent of the magnetic field.
The Planck maps of thermal dust emission display filamentsdistributed over the whole sky (Planck Collaboration XI 2014).The filamentary structure of the diffuse interstellar matter is alsoa striking feature of dust observations at higher angular reso-lution, performed by Herschel, and of spectroscopic H obser-vations (e.g. Miville-Deschênes et al. 2010; André et al. 2014;Clark et al. 2014). The analysis of Planck dust polarization datain the diffuse interstellar medium (ISM), at low and intermedi-ate Galactic latitudes, indicates that the structures of interstel-lar matter tend to be aligned with the plane of the sky (POS)projection of the magnetic field (BPOS, Planck CollaborationInt. XXXII 2016). This preferential relative orientation is alsoobserved in simulations of magneto-hydrodynamic (MHD) tur-bulence of the diffuse ISM (Hennebelle 2013; Soler et al. 2013).Such a coupling between the structure of interstellar matter andBPOS is not included in the PSM or FGPol models of the dustpolarization sky (Delabrouille et al. 2013; O’Dea et al. 2012).
The goal of this paper is to test whether the correlation be-tween the filamentary structures of the intensity map and BPOSin the diffuse ISM accounts for the observed E−B asymmetry.Zaldarriaga (2001) describes the E- and B-modes decomposi-tion of simple patterns of polarized emission, including filamentswith a homogeneous polarization degree and orientation. Thepresence of E-modes is related to invariance by parity of the po-larization pattern. There is E-only power if BPOS is either paral-lel or perpendicular to the filaments. If BPOS is oriented at +45◦or −45◦ with respect to the filaments, there is B-only power.
In this paper, we filter the Planck intensity and polarizationmaps to isolate filaments in the range of angular scales where theE−B asymmetry is observed. We identify coherent elongated fil-aments within regions of low column density at high Galacticlatitude using a filament-finding algorithm. We evaluate themean polarization angle in each of these filaments and compareit to the mean orientation of each filament. In doing so, we ex-tend the analysis presented in Planck Collaboration Int. XXXII(2016) to the relevant region of the sky for CMB polarization
1 Planck (http://www.esa.int/Planck) is a project of theEuropean Space Agency (ESA) with instruments provided by two sci-entific consortia funded by ESA member states and led by PrincipalInvestigators from France and Italy, telescope reflectors providedthrough a collaboration between ESA and a scientific consortium ledand funded by Denmark, and additional contributions from NASA(USA).
observations at high Galactic latitude. In order to achieve a highsignal-to-noise ratio and enhance the contrast with respect to thelocal background dust emission, we stack the Stokes I, Q, U, andalso E and B maps, for the filaments we select in the Planck dustintensity map. We use the stacked images to quantify the powerasymmetry in E- and B-modes associated with the filaments.
This paper is organized as follows. In Sect. 2, we introducethe Planck 353 GHz data used in this study. The filament-findingalgorithm is presented in Sect. 3. Section 4 presents the study ofrelative orientation between the filaments, BPOS, and the POScomponent of the large-scale GMF (Bm,POS) at high Galacticlatitude. In Sect. 5, we present the stacking of both intensityand polarization maps and derive the mean polarization frac-tion of the filaments. In Sect. 6, we discuss the relation betweenthe relative orientation of the filaments and BPOS and the E−Basymmetry. Section 7 presents our results in the context of ear-lier studies and its relation to Galactic astrophysics. Finally, wepresent our conclusions in Sect. 8. This paper has four appen-dices. Appendix A details the Hessian analysis implemented toidentify the filaments in the dust intensity map. The applicationof the filament-finding algorithm to a simulated Gaussian dustsky is detailed in Appendix B. In Appendix C, we study the im-pact of our selection of the filaments on the main results of thepaper. The computation of all the angle uncertainties that we usein our analysis is presented in Appendix D.
2. Planck data
The Planck satellite has observed the sky polarization in sevenfrequency bands from 30 to 353 GHz (Planck Collaboration I2015). In this paper, we only use the 2015 (“DX11d”) datafrom the High Frequency Instrument (HFI, Lamarre et al.2010) at 353 GHz, since they are best suited to study thestructure of dust polarization (Planck Collaboration Int. XIX2015; Planck Collaboration Int. XX 2015; Planck CollaborationInt. XXI 2015; Planck Collaboration Int. XXII 2015). Thedata processing, map-making, and calibration of the HFI dataare described in Planck Collaboration VII (2016) and PlanckCollaboration VIII (2016). In our analysis, we ignore the dustand CO spectral mismatch leakage from intensity to polariza-tion (Planck Collaboration VIII 2016). Planck Collaboration Int.XXX (2016) has shown that the amplitude of the dust spectralmismatch leakage at high latitude ( fsky = 0.5) is small comparedto the total polarization signal in E and B modes. No CO emis-sion is detected at 353 GHz away from the Galactic plane andthe brightest molecular clouds (Planck Collaboration XIII 2014)and so we do not consider it in our analysis.
To quantify the statistical noise and systematic effects onthe results presented in this paper, we use the two HalfMission(HM), two DetSet (DS), and two HalfRing (HR) Planck353 GHz polarization maps (Planck Collaboration VII 2016).The two HM maps are made from the two halves of the full-mission Planck data, whereas the two HR maps are produced bysplitting each ring (also called stable pointing period) into twoequal duration parts. The two DS maps are constructed usingtwo subsets of polarization-sensitive bolometers at a given fre-quency. The noise is uncorrelated between the two HM, HR, andDS maps. We only use them to compute the error bars on therelevant quantities that we measure in this paper.
A141, page 2 of 17
Planck Collaboration: E- and B-modes of dust polarization
The total polarization intensity (P353) and the polarizationangle (ψ) are derived from the full-mission Stokes Q353 and U353maps at 353 GHz using the relations
P353 =
√Q2
353 + U2353, (1)
ψ = 0.5 × atan2(−U353,Q353), (2)
where the two-argument function atan2(−U353,Q353) is usedto compute atan(−U353/Q353) avoiding the π ambiguity. To re-cover the correct full range of polarization angles ([−π/2, π/2]as used for ψ here), attention must be paid to the signs of bothU353 and Q353, not just their ratio. We use the IAU conven-tion for ψ, which is measured from the Galactic North (GN)and positive to the East. The minus sign in Eq. (2) converts theconvention provided in the Planck data to that of the IAU (seePlanck Collaboration Int. XIX 2015). The orientation angle (χ)of BPOS is defined within the π ambiguity by adding π/2 to thepolarization angle
χ = ψ +π
2· (3)
For the dust intensity at 353 GHz, we use the modelmap D353, computed from a modified blackbody fit to thePlanck data at ν ≥ 353 GHz and the IRAS 100 µm map(Planck Collaboration XI 2014). This map has lower noise thanthe 353 GHz Stokes I map and is corrected for zodiacal lightemission, CMB anisotropies, and the cosmic infrared back-ground monopole. We neglect the contribution of the CMB po-larization at 353 GHz for this study.
The full-mission Planck Stokes Q353 and U353 maps are pro-vided in HEALPix2 format (Górski et al. 2005) at 4.′8 resolu-tion and D353 at 5′. To increase the signal-to-noise ratio, wesmooth the three maps to a common resolution of 15′, takinginto account the effective beam response of each map, and re-duce to a HEALPix resolution of Nside = 512. For the polariza-tion data, we decompose the Stokes Q353 and U353 maps intoE353 and B353 a`ms (E`m and B`m) using the “ianafast” routineof HEALPix, apply the Gaussian smoothing in harmonic space(after deconvolving the effective azimuthally symmetric beamresponse of each map), and transform the smoothed E353 andB353 a`ms back to Q353 and U353 maps using the “isynfast”routine at Nside = 512. We also transform the E353 and B353 a`msto E353 and B353 maps at Nside = 512 using the relations
E353(n) =∑
E`mY`m(n), B353(n) =∑
B`mY`m(n). (4)
All the maps that we use are in thermodynamic units (µKCMB).In this paper, we work with the bandpass-filtered dust in-
tensity map, Db353, to identify and isolate filaments over the fil-
tering scale using a filament-finding algorithm. By filtering outlarge-scale and small-scale modes, we enhance the contrast ofthe filaments with respect to the diffuse background and reducethe instrumental noise, which is critical for accurately measur-ing the polarization orientations of the filaments within regionsof low column density at high Galactic latitude.
