Planck intermediate results

A&A 586, A133 (2016)

DOI: 10.1051

0004-6361

201425034

ESO 2016

Astronomy

Astrophysics

Planck

intermediate results

XXX. The angular power spectrum

of polarized dust emission at intermediate and high Galactic latitudes

Planck Collaboration: R. Adam

, P. A. R. Ade

, N. Aghanim

, M. Arnaud

, J. Aumont

, C. Baccigalupi

, A. J. Banday

R. B. Barreiro

, J. G. Bartlett

, N. Bartolo

, E. Battaner

102

103

, K. Benabed

, A. Benoit-Lévy

, J.-P. Bernard

, M. Bersanelli

P. Bielewicz

, A. Bonaldi

, L. Bonavera

, J. R. Bond

, J. Borrill

, F. R. Bouchet

, F. Boulanger

, A. Bracco

, M. Bucher

C. Burigana

, R. C. Butler

, E. Calabrese

, J.-F. Cardoso

, A. Catalano

, A. Challinor

, A. Chamballu

, R.-R. Chary

H. C. Chiang

, P. R. Christensen

, D. L. Clements

, S. Colombi

, L. P. L. Colombo

, C. Combet

, F. Couchot

, A. Coulais

B. P. Crill

, A. Curto

, F. Cuttaia

, L. Danese

, R. D. Davies

, R. J. Davis

, P. de Bernardis

, G. de Zotti

, J. Delabrouille

J.-M. Delouis

, F.-X. Désert

, C. Dickinson

, J. M. Diego

, K. Dolag

101

, H. Dole

, S. Donzelli

, O. Doré

, M. Douspis

A. Ducout

, J. Dunkley

, X. Dupac

, G. Efstathiou

, F. Elsner

, T. A. Enßlin

, H. K. Eriksen

, E. Falgarone

, F. Finelli

O. Forni

, M. Frailis

, A. A. Fraisse

, E. Franceschi

, A. Frejsel

, S. Galeotta

, S. Galli

, K. Ganga

, T. Ghosh

, M. Giard

Y. Giraud-Héraud

, E. Gjerløw

, J. González-Nuevo

, K. M. Górski

104

, S. Gratton

, A. Gregorio

, A. Gruppuso

, V. Guillet

F. K. Hansen

, D. Hanson

, D. L. Harrison

, G. Helou

, S. Henrot-Versillé

, C. Hernández-Monteagudo

, D. Herranz

E. Hivon

, M. Hobson

, W. A. Holmes

, K. M. Hu

enberger

, G. Hurier

, A. H. Ja

, T. R. Ja

, J. Jewell

, W. C. Jones

M. Juvela

, E. Keihänen

, R. Keskitalo

, T. S. Kisner

, R. Kneissl

, J. Knoche

, L. Knox

, N. Krachmalnico

, M. Kunz

H. Kurki-Suonio

, G. Lagache

, J.-M. Lamarre

, A. Lasenby

, M. Lattanzi

, C. R. Lawrence

, J. P. Leahy

, R. Leonardi

J. Lesgourgues

, F. Levrier

, M. Liguori

, P. B. Lilje

, M. Linden-Vørnle

, M. López-Caniego

, P. M. Lubin

, J. F. Macías-Pérez

B. Ma

, D. Maino

, N. Mandolesi

, A. Mangilli

, M. Maris

, P. G. Martin

, E. Martínez-González

, S. Masi

, S. Matarrese

P. Mazzotta

, P. R. Meinhold

, A. Melchiorri

, L. Mendes

, A. Mennella

, M. Migliaccio

, S. Mitra

, M.-A. Miville-Deschênes

A. Moneti

, L. Montier

, G. Morgante

, D. Mortlock

, A. Moss

, D. Munshi

, J. A. Murphy

, P. Naselsky

, F. Nati

, P. Natoli

C. B. Netterfield

, H. U. Nørgaard-Nielsen

, F. Noviello

, D. Novikov

, I. Novikov

, L. Pagano

, F. Pajot

, R. Paladini

, D. Paoletti

B. Partridge

, F. Pasian

, G. Patanchon

, T. J. Pearson

, O. Perdereau

, L. Perotto

, F. Perrotta

, V. Pettorino

, F. Piacentini

, M. Piat

E. Pierpaoli

, D. Pietrobon

, S. Plaszczynski

, E. Pointecouteau

, G. Polenta

, N. Ponthieu

, L. Popa

, G. W. Pratt

, S. Prunet

J.-L. Puget

, J. P. Rachen

, W. T. Reach

100

, R. Rebolo

, M. Remazeilles

, C. Renault

, A. Renzi

, S. Ricciardi

I. Ristorcelli

, G. Rocha

, C. Rosset

, M. Rossetti

, G. Roudier

, B. Rouillé d’Orfeuil

, J. A. Rubiño-Martín

, B. Rusholme

M. Sandri

, D. Santos

, M. Savelainen

, G. Savini

, D. Scott

, J. D. Soler

, L. D. Spencer

, V. Stolyarov

, R. Stompor

R. Sudiwala

, R. Sunyaev

, D. Sutton

, A.-S. Suur-Uski

, J.-F. Sygnet

, J. A. Tauber

, L. Terenzi

, L. To

olatti

M. Tomasi

, M. Tristram

, M. Tucci

, J. Tuovinen

, L. Valenziano

, J. Valiviita

, B. Van Tent

, L. Vibert

, P. Vielva

, F. Villa

L. A. Wade

, B. D. Wandelt

, R. Watson

, I. K. Wehus

, M. White

, S. D. M. White

, D. Yvon

, A. Zacchei

, and A. Zonca

ffi

liations can be found after the references)

Received 19 September 2014

Accepted 1 December 2014

ABSTRACT

The polarized thermal emission from di

use Galactic dust is the main foreground present in measurements of the polarization of the cosmic

microwave background (CMB) at frequencies above 100 GHz. In this paper we exploit the uniqueness of the

Planck

HFI polarization data from 100

to 353 GHz to measure the polarized dust angular power spectra

and

over the multipole range 40

< ` <

600 well away from the Galactic

plane. These measurements will bring new insights into interstellar dust physics and allow a precise determination of the level of contamination for

CMB polarization experiments. Despite the non-Gaussian and anisotropic nature of Galactic dust, we show that general statistical properties of the

emission can be characterized accurately over large fractions of the sky using angular power spectra. The polarization power spectra of the dust are

well described by power laws in multipole,

∝

, with exponents

−

02. The amplitudes of the polarization power spectra vary

with the average brightness in a way similar to the intensity power spectra. The frequency dependence of the dust polarization spectra is consistent

with modified blackbody emission with

59 and

6 K down to the lowest

Planck

HFI frequencies. We find a systematic di

erence

between the amplitudes of the Galactic

- and

-modes,

5. We verify that these general properties are preserved towards high

Galactic latitudes with low dust column densities. We show that even in the faintest dust-emitting regions there are no “clean” windows in the sky

where primordial CMB

-mode polarization measurements could be made without subtraction of foreground emission. Finally, we investigate the

level of dust polarization in the specific field recently targeted by the BICEP2 experiment. Extrapolation of the

Planck

353 GHz data to 150 GHz

gives a dust power

≡

(

) of 1

−

CMB

over the multipole range of the primordial recombination bump (40

< ` <

120);

the statistical uncertainty is

−

CMB

and there is an additional uncertainty (

−

24)

−

CMB

from the extrapolation. This

level is the same magnitude as reported by BICEP2 over this

range, which highlights the need for assessment of the polarized dust signal even in

the cleanest windows of the sky.

