1409_5738.pdf

Astronomy & Astrophysics

manuscript no. Dust ̇Polarized ̇Cl ̇astro-ph

ESO 2014

September 22, 2014

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

, K. Benabed

, A. Benoit-L

evy

, 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

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. D

esert

, C. Dickinson

, J. M. Diego

, K. Dolag

, H. Dole

, S. Donzelli

, O. Dor

, 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-H

eraud

, E. Gjerløw

J. Gonz

alez-Nuevo

, K. M. G

orski

, S. Gratton

, A. Gregorio

, A. Gruppuso

, V. Guillet

, F. K. Hansen

, D. Hanson

D. L. Harrison

, G. Helou

, S. Henrot-Versill

, C. Hern

andez-Monteagudo

, D. Herranz

, E. Hivon

, W. A. Holmes

K. M. Hu

enberger

, G. Hurier

, A. H. Ja

, T. R. Ja

, J. Jewell

, W. C. Jones

, M. Juvela

, E. Keih

anen

, 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. L

opez-Caniego

, P. M. Lubin

, J. F. Mac

ıas-P

erez

, B. Ma

, D. Maino

, N. Mandolesi

, A. Mangilli

M. Maris

, P. G. Martin

, E. Mart

ınez-Gonz

alez

, S. Masi

, S. Matarrese

, P. Mazzotta

, A. Melchiorri

, L. Mendes

, A. Mennella

M. Migliaccio

, S. Mitra

, M.-A. Miville-Desch

enes

, 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

R. Rebolo

, M. Remazeilles

, C. Renault

, A. Renzi

, S. Ricciardi

, I. Ristorcelli

, G. Rocha

, C. Rosset

, M. Rossetti

G. Roudier

, B. Rouill

e d’Orfeuil

, J. A. Rubi

no-Mart

ın

, 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

, 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)

Preprint online version: 19 September 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. The present uncertainties are large and will be reduced through an ongoing, joint analysis of the

Planck

and

BICEP2 data sets.

Key words.

Submillimetre: ISM – Radio continuum: ISM – Polarization – ISM: dust, magnetic fields – cosmic background radiation

arXiv:1409.5738v1 [astro-ph.CO] 19 Sep 2014

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

size about 0.1

m. 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;

Beno

ıt 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, cosmo-

logical

-mode polarization signals may be present that were im-

printed during the epoch of inflation. The discovery of a primor-

dial

-mode polarization signature is a major scientific goal of a

large number of CMB experiments. These include ground-based

experiments (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, Rubi

no-Mart

ın 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 2014; Planck

Collaboration Int. XX 2014; Planck Collaboration Int. XXI

2014; Planck Collaboration Int. XXII 2014). 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

∗

Corresponding author: Jonathan Aumont, jonathan.aumont@ias.u-

psud.fr

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.

explore statistically the potential existence of “clean” patches on

the sky that might be suitable for cosmology.

Our analysis of dust polarization is relevant to the present

generation of CMB polarization observations, as well as the de-

sign 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 polarization 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 describe the

general properties of the polarization maps in terms of emis-

sion components and systematic e

ects. In Sect. 3, we describe

our method for computing the dust

and

angular power

spectra, including the selected science regions of interest on the

sky. We assess and compare the two methods we use to com-

pute the power spectra in Appendix A. In Sect. 4, we present

power spectra of dust polarization for multipoles

` >

40, com-

puted with high signal-to-noise ratio (S

N) on large fractions of

the sky, and characterize their shape and amplitude. We extend

this analysis to smaller sky patches at high Galactic latitude in

Sect. 5. In Appendix B 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 C.

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 (LFI, Low Frequency Instrument, Mennella

et al. 2011) and (HFI, High Frequency Instrument, 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),

Planck Collaboration IX (2014), and Planck Collaboration X

While this paper was in preparation two papers have used publicly

available polarization information from

Planck

to infer potentially high

levels of dust contamination in the BICEP2 field (Mortonson & Seljak

2014; Flauger et al. 2014).

Planck Collaboration: Dust polarization at high latitudes

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

eral processing applied to the data to measure polarization. In

this paper, we make use of unreleased, full-mission (Surveys 1

to 5, 30 months, Planck Collaboration I 2014), polarization maps

of the

Planck

HFI (internal data release “DX11d”), projected

into the

HEALPix

pixelization scheme (G

orski et al. 2005). This

is one of the first publications to use these maps, which will be

described in the

Planck

cosmology 2014 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 of 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 additionally use maps

made from other data subsets (Planck Collaboration VI 2014).

We use the ring halves, (hereafter “HalfRing”), where the ap-

proximately 60 circles performed for each

Planck

telescope ring

(also called a stable pointing period) are divided into two inde-

pendent subsets of 30 circles. Additionally 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 further

maps with independent noise.

