Planck Early Results XVIII: The power spectrum of cosmic infrared background anisotropies

Abstract

Using Planck maps of six regions of low Galactic dust emission with a total
area of about 140 square degrees, we determine the angular power spectra of
cosmic infrared background (CIB) anisotropies from multipole l = 200 to l =
2000 at 217, 353, 545 and 857 GHz. We use 21-cm observations of HI as a tracer
of thermal dust emission to reduce the already low level of Galactic dust
emission and use the 143 GHz Planck maps in these fields to clean out cosmic
microwave background anisotropies. Both of these cleaning processes are
necessary to avoid significant contamination of the CIB signal. We measure
correlated CIB structure across frequencies. As expected, the correlation
decreases with increasing frequency separation, because the contribution of
high-redshift galaxies to CIB anisotropies increases with wavelengths. We find
no significant difference between the frequency spectrum of the CIB
anisotropies and the CIB mean, with Delta I/I=15% from 217 to 857 GHz. In terms
of clustering properties, the Planck data alone rule out the linear scale- and
redshift-independent bias model. Non-linear corrections are significant.
Consequently, we develop an alternative model that couples a dusty galaxy,
parametric evolution model with a simple halo-model approach. It provides an
excellent fit to the measured anisotropy angular power spectra and suggests
that a different halo occupation distribution is required at each frequency,
which is consistent with our expectation that each frequency is dominated by
contributions from different redshifts. In our best-fit model, half of the
anisotropy power at l=2000 comes from redshifts z<0.8 at 857 GHz and z<1.5 at
545 GHz, while about 90% come from redshifts z>2 at 353 and 217 GHz,
respectively.

Full text

Page 1

arXiv:1101.2028v1 [astro-ph.CO] 11 Jan 2011
Astronomy & Astrophysics manuscript no. Planck2011-6.6˙astroph
January 12, 2011
c ? ESO 2011
Planck Early Results: The Power Spectrum Of Cosmic Infrared
Background Anisotropies
Planck Collaboration: P. A. R. Ade70, N. Aghanim46, M. Arnaud57, M. Ashdown55,76, J. Aumont46, C. Baccigalupi68, A. Balbi28,
A. J. Banday74,7,62, R. B. Barreiro52, J. G. Bartlett3,53, E. Battaner78, K. Benabed47, A. Benoˆ ıt47, J.-P. Bernard74,7, M. Bersanelli26,41, R. Bhatia33,
K. Blagrave6, J. J. Bock53,8, A. Bonaldi37, L. Bonavera68,5, J. R. Bond6, J. Borrill61,72, F. R. Bouchet47, M. Bucher3, C. Burigana40, P. Cabella28,
J.-F. Cardoso58,3,47, A. Catalano3,56, L. Cay´ on19, A. Challinor77,55,10, A. Chamballu44, L.-Y Chiang49, C. Chiang18, P. R. Christensen65,29,
D. L. Clements44, S. Colombi47, F. Couchot60, A. Coulais56, B. P. Crill53,66, F. Cuttaia40, L. Danese68, R. D. Davies54, R. J. Davis54, P. de
Bernardis25, G. de Gasperis28, A. de Rosa40, G. de Zotti37,68, J. Delabrouille3, J.-M. Delouis47, F.-X. D´ esert43, H. Dole46, S. Donzelli41,50,
O. Dor´ e53,8, U. D¨ orl62, M. Douspis46, X. Dupac32, G. Efstathiou77, T. A. Enßlin62, H. K. Eriksen50, F. Finelli40, O. Forni74,7, P. Fosalba48,
M. Frailis39, E. Franceschi40, S. Galeotta39, K. Ganga3,45, M. Giard74,7, G. Giardino33, Y. Giraud-H´ eraud3, J. Gonz´ alez-Nuevo68,
K. M. G´ orski53,80, J. Grain46, S. Gratton55,77, A. Gregorio27, A. Gruppuso40, F. K. Hansen50, D. Harrison77,55, G. Helou8, S. Henrot-Versill´ e60,
D. Herranz52, S. R. Hildebrandt8,59,51, E. Hivon47, M. Hobson76, W. A. Holmes53, W. Hovest62, R. J. Hoyland51, K. M. Huffenberger79,
A. H. Jaffe44, W. C. Jones18, M. Juvela17, E. Keih¨ anen17, R. Keskitalo53,17, T. S. Kisner61, R. Kneissl31,4, L. Knox21, H. Kurki-Suonio17,35,
G. Lagache46⋆, J.-M. Lamarre56, A. Lasenby76,55, R. J. Laureijs33, C. R. Lawrence53, S. Leach68, R. Leonardi32,33,22, C. Leroy46,74,7, P. B. Lilje50,9,
M. Linden-Vørnle12, F. J. Lockman34, M. L´ opez-Caniego52, P. M. Lubin22, J. F. Mac´ ıas-P´ erez59, C. J. MacTavish55, B. Maffei54, D. Maino26,41,
N. Mandolesi40, R. Mann69, M. Maris39, P. Martin6, E. Mart´ ınez-Gonz´ alez52, S. Masi25, S. Matarrese24, F. Matthai62, P. Mazzotta28,
A. Melchiorri25, L. Mendes32, A. Mennella26,39, S. Mitra53, M.-A. Miville-Deschˆ enes46,6, A. Moneti47, L. Montier74,7, G. Morgante40,
D. Mortlock44, D. Munshi70,77, A. Murphy64, P. Naselsky65,29, P. Natoli28,2,40, C. B. Netterfield14, H. U. Nørgaard-Nielsen12, D. Novikov44,
I. Novikov65, I. J. O’Dwyer53, S. Oliver16, S. Osborne73, F. Pajot46, F. Pasian39, G. Patanchon3, O. Perdereau60, L. Perotto59, F. Perrotta68,
F. Piacentini25, M. Piat3, D. Pinheiro Gonc ¸alves14, S. Plaszczynski60, E. Pointecouteau74,7, G. Polenta2,38, N. Ponthieu46, T. Poutanen35,17,1,
G. Pr´ ezeau8,53, S. Prunet47, J.-L. Puget46, J. P. Rachen62, W. T. Reach75, M. Reinecke62, M. Remazeilles3, C. Renault59, S. Ricciardi40, T. Riller62,
I. Ristorcelli74,7, G. Rocha53,8, C. Rosset3, M. Rowan-Robinson44, J. A. Rubi˜ no-Mart´ ın51,30, B. Rusholme45, M. Sandri40, D. Santos59, G. Savini67,
D. Scott15, M. D. Seiffert53,8, P. Shellard10, G. F. Smoot20,61,3, J.-L. Starck57,11, F. Stivoli42, V. Stolyarov76, R. Stompor3, R. Sudiwala70,
R. Sunyaev62,71, J.-F. Sygnet47, J. A. Tauber33, L. Terenzi40, L. Toffolatti13, M. Tomasi26,41, J.-P. Torre46, M. Tristram60, J. Tuovinen63,
G. Umana36, L. Valenziano40, P. Vielva52, F. Villa40, N. Vittorio28, L. A. Wade53, B. D. Wandelt47,23, M. White20, D. Yvon11, A. Zacchei39, and
A. Zonca22
(Affiliations can be found after the references)
Preprint online version: January 12, 2011
Abstract
Using Planck maps of six regions of low Galactic dust emission with a total area of about 140 deg2, we determine the angular power spectra of
Cosmic Infrared Background (CIB) anisotropies from multipole ℓ = 200 to ℓ = 2000 at 217, 353, 545 and 857GHz. We use observations of Hi
emission as a tracer of thermal dust emission in order to reduce the already low level of Galactic dust emission and use the 143GHz Planck maps
in these fields to clean out cosmic microwave background anisotropies. Both of these cleaning processes are necessary in order to avoid significant
contamination of the CIB signal. We measure correlated CIB structure across frequencies. As expected, the correlation decreases with increasing
frequency separation as the contribution of high-redshift galaxies to CIB anisotropies increases with wavelengths. We find no significant difference
between thefrequency spectrum of theCIB anisotropies and theCIBmean, with∆I/I=15%from217 to 857GHz. Intermsof clustering properties,
the Planck data alone ruled out the linear scale- and redshift- independent bias model. Non-linear corrections are important. Consequently, we
develop an alternative model that couples a dusty galaxy, parametric evolution model with a simple halo model approach. It provides an excellent
fit to the measured anisotropy angular power spectra and suggests that a different halo occupation distribution is required at each frequency, which
is consistent with the fact that we expect each frequency to be dominated by contributions from different redshifts. In our best-fit model, half of
the anisotropies power at ℓ=2000 comes from redshifts z < 0.8 at 857GHz and z < 0.9 at 545GHz, while about 1/5 and 2/3 come from redshifts
z >3.5 at 353 GHz and 217GHz, respectively.