Key words.

cosmic background radiation – cosmology: observations – ISM: structure – ISM: magnetic fields – polarization

Corresponding author: J. Aumont,

e-mail:

jonathan.aumont@ias.u-psud.fr

Article published by EDP Sciences

A133, page 1 of 25

A&A 586, A133 (2016)

1. Introduction

The sky at high Galactic latitude and frequencies above about

100 GHz is dominated by thermal emission from the Galactic

interstellar medium, specifically arising from dust grains of

about 0.1

m in size. Asymmetrical dust grains align with the

Galactic magnetic field to produce polarized emission. This po-

larized submillimetre emission has been measured from ground-

based and balloon-borne telescopes (e.g., Hildebrand et al. 1999;

Benoît et al. 2004; Ponthieu et al. 2005; Vaillancourt 2007;

Matthews et al. 2014). The observed polarization relates to the

nature, size, and shape of dust grains and the mechanisms of

alignment, discussed for example by Draine (2004) and Martin

(2007). It also probes the structure of the Galactic magnetic field,

which is an essential component of models of Galactic dust po-

larization (Baccigalupi 2003; Fauvet et al. 2011, 2012; O’Dea

et al. 2012; Ja

e et al. 2013; Delabrouille et al. 2013).

The polarized emission from dust is also of interest in the

context of foregrounds (Tucci et al. 2005; Dunkley et al. 2009a;

Gold et al. 2011) to the cosmic microwave background (CMB).

On angular scales between 10

′

and a few tens of degrees, cos-

mological

-mode polarization signals may be present that were

imprinted during the epoch of inflation. The discovery of a pri-

mordial

-mode polarization signature is a major scientific goal

of many CMB experiments. These include ground-based ex-

periments (ACTPol, Niemack et al. 2010; BICEP2, BICEP2

Collaboration 2014a; Keck-array, Staniszewski et al. 2012;

POLARBEAR, Arnold et al. 2010; QUBIC, Ghribi et al. 2014;

QUIJOTE, Rubiño-Martín et al. 2010; and SPTpol, Austermann

et al. 2012), stratospheric balloon missions (EBEX, Grainger

et al. 2008; and SPIDER, Fraisse et al. 2013), and the ESA

Planck

satellite (Tauber et al. 2010). Accurate assessment and,

if necessary, subtraction of foreground contamination is critical

to the measurement of CMB

- and

-mode polarization be-

cause the expected signals from inflation and late-time reioniza-

tion are expected to be small.

Planck

has measured the all-sky dust polarization at

353 GHz, where the dust emission dominates over other po-

larized signals. These data have been presented in a first set

of publications in which the focus was on the structure of the

Galactic magnetic field and the characterization of dust polar-

ization properties (Planck Collaboration Int. XIX 2015; Planck

Collaboration Int. XX 2015; Planck Collaboration Int. XXI

2015; Planck Collaboration Int. XXII 2015). Here, we use the

Planck

polarized data to compute the

and

power spec-

tra of dust polarization over the multipole range 40

< ` <

600,

on large fractions of the sky away from the Galactic plane. We

also investigate dust polarization in sky patches at high Galactic

latitude with sizes comparable to those surveyed by ground-

based CMB experiments. We derive statistical properties of dust

polarization from these spectra, characterizing the shape of the

spectra and their amplitude with respect to both the observing

frequency and the mean dust intensity of the sky region over

which they are computed. We verify that these properties hold

in low-column-density patches at high Galactic latitude and we

explore statistically the potential existence of “clean” patches on

the sky that might be suitable for cosmology.

Planck

(

http://www.esa.int/Planck

) is a project of the

European Space Agency (ESA) with instruments provided by two sci-

entific consortia funded by ESA member states (in particular the lead

countries France and Italy), with contributions from NASA (USA) and

telescope reflectors provided by a collaboration between ESA and a sci-

entific consortium led and funded by Denmark.

Our analysis of dust polarization is relevant to the present

generation of CMB polarization observations, and to the design

of future experiments. It gives a statistical description of Galactic

dust polarization, providing input for the modelling of Galactic

dust as part of component separation methods and for CMB po-

larization likelihood analysis parameterization.

The BICEP2 collaboration has recently reported a significant

detection of the

-mode power spectrum around the expected

angular scale of the recombination bump, namely a few degrees

(BICEP2 Collaboration 2014a,b). Their analysis was based on

dust polarization models that predicted subdominant contamina-

tion of their

-mode signal by dust polarization. We use infor-

mation from our detailed analysis of

Planck

polarization data at

353 GHz to assess the potential dust contamination.

The paper is organized as follows. In Sect. 2, we present

the

Planck

HFI polarization data used in this work and de-

scribe the general properties of the polarization maps in terms

of emission components and systematic e

ects. In Sect. 3, we

describe our method for computing the dust

and

an-

gular power spectra, including the selected science regions of

interest on the sky. We assess and compare the two methods we

use to compute the power spectra in Appendix A. In Sect. 4,

we present power spectra of dust polarization for multipoles

` >

40, computed with high signal-to-noise ratio (S

N) on large

fractions of the sky, and characterize their shape and ampli-

tude. Complementary angular power spectra involving temper-

ature and polarization,

and

T B

, and cross-polarization,

are given in Appendix B. We extend this analysis to smaller sky

patches at high Galactic latitude in Sect. 5. In Appendix C we

discuss some complementary aspects of this analysis of patches.

These results are used specifically in Sect. 5.3 to build a map

of the expected dust contamination of the

power spectrum

at 150 GHz and

80. In Sect. 6 we present a study of the

polarized dust emission in the vicinity of the BICEP2 region.

Systematic e

ects relating to the

Planck

angular power spectrum

estimates are assessed in Appendix D and we discuss the decor-

relation of the dust signal between frequencies in Appendix E.

Section 7 summarizes the main conclusions and discusses the

implications of this work for future CMB experiments

Planck

polarization maps

2.1. Planck data

The Planck Collaboration recently released the

Planck

satellite

nominal mission temperature data and published a set of papers

describing these data and their cosmological interpretation (e.g.,

Planck Collaboration I 2014; Planck Collaboration XVI 2014).

These results are based on the data from the two instruments on

board the satellite (Low Frequency Instrument, LFI, Mennella

et al. 2011, and High Frequency Instrument, HFI, Planck HFI

Core Team 2011). The data processing of the nominal mission

data (Surveys 1 and 2, 14 months) was summarized in Planck

Collaboration II (2014) and Planck Collaboration VI (2014).