The

Planck

maps we use are in thermodynamic units

CMB

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

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

(

)

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 interme-

diate Galactic latitudes, the colour correction factor is 1.12 at

353 GHz (see Table 3 in Planck Collaboration Int. XXII 2014).

As well as 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; Beno

ıt et al. 2004; Ponthieu et al. 2005;

Vaillancourt 2007). It is the dominant polarized foreground sig-

nal in the high frequency

Planck

bands (Tucci et al. 2005;

The conversion factor from K

CMB

to MJy sr

−

is computed for

a specific intensity

∝

−

. The colour correction depends on the

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 equation 19 in Planck Collaboration Int.

XXII 2014). The conversion factors and the colour corrections are

computed via equation (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.

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

Planck Collaboration Int. XXII 2014).

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 2014; Planck Collaboration

Int. XX 2014). 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

(2014) and Planck Collaboration Int. XX (2014) report an anti-

correlation between

and the local dispersion of

, which indi-

cates that variations in

arise mainly from depolarization asso-

ciated with changes in the magnetic field orientation within the

beam, rather than from changes in the e

ffi

ciency of grain align-

ment.

Planck Collaboration Int. XXII (2014) showed that the SED

of dust emission in polarization over the four

Planck

HFI fre-

quencies 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 function. About 39 % of the sky at intermediate Galactic

latitudes was analysed.

Among 400 circular patches with 10

◦

radius (equivalent to a sky fraction

sky

0076) the 1

dis-

persion of

was 0.17 for constant

6 K. We scale

this uncertainty on

to larger sky areas by using the factor

0076

sky

)

. This is a conservative choice because this un-

certainty includes the e

ects 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 quan-

tity the S

N of the data is higher than for polarization and Planck

Collaboration Int. XXII (2014) report a dispersion of 0.07 (1

)

over the same sized circular patches. Planck Collaboration Int.

XVII (2014) extend this analysis for intensity to high Galactic

latitudes in the southern Galactic cap, using the dust-H

i

correla-

tion 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 intensity, unless the dust emission has a signifi-

cant 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 preliminary

Planck

results at

&

50; see Barkats et al. 2014 for a recent com-

pendium). For 353 GHz data at small angular scales (

&

400),

the

-mode CMB polarization is comparable to the power of

dust polarization at high Galactic latitudes.

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

sky

8 minus that with

sky

Planck Collaboration: Dust polarization at high 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, when studying the

Planck

HFI bands, because the

CMB

-mode polarization is significant with respect to the dust

at 353 GHz at high multipoles (and even at lower multipoles for

the lower frequencies), we subtract from the dust power spec-

tra the

Planck

best-fit

CDM

model (column 2 of table 2

in Planck Collaboration XVII 2014), paying the price of an in-

creased 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 2014). 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 also

see Appendix C.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 spurious polariza-

tion, due to the di

erences in spectral transmission at the CO fre-

quencies 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 observations (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 anal-

ysis is not contaminated by CO by examining the frequency de-

pendence 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 con-

sistent with the cosmological model derived from temperature

anisotropies. Systematic e

ects in the data have so far limited

the use of

Planck

HFI polarization data on large angular scales.

The polarization systematics in the 2013 data were discussed

and estimated in Planck Collaboration VI (2014) (see the power

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

the

Planck

Galactic polarization papers (Planck Collaboration

Int. XIX 2014; Planck Collaboration Int. XX 2014; Planck

Collaboration Int. XXI 2014; Planck Collaboration Int. XXII

2014) 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

2014 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, then over the

multipole range 40

< ` <

600 relevant to this analysis, the main

systematic e

ects 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 leakage pat-

terns are fully determined by the scanning strategy. They repre-

sent 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 polarization leakage:

(i) monopole di

erences between detectors not corrected by data

destriping (the intensity source term is constant over the sky); (ii)

bolometer inter-calibration mismatch (the source term is the full

intensity map, including the CMB dipole and the Galactic emis-

sion); and (iii) a dust spectral mismatch term. The spectral band-

pass varies from one bolometer to another for a given band. As

the bolometer gain is calibrated on the CMB dipole, the di

eren-

tial gains on the dust emission produce the bandpass mismatch

term (for which the source term is the dust intensity map).

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 allow us to compute angular power

Planck Collaboration: Dust polarization at high latitudes

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 B.1 and C.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

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

assess the level of dust in the framework of CMB data analysis,

we follow the approximation that is generally made when pro-

cessing 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

“ambiguous” 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 account and

correct analytically for the incomplete sky coverage. However,

the presence of the ambiguous modes yields a biased estimate of

the variance of the spectra, unless so-called “pure” power spec-

trum 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-

estima-

tor, and, for comparison,

Xpure

, a pure pseudo-

estimator.