For filtering, we apply the three-dimensional spline waveletdecomposition based on the undecimated wavelet transform, asdescribed by Starck et al. (2006). We use the publicly avail-able package Interactive Sparse Astronomical Data AnalysisPackages (ISAP3) to compute the Db
353 map at Nside = 512 reso-lution. The spline wavelet used in this analysis provides less os-cillation in position space compared to Meyers or needlet ones2 http://healpix.jpl.nasa.gov3 http://www.cosmostat.org/isap.html
0 50 100 150 200 250 300 350 400
`
0.00
0.25
0.50
0.75
1.00
Am
plit
ud
e
spline filter
Fig. 1. The representative bandpass filter, retaining only the scales be-tween ` = 30 and 300.
(Lanusse et al. 2012). The filtering is done in pixel space; thecorresponding bandpass filter in harmonic space varies a littleover the sky. Figure 1 presents the typical shape of this band-pass filter, which selects the scales between ` = 30 and 300. Thefiltering scale is chosen in such a way that it highlights all thebright filaments present in the Planck D353 map. We also com-pute the bandpass-filtered polarization maps, Qb
353, Ub353, Eb
353,and Bb
353.
3. Filament-finding algorithm
3.1. Methodology
Identification of filaments as coherent structures is a cru-cial part of this analysis. Previous studies used algorithmssuch as DisPerSE (Sousbie 2011; Arzoumanian et al. 2011),getfilaments (Men’shchikov 2013), and the rolling Houghtransform (Clark et al. 2014). Hennebelle (2013) and Soler et al.(2013) used the inertia matrix and the gradient of the density andcolumn density fields to identify filaments in numerical simula-tions of MHD turbulence.
In this paper, we employ the Smoothed Hessian Major AxisFilament Finder (SMAFF, Bond et al. 2010a) algorithm, whichhas been used to identify filaments in the three-dimensionalgalaxy distribution (Bond et al. 2010b). SMAFF is primarily basedon the Hessian analysis. The Hessian analysis has also beenused to analyse the Planck dust total intensity map in PlanckCollaboration Int. XXXII (2016), Herschel images of the L1641cloud in Orion A (Polychroni et al. 2013), and large-scale struc-ture in simulations of the cosmic web (Colombi et al. 2000;Forero-Romero et al. 2009). Planck Collaboration Int. XXXII(2016) has reported good agreement between the filament orien-tations derived from the Hessian and inertia matrix algorithms.
3.2. Implementation
In this study, we apply the two-dimensional version of SMAFFto the Db
353 map, which is shown in the upper right panel ofFig. 2. From the Hessian matrix, we compute an all-sky mapof the lower eigenvalue λ− and the orientation angle θ− of theperpendicular to the corresponding eigenvector, measured withrespect to the GN. The details of the Hessian analysis are pro-vided in Appendix A. The map of λ− is presented in the lower
A141, page 3 of 17
A&A 586, A141 (2016)
D353
−2000 2000µKCMB
Db353
−80 80µKCMB
λ−
−2 2KCMB rad−2
120◦ 60◦ 0◦ 300◦ 240◦
−60◦
−30◦
0◦
30◦
60◦λ− filaments
Fig. 2. Data processing steps implemented to identify filaments from the Planck data. We start with the Planck D353 map (upper left panel)smoothed at 15′ resolution. The bandpass-filtered Db
353 map (upper right panel) is produced using the spline wavelet decomposition, retaining onlythe scales between ` = 30 and 300. The lower eigenvalue map of the Hessian matrix, λ−, is shown in the lower left panel. Structures identifiedin the high-latitude sky λ− map are shown in the lower right panel. The superimposed graticule is plotted in each image and labelled only on thelower right panel. It shows lines of constant longitude separated by 60◦ and lines of constant latitude separated by 30◦. The same graticule is usedin all plots of the paper.
left panel of Fig. 2. For the subsequent analysis, we consideronly the high-latitude sky, defined as |b| > 30◦, with the LargeMagellanic Cloud and Small Magellanic Cloud regions maskedout.
The map of λ− highlights filaments in the Db353 map with
an orientation angle θ−, which we refer to as the Hessian an-gle hereafter. The distribution of λ− over the unmasked pixelsis shown in Fig. 3. This distribution of λ− is non-Gaussian withan extended tail. We use the median absolute deviation (MAD,Hampel 1974; Komm et al. 1999) to measure the width, σλ− , ofthe distribution, as given by
σλ− = 1.4826 ×median(|λ− − mλ− |), (5)
where mλ− is the median of the λ− distribution. We select fila-ments using an upper threshold (K−) on λ− given by
K− = mλ− − 3σλ− . (6)
Hereafter, we refer to the filaments satisfying λ− < K− as“strong”. This threshold K− separates the strong filaments fromthe weak ones, as detailed in Appendices B and C. By construc-tion, the threshold K− rejects pixels where λ− is positive, sincethose pixels do not correspond to local maxima.
We seek coherent elongated structures in the map. In SMAFF,this is achieved by placing an upper limit C on the difference
-2.0 -1.5 -1.0 K− -0.5 0.0 0.5 1.0
λ− [KCMB rad−2]
0.00
00.
004
0.00
80.
012
0.01
60.
020
PD
F
3σλ−
Fig. 3. Distribution of the eigenvalues λ− over the unmasked pixels inthe high-latitude sky. The grey region represents the pixels that wereused in the SMAFF algorithm to find strong filaments.
between Hessian angles within a given structure. For our pur-pose, we set the value of C = 15◦ to identify relatively straightfilaments.
A141, page 4 of 17
Planck Collaboration: E- and B-modes of dust polarization
0 1 2 3 4 5
NbH [1020cm−2]
015
3045
60
NF
Fig. 4. Histogram of the mean column density of the filament sample.The column density is computed from the Db
353 map using the conver-sion factor derived in Planck Collaboration Int. XVII (2014).
We start with the pixel having the most negative λ− anddenote the corresponding Hessian orientation angle by θs
−. Weidentify its neighbouring pixels using the “neighbours_ring”routine of HEALPix and look for pixels with λ− < K− and orien-tation angle such that |θ− − θs
−| ≤ C. If both conditions are sat-isfied, we count that neighbouring pixel as a part of the filamentand move on to that neighbouring pixel. The neighbouring pixelbecomes the new reference point and we search for its neigh-bours that satisfy both conditions. In our algorithm, θs
− is fixedby the starting pixel, which has the most negative λ−. We con-tinue this friend-of-friend algorithm to connect pixels until oneof the conditions is no longer satisfied. We limit our selectionto filaments with a length (L, defined as the maximum angu-lar distance between pixels within a given structure) larger thanor equal to the threshold length L0, which we choose to be 2◦.This process yields a set of 259 elongated filaments, as shownin the lower left panel of Fig. 2. Hereafter, we refer to this set asour filament sample. Selected sky pixels represent 2.2% of thehigh-latitude sky considered in our analysis. There is no overlapbetween the filaments in our sample.
The column density is computed from the Db353 map using
the conversion factor, 0.039 MJy sr−1 per 1020 H cm−2; this wasderived in Planck Collaboration Int. XVII (2014) by correlatingthe Planck 353 GHz dust total emission map with an H columndensity map over the southern Galactic polar cap. We averagethe column density along each filament and assign one meancolumn density, Nb
H, to each. This column density is computedon the filtered intensity map. The histogram of Nb
H for the fila-ment sample is presented in Fig. 4. The number of filaments perNb
H is represented by NF.
4. Interplay between the filament orientationand the magnetic field
In this section, we study the orientations of matter structuresand BPOS in our filament sample (Sect. 3). The orientation an-gle of BPOS is derived from the observed Stokes Q353 and U353maps using Eqs. (2) and (3). We also consider the orienta-tion angle (χm) of Bm,POS, as estimated from starlight polariza-tion observations (Heiles 1996) and pulsar rotation measures
North
South
WestEast
+x
-x
-y+y
Bm,POS
χm
ψ
BPOS
χ
θ−
Fig. 5. Sketch of the mean orientation angle of the filament (θ−), themagnetic field (χ), the polarization angle (ψ), and the large-scale GMF(χm) along the filament. All the angles are defined with respect to theGN and follow the IAU convention.