Planck

HFI measures the linear polarization at 100, 143, 217,

and 353 GHz (Rosset et al. 2010). The properties of the detectors

(sensitivity, spectral response, noise properties, beams, etc.) are

described in detail in Lamarre et al. (2010) and their in-flight per-

formance is reported in Planck HFI Core Team (2011), Planck

Collaboration VII (2014), Planck Collaboration VIII (2014),

While this paper was in preparation three papers have used publicly

available polarization information from

Planck

to infer potentially high

levels of dust contamination in the BICEP2 field (Flauger et al. 2014;

Mortonson & Seljak 2014; Colley & Gott 2015).

A133, page 2 of 25

Planck Collaboration: Dust polarization at high latitudes

Planck Collaboration IX (2014), and Planck Collaboration X

(2014), while Planck Collaboration VI (2014) describes the

general processing applied to the data to measure polarization.

In this paper, we make use of full-mission (Surveys 1 to 5,

30 months, Planck Collaboration I 2014) polarization maps of

Planck

HFI (internal data release “DX11d”), projected into the

HEALPix

pixelization scheme (Górski et al. 2005). This is one of

the first publications to use these maps, which will be described

in the

Planck

cosmology 2015 release.

To compute polarization angular power spectra, we use

and

maps at 100, 143, 217, and 353 GHz. Specifically, we

calculate power spectra using the so-called “Detector-Set” maps

(hereafter “DetSets”), constructed using two subsets of polar-

ization sensitive bolometers (PSBs) at a given frequency (see

Table 3 in Planck Collaboration VI 2014). Each DetSet polariza-

tion map is constructed using data from two pairs of PSBs, with

the angle between the two PSBs in a pair being 90

◦

, and the an-

gle between pairs being 45

◦

. In this paper we concentrate on the

and

maps at 353 GHz. The Stokes

and

maps at lower

frequencies (100, 143, and 217 GHz) are only used to determine

the spectral energy distribution (SED) of the dust emission in

polarization.

To quantify systematic e

ects, we also use maps made from

other data subsets (Planck Collaboration VI 2014). We use the

ring halves (hereafter “HalfRing”), where the approximately

60 circles performed for each

Planck

telescope ring (also called a

stable pointing period) are divided into two independent subsets

of 30 circles. We use observational years (hereafter “Years”),

consisting of Surveys 1 and 2 on the one hand and Surveys 3 and

4 on the other, to build two additional maps with independent

noise.

The

Planck

maps we use are in thermodynamic units

CMB

). To characterize the SED of the dust emission in po-

larization we express the data as the specific intensity (such as

that for Stokes

dust emission,

(

)) at the

Planck

reference

frequencies, using the conversion factors and colour corrections

from Planck Collaboration IX (2014)

. For the average dust SED

at intermediate Galactic latitudes, the colour correction factor is

1.12 at 353 GHz (see Table 3 in Planck Collaboration Int. XXII

2015).

In addition to these basic products, a

Planck

CO map from

Planck Collaboration XIII (2014), the so-called “type 3” map,

and the

Planck

857 GHz map, are also used in the selection of

the large intermediate latitude analysis regions (see Sect. 3.3.1).

2.2. Emission contributions to the Planck HFI polarization

maps

2.2.1. Polarized thermal dust emission

Thermal dust emission is partially linearly polarized (e.g.,

Hildebrand et al. 1999; Benoît et al. 2004; Ponthieu et al. 2005;

Vaillancourt 2007). It is the dominant polarized foreground

The conversion factor from K

CMB

to MJy sr

−

is computed for a

standard specific intensity

∝

−

. The colour correction depends on

the actual dust SED; it is the scaling factor used to transform from the

specific intensity of the dust emission, at the reference frequency, to

the

Planck

brightness in MJy sr

−

(see Eq. (19) in Planck Collaboration

Int. XXII 2015). The conversion factors and the colour corrections are

computed via Eq. (32) in Planck Collaboration IX (2014) using the

Planck

HFI filters and the

Planck

UcCC

software available through the

Planck

Explanatory Supplement (

http://www.sciops.esa.int/

wikiSI/planckpla/index.php?title=Unit_conversion_and_

Color_correction&instance=Planck_Public_PLA

); we use the

band-average values.

signal in the high frequency

Planck

bands (Tucci et al. 2005;

Dunkley et al. 2009b; Fraisse et al. 2009; Fauvet et al. 2011;

Planck Collaboration Int. XXII 2015).

Dust polarization arises from alignment of non-spherical

grains with the interstellar magnetic field (e.g., Hildebrand 1988;

Draine 2004; Martin 2007). The structure of the dust polarization

sky has already been described using maps of the polarization

fraction (

) and angle (

) derived from the

Planck

HFI 353 GHz

data (Planck Collaboration Int. XIX 2015; Planck Collaboration

Int. XX 2015). The map of

shows structure on all scales,

with polarization fractions ranging from low (less than 1%) to

high values (greater than 18%). Planck Collaboration Int. XIX

(2015) and Planck Collaboration Int. XX (2015) report an anti-

correlation between

and the local dispersion of

, which in-

dicates that variations in

arise mainly from depolarization as-

sociated with changes in the magnetic field orientation within

the beam, rather than from changes in the e

ffi

ciency of grain

alignment.

Planck Collaboration Int. XXII (2015) showed that the SED

of polarized dust emission over the four

Planck

HFI frequencies

from 100 to 353 GHz is consistent with a modified blackbody

emission law of the type

(

)

∝

(

), with spectral index

59 for

6 K

, and where

is the Planck func-

tion. About 39% of the sky at intermediate Galactic latitudes was

analysed

. Among 400 circular patches with 10

◦

radius (equiv-

alent to a sky fraction

sky

0076) the 1

dispersion of

was 0.17 for constant

6 K. We scale this uncertainty on

to larger sky areas by using the factor (0

0076

sky

)

. This

is a conservative choice because this uncertainty includes the ef-

fects of noise in the data and so is an upper limit to the true

regional variations of

on this scale. This polarization spectral

index can be compared to variations in the spectral index

for the intensity SED. For that quantity the S

N of the data is

higher than for polarization and Planck Collaboration Int. XXII

(2015) report a dispersion of 0.07 (1

) over the same sized cir-

cular patches. Planck Collaboration Int. XVII (2014) extend this

analysis for intensity to high Galactic latitudes in the southern

Galactic cap, using the dust-H



correlation to separate the faint

emission of dust from the anisotropies of the cosmic infrared

background, and find a dispersion of about 0.10 in

. We

expect spectral variations to be correlated in polarization and in-

tensity, unless the dust emission has a significant component that

is unpolarized.