Key words. Cosmology: observations
1. Introduction
In addition to instrument noise, deep cosmological surveys in
the far-infrared to millimeter spectral range are limited in depth
by confusion from extragalactic sources. This limitation arises
from the high density of faint, distant galaxies that produces sig-
nal fluctuations within the telescope beam. As a consequence,
⋆Corresponding author: G. Lagache, guilaine.lagache@ias.u-psud.fr
the Cosmic Infrared Background (CIB), which records much of
the radiant energy released by processes of structure formation
that have occurred since the decoupling of matter and radia-
tion following the Big Bang (Puget et al. 1996; Hauser & Dwek
2001; Dole et al. 2006), is barely resolved into its constituents.
Indeed,less than10%oftheCIBisresolvedbySpitzerat160µm
(B´ ethermin et al. 2010a), about 10% by Herschel at 350 µm
(Oliver et al. 2010) and a negligible fraction is resolved by
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Planck Collaboration: CIB anisotropies with Planck
Planck1(Fernandez-Conde et al. 2008). Thus, in the absence of
foreground (Galactic dust) and background (cosmic microwave
background,CMB) emissions, and when the instrument noise is
subdominant, maps of the diffuse emission at the angular reso-
lution probed by the current surveys reveal a web of structures,
characteristic of CIB anisotropies. With the advent of large area
far-infrared to millimeter surveys (Herschel, Planck, SPT and
ACT), CIB anisotropies constitute a new tool for structure for-
mation and evolution.
CIB anisotropies are expected to trace large-scale struc-
ture and to probe the clustering properties of galaxies which
in turn are linked to those of their hosting dark matter halos.
As the clustering of dark matter is well understood, observa-
tions of anisotropies in the CIB constrain the relationship be-
tween dusty, star-forming galaxies and the dark matter distribu-
tion. The connection between a population of galaxies and dark
matter halos can be described by its halo occupation distribution
(HOD; Peacock & Smith 2000; Seljak 2000; Benson et al. 2000;
White et al. 2001; Berlind & Weinberg 2002; Cooray & Sheth
2002), which specifies the probabilitydistribution of the number
of objectswith a givenproperty(e.g.,luminosity,stellar mass, or
starformationrate)withinadarkmatterhaloofagivenmassand
their radial distribution within the halo. The HOD and the halo
model provide a powerful theoretical framework for describing
the connection between galaxies and dark matter halos. Once
decisions are made about which properties of the halos and their
environment the HOD depends upon, what the moments of the
HOD are and what the radial profile of objects within halos is,
the halo model can be used to predict any clustering-related ob-
servable. In particular, the halo model predicts that the bias, de-
scribing the clustering of galaxies in relation to the dark matter,
becomes scale-independentat large scales. This assumption of a
scale-independent bias is often made in modeling the CIB.
The way galaxies populate dark matter halos is not the
only ingredient that enters into the CIB anisotropies modeling.
Correlated anisotropies also depend on the mean emissivity
per comoving unit volume of dusty, star-forming galaxies, that
results from dusty galaxies evolution models. Such models are
more and more constrained thanks to the increasing number
of observations (mainly galaxies number counts and lumi-
nosity functions), but remain largely empirical. So far, CIB
anisotropies models have combined (i) a scale-independent
bias clustering with, a very simple emissivity model based
on the CIB mean (Knox et al. 2001; Hall et al. 2010) or an
empirical model of dusty galaxy evolution (Lagache et al.
2007) or the predictions of the physical model by Granato et al.
(2004) for the formation and evolution of spheroidal galaxies
(Negrello et al. 2007) (ii) a HOD with the Lagache et al. (2003)
dusty galaxies evolution model (Amblard & Cooray 2007;
Viero et al. 2009) (iii) a merger model of dark matter halos with
a very simple dust evolution model (Righi et al. 2008).
The angular power spectrum of CIB anisotropies has two
contributions, a white-noise component due to shot noise and
an additional component arising due to spatial correlations be-
tween the sources of the CIB. Correlated CIB anisotropies
have been measured at 3330 GHz by Akari (Matsuura et al.
1Planck (http://www.esa.int/Planck ) is a project of the European
Space Agency (ESA) with instruments provided by two scientific con-
sortia funded by ESA member states (in particular the lead countries
France and Italy), with contributions from NASA (USA) and telescope
reflectorsprovided by acollaboration between ESAand ascientificcon-
sortium led and funded by Denmark.
2010), 3000 GHz by IRAS/IRIS (Penin et al. 2011), 1875 GHz
by Spitzer (Lagache et al. 2007; Grossan & Smoot 2007), 1200,
857, 600 GHz by BLAST (Viero et al. 2009) and 220 GHz
by SPT (Hall et al. 2010) and ACT (Dunkley et al. 2010).
Depending on the frequency, angular resolution and size of the
survey these measurements can probe two different clustering
regimes. On small angular scales (ℓ ≥ 2000), they measure
the clustering within a single dark matter halo, and hence the
physics governing how dusty, star-forming galaxies form within
a halo. On large angular scales, CIB anisotropies measure clus-
teringbetweengalaxiesindifferentdarkmatterhalos.Suchmea-
surements primarily constrain the large-scale, linear bias, b, of
dustygalaxies,usuallyassumedto bescale-independentoverthe
relevant range. Given their limited dynamic range in scale, cur-
rent measurements are equally consistent with an HOD model,
a power-law correlation function or a scale-independent, linear
bias. All models return a value for the large-scale bias which
is 2–4 times higher than that measured for local, dusty, star-
forming galaxies (where b ≃ 1).
Due to its frequencycoveragefrom 100 to 857GHz,the HFI
instrument on board Planck is ideally suited to probe the dark
matter – star-formation connection. Planck (Tauber et al. 2010;
Planck Collaboration 2011a) is the third generation space mis-
sion to measure the anisotropy of the cosmic microwave back-
ground (CMB). It observes the sky in nine frequency bands
covering 30–857GHz with high sensitivity and angular reso-
lution from 31′to 5′. The Low Frequency Instrument (LFI;
Mandolesi et al. 2010; Bersanelli et al. 2010; Mennella et al.
2011) covers the 30, 44, and 70GHz bands with ampli-
fiers cooled to 20K. The High Frequency Instrument (HFI;
Lamarre et al. 2010; Planck HFI Core Team 2011a) covers the
100, 143, 217, 353, 545, and 857GHz bands with bolome-
ters cooled to 0.1K. Polarization is measured in all but the
highest two bands (Leahy et al. 2010; Rosset et al. 2010). A
combination of radiative cooling and three mechanical cool-
ers produces the temperatures needed for the detectors and op-
tics (Planck Collaboration2011b). Two Data Processing Centres
(DPCs) check and calibrate the data and make maps of the sky
(Planck HFI Core Team 2011a; Zacchei et al. 2011). Planck’s
sensitivity, angular resolution, and frequency coverage make
it a powerful instrument for galactic and extragalactic astro-
physics as well as cosmology. Early results are given in Planck
Collaboration (2011a–u).