(Rand & Lyne 1994; Han et al. 1999). We compare thesethree orientations, as represented in Fig. 5. Our analysis followsPlanck Collaboration Int. XXXII (2016), which used a set of pix-els representing approximately 4% of the sky at low and inter-mediate Galactic latitudes. Only 25% of the pixels in our currentfilament sample were considered in this earlier study.
4.1. Relative orientation of the filaments and the magneticfield
We study the angle difference between the orientations of the fil-aments and BPOS in our sample. First we associate one POS ori-entation angle with each of the filaments with respect to the GN.By construction, due to our selection criteria on the angles, thefilaments are fairly straight and, hence, they may be describedwith a single orientation angle. Given one filament, we measurethe mean orientation angle, θ−, over the n pixels that belongs toit. We make use of the pseudo-vector field with unit length com-puted from the values of θ− for each pixel. This pseudo-vectorhas components Q− = cos 2θ− and U− = − sin 2θ− (followingthe HEALPix convention for the Q− and U− components). Themean POS orientation angle θ− of the filament is obtained byfirst averaging Q− and U− over all n pixels and then calculatingthe position angle of this averaged pseudo-vector. It is given by
θ− = 0.5 × atan2
−1n
n∑i=1
U−,1n
n∑i=1
Q−
. (7)
If we rotate the Stokes Q353 and U353 maps by θ−, i.e., into theframe where the axis of the filament is in the North-South direc-tion, the rotated Q353 and U353 can be written as
Q′353 = Q353 cos 2θ− − U353 sin 2θ−, (8)
U′353 = Q353 sin 2θ− + U353 cos 2θ−. (9)
Combining Eqs. (8) and (9) with Eqs. (1) and (2), we get
Q′353 = P353 cos 2(ψ − θ−) = − P353 cos 2(χ − θ−), (10)
U′353 = − P353 sin 2(ψ − θ−) = P353 sin 2(χ − θ−), (11)
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where the orientation angle χ is defined in Eq. (3). Similarto the computation of θ−, we average Q′353 and U′353 over alln pixels and then calculate the position angle of this averagedpseudo-vector
Q′353 =1n
n∑i=1
Q′353 ≡ − P353 cos 2∆χ−θ− , (12)
U′353 =1n
n∑i=1
U′353 ≡ P353 sin 2∆χ−θ− , (13)
where
∆χ−θ− = 0.5 × atan2(U′353,−Q′353), (14)
P353 =
√Q′2
353 + U ′2353. (15)
The angle difference ∆χ−θ− measures the weighted mean of theangle difference per pixel between the orientations of the givenfilament and BPOS. The index χ refers to the mean orientationangle of BPOS along the filament. Note that we directly mea-sure the angle difference between the filament and BPOS, withoutcomputing χ for each filament.
The histogram of relative orientation (HRO) between the fil-ament and BPOS for our filament sample is presented in the upperpanel of Fig. 6. The mean value of the histogram is 2.◦3 com-puted using the equivalent of Eq. (7). Our histogram agrees withthe pixel-by-pixel analysis at intermediate and low Galactic lati-tudes presented in Planck Collaboration Int. XXXII (2016). Likein this earlier study, we find that the filaments are statisticallyaligned with BPOS. A similar alignment between the filamentsin the intensity map and BPOS has been reported for synchrotronemission observed by WMAP at 23 GHz (Vidal et al. 2015).
To quantify the shape of the histogram of ∆χ−θ− , we fit it witha Gaussian plus a constant. The Gaussian has a 1σ dispersionof 19◦. The constant may be accounted for by the projection ofthe magnetic field and filament orientations on the POS as dis-cussed in Planck Collaboration Int. XXXII (2016).
4.2. Relative orientation of the magnetic fieldand the large-scale Galactic magnetic field
Here, we compare the orientation of BPOS on the filaments withthat of Bm,POS. Heiles (1996) derived the orientation of Bm,POSpointing towards l0 = 82.◦8 ± 4.◦1 and b0 = 0.◦4 ± 0.◦5 from thepolarization pseudo-vectors of stars more distant than 500 par-secs. Slightly different l0 values have been reported in other stud-ies. From the rotation measures of nearby pulsars within a fewhundred parsecs of the Sun, Rand & Lyne (1994) found the di-rection of Bm,POS pointing towards l0 = 88◦ ± 5◦. In anotherstudy of pulsar rotation measures, Han et al. (1999) derived thedirection of Bm,POS as l0 ' 82◦. These two studies do not reportvalues for b0, which is assumed to be zero. Based on these ob-servations, we assume that the mean orientation of Bm,POS in thesolar neighbourhood is l0 = 84◦ ± 10◦ and b0 = 0◦ ± 10◦, withthe same uncertainty on l0 and b0.
We construct a pseudo-vector field with unit length basedon the uniform orientation of Bm,POS. This pseudo-vector hascomponents: Qm = cos 2ψm = cos 2(χm − π/2) and Um =− sin 2ψm = − sin 2(χm − π/2) (following the HEALPix con-vention for the Qm and Um maps), where ψm is the polarizationangle of Bm,POS. The procedure to go from the uniform Bm,POSpointing towards (l0, b0) to ψm is detailed by Heiles (1996). Themean orientation angle (χm) of Bm,POS for each filament is ob-tained by first averaging Qm and Um over all n pixels within a
−80 −60 −40 −20 0 20 40 60 80∆χ−θ− [deg]
08
1624
32
NF
−80 −60 −40 −20 0 20 40 60 80∆χ−χm
[deg]
06
1218
2430
NF
−80 −60 −40 −20 0 20 40 60 80∆χ
m−θ− [deg]
08
1624
32
NF
Fig. 6. Upper panel: HRO between the filaments and BPOS. Middlepanel: HRO between BPOS and Bm,POS. Lower panel: HRO between thefilaments and Bm,POS.
filament and then calculating the position angle of this averagedpseudo-vector. We compute the angle difference, ∆χ−χm
, betweenthe orientations of BPOS and Bm,POS on the filament in a similarmanner to the method described in Sect. 4.1.
A141, page 6 of 17
Planck Collaboration: E- and B-modes of dust polarization
−2 −1 0 1 2
−3−2
−10
12
3
yoff
set
[deg
]
〈D353〉
950
1080
1210
1340
1470
1600
−2 −1 0 1 2
x offset [deg]
〈Q ′353〉
−65
−56
−47
−38
−29
−20
−2 −1 0 1 2
〈U ′353〉
0
4
8
12
16
20
µK
CM
B
Fig. 7. Mean images of the Planck 〈D353〉, 〈Q′353〉, and 〈U′353〉 maps over the λ− filaments at 15′ resolution.
The HRO between BPOS and Bm,POS for our filament sampleis presented in the middle panel of Fig. 6. The mean value ofthe histogram is 1.◦3± 3.◦7 where the uncertainty is computed bychanging the mean orientation of Bm,POS within its quoted uncer-tainties. This HRO has a larger dispersion than that between thefilaments and BPOS shown in the upper panel of Fig. 6. To quan-tify the shape of the histogram of ∆χ−χm
, we fit it with a Gaussianplus a constant. The Gaussian has a 1σ dispersion of 36◦.
We conclude that BPOS of our filament sample is statisticallyaligned with Bm,POS. Planck Collaboration Int. XXXII (2016) re-ports a similar correlation, for the low and intermediate Galacticlatitudes, when comparing the polarization measured on the fila-ments with their background polarization maps. The scatter mea-sured by the HRO may be interpreted considering both the tur-bulent component of the magnetic field and projection effects.
4.3. Relative orientation of the large-scale Galactic magneticfield and the filaments
We combine the results obtained in Sects. 4.1 and 4.2 to assessstatistically the orientation of Bm,POS in the solar neighbourhoodwith respect to the filaments. BPOS is statistically aligned withthe filaments in our sample and with Bm,POS. From both results,one would intuitively expect Bm,POS to be statistically alignedwith the filaments. To test this expectation, for each filament, wecompute the angle difference, ∆χm−θ− , between the orientationsof Bm,POS and the filament. The angle difference ∆χm−θ− is com-puted in a similar manner to the method described in Sect. 4.1.