2.2.2. CMB

The CMB temperature anisotropies have been measured with

unprecedented accuracy by the Planck Collaboration (Planck

Collaboration I 2014; Planck Collaboration XV 2014), and pre-

liminary

Planck

polarization results have been demonstrated to

be in very good agreement with the cosmology inferred from

temperature measurements (Planck Collaboration I 2014; Planck

Collaboration XVI 2014).

For

, the

CDM concordance model has been shown to

be a very good fit to all the available data (including prelimi-

nary

Planck

results at

&

50; see Barkats et al. 2014, for a

recent compendium). For 353 GHz data at small angular scales

This spectral index was called

in that paper, but we adopt a

more compact notation here.

More specifically, for the latitude range 10

◦

, with patches

contained within the region in Fig. 1 (below) defined by including that

with

sky

8 and then removing that with

sky

A133, page 3 of 25

A&A 586, A133 (2016)

(

&

400), the

-mode CMB polarization is comparable to the

power of dust polarization at high Galactic latitudes.

The CMB

-mode power, even for the highest primordial

tensor perturbation models, is negligible with respect to the dust

polarization at 353 GHz. Since no reliable published CMB po-

larization maps are available, we have chosen not to remove

the CMB polarization from the

Planck

HFI

and

maps.

Nevertheless, because the CMB

-mode polarization is signif-

icant with respect to the dust at 353 GHz at high multipoles

(and even at lower multipoles for the lower frequencies), when

studying the

Planck

HFI bands we subtract the

Planck

best-fit

CDM

model (Col. 2 of Table 2 in Planck Collaboration

XVII 2014) from the dust power spectra, paying the price of an

increased error due to sample variance. No CMB is removed in

this work when computing the dust

spectra.

2.2.3. Synchrotron emission

Synchrotron emission is known to be significantly polarized

(up to 75% for typical relativistic electron spectra, Rybicki &

Lightman 1979). Since its specific intensity scaling with fre-

quency follows a power law with a spectral index close to

−

(Gold et al. 2011; Macellari et al. 2011; Fuskeland et al. 2014),

synchrotron polarized emission is expected to be subdominant

in the

Planck

HFI channels in general and negligible at 353 GHz

(Tucci et al. 2005; Dunkley et al. 2009a; Gold et al. 2011;

Fauvet et al. 2011; Fuskeland et al. 2014; Planck Collaboration

Int. XXII 2015). Hence, we neither subtract nor mask any syn-

chrotron contribution before estimating the angular power spec-

tra of dust polarization. The justification of this assumption will

be demonstrated below by studying the frequency dependence

of the polarized dust power between 100 and 353 GHz (but see

also Appendix D.4).

2.2.4. Polarized point sources

Radio sources have been shown to have a fractional polarization

of a few percent (e.g., Battye et al. 2011; Massardi et al. 2013).

Their contribution to the polarization angular power spectra in

the

Planck

HFI bands is expected to be negligible at low and

intermediate multipoles (Battye et al. 2011; Tucci & To

olatti

2012). Upper limits have been set on the polarization of infrared

galaxies, and their contribution to the polarization power spec-

tra is also expected to be negligible (e.g., Sei

ert et al. 2007).

However, the brightest of the polarized point sources can be re-

sponsible for ringing in the angular power spectra estimation,

and therefore need to be masked (see Sect. 3.3.1).

2.2.5. CO emission

The first three carbon monoxide (CO) Galactic emission lines

at 115 GHz (

→

0), 230 GHz (

→

1), and 345 GHz

(

→

2), contribute significantly to the power in the

Planck

HFI bands at 100, 217, and 353 GHz, respectively (Planck

Collaboration IX 2014). The

Planck

data were used to pro-

duce the first all-sky maps of Galactic CO emission (Planck

Collaboration XIII 2014). It is known that CO emission can

be intrinsically polarized (Goldreich & Kylafis 1982; Li &

Henning 2011). Furthermore, CO emission can induce spuri-

ous polarization because of the di

erences in spectral trans-

mission at the CO frequencies between

Planck

HFI detectors

(Planck Collaboration IX 2014). For these reasons, we mask

CO-emitting regions (Sect. 3.3.1). Outside this mask, as has been

shown in Planck Collaboration XIII (2014) by comparing the

Planck

CO maps to high Galactic latitude ground-based CO ob-

servations (Hartmann et al. 1998; Magnani et al. 2000), the CO

emission is negligible in the

Planck

channels (lower than one

fourth of each channel noise rms at 95% CL). We will check that

our polarization analysis is not contaminated by CO by examin-

ing the frequency dependence of the polarized angular power

spectra of the dust (see Sect. 4.5).

2.3. Systematics of the Planck HFI polarization maps

The first CMB polarization results from

Planck

were presented

in Planck Collaboration I (2014) and Planck Collaboration XVI

(2014). The

power spectrum at

` >

200 was found to be

consistent with the cosmological model derived from tempera-

ture anisotropies. Systematic e

ects in the data have so far lim-

ited the use of

Planck

HFI polarization data on large angular

scales. The polarization systematics in the 2013 data were dis-

cussed and estimated in Planck Collaboration VI (2014; see the

power spectra shown in their Fig. 27). The same data were used

in the

Planck

Galactic polarization papers (Planck Collaboration

Int. XIX 2015; Planck Collaboration Int. XX 2015; Planck

Collaboration Int. XXI 2015; Planck Collaboration Int. XXII

2015) and there the low brightness regions of the 353 GHz sky

were masked.

In this paper, we use a new set of

Planck

polarization maps

for which the systematic e

ects have been significantly reduced.

Corrections will be fully described and applied in the

Planck

2015 cosmology release, as well as the remaining systematic ef-

fects that we describe briefly here.

Two main e

ects have been corrected in the time-ordered

data prior to mapmaking. The correction for the nonlinearity of

the analogue-to-digital converters was improved, and we have

also corrected the data for very long time constants that were not

previously identified (see Planck Collaboration VI 2014; Planck

Collaboration X 2014). After these corrections, the main sys-

tematic e

ects over the multipole range 40

< ` <

600 relevant

to this analysis result from leakage of intensity into polariza-

tion maps. E

ects arising from polarization angle and polariza-

tion e

ffi

ciency uncertainties have been shown to be second order

(Planck Collaboration VI 2014). The leakage e

ect can be ex-

pressed as

∆

{

}

(

n

)

∑

→{

}

(

n

)

(

n

)

(1)

where

is the frequency, and

→{

}

, the leakage pattern

for bolometer

, is multiplied by the di

erent

leakage source

maps

and their associated scaling coe

ffi

cients

. The leak-

age patterns are fully determined by the scanning strategy. They

represent the cumulative result of all the systematic e

ects that

lead to a leakage of intensity to polarization. In the

Planck

HFI bands, there are three main sources of intensity to polar-

ization leakage: (i) monopole di

erences between detectors not

corrected by data destriping (the intensity source term is con-

stant over the sky); (ii) bolometer inter-calibration mismatch (the

source term is the full intensity map, including the CMB dipole

and the Galactic emission); and (iii) a dust spectral mismatch

term. The spectral bandpass varies from one bolometer to an-

other for a given band. As the bolometer gain is calibrated on

the CMB dipole, the di

erential gains on the dust emission pro-

duce the bandpass mismatch term (for which the source term is

the dust intensity map).