The primary objective of this paper is to measure with
Planck HFI the CIB anisotropies due to the clustering of star-
forming galaxies. To achieve this we analyze small regions of
sky, with a total area of about 140 square degrees, where we
are able to cleanly separate the foreground (Galactic cirrus) and
background (CMB) components from the signal. Unlike pre-
vious CIB anisotropies studies (but see Penin et al. 2011), we
do not remove the cirrus by fitting a power law power spectra
at large scales, but use an independent, external tracer of dif-
fuse dust emission (the Hi gas). We accurately measure the in-
strumental contributions (noise, beam) to the power spectra of
CIB anisotropies and use a dedicated optimal method to mea-
sure power spectra (Ponthieu et al. 2011). All of these steps al-
low us to recover, for the first time, the power spectra of CIB
anisotropies from 200 ≤ ℓ ≤ 2000 at four frequencies simulta-
neously: 217, 353, 545 and 857GHz.
The paper is organized as follow: in Sect. 2, we present
the data we are using, the field selection and the removal of
foregroundand backgroundcomponents (CMB, Galactic cirrus,
bright point sources) from the CIB. In Sect. 3 we discuss the dif-
ferent contributions to the power spectra of the residual maps.
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Planck Collaboration: CIB anisotropies with Planck
Section 4 describes how we estimated the power spectrum, its
bias and errors. Our main results are presented in Sect. 5. This
section also describes our modeling and discusses the clustering
of high-redshift, dusty galaxies. We conclude in Sect. 6.
Throughout the paper we use the WMAP7 cosmological pa-
rameters for standard ΛCDM cosmology (Larson et al. 2010).
2. Selected fields and data cleaning
2.1. Planck Data
We use Planck channel maps of the HFI at 5 frequencies: 143,
217, 353, 545 and 857GHz. Their characteristics and how they
were created is described in detail in the companion paper on
HFI Early Processing (hereafter HEP; Planck HFI Core Team
2011a). In summary, the channel maps correspond to temper-
ature observations for the two first sky surveys by Planck. The
data are organized as Time-Ordered-Information,hereafter TOI.
The attitude of the satellite as a function of time is provided
by two star trackers on the spacecraft. The pointing for each
bolometer is computed by combining the attitude with the lo-
cation of the bolometer in the focal plane, as determined by
planet observations (see below). TOIs of raw bolometer data
are first processed to produce cleaned timelines and to set flags
to mark data we do not currently fit. This TOI processing in-
cludes (1) signal demodulation and filtering, (2) deglitching,
which flags the strong part of any glitch and subtracts the tails,
(3) conversion from instrumental units (volts) to physical units
(watts of absorbed power, after a correction for the weak non-
linearity of the response), (4) decorrelation of thermal stage
fluctuations, (5) removal of the systematic effects induced by
4K cooler mechanical vibrations, and (6) deconvolution of the
bolometer time constant. Focal plane reconstruction and beam
shape estimation is made using observations of Mars. The sim-
plest description of the beams, an elliptical Gaussian, leads to
full-width half-maximum (FWHM) values, θS, given in Table 3
of the HEP (i.e. 9.53′,7.08′,4.71′,4.50′,4.72′and 4.42′respec-
tively, with an uncertainty between 0.10′and 0.28′). From the
cleaned TOI and the pointing, channel maps have been com-
puted using all the bolometers at a given frequency. The path
from TOI to maps in the HFI DPC is schematically divided
into three steps: ring-making, ring offset estimation and map-
making. The first step combines the data within a stable point-
ing period, during which the same circle on the sky is scanned
repeatedly, in order to create rings with higher signal-to-noise
ratio, taking full advantage of the redundancy of observations
provided by the Planck scanning strategy. The low frequency
component of the noise is accounted for in a second step by us-
ing a destriping technique which models this component as a
global offset of the ring values. Finally, cleaned maps are pro-
duced by coadding the offset-corrected rings. The maps are pro-
duced in Galactic coordinates, using the HEALPix pixelisation
scheme(seehttp://healpix.jpl.nasa.govandG´ orski et al.(2005)).
Photometric calibration is performed either at ring level (using
the CMB dipole) for the lower frequency channels or at the map
level (using FIRAS data) for the higher frequencychannels (545
and 857GHz). The absolute gain calibration of the HFI Planck
maps is known to better than 2% for the lower frequencies (143,
217 and 353GHz) and 7% for the higher frequencies (545 and
857GHz), as summarised in the HEP Table 3. Inter-calibration
accuracy between channels is better than absolute calibration.
We make use of the so-called DX4 HFI data release, a
dataset from which the CMB has not been removed. We use the
217, 353, 545, and 857GHz channels for CIB analysis, and the
Figure2. Wiener filter applied to the 143GHz map for CMB
subtraction.Thefilteressentially cuts outhighmultipoles,where
the CMB to noise ratio of the 143GHz map is low. Whereas this
filter has tobe knownforestimatingandsubtractingcontribution
of residual CMB and 143GHz noise to the power spectrum of
CMB-cleaned channels, the exact value of the filter is not really
critical.
143GHz channel for CMB removal. Maps are given either in
MJysr−1(with the photometric convention νIν=cst2) or µKCMB,
the conversion between the two being exactly computed using
the bandpass filters (see Planck HFI Core Team 2011a).
2.2. Extragalactic fields with high angular resolution Hi data
Although Planck is an all-sky survey, we restrict our first CIB
anisotropy measurements to a few fields at high Galactic lati-
tude, to minimize the Galactic dust contamination. The choice
of the fields was driven by the availability of Hi data at an angu-
lar resolution close to HFI.
The 21-cm Hi spectra used here were obtained with the
100-meter Green Bank Telescope (GBT) over the period 2005
to 2010. Details of this high-latitude survey are presented by
Martinet al. (in prep).The total area mappedis about825square
degrees.
The spectra were taken with on-the-fly mapping. The pri-
mary beam of the GBT at 21cm has a FWHM of 9.1′, and the
integration time (4s) and telescope scan rate were chosen to
sample every 3.5′, more finely than the Nyquist interval, 3.86′.
The beam is only slightly broadened to 9.4′in the in-scan di-
rection. Scans were made moving the telescope in one direc-
tion (Galactic longitude or Right Ascension), with steps of 3.5′
in the orthogonal coordinate direction before the subsequent re-
verse scan.
Data were recorded with the GBT spectrometer by in-band
frequency switching yielding spectra with a velocity coverage
−450 ≤ VLSR ≤ +355 kms−1at a resolution of 0.80 kms−1.
Spectra were calibrated, corrected for stray radiation, and
placed on a brightness temperature (Tb) scale as described in
Blagrave et al. (2010), Boothroyd et al. (in prep), and Martin et
al. (in prep). A third-order polynomial was fit to the emission-
free regions of the spectra to remove any residual instrumental
baseline. The spectra were gridded on the natural GLS (SFL)
projection to produce a data cube. Some regions were mapped
2The convention νIν=cst means that the MJysr−1are given for a
source with a spectral energy distribution Iν ∝ ν−1. For a source with
a different spectral energy distribution a color correction has to be ap-
plied (see Planck HFI Core Team 2011a).
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Planck Collaboration: CIB anisotropies with Planck
FieldGalactic Longitude
degrees
85.33
164.84
132.37
152.38
61.29
58.02
Galactic Latitude
degrees
44.28
65.50
47.50
53.30
72.32
68.42
Size Mean N(Hi)
1020at/cm2
1.2
1.8
1.2
0.7
1.2
1.1
σ N(Hi)
1020at/cm2
0.3
0.6
0.3
0.2
0.2
0.2
arcmin×arcmin
308×308
308×308
308×308
241.5×241.5
283.5×283.5
283.5×283.5
N1
AG
SP
LH2
Bootes 1
Bootes 2
Table 1. CIB field description: center (Galactic coordinates), size, mean and dispersion of Hi column density.