The HRO between the filament and Bm,POS for our filamentsample is presented in the lower panel of Fig. 6. A correlationbetween the orientation angles of the filament and Bm,POS ispresent, but the HRO shows more scatter than the HRO betweenthe filaments and BPOS and that between BPOS and Bm,POS. Thehistogram has a mean value of −3.◦1±2.◦6. To quantify the shapeof the histogram of ∆χ−χm
, we fit with a Gaussian plus a constant.The Gaussian has a 1σ dispersion of 54◦. Planck CollaborationInt. XXXII (2016) reported a similar loss of correlation whencomparing the orientations of the filaments with that of BPOSderived from their local background polarization maps.
5. Mean polarization properties of filaments
In this section, we present stacked images of the filaments inStokes I, Q, and U, after rotation to align the filaments and to
compute Q and U with respect to their orientation. The imagesare used to compute the average polarization fraction of our sam-ple of filaments.
5.1. Stacking filaments
Over the high-latitude sky, the signal-to-noise ratio of the353 GHz Planck polarization maps is low and it is not possibleto measure the polarization fraction of individual dust intensityfilaments in our sample. In order to increase the signal-to-noiseratio, we therefore stack images of the 259 filaments and theirsurroundings.
For each filament in the sample, using the gnomview rou-tine of HEALPix, we extract from the Planck maps a local, flat-sky, image (7◦ × 5◦ patch) centred on the filament centre androtated by θ− in the clockwise direction to align the filament inthe North-South direction. We stack the images of the filamentsin D353, Q′353, and U′353 (as defined in Eqs. (8) and (9)) afteraligning all the maps in the North-South direction. We producemean stacked images, denoted with angle brackets 〈...〉, by di-viding the sum of the individual images by the total number offilaments in our sample; they are presented in Fig. 7. The 1σerrorbar both on the 〈Q′353〉 and 〈U′353〉 images is 1.3 µKCMB,as computed from the difference of two polarization HM maps.All the features presented in Fig. 7 are significant compared tothe data systematics and statistical noise. The average filamentappears as a negative feature with respect to the background inthe 〈Q′353〉 image and is not seen in the 〈U′353〉 image. This resultis a direct consequence of the alignment between the filamentsand BPOS (Sect. 4.1 and Zaldarriaga 2001). The background inboth the 〈Q′353〉 and 〈U′353〉 images is rather homogeneous. Thisreflects the smoothness of BPOS within the 7◦ × 5◦ patches.
We perform a null test to assess the significance of the stack-ing of filaments. This test is made by stacking 259 randomlychosen 7◦ × 5◦ patches in the high-latitude sky. Each patch isrotated in the clockwise direction, with the orientation angle θ−of the central pixel. The images of 〈D353〉, 〈Q′353〉, and 〈U′353〉
for random patches are consistent with noise. The amplitude of〈Q′353〉 and 〈U′353〉 images is comparable to that of the differencebetween stacked images obtained when applying the same anal-ysis to each of the two polarization HM maps. This confirms thehypothesis that the filaments detected in Fig. 7 are indeed realand are rotated with a well-determined angle θ−.
A141, page 7 of 17
A&A 586, A141 (2016)
−2 −1 0 1 2
050
010
0015
00
RD
35
3[µ
KC
MB
]
RDB353
RDB353
+ RDF353
−2 −1 0 1 2
−60
−40
−20
0
RQ′ 35
3[µ
KC
MB
] RQ′B353RQ′B353
+ RQ′F353
−2 −1 0 1 2
Distance [deg]
015
3045
RU′ 35
3[µ
KC
MB
]
RU′B353
Fig. 8. Radial profiles of the mean stacked Planck D353, Q′353, and U′353images as functions of distance from the centre of the filament (blueline). The grey shaded region shows the 1σ dispersion from the datavalues at a given radial distance from the filament axis that we average.The dashed line is the Gaussian fit to the filament profile plus a constantbackground emission.
5.2. Polarization fraction
Instead of using individual pixels, we collapse the mean stackedimages in the filament direction to draw the radial profiles (R) ofthe D353, Q′353, and U′353 images, which are presented in Fig. 8.The shaded area in Fig. 8 represents the 1σ dispersion from thedata values at a given radial distance from the filament axis thatwe average. We clearly identify the profile of the filament on topof the constant background emission in the D353 and Q′353 ra-dial profiles, while the radial profile of U′353 is consistent with aconstant background emission. The radial profiles of D353, Q′353,and U′353 can be decomposed into the filament (F) and the back-ground (B) contributions as
RD353 = RDF353
+ RDB353, (16)
RQ′353= RQ′F353
+ RQ′B353, (17)
RU′353= RU′F353
+ RU′B353. (18)
We fit the radial profiles of D353 and Q′353 in Fig. 8 with aGaussian profile for the filament emission plus a constant for thebackground emission. We find that the centre of the Gaussianprofile is zero and that their 1σ dispersion is 27′±1′ for both theD353 and Q′353 radial profiles.
Following Eqs. (12) and (13), we can express the averageStokes Q′353 and U′353 for one given filament as
Q′F353 = − PF353 cos 2∆F
χ−θ−= − pF DF
353 cos 2∆Fχ−θ−
, (19)
U′F353 = PF353 sin 2∆F
χ−θ−= pF DF
353 sin 2∆Fχ−θ−
, (20)
where DF353 and PF
353 are the average specific intensity and po-larization intensity of the filament. The superscript F representsthe contribution from the filament only. The polarization frac-tion (pF) of a filament is defined by
pF =PF
353
DF353
· (21)
From Fig. 4, we know that most filaments in our sample havecomparable column densities and hence roughly the same DF
353.The mean stacked Stokes Q′F353 and U′F353 values for all the fila-ments can be approximated as
〈Q′F353〉 ' −〈pF cos 2∆F
χ−θ−〉〈DF
353〉 ' −〈pF〉〈cos 2∆F
χ−θ−〉〈DF
353〉,
(22)
〈U′F353〉 ' 〈pF sin 2∆Fχ−θ−〉〈DF
353〉 ' 〈pF〉〈sin 2∆Fχ−θ−〉〈DF
353〉,
(23)
where 〈 pF〉 is the mean polarization fraction of our filament sam-ple. For the radial profiles of the filament emission, we have
RQ′F353' −〈pF〉〈cos 2∆F
χ−θ−〉RDF
353, (24)
RU′F353' 〈pF〉〈sin 2∆F
χ−θ−〉RDF
353. (25)
The angle difference between the filaments and BPOS (Eq. (14))is used for ∆F
χ−θ−. The histogram of ∆F
χ−θ−is roughly symmetric
around 0◦, implying RU′F353� RQ′F353
.Similarly, the radial profiles from the background emission
can be written as
RQ′B353' −〈pB〉〈cos 2∆B
χ−θ−〉RDB
353, (26)
RU′B353' 〈pB〉〈sin 2∆B
χ−θ−〉RDB
353, (27)
where 〈 pB〉 is the average polarization fraction of the back-ground emission and ∆B
χ−θ−is the angle difference between a
given filament and BPOS from its local background polarization.The observed radial profile of the stacked Q′353 image in
Fig. 8 can be written as
RQ′353= RQ′F353
+ RQ′B353
' −〈pF〉〈cos 2∆Fχ−θ−〉RDF
353− 〈 pB〉〈cos 2∆B
χ−θ−〉RDB
353
= −〈pF〉〈cos 2∆Fχ−θ−〉RD353
+[〈pF〉〈cos 2∆F
χ−θ−〉 − 〈 pB〉〈cos 2∆B
χ−θ−〉]
RDB353
= a RD353 + b, (28)
where a = −〈 pF〉〈cos 2∆Fχ−θ−〉 is the scaling parameter and
b =[〈pF〉〈cos 2∆F
χ−θ−〉 − 〈 pB〉〈cos 2∆B
χ−θ−〉]
RDB353
is the offset ofthe linear fit between the RQ′353
and RD353 profiles. The linear fit
A141, page 8 of 17
Planck Collaboration: E- and B-modes of dust polarization
900 1000 1100 1200 1300 1400 1500
RD353 [µKCMB]
−60−5
4−4
8−4
2−3
6−3
0
RQ′ 35
3[µ
KC
MB
]
Fig. 9. Linear fit (dashed line) to the correlation between the mean radialprofiles of the stacked Q′353 and D353 images. The grey shaded regionshows the 1σ dispersion from the data values at a given radial distancefrom the filament axis that we average.
between RQ′353and RD353 is shown in Fig. 9. The best-fit parame-
ter values from the linear fit are⟨pF⟩ ⟨
cos 2∆Fχ−θ−
⟩= 5.3%, (29)[⟨
pF⟩ ⟨
cos 2∆Fχ−θ−
⟩−⟨pB⟩ ⟨
cos 2∆Bχ−θ−
⟩]RDB
353=19.8 µKCMB. (30)
In our estimate of 〈pF〉〈cos 2∆Fχ−θ−〉, the noise bias is negligi-
ble, because the noise averages out in the stacking of the Q′353and U′353 maps. The HRO between the filament and BPOS (upperpanel in Fig. 6) is used to compute 〈cos 2∆F
χ−θ−〉, which is 0.48.