A133, page 4 of 25

Planck Collaboration: Dust polarization at high latitudes

Results from a global fit of the

and

maps with these

three leakage terms

→{

}

are used to quantify the leakage.

This fit yields estimates of the scaling coe

ffi

cients at each fre-

quency

in Eq. (1), which allows us to compute angular power

spectra of the leakage terms in Sect. 4. We point out that the fit

captures any emission in the maps that has a pattern on the sky

similar to one of the leakage patterns. These global fit maps are

used to assess the level of systematics, but are not removed from

the data.

An independent and complementary estimate of systematic

ects in the data is provided by the null tests that we can

build from the DetSets, HalfRings, and Years data subsets (see

Sect. 2.1). These null tests are a good way of determining the

level of any systematics other than intensity to polarization leak-

age. These are pursued in Sect. 4.1 and Appendices C.1 and D.3.

3. Computation of angular power spectra

of polarized dust emission

3.1. Methods

We can use the

Planck

data to compute the polarization angu-

lar power spectra (

and

) of the polarized dust emission

within selected sky regions. Even if the statistical properties of

the dust emission on the sky might not be entirely captured by a

two-point function estimator, because the scope of this paper is

to assess the level of dust in the framework of CMB data anal-

ysis, we follow the approximation that is generally made when

processing such data, i.e., that a large fraction of the information

is contained in power spectra.

On an incomplete sky, a polarization field can be divided into

three classes of modes: pure

-modes; pure

-modes; and “am-

biguous” modes, which are a mixture of the true

- and

-modes

(Bunn et al. 2003).

The ambiguous modes represent a cross-talk between

and

, which is often referred to as “

-to-

leakage” for the

CMB (because for the CMB

). Methods used to

estimate the CMB angular power spectrum for polarization ac-

count for and correct analytically for the incomplete sky cov-

erage. However, the presence of the ambiguous modes yields a

biased estimate of the variance of the spectra, unless so-called

“pure” power spectrum estimators are used (Smith 2006). For

dust polarization, the power in

- and

-modes are comparable,

and we do not expect a significant variance bias.

The two specific approaches we use are

Xpol

, our main

method, which we describe as a “classical” pseudo-

estimator

and, for comparison,

Xpure

, a pure pseudo-

estimator. Since

they give similar results for the present study, as we demonstrate

below, the former is chosen as our main method because it is

computationally less expensive.

3.1.1. Xpol

Xpol

is an extension of the

Xspect

method (Tristram et al.

2005) to polarization.

Xspect

computes the pseudo power

spectra and corrects them for incomplete sky coverage, filter-

ing e

ects, and pixel and beam window functions. Correction

for incomplete sky coverage is performed using a

Master

-like

algorithm (Hivon et al. 2002), consisting of the inversion of the

mode-mode coupling matrix

′

that describes the e

ect of the

Pseudo-

estimators compute an estimate of the angular power

spectra directly from the data (denoted the “pseudo-power spectrum”)

and then correct for sky coverage, beam smoothing, data filtering, etc.

partial sky coverage;

′

is computed directly from the power

spectrum of the mask that selects the data in the analysis re-

gion of interest (Sect. 3.3). The

HEALPix

pixel window functions

(Górski et al. 2005) are used to correct for pixelization e

ects.

For the beams we use the

Planck

HFI individual detector beam

transfer functions described in Planck Collaboration VII (2014).

Xpol

estimates the error bars analytically, without requiring

Monte Carlo simulations. Using simulated data comprising in-

homogeneous white noise and a Gaussian map with a dust power

spectrum, we have checked that this analytical estimate is not bi-

ased for multipoles

` >

40. The analytical error bars combine the

contributions from instrumental noise and sample variance. The

Gaussian approximation of the sample variance is

var

(

bin

)

bin

sky

∆

bin

(

bin

)

(2)

where

{

}

sky

is the retained sky fraction (which can be

sky

if the sky field is apodized), and

∆

bin

is the size of the mul-

tipole bin

bin

. To estimate the error relevant to the polarized dust

signal measured within a given region, we subtract quadratically

this estimate of the contribution from sample variance. The per-

formance of

Xpol

on Gaussian simulations that have

and

dust-like angular power spectra is presented in Appendix A.

3.1.2. Xpure

Xpure

is a numerical implementation of the pure pseudo-

spectral approach described and validated in Grain et al. (2009).

The method is optimized for computing CMB

-mode power

spectra over small sky patches. It uses a suitably chosen sky

apodization that vanishes (along with its first derivative) at the

edges of the patch in order to minimize the e

ects of

-to-

leakage. For the estimation of the angular power spectra of the

Planck

data we compute the cross-correlation of two di

erent

DetSets. Uncertainties are obtained by performing Monte Carlo

inhomogeneous white noise simulations, considering the diago-

nal terms of the pixel-pixel covariance for the two data sets.

We have applied this algorithm to

Planck

353 GHz maps

and simulations to estimate the

-mode power spectrum of the

Galactic thermal dust emission in regions ranging from 1% to

30% of the sky. We have used

Xpure

here as a cross-check for

the robustness of the

Xpol

method presented in the previous sec-

tion. Validation and comparison of the performance of the two

algorithms is presented in Appendix A on simulations and in

Appendix D.1 on application to the data.

3.2. Computing cross-spectra

To avoid a bias arising from the noise, we compute all the

Planck

power spectra from cross-correlations of DetSets maps

(see Sect. 2). The noise independence of the two DetSet maps

at a given frequency was quantified in Planck Collaboration XV

(2014) and the resulting level of noise bias in the cross-power

spectra between them has been shown to be negligible. The

cross-power spectrum at a given frequency

(

)

≡

(

)

(3)

where

and

are the two independent DetSet 1 and DetSet 2

maps at the frequency

. The

Planck

cross-band spectrum

A133, page 5 of 25

A&A 586, A133 (2016)

Fig. 1.

Masks and complementary selected large regions that retain frac-

tional coverage of the sky

sky

from 0.8 to 0.3 (see details in Sect. 3.3.1).

The grey is the CO mask, whose complement is a selected region with

sky

8. In increments of

sky

1, the retained regions can be iden-

tified by the colours yellow (0.3) to black (0.8), inclusively. Also shown

is the (unapodized) point source mask used.

between the frequencies

and

′

(

′

)

≡

[

(

′

)

(

′

)

(

′

)

(

′

)]

(4)

or equivalently the cross-spectrum between

and

′

of the aver-

aged frequency maps (

)

As we noted in Sect. 2.2.2, from each computed

spec-

trum we subtract the

Planck

best-fit

CDM

model (Planck

Collaboration XVI 2014), i.e., the theoretical

model ob-

tained from the temperature data fit, while

spectra are kept

unaltered.