Figure1.
(Miville-Deschˆ enes & Lagache 2005). Fields Bootes 1 and 2 are both included in the large rectangle. All IRIS images have the
same dynamic range, with a linear color scale ranging from dark red to white from 0 to 2 MJy/sr.
From Left to right and top to bottom: N1, AG, SP, LH2 and bootes fields overlaid on IRIS 100 µm map
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Planck Collaboration: CIB anisotropies with Planck
two or three times. With the broadspectral coverage,all Hi com-
ponents from local gas to high velocity clouds are accessible.
We select from this GBT cirrus survey the 6 faintest fields
in terms of Hi column densities. Their main characteristics are
given in Table 1 and the IRAS 100µm maps are shown in Fig.
1. They have all very low dust contamination, and thus Hi col-
umn densities, including the faintest all-sky sight line (refer-
enced as LH2 in the Table). The field areas are between 16 to 25
square degrees for a total coverage of about 140 square degrees.
Going to higher average Hi column densities (N(Hi)> 2 × 1020
cm−2) is not recommended as dust emission associated with
molecular regions starts to contaminate the signal (see Fig. 9 of
Planck Collaboration2011t) and Hi is no longer a good tracer of
dust emission.
The HEALPix HFI maps are reprojected onto the small
Hi maps by binning the original HEALPix data (sampled with
HEALPix Nside of 2048, correspondingto a pixel size of 1.72′)
into Hi map pixels (pixel size 3.5′for all fields). An average of
slightlymorethan4HEALPixpixelsareaveragedforeachsmall
map pixel.
2.3. Removing the bright sources from HFI maps
We remove from the maps all sources listed in the
Planck Early Release Compact Source Catalog (ERCSC)
(Planck Collaboration 2011c). This represents only a few
sources per field (if any), but the bright source removal is im-
portant for both power spectrum analysis and CMB map con-
struction. It is also important to know the flux limit in order to
compute the radio and dusty galaxy shot-noise contribution to
the power spectra. Since our fields have roughly the same, very
low dust contamination, source detection is not limited by cir-
rus. Indeed, the flux cut is set by extragalactic source confusion
at high frequencies and the CMB at low frequencies. The same
flux cut can thus be applied to all our fields. We take the mini-
mum ERCSC flux densities in our fields as the flux cuts. They
are given in Table 3.
In practice, point source removalis performedin the original
HFI HEALPix data, priorto reprojection.Foreach source,a disc
of size equal to the FWHM of the beam centered on the source
positionis blanked.Holes dueto missingdata are thenfilled bya
gap-filling process, which interpolates/extrapolatesinto the hole
the values of neighbouringpixels.
2.4. Removing the CMB contamination from HFI maps
CMB anisotropies contribute significantly to the total HFI map
variance in all channels at frequencies up to and including
353GHz.ThedetectionandcharacterizationofCIB anisotropies
at these frequencies requires separation of the contribution from
the CMB.
The present work focuses on very clean regions of the sky,
for which Galactic foregrounds are very faint, and are moni-
tored using ancillary Hi observations. To remove CMB in the
fields retained for our analysis we use a simple subtraction tech-
nique. While this simple method could be improved in future,
it makes it possible to reliably evaluate CMB residuals, noise
contamination, and to propagate errors due to imperfect instru-
mental knowledge. It also guarantees that high frequency CIB
anisotropy signals will not leak into lower frequency, CMB-free
maps.
We remove CMB contamination in the 217 and 353GHz
channels by subtracting a CMB template obtained from the
Figure3. Power spectra of the different components for field
SP (the figure is similar for the other fields). Power spectra of
the 217, 353, 545 and 857GHz Planck maps (continuous black
line) are compared to the noise power spectra (diamonds), to
the CMB-cleaned power spectra (red), and to the CMB- and
interstellar dust-cleaned power spectra (green). In this plot sig-
nal power spectra have not been corrected for the beam window
function. Noise power spectra are computed using half-pointing
period maps, as explained in Sect. 3.3.
lower frequency data. We model the data, for each frequency
ν, as
xℓm(ν) = bℓ(ν)
?
aCMB
ℓm
+ aCIB
ℓm(ν)
?
+ nℓm(ν) (1)
5

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Planck Collaboration: CIB anisotropies with Planck
Figure4. Maps of the 26 Sq. Deg. of the N1 field, from left to right: 217, 353, 545 and 857GHz. From top to bottom: raw HFI
maps; CMB- and ERCSC source-cleanedmaps; residual maps (CMB-, sources-, and cirrus-cleaned);residual maps smoothedat 10′
to highlight the CIB anisotropies. The joint structures clearly visible (bottom row) correspond to the anisotropies of CIB. Residual
point sources are also visible. They have fluxes lower than the fluxes of the ERCSC removed sources. They have no impact on our
analysis.
where xℓm(ν) represents the channel data at frequency ν, aCMB
the CMB map, aCIB
comprising(if needed)anyotherastrophysicalcontaminant.The
effect of the beam is accounted for with a (channel dependent)
multiplicative factor, bℓ(ν) (see Sect. 3.2). For the purpose of
CMB removal, bℓ(ν) is obtained from the Gaussian best fit to the
effective HFI beam of the channel maps.
ℓm
ℓm(ν) the CIB map and nℓm(ν) is a noise term
At 100and143GHz,we assume that,in the fields ofinterest,
only CMB and noise is present. CIB anisotropies are very small,
and in the selected fields the contamination by other sources
(e.g., cirrus) is negligible(see Sect. 3.4). In principle, both chan-
nels can be used to make a template of CMB emission. However
the 100GHz channel is significantly less sensitive than 143GHz
and has an angular resolution two times worse than the 217 and
353GHz channels. Therefore, we only use the 143GHz channel
as a CMB template.
6

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Planck Collaboration: CIB anisotropies with Planck
Figure5. Hi and dust maps for two fields: SP (top) and AG (bottom). The first two maps on the left show the Hi components (Local
and IVC for SP, IVC and HVC for AG), the third maps show the 857GHz emission associated with Hi (?
the right side show the HFI 857GHz maps. Those HFI maps have been convolved by the GBT beam to allow a better comparison
by eye. Hi maps are given in units of 1020atoms cm−2. Note the correlation of the dust emission with the different Hi velocity
components and its variation from field to field.
iαi
νNi
HI) and the maps on
We correct the 217GHz maps for CMB contamination as
follows:
yℓm(ν217) = xℓm(ν217) −bℓ(ν217)
= bℓ(ν217)
+ nℓm(ν217) −bℓ(ν217)
bℓ(ν143)wℓxℓm(ν143)
aCIB
?
ℓm(ν217) + (1 − wℓ)aCMB
bℓ(ν143)wℓnℓm(ν143),
ℓm
?
(2)
where wℓis a Wiener filter, designed to minimize the total con-
tamination of the new map, y(ν217), by CMB and instrument
noise. The 353GHz map is corrected from CMB contamination
in a similar way. Note that this cleaning is performed on a large
region comprising all the small fields used in the present analy-
sis. The Wiener filter is obtained as:
wℓ=bℓ(ν143)CCMB
Yℓ(ν143)
ℓ
(3)
where CCMB
Yℓ(ν143) is the power spectrum of the 143GHz map. The Wiener
filter wℓis close to 1 at low ℓ, and close to 0 at large ℓ (see Fig.
2). Note that errors on the beam estimate, on the assumed CMB
power spectrum, or on the estimation of the 143GHz power
spectrum would result in sub-optimal filtering rather than in bi-
ases. We checked that the CMB remaining in the CMB-cleaned
maps does not change significantly with different assumptions
leading to different wℓ.