Taking this factor into account, the mean polarization fraction ofour filament sample is
〈pF〉 = 11%. (31)
We have computed 〈pF〉 using the two independent subsets ofthe Planck data (HM maps). For both data sets, we find thesame mean value of 11%. This is expected, because the HRObetween the filaments and BPOS is determined with high accu-racy (Appendix D). It shows that the measurement error on 〈pF〉
is small. Obviously 〈 pF〉 may be different for another set of fil-aments, because it depends on the angles of the filaments withrespect to the plane of the sky.
The mean polarization fraction of the filaments determinedfrom the filament sample is smaller than the maximum degree ofpolarization reported in Planck Collaboration Int. XIX (2015),which is pmax = 19.6%. This result suggests that there is somedepolarization due to changes in the magnetic field orienta-tion within the filaments (Planck Collaboration Int. XX 2015;Planck Collaboration Int. XXXIII 2016).
The offset b from the linear fit to the radial profiles of theQ′353 and D353 images is positive. This means that⟨pB⟩ ⟨
cos 2∆Bχ−θ−
⟩<⟨pF⟩ ⟨
cos 2∆Fχ−θ−
⟩. (32)
The distribution of ∆Bχ−θ−
for our filament sample is needed tocompute 〈 pB〉. We make a simple approximation using the HRObetween the filament and Bm,POS (lower panel in Fig. 6) as aproxy for ∆B
χ−θ−. We then find 〈cos 2∆B
χ−θ−〉 = 0.25 ± 0.14 for
our filament sample. Combining Eqs. (29) and (32), we put a
upper limit on the mean polarization fraction of the backgroundemission as
〈 pB〉 < 21%+27%−7% . (33)
This upper limit on 〈pB〉 depends on the mean orientation ofBGMF and it ranges from 48% to 14% for the 10◦ uncertainty onl0 and b0 (Sect. 4.2). The profile of RU′B353
in the lower panel ofFig. 8 is roughly constant and positive. This result indicates thatBPOS is smooth within the 7◦ × 5◦ patches and 〈sin 2∆B
χ−θ−〉 is
positive (Eq. (27)), which follows if the distribution of ∆Bχ−θ−
isnot symmetric with respect to 0◦. We point out that the histogramin the lower panel in Fig. 6 may not be used to estimate the signof 〈sin 2∆B
χ−θ−〉 because the uncertainty on l0 and b0 is too large.
6. E–B asymmetry
In this section, we quantify the E−B asymmetry of dust polar-ization, over the ` range 30 to 300, for our filament sample, andrelate it to the relative orientation between filaments and BPOS.First, we present the stacked images of filtered maps that weuse to quantify the E−B asymmetry. Second, we compute thecontribution of the pixels used in the stacking to the variance inthe high-latitude sky. Last, we present an analytical approxima-tion that relates the HRO between the filaments and BPOS to theE−B asymmetry.
6.1. Stacked images of filtered maps
Here we use the Planck Db353, Qb
353, Ub353, Eb
353, and Bb353 maps to
analyse the ` range, between 30 and 300, over which we identifythe filaments in the dust intensity map. Our choice of ` rangehas a large overlap with the angular scales, 40 < ` < 600,where the E−B asymmetry has been measured with the powerspectra of dust polarisation. We stack all the bandpass-filteredmaps, rotated by the mean orientation angle θ− of the filament,as described in Sect. 5.1. The Q′b353 and U′b353 maps are the fil-tered Stokes Q353 and U353 maps computed with respect to theaxis of the filament. The mean stacked images of the bandpass-filtered maps are presented in Fig. 10. The sidelobes that appearin the images of 〈Db
353〉, 〈Q′b353〉, and 〈Eb
353〉 of Fig. 10 on bothsides of the filament centre are coming from the filtering. The1σ errorbar on the 〈Q′b353〉, 〈U
′b353〉, 〈E
b353〉, and 〈Bb
353〉 images is0.2 µKCMB, as computed from the difference of two polariza-tion HM maps. The filaments appear as a negative feature in the〈Q′b353〉 image and a positive feature in that of 〈Eb
353〉. The 〈U′b353〉
and 〈Bb353〉 images are consistent with a mean value zero.
Next, we stack the products of two quantities, i.e.,〈Db
353Eb353〉, 〈D
b353Bb
353〉, 〈Eb353Eb
353〉, 〈Eb353Bb
353〉, and 〈Bb353Bb
353〉.We follow the same methodology as discussed in Sect. 5.1with oriented stacking at the centre of the filament. To avoid anoise bias in the square quantities, 〈Eb
353Eb353〉 and 〈Bb
353Bb353〉,
we compute the cross-product of the two HalfMission maps(HM1 and HM2). For other quantities, 〈Db
353Eb353〉, 〈D
b353Bb
353〉,and 〈Eb
353Bb353〉, we use the full-mission maps. The images pro-
duced by stacking the products are presented in Fig. 11. Notethat the maps shown in Figs. 10 and 11 are not used for dataanalysis, but still we comment on these images. We computedthe mean stacked images from each of the two polarization DSor HR maps and all the features presented in Fig. 11 appear forboth independent subset of Planck data. In the idealized descrip-tion of the filaments in Zaldarriaga (2001), we will expect the av-erage filament to appear similarly in the EE, BB, and T E maps.
A141, page 9 of 17
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−3
−2
−1
0
1
2
3 〈Q ′b353〉
−7
−5
−3
−1
1
3
µK
CM
B
〈U ′b353〉
−2.0
−1.2
−0.4
0.4
1.2
2.0
µK
CM
B
−2 −1 0 1 2
x offset [deg]
−3
−2
−1
0
1
2
3
yoff
set
[deg
]
〈Db353〉
−50
−10
30
70
110
150
µK
CM
B
−2 −1 0 1 2
−3
−2
−1
0
1
2
3 〈E b353〉
−5
−3
−1
1
3
5
µK
CM
B
−2 −1 0 1 2
〈Bb353〉
−2.0
−1.2
−0.4
0.4
1.2
2.0
µK
CM
B
x offset [deg]
Fig. 10. Mean images of the Planck 〈Db353〉, 〈Q
′b353〉, 〈U
′b353〉, 〈E
b353〉, and 〈Bb
353〉 maps over the λ− filaments.
−3−2
−10
12
3 〈Bb353Bb
353〉
60
95
130
165
200
235
µK
2 CM
B
〈Db353E b
353〉
0
360
720
1080
1440
1800
µK
2 CM
B
−2 −1 0 1 2
x offset [deg]
−3−2
−10
12
3
yoff
set
[deg
]
〈E b353E b
353〉
100
140
180
220
260
300
µK
2 CM
B
−2 −1 0 1 2
−3−2
−10
12
3 〈Db353Bb
353〉
−1000
−600
−200
200
600
1000
µK
2 CM
B
−2 −1 0 1 2
〈E b353Bb
353〉
−50
−30
−10
10
30
50
µK
2 CM
B
x offset [deg]
Fig. 11. Mean images of the Planck 〈Eb353Eb
353〉, 〈Bb353Bb
353〉, 〈Db353Eb
353〉, 〈Db353Bb
353〉, and 〈Eb353Bb
353〉 maps over the λ− filaments.
A141, page 10 of 17
Planck Collaboration: E- and B-modes of dust polarization
The differences between these three images shows that the real-ity of the dust sky is more complex than the idealized model.