3.3. Selection of regions

To measure the dust polarization power spectra with high S

we select six large regions, the analysis regions of interest at

intermediate Galactic latitude, which have e

ective coverage of

the sky from 24 to 72% (see Sect. 3.3.1)

. For statistical studies

at high Galactic latitude, we compute spectra on a complete set

of smaller regions or patches (Sect. 3.3.2), similar in size to the

patches observed in typical CMB experiments.

3.3.1. Large regions

For selection of all of the large regions, we used the

Planck

CO map from Planck Collaboration XIII (2014), smoothed to

a 5

◦

resolution, to mask the sky wherever the CO line brightness

≥

4 K km s

−

1 8

. This mask is shown in Fig. 1. The com-

plement to this mask by itself defines a preliminary region that

retains a sky fraction

sky

We then mask the sky above successively lower thresholds of

857

in the

Planck

857 GHz intensity map, smoothed to a 5

◦

res-

olution, chosen such that together with the CO mask we select

five more preliminary regions that retain

sky

from 0.7 to 0.3 in

steps of 0.1. These six regions are shown in Fig. 1.

Although the selection process is similar to that in Planck

Collaboration XV (2014), there are di

erences in detail.

We use the CO (

→

0) type 3 map, which has the highest S

At this resolution and for this map, the cut we apply corresponds to

To avoid power leakage, these six masks are then apodized

by convolving with a 5

◦

FWHM Gaussian which alters the win-

dow function by gradually reducing the signal towards the edges

of the retained regions and thus lowers the e

ective retained sky

coverage. The

sky

value is simply defined as the mean sky cov-

erage of the window function map.

Finally, we mask data within a radius 2

beam

of point sources

selected from the

Planck

Catalogue of Compact Sources (PCCS,

Planck Collaboration XXVIII 2014) at 353 GHz. The selected

sources have

7 and a flux density above 400 mJy

Selection of spurious infrared sources in bright dust-emitting re-

gions is avoided by using contamination indicators of infrared

cirrus listed in the PCCS description (Planck Collaboration

XXVIII 2014). This point-source masking is done to prevent

the brightest polarized sources from producing ringing in the

power spectrum estimation, while avoiding the removal of dust

emitting regions and their statistical contribution to the an-

gular power spectra. The details of this source selection will

be presented in the

Planck

2014 release papers. The edges

of the masks around point sources were apodized with a 30

′

FWHM Gaussian, further reducing the retained net e

ective sky

coverage.

In combination these masking and apodization procedures

result in six large retained (LR) regions, which we label using

the percentage of the sky retained (the net e

ective fractional

sky coverages,

sky

, are listed in Table 1), e.g., LR72 for the

largest region and LR24 for the smallest.

Table 1 also lists other properties of the regions, including

〈

353

〉

, the mean specific intensity at 353 GHz within the region

in MJy sr

−

, and



, the mean H



column density in units of

−

computed on the LAB H



survey data cube (Kalberla

et al. 2005).

3.3.2. Small patches at high Galactic latitude

To examine the statistics of the angular power spectra at

high Galactic latitude, we also analyse the

Planck

polariza-

tion maps within patches with a size similar to those of typ-

ical ground-based and balloon-borne CMB experiments (e.g.,

QUIET Collaboration et al. 2012; Hanson et al. 2013; BICEP2

Collaboration 2014b; Ade et al. 2014; Naess et al. 2014).

Specifically, we consider 400 deg

circular areas (radius 11.

◦

centred on the central pixel positions of the

HEALPix

side

grid that have Galactic latitude

◦

. This results in 352 such

patches. These are apodized with a 2

◦

FWHM Gaussian, which

reduces the retained sky fraction to

sky

0080 for each patch.

For these small patches, we do not mask point sources as was

done in selecting the large regions, since we want to preserve

the same

sky

for each mask, nor is the CO mask needed, since

we use these masks only on the 353 GHz high Galactic latitude

data. We note that for

side

8, pixels have an area of about

54 deg

and characteristic centre-to-centre spacing of about 7.

◦

Therefore, this

HEALPix

grid oversamples the sky relative to this

patch size, so that the patches overlap and are not independent.

This included the brightest point sources in the Large and Small

Magellanic Clouds (LMC and SMC). We tested that masking the en-

tirety of the LMC and SMC has no significant e

ect on the spectra or

on the conclusions that we derive from them.

A133, page 6 of 25

Planck Collaboration: Dust polarization at high latitudes

Table 1.

Properties of the large retained (LR) science regions described in Sect. 3.3.1.

LR24

LR33

LR42

LR53

LR63

LR72

sky

. . . . . . . . . . . . . . . . . . . . . .

0.3

0.4

0.5

0.6

0.7

0.8

sky

. . . . . . . . . . . . . . . . . . . . . .

0.24

0.33

0.42

0.53

0.63

0.72

〈

353

〉

MJy sr

−

. . . . . . . . . . . . . .

0.068

0.085

0.106

0.133

0.167

0.227



−

. . . . . . . . . . . . . .

1.65

2.12

2.69

3.45

4.41

6.05

. . . . . . . . . . . . . . . . . . . . . .

−

. . . . . . . . . . . . . . . . . . . . . .

−

(

−

42,

21) . .

(

−

21) . .

(

80) . . . . . . . . . . . . . . .

124

197

328

〈

〉

. . . . . . . . . . . . . . . . .

Notes.

For each region,

sky

is the initial sky fraction,

sky

its value after point source masking and apodization,

〈

353

〉

the mean specific intensity at

353 GHz within the region, in MJy sr

−

, and



the mean H



column density, in units of 10

−

(Kalberla et al. 2005). For the power-law fits

in multipole

, we also list the exponents

and

(Sect. 4.2), the

of the fits with fixed exponents

−

42, the value

of the

fitted

amplitude at

80 (in

CMB

at 353 GHz, Sect. 4.3), and the mean of the amplitude ratio

〈

〉

(see Sect. 4.4).

4. Dust polarized angular power spectra

at intermediate Galactic latitude

In this section, we quantify the 353 GHz dust polarization in the

power spectrum domain, achieving a high S

N through the use

of LR regions. For convenience we present results for

and

, where

≡

(

4.1. Description of the spectra

Using

Xpol

we have computed

and

from the two

DetSets at 353 GHz, as a function of the multipole

in the range

40–600

. These represent the first measurements of the thermal

dust

and

power spectra on large fractions of the sky

for

` >

40 (see Ponthieu et al. 2005; Gold et al. 2011, for earlier

related studies).

In Fig. 2 we present the results for

sky

{

}

The amplitudes of the spectra increase with increasing

sky

be-

cause the polarized emission is brighter on average when more

sky is retained. We point out that the LR regions presented in

Sect. 3.3.1 overlap because they form a nested set. However, the

power spectra derived from them are almost independent of each

other because each spectrum is dominated by the brightest ar-

eas in the corresponding region, namely the parts closest to the

Galactic plane and hence the areas in which each LR region dif-

fers from those nested inside it.