Errors in photometric calibration between channels are an
issue. Although these errors are estimated to be small (2% at
143, 217, and 353GHz), they may result in residual CMB at low
ℓ. They are accounted for in the processing, as detailed in Sect.
4.2.1.
ℓ
is the current best fit CMB model spectrum, and
Fig. 3 shows the HFI power spectra of the raw and CMB-
cleaned maps, for one of the 6 fields. The CMB correction
is very large at 217GHz: the residual is a factor ∼100 below
the raw power spectrum at ℓ ≃ 430 (it is a factor ∼2 below at
353GHz). Note that whereas this illustrates the effectiveness of
CMB removal, it is also a source of worry about the impact of
relative calibration errors for the 217GHz channel. However,
the power spectrum after CMB cleaning is ∼1% of the original
map power spectrum only for ℓ ≤ 600. CMB-cleaned maps
are shown on Fig 4. We see that the CMB has been efficiently
removed.
Finally we remark that an alternative method of removing
CMB contamination, based on an internal linear combination of
frequencymapsandaneedletanalysis(Delabrouille et al.2009),
was extensively studied and used in some of the Planck Early
Papers, but it was not well suited to our purposes. The method
tended to perform well over large patches of sky but left visible,
large-scale residuals in the sky patches of interest, and had leak-
age between the faint CIB and the CMB in the presence of other
components (noise and Galactic cirrus).
2.5. Removing the cirrus contamination from HFI maps
From 100 µm to 1mm, at high Galactic latitude and outside
molecular clouds a tight correlation is observed between far-
infrared emission from dust and the 21-cm emission from gas3
(e.g. Boulanger et al. 1996; Lagache et al. 1998). Hi can thus be
used as a tracer of cirrus emission in our fields, and indeed it is
the best tracer of diffuse interstellar dust emission.
3The Pearson correlation coefficient is > 0.9 (Lagache et al. 2000).
7

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Planck Collaboration: CIB anisotropies with Planck
Hi components –The Hi data in each field show different veloc-
ity components: a local component, typical of high-latitude Hi
emission, intermediate-velocityclouds(IVCs) and high-velocity
clouds (HVCs). These clouds are typically defined as concentra-
tions of neutral hydrogenat velocities inconsistent with a simple
model of differential Galactic rotation. The distinction between
IVCs and HVCs is loosely based on the observed radial veloc-
ities of the clouds; IVCs have radial velocities with respect to
the local standard of rest (LSR) of 30 ≤ |VLSR| ≤ 90km s−1,
while HVCs have velocities |VLSR| > 90km s−1. HVCs might
be infalling clouds fueling the Galaxy with low-metallicity gas
whereas IVCs might have a Galactic origin (e.g. Richter et al.
2001). For each field, we construct integrated Hi emission maps
of the three velocity components. The selection of the veloc-
ity range for each component is based on inspection of the me-
dian 21cm spectrum and of the rms 21cm spectrum (i.e., the
standard deviation of each channel map). It is fully described in
Planck Collaboration (2011t). The Hi maps are then converted
to Hi column density using the optically thin approximation:
N(HI)(x,y) = 1.823× 1018?
v
TB(x,y,v)δv,
(4)
where TB is the 21cm brightness temperature and v the
velocity. Corrections have been applied for opacity (see
Planck Collaboration2011t), theyarelowerthan5%forourCIB
fields. As illustrated in Fig. 5, the different fields have clearly
distinct Hi contributions, with e.g., no local component in the
direction of the AG field.
Hi-dust correlation –To remove the cirrus contamination from
the HFI maps, we need to determine the far-IR to mm emission
of the different Hi components identified with the 21cm obser-
vations. We assume that HFI maps, Iν(x,y), at frequency ν can
be represented by the following model:
Iν(x,y) =
?
i
αi
νNi
HI(x,y) + Cν(x,y)(5)
where Ni
αi
i at frequency ν and Cν(x,y) is a residual. The correlation co-
efficients αi
minimization given the Hi and HFI data and the model (Eq. 5).
Although the Hi column densities of the different components
arequitesimilar (seeFig.5),the emissivitiesmayvarybyfactors
of more than 10 between the local/IVC and HVC components
(see Planck Collaboration 2011t) so it is important to consider
them separately. The emissivities can be used to characterize
the opacity and temperature of the dust emission in the differ-
ent components. This is beyond the scope of this paper, being
extensively discussed in Planck Collaboration (2011t).
HI(x,y) is the column density of the ithHi component,
νis the far-IR to mm – Hi correlation coefficient of component
ν(often called emissivities) are estimated using a χ2
Cirrus contamination removal –We remove from the HFI maps
the Hi velocity maps multiplied by the correlation coefficients.
Maps are shown on the last two columns of Fig. 5 for two
fields. The removal is done at the HFI angular resolution,
even though the Hi map is of lower resolution. This is not a
problem since cirrus, with a k−3power-law power spectrum
(Miville-Deschˆ enes et al. 2007), has negligible power between
the GBT and HFI angular resolutions, in comparison to the
power in the CIB.
Figure6. Contribution to the CIB per redshift slice, extracted
from B´ ethermin et al. (2010c). The black solid line is the CIB
spectrum predicted by the model. The contribution to the CIB
from 0 < z < 0.3, 0.3 < z < 1, 1 < z < 2 and z > 2
galaxies is given by the red short-dash, green dot-dash, blue
three dot-dash and purplelong-dashedlines, respectively.Lower
limits coming from the stacking analysis at 100 µm, 160 µm
(Berta et al. 2010), 250 µm, 350 µm, 500 µm (Marsden et al.
2009), 850 µm (Greve et al. 2009) and 1.1 mm (Scott et al.
2010) are shown as black arrows. The black diamonds give the
Matsuura et al. (2010) absolute measurements with Akari. The
black square the Lagache et al. (2000) absolute measurements
with DIRBE/WHAM and the cyan line the Lagache et al. (2000)
FIRAS measurement.
Residual maps and power spectra –Fig. 3 shows the HFI
power spectra before and after the dust removal in the SP field.
Cirrus removal has more impact for the two high-frequency
channels. At 217GHz, the correction is very small (13% at
ℓ=500). Such a method of using Hi data to remove the cirrus
contamination from power spectra has also been successfully
applied at higher frequencies than ours, where the cirrus con-
tamination is higher,by Penin et al. (2011). They have been able
to isolate precisely the CIB anisotropies power spectra at 1875
and 3000GHz with Spitzer and IRAS/IRIS, in the N1 field.
The residual maps, at the HFI angular resolution, are shown
in Fig. 4 for the N1 field. We clearly see that the cirrus has
been efficiently removed. The bottom row shows the residual
maps, smoothed at 10′. Common structures, corresponding to
the CIB anisotropies, are clearly visible at the four frequencies.
Table 2 gives the Pearson correlation coefficients between the
CIB anisotropy maps. They are about 0.9 between the 545 and
857GHz maps and0.5 betweenthe 217and857GHz CIB maps.
The decrease when the frequency difference between the maps
is largeris expected,as the contributionof high-redshiftgalaxies
to the CIB (and its anisotropies)increases with wavelength.This
is illustrated in Fig. 6, extracted from B´ ethermin et al. (2010c),
where we show the redshift distribution of the CIB. The red-
shift distribution of correlated CIB anisotropies are discussed in
Fernandez-Conde et al. (2008, 2010) and Penin et al. (in prep).
8

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Planck Collaboration: CIB anisotropies with Planck
N1
217GHz
1
353GHz
0.56
1
545GHz
0.53
0.84
1
545 GHz
0.39
0.75
1
857GHz
0.49
0.77
0.91
857 GHz
0.39
0.74
0.89
217 GHz
353 GHz
545 GHz
Bootes 1
217 GHz
353 GHz
545 GHz
217 GHz
1
353 GHz
0.44
1
Table 2. Pearson correlationcoefficient between CIB anisotropymaps (values are given for the N1 and Bootes 1 fields to illustrate
the range of coefficients). The high-frequencymaps are highly correlated. A decorrelation is seen when going to lower frequencies.