6.2. Measured E–B asymmetry
We measure the E−B asymmetry at the filtering scale over thehigh-latitude (HL) region. We compute the variances (V) of theEb
353 and Bb353 maps using the relations
VEE (HL) =1
NHL
NHL∑i=1
Eb353,HM1Eb
353,HM2 = (46.6 ± 1.1) µK2CMB,
(34)
VBB (HL) =1
NHL
NHL∑i=1
Bb353,HM1Bb
353,HM2 = (29.1 ± 1.0) µK2CMB,
(35)
where NHL is the total number of pixels in the HL region. Theratio of the filtered B353 and E353 variances is
VBB (HL)VEE (HL)
= 0.62 ± 0.03. (36)
The uncertainty on VEE is computed by repeating the calcu-lation of Eq. (34) using the different cross-products, i.e., thetwo HalfRing (HR1 and HR2) and the two DetSet (DS1 andDS2) maps. We use the cross-HalfMissions as a reference formean VEE . The 1σ uncertainty on VEE comes from the differ-ences of these cross-products (DetSets minus HalfMissions andHalfRings minus HalfMissions). This 1σ uncertainty is domi-nated by data systematics rather than statistical noise. This is inagreement with the uncertainties on power spectra over the same` range, as shown for fsky = 0.5 in Fig. 2 of Planck CollaborationInt. XXX (2016). The statistical noise on VEE is estimatedfrom the cross-HalfMissions between their two HalfRing half-differences, NEE = (Eb
353,HR1 − Eb353,HR2)HM1/2 × (Eb
353,HR1 −
Eb353,HR2)HM2/2 as
σVEE (HL) =σNEE
√NHL
= 0.02 µK2CMB. (37)
A similar procedure is applied to compute the uncertainty onVBB. The ratio of the B353 and E353 variances differs slightlyfrom the measurement at the power spectra level (PlanckCollaboration Int. XXX 2016), probably because of the multi-pole range over which the ratio of the B353 and E353 variancesare computed.
We compute the covariances of the three maps over the high-latitude sky using the relations
VT E (HL) =1
NHL
NHL∑i=1
Db353 Eb
353 = (124.1 ± 1.4) µK2CMB, (38)
VT B (HL) =1
NHL
NHL∑i=1
Db353 Bb
353 = (3.0 ± 1.5) µK2CMB, (39)
VEB (HL) =1
NHL
NHL∑i=1
Eb353 Bb
353 = (−0.2 ± 0.3) µK2CMB. (40)
The above covariances divided by VEE are listed in Table 1. Theratio VT E/VEE that we find is consistent with the measurementsof Planck Collaboration Int. XXX (2016) at the power spectrumlevel.
Fig. 12. Map of the selected pixels (grey colour) used in the stackinganalysis. It covers 28% of the high-latitude sky. Each tile in the imageis a 7◦ × 5◦ patch around the filament centre and rotated by θ−.
Table 1. Ratios of the variances computed from the selected pixels (SP)used in the stacking analysis and the rest (O) of the high-latitude sky(HL).
Ratio SP O HL
VBB/VEE 0.66 ± 0.01 0.51 ± 0.05 0.62 ± 0.03
VT E/VEE 2.74 ± 0.04 2.48 ± 0.15 2.67 ± 0.07
VT B/VEE −0.07 ± 0.04 0.12 ± 0.04 0.06 ± 0.03
VEB/VEE −0.01 ± 0.02 0.02 ± 0.02 0.00 ± 0.01
6.3. Contribution of filaments to the variance of the Eand B maps
In this section, we compute the variance of the Eb353 and Bb
353maps over the sky pixels used to produce the stacked images inFig. 10, and compare the values with those measured over theHL region. These pixels are within the 7◦ × 5◦ patches, with anorientation angle θ−, centred on the filaments (Sect. 5.1). Thesepixels define the grey regions in Fig. 12. We label them as SPand the rest of the high-latitude sky as O. The stacking procedureincludes the filaments along with their surrounding backgroundemission and, hence, effectively increases the selected fraction ofthe high-latitude sky, f1, from 2.2% (filament pixels as describedin Sect. 3.2) to 28%.
We compute the variance from the SP pixels using the rela-tion given in Eq. (34),
VEE (SP) = (137.5 ± 1.4) µK2CMB. (41)
The sky variance of the Eb353 map in the high-latitude sky can
be written as the sum of contributions from SP ( f1 = 0.28) andO (1 − f1 = 0.72) regions. It is given by
VEE (HL) = f1 × VEE (SP) + (1 − f1) × VEE (O). (42)
The ratio (RSP) of the variance from the stacked pixels to thetotal sky variance is given by
RSP =f1 × VEE (SP)
VEE (HL)= 0.83. (43)
The value of RSP is expected to be high, since the filaments arebright structures on the sky. The pixels we used for stacking con-tribute 83% of the total sky variance in the high-latitude sky. Asimilar result has been reported for the synchrotron emission,where bright filaments/shells are also measured to contributemost of the sky variance in polarization (Vidal et al. 2015).
A141, page 11 of 17
A&A 586, A141 (2016)
It has been noted that the structure in the Planck 353 GHzdust polarization maps is not fully accounted for by the fila-ments seen in the total dust intensity map. In particular, thelocal dispersion of the polarization angle shows structures inthe polarization maps that have no counterpart in total inten-sity (Planck Collaboration Int. XIX 2015). These structures arethought to trace morphology of BPOS uncorrelated with matterstructures (Planck Collaboration Int. XX 2015). However, as ourRSP value shows, these polarization structures do not contributemuch to the variance of the dust polarization.
In the same way as in Sect. 6.2, we compute the variancesVEE , VBB, VT E , VT B, and VEB over the SP and O regions. Table 1presents the ratios of the variances computed over different skyregions.
6.4. Analytical approximation
Following Zaldarriaga (2001) and the description of the fila-ments in Eqs. (19) and (20), we can express the E- and B-modesof a given filament as
EF353 ' − Q′F353 = PF
353 cos 2∆Fχ−θ−
, (44)
BF353 ' − U′F353 = − PF
353 sin 2∆Fχ−θ−
, (45)
where PF353 is the mean polarization intensity of the filament.
The direct relation between EF353 and Q′F353, and BF
353 and U′F353only holds for an idealised filament. For N idealized filamentsoriented arbitrarily on the sky with respect to the GN, the ratiobetween the variances of the B and E maps is given by
VBB
VEE =〈BF
353BF353〉
〈EF353EF
353〉=
∑ 2`+14π CBB
` w2`∑ 2`+1
4π CEE`
w2`
, (46)
which is expanded in terms of power spectra CBB` and CEE
` un-der the assumption of statistical isotropy and homogeneity. Thebeam window function (w`) is the filter function. Both the ob-served CBB
` and CEE` dust power spectra follow a power-law
model with the same slope α (Planck Collaboration Int. XXX2016). This reduces Eq. (46) to
VBB
VEE =
∑(2` + 1) ABB `−α w2
`∑(2` + 1) AEE `−α w2
`
=ABB
AEE · (47)
From the histogram of the upper panel of Fig. 6, the distributionof the angle ∆F
χ−θ−is known for our filament sample. Similar to
the assumption made in Sect. 5.2, we assume that all the fila-ments have the same polarized intensity and therefore
ABB
AEE '〈sin2 2∆F
χ−θ−〉
〈cos2 2∆Fχ−θ−〉
= 0.66. (48)
We have computed the ratio ABB/AEE using the two independentsubsets of the Planck data (HM maps) and find the same meanvalue of 0.66. This value of the ABB/AEE ratio based on this an-alytical model matches the observed mean value of 0.62 ± 0.03(Sect. 6.2). We note that the model value is directly inferred fromthe distribution of ∆F
χ−θ−for our filament sample. If the HRO of
∆Fχ−θ−
was flat with uniform probability between −90◦ and +90◦,we would have found equal variances in both the E353 and B353maps.
In summary, we propose that the alignment between BPOSand the filament orientations accounts for the E−B asymmetry
in the range of angular scales 30 < ` < 300. The mean valueof VBB/VEE for O region is consistent with the HL value within1.9σ (Table 1). This shows that a similar alignment between thematter structures and BPOS can be inferred over the rest of thehigh-latitude sky. Some high Galactic latitude sky areas, such asthe BICEP2 field (BICEP2 Collaboration 2014), do not includeany of the strong filaments from our study. The Planck 353 GHzpolarization maps do not have the required signal-to-noise ratioto measure the ABB/AEE ratio for individual BICEP2-like fields(Planck Collaboration Int. XXX 2016). Therefore more sensitiveobservations will be needed to test whether our interpretation isrelevant there.