The

and

spectra are characterized by a power-law

dependence on multipole

over the range

40 to 600; further-

more, the slope is similar for di

erent regions that retain from 24

to 63% of the sky.

Figure 2 also shows the

power spectrum computed from

the

Planck

2013 best-fit

CDM model of the CMB temperature

data (Planck Collaboration XVI 2014), and the much lower ex-

pectation for

from the CMB model with primordial gravita-

tional waves with amplitude

2. At 353 GHz, the

angu-

lar power spectra of the dust are about 3

−

4 orders of magnitude

larger than the CMB model at

30, 1

−

2 orders of magnitude

The spectra are in units of

CMB

at 353 GHz.

larger at

100, and about the same order of magnitude as the

CMB at

` >

300. At 353 GHz, the

angular power spectra for

dust are much greater than the CMB model power spectrum for

all

values in Fig. 2. The dust power spectra are larger than the

2 CMB spectrum by 4

−

5 orders of magnitude at

30,

and by 3

−

4 orders of magnitude at

100. At

500, where

the lensing of CMB anisotropies is the dominant contribution to

the CMB model spectrum, the dust is still 2

−

3 orders of magni-

tude higher.

As discussed in Appendix A, we do not expect any signifi-

cant bias, or

-to-

leakage, from the computation of the dust

angular power spectra using

Xpol

. Figure 2 also includes the

and

spectra at 353 GHz, computed from our estimate

of the leakage terms from intensity to polarization (discussed

in Sect. 2.3), for the same three LR regions. The dust spectra

are much higher than the corresponding spectra for the leakage,

which represent the main systematic e

ects over the

range of

interest for this work. The largest contamination of the dust sig-

nal by leakage is about 3.5% at

50, for both the

and

spectra. Because we consider that our estimate of the leak-

age maps is conservative, we conclude that contamination of the

dust

and

power spectra by systematic e

ects amounts

to a maximum of 4% at

50, and is less at higher multipoles.

Therefore, we have not corrected the

Planck

data for intensity-

to-polarization leakage in this work.

Finally, in Fig. 2 we present for the three LR re-

gions the absolute value of the null-test spectra anticipated

in Sect. 2.3, here computed from the cross-spectra of the

HalfRing

DetSet di

erences, i.e., (353

DS1

HR1

−

353

DS1

HR2

)

(353

DS2

HR1

−

353

DS2

HR2

)

2. These

and

spectra show a

behaviour that is close to what is expected from a white-noise

dominated (thus

) spectrum. The amplitudes of these error es-

timates are consistent with the noise expectations; there is no

evidence for any e

ects of systematics.

For completeness, in Appendix B we present a further quan-

tification of the power spectrum of thermal dust emission at

353 GHz via the spectra involving temperature and polarization,

and

T B

, and the polarization cross-spectrum,

A133, page 7 of 25

A&A 586, A133 (2016)

Fig. 2.

Planck

HFI 353 GHz

(red,

top

) and

(blue,

bottom

) power spectra (in

CMB

) computed on three of the selected LR analysis

regions that have

sky

3 (circles, lightest),

sky

5 (diamonds, medium) and

sky

7 (squares, darkest). The uncertainties shown are

The best-fit power laws in

are given for each spectrum as a dashed line of the corresponding colour. The

Planck

2013 best-fit

CDM

expectation (Planck Collaboration XVI 2014) and the corresponding

CMB model are shown as solid black lines; the rise for

` >

200

is from the lensing contribution. In the lower parts of each panel, the global estimates of the power spectra of the systematic e

ects responsible

for intensity-to-polarization leakage (Sect. 2.3) are shown in di

erent shades of grey, with the same symbols to identify the three regions. Finally,

absolute values of the null-test spectra anticipated in Sect. 2.3, computed here from the cross-spectra of the HalfRing

DetSet di

erences (see text),

are represented as dashed-dotted, dashed, and dotted grey lines for the three LR regions.

A133, page 8 of 25

Planck Collaboration: Dust polarization at high latitudes

Fig. 3.

Best-fit power-law exponents

(red squares) and

(blue

circles) fitted to the 353 GHz dust

and

power spectra for the

erent LR regions defined in Sect. 3.3.1, distinguished here with

sky

Although the values in the regions are not quite independent, simple

means have been calculated and are represented as red and blue dashed

lines.

4.2. Power-law fit

To assess the apparent power-law dependence

∝

quan-

titatively, we made a

fit to the spectra at 353 GHz using the

form

(

80)

, where

∈ {

}

. For

and

we fit the 22 band-powers in the range 60

< ` <

500. For

both power spectra we restricted the fit to

` >

60 to avoid a pos-

sible bias from systematic e

ects in the data, and also because

the angular power spectra of the dust polarization exhibit more

spatial variation than is expected for a Gaussian random field,

particularly on large angular scales.

The exponents of the power-law fits,

, are plotted in

Fig. 3 for each of the six LR regions identified with

sky

. All

exponents are consistent with constant values of

−

02 and of

−

03. While there is a slight indication

of a steeper slope for

than for

, hereafter we adopt the

mean exponent

−

02. This exponent is consistent with

the value

fitted to the

Planck

353 GHz dust intensity power

spectra in this range of

(Planck Collaboration XV 2014; Planck

Collaboration Int. XXII 2015) on similar-sized regions at inter-

mediate Galactic latitude, but slightly flatter than the

fitted

at higher

and higher frequency (Miville-Deschênes et al. 2007,

2010; Planck Collaboration XV 2014).

For the fits with fixed exponent

−

42, the values

of the

(with number of degrees of freedom,

21) are

listed in Table 1 for the six LR regions. For the

spectra, the

values range from 26.3 (probability to exceed, PTE

2) to

44.8 (PTE

002), with a trend for a quality of fit that degrades

with increasing

sky

. For the

spectra, the

values range

from 14.0 (PTE

87) to 22.1 (PTE

39), with a trend

related to

sky

Possible explanations for this di

erence between

and

spectrum shape descriptions, are the chance correlation be-

tween dust and CMB polarization, which is not taken into ac-

count in the subtraction of the CMB

spectrum, and the in-

creasing S

N degrading the overall quality of the fit when going

from

and for

from

sky

3 to

sky

8 as

the amplitude of the dust polarized signal increases. We stress

that while the power law in

is a good general description of the

shape of the dust polarized power spectra, a full characterization

would have to consider the detailed features that can be seen in

Fig. 2.

Fig. 4.

Amplitude of the dust

(red squares) and

(blue circles)

power spectra, normalized with respect to the largest amplitude for each

mode. These are plotted versus the mean dust intensity

〈

353

〉

for the

six LR regions (

top panel

). A power-law fit of the form

(

〈

353

〉

)

〈

353

〉

∈ {

}

, is overplotted as a dashed line of the corre-

sponding colour (these almost overlap). The

bottom panel

presents the

ratio of the data and the fitted

〈

353

〉

power law; the range associated

with the

uncertainty in the power-law exponent of 1.9 is shown in

grey. For details see Sect. 4.3.