We interpret this decorrelation as reflecting the redshift distribution of CIB anisotropies (see text, Fernandez-Conde et al. (2008)
and Penin et al. (in prep).
3. Astrophysical and instrumental components of
residual HFI maps power spectra
Once the CMB and cirrus have been removed, there are three
main astrophysical contributors to the power spectrum at the
HFI frequencies: dusty star-forming galaxies (with both cluster-
ing, Cd,clust
ℓ
galaxies (with only a shot-noise component,Cr,shot
strument noise and the signal are not correlated, the measured
power spectrum Cℓ(ν) is:
(ν), and shot noise, Cd,shot
ℓ
(ν), components) and radio
(ν)). If the in-
ℓ
Cℓ(ν) = b2
ℓ(ν)
?
Cd,clust
ℓ
(ν) + Cd,shot
ℓ
(ν) +Cr,shot
ℓ
(ν)
?
+ Nℓ(ν)
(6)
where bℓ(ν) is the beam window function, and Nℓ(ν) the power
spectrum of the instrument noise. Note that we neglect here the
Sunyaev-Zeldovich (SZ; Sunyaev & Zeldovich 1980) contribu-
tion to the power spectra. Extrapolation of SPT (Lueker et al.
2010) and ACT (Dunkley et al. 2010) constraints show that SZ
is negligible compared to CIB anisotropies at ν ≥ 217GHz. Our
goal is to accurately measure Cd,clust
extensively discuss in Sect. 5. We begin by discussing all of the
other components of Eq. 6 in this section.
ℓ
(ν), which we present and
3.1. Shot noise
The shot noise arises due to sampling of a background com-
posed of a finite number of sources. We assume the distribution
is Poisson, so that its power spectrum is independent of ℓ. If we
identify and remove all sources brighter than Scut, the shot noise
from the remaining sources fainter than Scutis given by (e.g.,
Scott & White 1999):
Cshot
ℓ
=
?Scut
0
S2dN
dSdS
(7)
where S is the source flux and dN/dS the differential number
counts. Such counts can be directly measured or derived from
evolution models of the relevant population of galaxies (dusty,
star-forming and radio galaxies in our case).
3.1.1. Star-forming, dusty galaxy shot noise, Cd,shot
ℓ
We use the recent model of B´ ethermin et al. (2010c) to compute
the IR galaxy shot-noise power. This is an updated version of
the Lagache et al. (2004) model that better reproduces new ob-
servationalconstraints(e.g.,fromHerschel).Thisnew,empirical
model uses the same galaxy spectral energy distribution (SED)
templates as Lagache et al. (2004), but a fully parametric evolu-
tion of the luminosity function. The parameters of the model are
determined by fitting the infrared/sub-mm number counts, and
some mid-IR luminosity functions, with a Monte-Carlo Markov
Chain (MCMC). The derivedpower spectra are givenin Table 3,
with uncertainties computedfrom the MCMC. The quoted num-
bers includestatistical and photometriccalibrationuncertainties.
This model has less energy output at high redshift (z ≃ 2),
and thus lower shot-noise power at long wavelength, than the
Lagache et al. (2004) model. The quoted noise levels depend on
the flux cut, which itself has an uncertaintylinked to the flux un-
certainty in the ERCSC. If we change the flux cut Scutby 30%
in Eq. 7, based on the uncertainty in ERCSC fluxes, the power
spectra change by less than 5% at all frequencies (and less than
1% at 217GHz).
As we shall discuss in Sect. 5, the dusty galaxy shot-
noise level will be a major factor in the interpretation of CIB
anisotropypowerspectra.Since we are obtainingthis valuefrom
a model, not measuring it directly in this paper (see Sect. 5), we
briefly discuss here the constraints on the model and the plau-
sible range of values using the 857GHz channel as an example
(the same conclusions are reached for the other Planck chan-
nels). Fig. 7 shows a compilation of models from the literature,
superimposed on the latest number counts observed by BLAST
and Herschel, and the expected shot noise as a function of Scut.
First we see, as stated above, that a small variation in Scutleads
to only a small variation in shot-noise power. Second, we see
thatthehighestshot-noiselevelis around13500Jy2sr−1,a factor
∼ 2.5 above our nominal value, but it comes from a model that
overestimates the observed numbercounts by a large factor (3–4
for 50 ≤ S ≤ 300mJy). Models that agree reasonably well with
the number counts have a shot-noise level below 8000Jy2sr−1.
The B´ ethermin et al. (2010c) model has the lowest shot noise.
However, it is currently the model that best reproduces all of the
available constraints,fromthe mid-infraredto the millimeter,in-
cluding the differential contribution of the S24≥ 80µJy sources
to the CIB as a function of redshift, which is a difficult observa-
tion to predict. We shall return to this issue in Sect. 5.
3.1.2. Radio galaxy shot noise, Cr,shot
ℓ
The shot-noise power from radio galaxies is subdominant to
that from dusty sources at the frequencies relevant to CIB
anisotropy analysis. The radio galaxy shot-noise power can be
estimated from the model of de Zotti et al. (2005). At frequen-
cies ≤ 100GHz, the model is in agreement with the source
counts computed using the extragalactic radio sources from the
ERCSC. At 143and 217GHz and forfluxes below300mJy (i.e.,
9

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Planck Collaboration: CIB anisotropies with Planck
Frequency (GHz)
Flux cut (mJy)
IR shot noise1
(Jy2sr−1)
Radio shot noise2
(Jy2sr−1)
IR shot noise1
(µK2
Radio shot noise2
(µK2
143
245
217
160
353
325
545
540
857
710
1.6±0.3
7.1
13.8±2.9
4.0
159±22
<3.4
1078±92
<5.7
5646 ±367
<7.4
(1.2 ± 0.2) × 10−5
4.9 × 10−5
(6 ± 1) × 10−5
1.7 × 10−5
(1.9 ± 0.3) × 10−3
< 3.8 × 10−5
0.3 ± 0.03
< 1.7 × 10−3
1131 ± 74
< 3.6
CMB)
CMB)
Table 3. Flux cut from the ERCSC for our 6 fields, and the shot-noise power for dusty and radio galaxies appropriate to those cuts
(see text). Values for shot noise1are derivedfrom the dusty galaxyevolution model of B´ ethermin et al. (2010c), while those for shot
noise2are from the radio galaxy evolution model of de Zotti et al. (2005) (see text for more details).
Figure7. A numberof recent models of dusty galaxy evolution,
andtheirassociatedshotnoisefordifferentfluxcuts,at857GHz.
Top: Comparison of the models with the Herschel and BLAST
differential numbers counts. Models are from Lagache et al.
(2004);Negrello et al.(2007);
Patanchon et al. (2009); Pearson & Khan (2009); Valiante et al.
(2009); B´ ethermin et al. (2010c); Franceschini et al. (2010);
Lacey et al. (2010); Marsden et al. (2010); Rowan-Robinson
(2009); Wilman et al. (2010). Data points are from Oliver et al.
(2010); B´ ethermin et al. (2010b); Glenn et al. (2010). Bottom:
Shot-noise level as a function of the flux cut for the same
models (same color and line coding between the two figures).
The vertical and horizontal continuous dark lines show the
Planck flux cut and shot-noise level from Table 3, respectively.
The B´ ethermin et al. (2010c) model is shown by the continuous
dark line.