7. Relation to Galactic astrophysics
In this section we place the paper results in the context of earlierstudies about the filamentary structure of interstellar matter andits correlation with the Galactic magnetic field.
Over the last decades, observations of interstellar gas anddust have been revealing the filamentary structure of the ISMin increasing details. Before Planck and Herschel, the discov-ery of the infrared cirrus with the IRAS and H all-sky sur-veys was a main milestone in our perception of the structure ofthe diffuse ISM (Boulanger 1994; Kalberla & Kerp 2009). Athigh Galactic latitude, the Planck dust emission is tightly corre-lated with H emission at local velocities (Planck CollaborationInt. XVII 2014). In particular, all of the filaments in our sam-ple have an H counterpart. They are selected on the 353 GHzmap but are seen at far-infrared wavelengths, in particular theIRAS 100 µm map.
The interstellar filaments seen in the Planck 353 GHz dustintensity map are not all straight. With our filament-finding al-gorithm, we have identified the straight segments with lengthsL ≥ 2◦. Some of these segments are pieces of longer non-straightfilaments. The 259 filaments in our sample make most of the skyvariance in polarization as measured in our analysis. This is nota complete sample but other filaments at high Galactic latitudedo not contribute much to the dust power in E-modes.
A number of studies, starting with the pioneering work ofGoodman et al. (1990), have used the polarization of backgroundstarlight to investigate the relative orientation between the inter-stellar filaments and BPOS. While most studies have targeted fil-aments identified in extinction maps of molecular clouds in thesolar neighbourhood, a few studies have focussed on filamentsseen in H emission (McClure-Griffiths et al. 2006; Clark et al.2014). These last two papers report a preferred alignment be-tween the filaments and BPOS in the diffuse ISM. The analysis ofPlanck data has complemented earlier studies providing greaterstatistics and sensitivity. Planck Collaboration Int. XXXII (2016)compare the orientations of matter structures identified in thePlanck 353 GHz map with that of the Galactic magnetic fieldat intermediate latitudes. The alignment between the filamentsand BPOS reported in this paper becomes weaker for increasingcolumn density (see Fig. 15 of Planck Collaboration Int. XXXII2016). Towards molecular clouds the relative orientation is ob-served to change progressively from preferentially parallel inareas with the lowest column density to preferentially perpen-dicular in the areas with the highest column density (PlanckCollaboration Int. XXXV 2016). The transition occurs at a col-umn density of 1021.7 cm−2. All the filaments considered in ouranalysis are much below this transition limit and observed, as ex-pected from earlier studies, to be statistically aligned with BPOS.
A141, page 12 of 17
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Planck Collaboration Int. XXXII (2016) and PlanckCollaboration Int. XXXV (2016) discuss these observational re-sults in light of MHD simulations, which quantify the respectiveroles of the magnetic field, turbulence, and gas self-gravity inthe formation of structures in the magnetized ISM. The align-ment between the magnetic field and matter structures in the dif-fuse ISM is thought to be a signature of turbulence. Simulationsshow that turbulent flows will tend to stretch gas condensationsinto sheets and filaments, which appear elongated in columndensity maps (Hennebelle 2013). These structures will tend tobe aligned with the magnetic field where the gas velocity isdynamically aligned with the field (Brandenburg & Lazarian2013). Alignment also results from the fact that matter and themagnetic field are stretched in the same direction because thefield is frozen into matter. The change in relative orientationobserved within molecular clouds might be a signature of theformation of gravitationally bound structures in the presenceof a dynamically important magnetic field. Indeed, Soler et al.(2013) report a change in the relative orientation between matterstructures and the magnetic field, from parallel to perpendicular,for gravitationally bound structures in MHD simulations. Thischange is most significant for their simulation with the highestmagnetization.
8. Conclusion
We present a statistical study of the filamentary structure of the353 GHz Planck Stokes maps at high Galactic latitude, relevantto the study of dust emission as a polarization foreground to theCMB. The main results of our work are summarized as follows.
We filter the intensity and polarization maps to isolate struc-tures over the range of angular scales where the E−B powerasymmetry is observed. From a Hessian analysis of the Plancktotal dust intensity map at 353 GHz, we identify a sample of259 filaments in the high-latitude sky with lengths L ≥ 2◦. Wemeasure the mean orientation angle of each filament in this sam-ple and find that the filaments are statistically aligned with theplane of the sky component of the magnetic field, BPOS, inferredfrom the polarization angles measured by Planck. We also findthat the orientation of BPOS is correlated with that of Bm,POS inthe solar neighbourhood. Our results show that the correlationbetween the structures of interstellar matter and BPOS in the dif-fuse ISM reported in Planck Collaboration Int. XXXII (2016) forintermediate Galactic latitudes also applies to the lower columndensity filaments (a few 1019 cm−2) observed at high Galacticlatitude.
We present mean images of our filament sample in dust in-tensity and Stokes Q353 and U353 with respect to the filamentorientation (Q′353 and U′353), computed by stacking individual7◦ × 5◦ patches centred on each filament. The stacked imagesshow that the contribution of the filaments is a negative featurewith respect to the background in the Q′353 image and is not seenin the U′353 image. This result directly follows from the fact thatthe histogram of relative orientation between the filaments andBPOS peaks and is symmetric around 0◦. Combining the stackedimages and the histogram, we estimate the mean polarizationfraction of the filaments to be 11%.
We relate the E−B asymmetry discovered in the power spec-trum analysis of Planck 353 GHz polarization maps (PlanckCollaboration Int. XXX 2016) to the alignment between the fil-aments and BPOS in the diffuse ISM. The set of 7◦ × 5◦ patcheswe stack represents 28% of the sky area at high Galactic lati-tude. The power of the E-mode dust polarization computed overthis area amounts to 83% of the total dust polarization power in
the high-latitude sky. We show with an analytical approximationof the filaments (based on the work of Zaldarriaga 2001), thatthe HRO between the filaments and BPOS may account for theCBB` /CEE
` ratio measured over the high-latitude sky. Our inter-pretation could also apply to the E−B asymmetry reported forthe synchrotron emission (Planck Collaboration X 2015), sincethere is also a correlation between the orientation angle of BPOSand the filamentary structures of the synchrotron intensity map(Vidal et al. 2015; Planck Collaboration XXV 2015).
Present models of the dust polarization sky (e.g., O’Deaet al. 2012; Delabrouille et al. 2013) produce an equal amountof power in E- and B-modes for masks excluding the Galacticplane, because they ignore the correlation between the structureof the magnetic field and that of matter. Our work should mo-tivate a quantitative modelling of the polarized sky, which willtake into account the observed correlations between the Galacticmagnetic field and the structure of interstellar matter.
Acknowledgements. The Planck Collaboration acknowledges the support of:ESA; CNES, and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF(Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO,JA and RES (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG(Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland);RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); ERC and PRACE(EU). A description of the Planck Collaboration and a list of its mem-bers, indicating which technical or scientific activities they have been in-volved in, can be found at http://www.cosmos.esa.int/web/planck/planck-collaboration. The research leading to these results has re-ceived funding from the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007−2013)/ERC grant agree-ment No. 267934. Some of the results in this paper have been derived usingthe HEALPix package.
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1 APC, AstroParticule et Cosmologie, Université Paris Diderot,CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne ParisCité, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13,France
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32 Dipartimento di Matematica, Università di Roma Tor Vergata, viadella Ricerca Scientifica, 1, 00133 Roma, Italy
33 Discovery Center, Niels Bohr Institute, Blegdamsvej 17, 2100Copenhagen, Denmark
34 European Southern Observatory, ESO Vitacura, Alonso de Cordova3107, Vitacura, 19001 Casilla, Santiago, Chile
35 European Space Agency, ESAC, Planck Science Office, Caminobajo del Castillo, s/n, Urbanización Villafranca del Castillo, 28692Villanueva de la Cañada, Madrid, Spain
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Ricerca Scientifica, 1, 00185 Roma, Italy51 IPAG: Institut de Planétologie et d’Astrophysique de Grenoble,
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57 Institute of Astronomy, University of Cambridge, Madingley Road,Cambridge CB3 0HA, UK
58 Institute of Theoretical Astrophysics, University of Oslo, Blindern,0371 Oslo, Norway
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93 Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478Warszawa, Poland
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−80 80KCMB rad−2
λ−
−2.5 2.5KCMB rad−2
-2.0 -1.5 K− -0.8 -0.5 0.0 0.5 1.0 1.5
λ− [KCMB rad−2]
0.00
00.