4.3. Amplitude dependence on

〈

I

353

〉

Similarly to what has been done for intensity power spectra in,

e.g., Gautier et al. (1992) and Miville-Deschênes et al. (2007),

we investigate how the amplitude of the dust polarization power

spectrum scales with the dust intensity. To quantify the dust

emission at 353 GHz, we use the model map derived from a mod-

ified blackbody fit to the

Planck

data at

≥

353 GHz and IRAS

100

m, presented in Planck Collaboration XI (2014).

This map is corrected for zodiacal emission and the brightest ex-

tragalactic point sources are subtracted. The Galactic reference

sets of the underlying IRAS and

Planck

data were obtained

through a method based on correlation with 21 cm data from

the LAB H



survey (Kalberla et al. 2005) integrated in velocity,

ectively removing the CIB monopole in the model map. The

mean dust intensity,

〈

353

〉

, listed for each LR region in Table 1,

ranges from 0

068 to 0

227 MJy sr

−

for increasing

sky

. The

mean column density calculated from the LAB survey data is

also listed in Table 1, with values ranging from 1

−

to 6

−

For all of the LR regions, we fit the

and

spectra

with a power law in

, using the fixed exponent

−

42,

over the

-ranges defined in Sect. 4.2. The amplitudes

de-

rived from these fits are listed in Table 1; the

amplitudes

can be retrieved from the

ratio. These amplitudes are

plotted as a function of

〈

353

〉

in Fig. 4 after normalization by the

maximum value found for the largest region (LR72). We fit the

empirical dependence of these amplitudes on

〈

353

〉

as a power

law of the form

(

〈

353

〉

)

〈

353

〉

where

∈ {

}

The two fitted exponents are quite similar,

02 and

02. The exponent that we find for polarization

is close to the one observed in the di

use interstellar medium

for the dust intensity, consistent with

∝

〈

〉

, where

〈

〉

the mean value of the dust specific intensity (Miville-Deschênes

et al. 2007). Values close to 2 are expected, because we compute

angular power spectra, which deal with squared quantities.

Although the data points roughly follow this

〈

353

〉

depen-

dence, the empirical law fails to fully describe individual dust

amplitudes (e.g., the estimate is o

by about 20% for

A133, page 9 of 25

A&A 586, A133 (2016)

Fig. 5.

Ratio of the amplitudes of the

and

dust power spectra

at 353 GHz for the di

erent LR regions defined in Sect. 3.3.1, distin-

guished here with

sky

. The mean value

〈

〉

52 is plotted as

a dashed line.

LR33). The scaling can help to asses the order of magnitude of

the dust polarization level on a specific region, but is not a sub-

stitute for actually characterizing the polarized angular power

spectra.

4.4. Amplitude of

relative to

We examine the ratio of the amplitudes of the fitted power laws

found in Sect. 4.3. The

ratios are listed in Table 1, and

are plotted for di

erent values of

sky

in Fig. 5. For all of the

LR regions, we observe more power in the

dust spectrum

than in

. All ratios are consistent with a value of

03, significantly di

erent from unity, over various large

fractions of the intermediate latitude sky.

This result is not taken into account in existing models of po-

larized microwave dust emission that have been developed to test

component separation methods. For example, we have computed

the

and

spectra over the LR regions for the

Planck

Sky Model (Delabrouille et al. 2013) and the model of O’Dea

et al. (2012); for both models and all LR regions we find a ratio

close to 1. However, these two models are based on a

very simplified picture of the Galactic magnetic field geometry

and assumptions on how the polarized emission depends on it.

Further insight into the structure of the dust polarization sky is

required to account for the observed ratio.

4.5. Amplitude dependence on frequency

Finally, we explore the frequency dependence of the amplitude

of the angular power spectra. We compute the

and

angular power spectra from the

and

DetSet maps at 100,

143, 217, and 353 GHz (see Sect. 3.2). From these four sets of

polarization maps, we compute ten power spectra: 100

100;

100

143; 100

217; 100

353; 143

143; 143

217; 143

353;

217

217; 217

353; and 353

353.

The ten angular cross-power spectra are consistent with a

power law in

, with the exponent

−

42 measured at

353 GHz (Sect. 4.2). Therefore, to each of these spectra we fit

the amplitudes of a power-law function that has a fixed expo-

nent

−

42, in the range 40

< ` <

500, for

and

. As an illustration of the quality of the fit, for the small-

est region (LR24,

sky

3) the averages and dispersions of

Fig. 6.

Frequency dependence of the amplitudes

of the angular

power spectra, relative to 353 GHz (see details in Sect. 4.5). Results

for

(red squares) and

(blue circles) for the smallest region,

LR24. These include evaluations from cross-spectra involving polariza-

tion data at two frequencies, plotted at the geometric mean frequency.

The square of the adopted relative SED for dust polarization, which

is a modified blackbody spectrum with

59 and

6 K,

is shown as a black dashed line. The

uncertainty area from the

expected dispersion of

, 0.03 for the size of LR24 as inferred from

Planck Collaboration Int. XXII (2015) (see Sect. 2.2.1), is shown in

grey.

the

(21 degrees of freedom) of the fits are

and

9 for the ten cross-frequency spectra.

To compare the frequency dependence of the results of the

fits to that expected from the SED for dust polarization from

Planck Collaboration Int. XXII (2015), we converted the fitted

amplitudes

from

CMB

to units of (MJy sr

−

)

, taking

into account the

Planck

colour corrections

. For all regions, we

examined the frequency dependence by plotting the amplitudes

normalized to unity at 353 GHz versus the e

ective frequency

A representative example is shown in Fig. 6 for the smallest re-

gion, LR24 (

sky

3).

For all of the LR regions the frequency dependence found

is in good agreement with the square of the adopted dust SED,

which is a modified blackbody spectrum having

59 and

6 K (Planck Collaboration Int. XXII 2015). We note

that no fit was performed at this stage and that the square of

the adopted dust SED goes through the 353 GHz data point.

However, if we fit the amplitude of the dust frequency depen-

dence, with fixed

and

, the

(

9) is 13.1 for

(PTE

16) and 3.2 for

(PTE

96). This good

agreement supports the assumption made in the present work

about the faintness of the synchrotron and CO emission at high

Galactic latitude (see Sect. 2.2). The residuals to the fit do

not show any evidence for excess power at 100 GHz, which

might arise from polarized synchrotron emission. This result

is consistent with the study of synchrotron polarization at high

Galactic latitudes by Fuskeland et al. (2014), and confirmed in

Appendix D.4. Furthermore, we do not see any excess at ei-

ther 100 or 217 GHz, such as could arise from leakage and

Conversion factors were computed as described in Sect. 2.1, here

using colour corrections corresponding to a dust modified blackbody

spectrum with

59 and

6 K.

For a cross-spectrum between data at frequency

and frequency

the e

ective frequency is taken for convenience as the geometric mean

≡

√

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