Le Borgne et al.(2009);
Figure8. Effective beam window functions (bℓ) from FICSBell
(black) and FEBeCoP (red) at 545GHz. The 6 FEBeCoP
beam window functions from each field are superimposed. Also
shown for comparison is the Gaussian beam with a FWHM of
4.72±0.21′(green lines), which is the equivalent FWHM of the
beam determined on Mars.
the case listed in Table 3) the de Zotti et al. (2005) model is
in agreement with the source counts of Vieira et al. (2010). At
higher fluxes the model needs to be scaled in order to agree with
the number counts obtained using the ERCSC. The estimated
scaling factors are 2.03 and 2.65 at 143 and 217GHz respec-
tively (see Planck Collaboration 2011i). At even higher frequen-
cies the number counts by the ERCSC are no longer complete.
We thus use the 217GHz scaling factor to set upper limits for
the shot noise. It is negligible compared to Cd,shot
quencies. Changing the flux cut by 30% affects the shot noise by
30%, but since the radio contribution is subdominant at the fre-
quenciesrelevant forCIB anisotropyanalysis, it has little impact
on our results.
ℓ
at these fre-
3.2. The beam window function, bℓ(ν)
Since the HFI beams are not azimuthally symmetric, the scan-
ning strategy has to be taken into account in modeling the effec-
tive beam response. We used two different methods to compute
the effective beam: FEBeCoP and FICSBell. With FEBeCoP, we
compute one effective beam per field, with FICSBell, one effec-
tive beam for the entire sky.
10

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Planck Collaboration: CIB anisotropies with Planck
Figure9. Three independentnoise measurementsin the SP field
at 353GHz (red continuous: half pointing period, green dashed:
surveys I and II, black dash-dotted: half focal plane array).
FICSBell –The FICSBell method (Hivon et al, in prep) gen-
eralizes the approach of Hinshaw et al. (2007) and Smith et al.
(2007) to polarization and to include other sources of systemat-
ics. The different steps of the method used for this study can be
summarized as follows:
1. The scanning related information(i.e., statistics of the orien-
tation of each detector within each pixel) is computed first,
and only once for a given observation campaign. The hit
moments are only computed up to degree 4, for reasons de-
scribed below.
2. The (Marsbased)beammaporbeammodelofeach detector,
d, is decomposed into its spherical harmonic coefficients
bd
ℓs=
?
dr Bd(r)Yℓs(r),
(8)
where Bd(r) is the beam map, centered on the North pole,
and Yℓs(r) is a spherical harmonic. Higher s indexes de-
scribes higher degrees of departure from azimuthal sym-
metry and, for HFI beams, the coefficients bd
ing functions of s at most ℓ considered. It also appears
that, for ℓ < 3000, the coefficients with |s| > 4 account
for ≤ 1% of the beam throughput. For this reason, only
modes with |s| ≤ 4 are considered in the present analysis.
(Armitage-Caplan & Wandelt (2009) reached a similar con-
clusion in their analysis of Planck-LFI beams.)
3. The bd
spin weighted maps for a given CMB sky realization.
4. The spin weighted maps and hit moments of the same or-
der, s, are combined for all detectors involved, to provide an
“observed” map.
5. The power spectrum of this map can then be computed, and
compared to the input CMB power spectrum to estimate the
effective beam window function over the whole sky, or over
a given region of the sky.
lsare decreas-
ℓscoefficients computed above are used to generate s-
Monte-Carlo (MC) simulations in which the sky realisations are
changed can be performed by repeating steps 3, 4 and 5. The
impact of beam model uncertainties can be studied by including
step 2 into the MC simulations.
FEBeCoP –As mentioned in Sect. 2.1, map making reduces
time ordered data to pixelised maps. Each pixel of a map rep-
resents a convolutionof the true sky with the combined effect of
scanning beam and scan pattern. FEBeCoP computes this com-
bination of beams and scans—the effective beams—as is, in the
pixel space. The FEBeCoP methodologyand algorithmhas been
described in Mitra et al. (2010), and Planck HFI Core Team
(2011a). Below we list the essential steps made in computation
of the beam window functions, for completeness:
1. For each pixel i in the map (or CIB field), we compute the
Fourier-Legendre transform, Bl, of the pixel space effective
beams Bi(ˆΩ) using the formula
bl =
?
∆Ωi
dˆΩPℓ(ˆΩi·ˆΩ) Bi(ˆΩ),
(9)
whereˆΩiis the direction vector of the center of the ithpixel
on the sky, Pℓrepresents Legendre polynomials of order ℓ
and the integration is performed over the (small) solid angle
∆Ωi, outside which the beam can be taken as zero. This for-
mula can be readily transformed to a discretised form with a
careful correction for the “pixel window function” as
bℓWp
ℓ≈ Ωpix
?
j
Pℓ(ˆΩi·ˆΩj) Bi(ˆΩj),
(10)
where the summation is over pixels which fall inside the
beam solid angle ∆Ωi, Ωpixis the area of each (equal area)
pixel and Wp
sates for the systematic error that is introduced when inte-
gration over a pixel is replaced by the value of the integrand
at the pixel center times the area of the pixel.
2. We then compute bℓat uniformly sampled directions in each
field to find the average window functions. The samples are
chosen as the HEALPix pixel centers at a coarser resolution
(Nside = 128) to ensure uniform sampling. Thus we obtain
the average window functions for each frequency and field.
3. To validate the average window functions obtained using the
above prescription, we perform Monte-Carlo simulations,
separately for each field and each frequency. We simulate
16 realizations of the sky starting from a ∼ ℓ−2angular
power spectrum, which are convolvedin two ways - (1) with
FEBeCoP generated effective beams in pixel space and (2)
with analytical Gaussian beam in harmonicspace for a beam
size appropriate for the given frequency channel. The con-
volved maps are then “masked” using a function which is
unity in the given field and smoothly (in ∼ 25% of field ra-
dius) goes to zero outside the field. Finally we compute the
ratio of the angular power spectra of these two maps, multi-
ply the ratio by the theoretical window function for the same
beam size and average over the realizations. Though these
“transfer functions” suffer from ringing effects often seen in
Fourier transforms of a narrow function, they wiggle around
the average window functions conformingthe validity of the
latter.
ℓis the pixel window function which compen-
Fig. 8 shows the FICSBell and FEBeCoP effective beams at
545GHz. Also shown is the Gaussian beam with a FWHM of
4.72′± 0.21′. This is the average FWHM of the scanning beam,
determined on Mars obtained by unweighted averaging the indi-
vidual detectors FWHM. Each FWHM is that of the Gaussian
beam which would have the same solid angle as that deter-
mined by using a full Gauss-Hermite expansion on destriped
data (see Planck HFI Core Team 2011a, for more details). We
see quite good agreement between the FICSBell, all-sky and
FEBeCoP,small-fieldeffectivewindowfunctions,witha2% dif-
ferenceat ℓ ≃ 2000,the highestℓ that will be consideredforCIB
11

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Planck Collaboration: CIB anisotropies with Planck
anisotropyanalysis (seeSect. 4). We also see fromthe figurethat
the erroronthe inputscanningbeamis largerthanthis difference
and will dominate the uncertainties at high ℓ and high frequency
(Sect. 4.2.2). In the followingwe will use the FEBeCoP window
functions as they are exactly computed for each of our fields.
Figure10. Instrument noise power spectra of the six fields ob-
tained usinghalf pointingperiodmaps. From topto bottom:217,
353, 545 and 857GHz (continuous: N1, dotted: AG, dashed: SP,
dash-dotted: Bootes 1, long-dash: Bootes 2, dash-3 dotted:
LH2).
3.3. Instrument noise, Nℓ(ν)
We can use three different jack-knife difference maps to derive
noise powerspectra: maps madefromthe first andsecond halves
of each pointing period (a half-pointing period is of the order of
20 minutes), maps made using half of the focal plane array, and
maps using the two different surveys (survey I and II). In each
case the noise power spectrum, Nℓ, is obtained by measuring the
power spectrum of the difference maps. The three methods give
similar Nℓ, as it is illustrated for one frequency and one field in
Fig. 9. We choose however to use the half-pointing period maps
since (1) the two survey maps are only both fully covered for
the LH2 and SP fields and (2) there are only three bolometers at
545 and 857GHz making half focal plane maps less accurate.