Since (as we demonstrate below) they give similar results for

the present study, the former is chosen to be our main method

because it is less computationally expensive.

3.1.1.

Xpol

Xpol

is an extension to polarization of the

Xspect

method

(Tristram et al. 2005).

Xspect

computes the pseudo power spec-

tra and corrects them for incomplete sky coverage, filtering ef-

fects, and pixel and beam window functions. Correction for in-

complete sky coverage is performed using a

Master

-like al-

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

mode-mode coupling matrix

′

that describes the e

ect of the

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

orski 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

)

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 on simulations is presented in Appendix A and appli-

cation to the data in Appendix C.1.

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 be-

tween the frequencies

and

′

(

′

)

≡

[

(

′

)

(

′

)

(

′

)

(

′

)

]

(4)

or equivalently the cross-spectrum between

and

′

of the aver-

aged frequency maps (

)

We recall that from each computed

spectrum 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.

Planck Collaboration: Dust polarization at high latitudes

Table 1: Properties of the large retained (LR) science regions described in Sect. 3.3.1. 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

i

the mean H

i

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).

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

〉

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

0.068

0.085

0.106

0.133

0.167

0.227

i

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

1.65

2.12

2.69

3.45

4.41

6.05

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

−

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

−

(

−

42,

dof

21) . . .

(

−

dof

21) . . .

(

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

124

197

328

〈

〉

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

Fig. 1: Masks and complementary selected large regions that re-

tain fractional coverage of the sky

sky

from 0.8 to 0.3 (see details

in Sect. 3.3.1). The darkest blue is the CO mask, whose com-

plement is a selected region with

sky

8. In increments of

sky

1, the retained regions can be identified by the colours

dark red (0.3) to dark blue (0.8), inclusively. Also shown is the

(unapodized) point source mask used.

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

Although the selection process is similar to that in Planck

Collaboration XV (2014), there are di

erences in detail.

≥

4 K km s

−

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

857

in the

Planck

857 GHz intensity map, smoothed to a 5

◦

resolution, 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 displayed in Fig. 1.

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 PCCS catalogue (Planck Collaboration

XXVIII 2014) at 353 GHz. The selected sources have S

and a flux density above 400 mJy. Selection of spurious infra-

red sources in bright dust-emitting regions is avoided by using

contamination indicators of infrared cirrus listed in the PCCS

description (Planck Collaboration XXVIII 2014). This point-

source masking is done in order to prevent the brightest polar-

ized radio sources from producing ringing in the power spec-

trum estimation, while avoiding the removal of dust emitting re-

gions and their statistical contribution to the angular power spec-

tra. 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 re-

ducing the retained net e

ective sky coverage.

In combination these masking and apodization procedures

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

hereafter using the percentage of the sky retained (the net e

ec-

tive 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

We use the CO (

→

0) “Type 3” map, which has the highest

signal-to-noise ratio. At this resolution and for this map, the cut we

apply corresponds to

Planck Collaboration: Dust polarization at high latitudes

in MJy sr

−

, and

i

, the mean H

i

column density in units of

−

computed on the LAB H

i

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; Naess et al. 2014; Ade 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 pre-

serve the same

sky

for each mask, nor is the CO mask needed,

since we use these masks only on the 353 GHz high Galactic lat-

itude data. Note that for

side

8, pixels have an area of about

54 deg

and characteristic dimension about 7.

◦

4. Therefore, this

HEALPix

grid oversamples the sky relative to this patch size, so

that the patches overlap and are not independent.

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, since 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.

Fig. 2 also shows the

power spectrum computed from

the

Planck

2013 best-fit

CDM model of the CMB tempera-

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

expectation for

from the CMB model with primordial grav-

itational waves with amplitude

2. At 353 GHz, the

The spectra are in units of

CMB

at 353 GHz.

angular power spectra of the dust are about 3–4 orders of mag-

nitude larger than the CMB model at

30, 1–2 orders of mag-

nitude 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 magni-

tude 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 magnitude higher.

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

nificant bias, or

-to-

leakage, from the computation of the

dust angular power spectra using

Xpol

. Fig. 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 regions the

absolute value of the null-test spectra anticipated in Sect. 2.3,

here computed from the cross-spectra of the HalfRing

DetSet

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 estimates are

consistent with the noise expectations; there is no evidence for

any e

ects of systematics.

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 bandpowers in the range 40

< ` <

500. For

both power spectra we restricted the fit to

` >

40 to avoid a

possible bias from systematics e

ects in the data, and also be-

cause the angular power spectra of the dust polarization exhibit

larger spatial variation, particularly on large angular scales, than

is expected for a Gaussian random field.

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 2014) on similar-sized regions at inter-

mediate Galactic latitude, but slightly flatter than the

fitted

at higher

and higher frequency (Miville-Desch

enes 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,

dof

21) are

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

spectra,

Planck Collaboration: Dust polarization at high latitudes

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 best-fit power laws in

are

displayed 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 displayed 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 displayed 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.