003
0.00
60.
009
0.01
2
PD
F
3σλ−
Fig. A.1. Filtered realization of the Gaussian dust sky (upper panel) andthe corresponding eigenvalue λ− (middle panel). The distribution of theeigenvalue λ− (lower panel) over the unmasked pixels is Gaussian.
Appendix A: Hessian analysisIn this Appendix, we detail the implementation of the Hessiananalysis to determine the orientations of the filaments. We startwith the Db
353 map at HEALPix resolution Nside = 512. For eachpixel on the sky, we estimate the first and second derivatives ofDb
353 with respect to the Galactic longitude l and latitude b (asdescribed in Monteserín et al. 2005). The Hessian matrix of Db
353is defined as
H =
[Hxx HxyHxy Hyy
], (A.1)
where
Hxx =∂2Db
353
∂2b, (A.2)
Hxy =∂2Db
353
cos b ∂b ∂l, (A.3)
Hyy =∂2Db
353
cos2 b ∂2l· (A.4)
We decompose the Db353 map into a`m coefficients using the
ianafast routine of HEALPix and then use the isynfast rou-tine in HEALPix to compute the second partial derivatives at eachpixel. This method is computationally faster than the one usedin Planck Collaboration Int. XXXII (2016). The smallest eigen-value, λ−, of the Hessian matrix is calculated as
λ− =12
(Hxx + Hyy − α), (A.5)
where α =
√(Hxx − Hyy)2 + 4H2
xy. The map of λ−, displayed in
the lower left panel of Fig. 2, highlights the filaments in the Db353
map. The filament orientation angle, θ−, is calculated using therelation
θ− = atan[−
Hxx − Hyy + α
2Hxy
]· (A.6)
This follows the IAU convention, since it is measured from theGN and positive to the East direction. The formula for the fila-ment orientation angle θ− is equivalent to equation (9) of PlanckCollaboration Int. XXXII (2016).
Appendix B: Gaussian realization of the dust sky
Here we apply the SMAFF algorithm on a Gaussian realizationof the dust total intensity map using the same methodology asdescribed in Sect. 3.2. The power spectrum is modelled as CTT
` ∝
`α, where α is the slope of the power law. We choose α = −2.4in the multipole range from ` = 30 to 300 (Planck CollaborationInt. XXII 2015).
We compute a Gaussian realization of a sky map on aHEALPix grid with an `−2.4 power spectrum, and use the splinewavelet transform to filter it. Hereafter, we refer to the filteredsimulated Gaussian map as the “S b map”. We compute thesmallest eigenvalue λ− of the S b map using the Hessian analysisdescribed in Appendix A. The S b map and its corresponding λ−map are shown in the upper and middle panels of Fig. A.1.
We only consider the high-latitude sky. With the upperthreshold K− = mλ− − 3σλ− , we remove pixels as shown in thelower panel of Fig. A.1. Running our friend-of-friend algorithmon the S b map, we do not detect any filament with L ≥ 2◦. Forhigher values of the threshold K−, we detect such filaments inthe S b map. We call them “weak” filaments. The threshold fac-tor K−, used in our study, is a key factor to separate the strongfilaments from the weak ones. The main results of this paper fol-low from the statistical properties of strong filaments. However,in Appendix C we demonstrate that they still hold for the weakfilaments.
Appendix C: Effect of λ− thresholdingon the filament count
We apply the threshold K− = mλ− − 3σλ− in Sect. 3.2 to findstrong filaments in the Db
353 map. In this section, we changethe threshold K− to quantify its effect on the HRO between thefilaments and BPOS. We choose different thresholds and dividethe selected λ− coherent structures into two categories, namelystrong and weak filaments. The strong filaments are selectedfrom the sky pixels that have λ− < mλ− − 3σλ− . Whereas,the weak filaments are selected from the sky pixels that havemλ− − 3σλ− ≤ λ− < mλ− − pσλ− , where p is a factor to definedthe threshold K− = mλ− − pσλ− . In Table C.1, we list the numberof weak filaments for different values of p.
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Planck Collaboration: E- and B-modes of dust polarization
−80 −60 −40 −20 0 20 40 60 80∆χ−θ− [deg]
020
4060
80
NF
518 filaments
259 filaments
Fig. C.1. Same as the upper panel of Fig. 6, when including angle differ-ences ∆χ−θ− from the weak filaments. The width of the histogram doesnot change.
Table C.1. Total filament count, including the strong and weak fila-ments, as a function of the threshold K− on the λ− map derived from theDb
353 map.
Factor Filament countp weak strong total
mλ− − 3σλ− ≤ λ− < mλ− − pσλ− λ− < mλ− − 3σλ−
3 0 259 2592 53 259 3121.5 140 259 3991 259 259 518
By choosing a threshold K− = mλ− − 1σλ− and running ourfriend-of-friend algorithm on the Db
353 map, we double the num-ber of filaments with L ≥ 2◦. For this larger set, we compute theHRO between the filaments and BPOS, and present it in Fig. C.1.The 1σ dispersion of the angle difference remains the same asfor our nominal set. This shows that the alignment between thestructures of interstellar matter and BPOS holds even when in-cluding the weak filaments.
Including the weak filaments and their surrounding back-ground emission in the analysis of Sect. 6.3 effectively increasesthe sky fraction from 28% to 50% of the high-latitude sky. Withmore sky coverage, the ratio of the variances, i.e., VBB/VEE ,VT E/VEE , VT B/VEE , and VEB/VEE is close to the HL region val-ues quoted in Table 1.
Appendix D: Uncertainties on the angles θ− and χ
In Sect. 4 we present our statistical analysis of the relative orien-tation between the filaments and BPOS. The filament orientationangle, θ−, is a single number computed from the Db
353 map.To estimate the error bar on θ−, we propagate the noise in theD353 map (ND353 ) through the Hessian analysis. We simulate aGaussian realization of the ND353 map and add it to the D353 map.
−30 −24 −18 −12 −6 0 6 12 18 24 30
[deg]
0.0
0.1
0.2
0.3
0.4
0.5
PD
F
∆θ
∆χ
∆Θ
Fig. D.1. Histogram of changes in θ− (red) and χ (blue) due to noisepresent in the Planck data. The histogram of changes in θ− from thedifference of the filtered and unfiltered D353 maps is shown in green.
We filter the noise added to the D353 map in the same manner asthe D353 map. For each filament in our sample, we compute thechange of its orientation angle from the added noise using theformula
∆θ− = θ− (Db353 + Nb
D353) − θ− (Db
353). (D.1)
The distribution of the angle difference is presented with redcolour in Fig. D.1. The mean of the distribution is 0◦ and the 1σdispersion is 0.2◦. This shows that the noise in the D353 map haslittle impact on our estimate of the filament orientation angle.
Next, we compute the uncertainty on the orientation angle ofBPOS. We use the two HM Planck polarization maps, and com-pute the orientation angle of BPOS using Eq. (2). For each fila-ment in our sample, we compute the difference between the twovalues obtained from the two HM maps,
∆χ = 0.5 × [χHM1 − χHM2]. (D.2)
The histogram of the angle difference is presented with bluecolour in Fig. D.1. The mean of the distribution is 0◦ and the1σ dispersion is 2◦. The error on the orientation angle of BPOSis small compared to the width of the HROs in Fig. 6.
Last, we assess the impact of the map filtering on our anal-ysis, computing the mean orientations of the filaments, over thesame set of selected pixels, on the D353 map. We compute theangle difference of the filament orientation derived from the fil-tered and unfiltered Planck 353 GHz maps
∆Θ = θ− (D353) − θ− (Db353). (D.3)
The histogram of ∆Θ is plotted with green colour in Fig. D.1.The mean of the distribution is 0◦ and the 1σ dispersion is 7.0◦.All the uncertainties measured in θ− and χ are small comparedto the dispersion of the HRO measured in Fig. 6. This means thatthe data noise (for the intensity and polarization maps) and thefiltering of the data are not critical for our study.
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