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 di

erent LR regions defined in Sect. 3.3.1, dis-

tinguished 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.

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

between 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. In any case,

we stress that while the power law in

is a good general de-

scription of the shape of the dust polarized power spectra, a full

characterization would have to consider the detailed features that

can be discerned in Fig. 2.

4.3. Amplitude dependence on

〈

353

〉

Similar to what has been done for intensity power spectra in e.g.,

Gautier et al. (1992) and Miville-Desch

enes et al. (2007), we in-

vestigate how the amplitude of the dust polarization power spec-

trum scales with the dust intensity. To quantify the dust emis-

sion at 353 GHz, we use the model map derived from a modi-

fied 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

i

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,

Fig. 4: Amplitude of the dust

(red squares) and

(blue

circles) power spectra, normalized with respect to the largest am-

plitude for each mode. These are plotted versus the mean dust in-

tensity

〈

353

〉

for the six LR regions (top panel). A power-law fit

of the form

(

〈

353

〉

)

〈

353

〉

∈ {

}

, is overplot-

ted as a dashed line of the corresponding colour (these almost

overlap). The bottom panel presents the ratio of the data and the

fitted

〈

353

〉

power law; the range associated with the

un-

certainty in the power-law exponent of 1.9 is displayed in grey.

For details see Sect. 4.3.

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), and plotted as a func-

tion of

〈

353

〉

in Fig. 4, after normalization by the maximum

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

cal 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-Desch

enes

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

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 plotted for di

erent values of

sky

in Fig. 5. For all of the

LR regions, we observe more power in the

dust spec-

trum 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

Planck Collaboration: Dust polarization at high latitudes

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, distinguished here with

sky

. The mean value

〈

〉

52 is plotted as a dashed line.

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. 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 exponent

−

42, in the range 40

< ` <

500, for

and

As an illustration of the quality of the fit, for the smallest region

(LR24,

sky

3) the averages and dispersions of the

(21

degrees of freedom) of the fits are

2 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 (2014), 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

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,

≡

√

Fig. 6: Frequency dependence of the amplitudes

the angular power spectra, relative to 353 GHz (see details in

Sect. 4.5). Results for

(red squares) and

(blue circles)

for the smallest intermediate latitude region, LR24, are shown in

the top panel. These include evaluations from cross-spectra in-

volving polarization data at two frequencies, plotted at the geo-

metric mean frequency. The square of the adopted relative SED

for dust polarization, which is a modified blackbody spectrum

with

59 and

6 K, is displayed as a black dashed

line. The bottom panel shows the relative discrepancy with re-

spect to this model spectrum. The

uncertainty area from

the expected dispersion of

for the size of LR24, as inferred

from Planck Collaboration Int. XXII (2014) (see Sect. 2.2.1), is

displayed in grey.

6 K (Planck Collaboration Int. XXII 2014). Note that no

fit was performed at this stage and that the square of the adopted

dust SED goes trough the 353 GHz data point. However, if we

fit the amplitude of the dust frequency dependence, with fixed

and

, the

(

dof

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

an 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 C.4. Furthermore, we

do not see any excess at either 100 or 217 GHz, such as could

arise from leakage, and

or polarization, associated with CO line

emission in these bands (Sect. 2.2.5).

5. Statistical study of the dust power spectra at

high Galactic latitudes

In Sect. 4, statistical properties of dust polarization were derived

from angular power spectra computed with the

Planck

353 GHz

data on large fractions of the sky (

sky

from 24 % to 72 %). Most

of the CMB experiments target fields at high Galactic latitude

with a smaller size than this. So now we evaluate the dust

mode power in such patches.

We perform a statistical analysis computing the

and

power spectra of the 353 GHz

Planck

polarization data for

high Galactic latitude patches of size 400 deg

, similar to the size

of the fields observed in ground-based and balloon-borne CMB

polarization experiments (see Sect. 3.3.2). We verify that the em-

Planck Collaboration: Dust polarization at high latitudes

Fig. 7: Fitted dust

(top panel) and

(bottom panel) amplitudes (

and

) at

80, in

for the 400 deg

patches

as a function of their mean

〈

353

〉

. The empirical scaling law,

∝

〈

353

〉

, adjusted in amplitude to the data points, is over-

plotted as a red line. The

statistical error on this relation from Monte Carlo simulations of

Planck

inhomogeneous noise (see

Appendix B.1) is represented as a light-blue shaded area and the total

error, including statistical noise plus Gaussian sample

variance, is represented as a light-red shaded area. Points are computed for all 352 patches, but note that, as described in Sect. 3.3.2,

the patches overlap and so their properties are not independent.