We also computed the noise power spectrum from the difference
between the auto- and cross- power spectrum of the two halves
maps. In the range of interest, 1500 ≤ ℓ ≤ 2100, where the con-
tribution from the noise becomes important, they agree at better
than 0.5, 1, 3, and 4% at 217, 353, 545 and 857GHz, respec-
tively. Fig. 10 shows the noise power spectra for all fields. They
are nearly flat, the deviation from flatness being due to the ef-
fect of deconvolutionfrom the instrumentalresponse at high fre-
quencyand residual low frequencynoise. Removingthe ERCSC
sources has no impact on the noise determination.
Fig. 3 shows the noise power spectra compared to the HFI
map power spectra for one illustrative field. It is evident from
this figure that we have a very high signal-to-noise ratio. At 545
and 857GHz, the signal is dominatingeven at the highest spatial
frequencies.At 217and353GHz,theresidualsignal(i.e.,CMB-
and cirrus-cleaned) is comparable to the noise at high ℓ (ℓ ≥
2000− 2500 depending on the field).
3.4. Additional corrections
Two additional corrections, linked to the CMB cleaning, have to
be done to the power spectra. First we remove the extra instru-
ment noise that has been introduced by CMB removal:
NCMB
ℓ
(ν) = Nℓ(ν143) × w2
ℓ×
?
bℓ(ν)
bℓ(ν143)
?2
,
(11)
withν equalto217or353GHz. Nℓ(ν143)is thenoisepowerspec-
trumofthe143GHz map.It is computedas thenoise inthe other
frequencychannels,using the half pointingperiodmaps, follow-
ing Sect. 3.3.
Second, due to the lower angular resolution of the 143GHz
channel compared to the 217 and 353GHz, we also have to re-
move the CMB contribution that is left close to the angular res-
olution of the 217 and 353GHz channels:
CCMBres
ℓ
(ν) = CCMB
ℓ
(ν) × F2
p× b2
ℓ(ν) × [1 − wℓ]2,
(12)
with Fpthe pixel and reprojection transfer function (detailed in
Sect. 4.1).
Finally, we have to assess the level of the astrophysical com-
ponents that have been removed from or added to the 217 and
353GHz channels,usingthe filtered 143GHz channelas a CMB
template. Cirrus emission is highly correlated between 143, 217
and 353GHz channels. Thus filtered cirrus emission has been
removed from the 217 and 353GHz. This has no impact on our
CIB anisotropy analysis as this extra cirrus removal only modi-
fies the emissivities, with no consequence on our residual maps
(it should be understood for further interpretation of the Hi-
correlateddust emission, whichis not the goalof this paper).For
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Planck Collaboration: CIB anisotropies with Planck
the shot-noise powers, we expect them to be quite decorrelated
for the (143,217)and (143, 353)sets of maps since the 143GHz
shot noise is dominated by radio sources whereas the 217 and
353GHz shot noise is dominated by dusty galaxies (see Table
3). To have an idea of the maximal effect (i.e., perfect decor-
relation between shot noise at 143, and 217 and 353GHz) we
compute the contamination by the 143GHz shot noise summing
the contribution of the radio and dusty galaxies and following:
Cℓ(ν) = Cshot(ν143) ×
?
bℓ(ν)
bℓ(ν143)wℓ
?2
(13)
The last term accounts for the filtering and ‘re-beaming’ of the
143GHz map. The contamination is the highest in the 217GHz
channel. It is a factor 1.2 smaller, equivalent, and a factor 122
smaller than the sum of the predicted radio and dusty galaxies
shot-noise powers at 217 GHz at ℓ ≃ 200, 1000 and 2000, re-
spectively. It is smaller by factors of 20, 2.9 and 325 than the
CIB anisotropies at 217GHz, at ℓ= 200, 1000 and 2000, respec-
tively. As this is the maximal contamination and as it is quite
small (and completely negligible at high ℓ), we do not apply any
correction to the CIB anisotropy power spectra.
We still have to consider the case of CIB correlated
anisotropiesat 143GHz. Theyhave beenmarginallyconstrained
at 150GHz by SPT and ACT at high ℓ. The power is < 5.2 ×
10−6µK2and < 9.8 × 10−6µK2at ℓ = 3000 in Dunkley et al.
(2010) and Hall et al. (2010), respectively. This contribution is
also completely negligible compared to the signal at 217GHz.
In conclusion, we can ignore the CIB and cirrus components
that are left in the CMB maps.
4. Angular power spectrum estimation
The angular power spectrum estimator used in this work is
POKER (Ponthieu et al. 2011) which is an adaptation to the flat
sky of the pseudospectrumtechniquedevelopedfor CMB analy-
sis (see e.g. MASTER, Hivon et al. 2002). In brief, POKER com-
putes the angular power spectrum of the masked data (a.k.a.,
the pseudo-power spectrum) and deconvolves it from the power
spectrum of the mask to obtain an unbiased estimate of the
binned signal angular power spectrum. We summarize the main
features of POKER in the following section and then detail how it
is used to producethe final estimate of the CIB anisotropypower
spectrum and its associated error bars.
In the following, the power spectrum associated to CIB
anisotropies will be denoted Cℓ and its unbiased estimator in
the flat-sky approximation P(ℓ). As already suggested, this final
estimate makes use of the power spectrum of the masked data.
This so-called pseudo-power spectrum will be denotedˆP(ℓ). In
the flat-sky approximation, the standard angular frequencies la-
beled by their zenithal and azimuthal numbers, usually called ℓ
and m respectively, are replaced by an ‘angular’ wave-vector ℓ;
its norm ℓ being equal to the zenithal number (see e.g., the ap-
pendix of White et al. 1999). Finally, we will assume that the
CIB anisotropy arises from a statistically isotropic process. As
is the case for the CMB, the CIB fluctuations are viewed as
isotropic and homogeneous stochastic variables on the celestial
sphere, leading to
?
a(ℓ)a⋆(ℓ′)
?
= (2π)2C(ℓ)δ2(ℓ − ℓ′),
(14)
with a(ℓ) the Fourier coefficients of CIB anisotropies. Such
an assumption is theoretically reasonable, moreover, we have
checked that |a(ℓ)|2computed from our CIB maps does not de-
pend on the direction of ℓ.
Figure11.
anisotropies estimate (illustrated here with the SP field at
353GHz). The bias induced by each dust and CMB component
is negligible comparedto both the CIB anisotropy signal and the
statistical noise(ingreen,includingcosmicvarianceonthenoise
estimate itself).
Contribution of residuals to the final CIB
4.1. POKER
The POKER implementation of the pseudospectrum approach
uses the Discrete Fourier Transform (hereafter DFT). For a map
of scalar quantity Djk(j,k denote pixel indices), it is defined as
Dmn =
1
NxNy
?
j,k
Djk× e−2πi(jm/Nx+kn/Ny),
(15)
Djk =
?
m,n
Dmn× e+2πi(jm/Nx+kn/Ny),
(16)
where Dmn is the set of discrete Fourier coefficients of
Djk. For a given wave-vector ℓ, labeled by the m and
n indices, its corresponding norm is denoted by ℓmn
(2π/∆θ)?(m′/Nx)2+ (n′/Ny)2with m′= m (resp. n′) if m ≤
Nx/2 and m′= Nx− m if m > Nx/2. The power spectrum of the
map is defined as the square-modulus of its Fourier coefficients,
i.e., P(ℓmn) = |Dmn|2.
The direct DFT of the masked data relates the true Fourier
coefficients to the pseudo-Fourier coefficients of the signal
=
ˆDmn=
?
m′n′
Wn,n′
m,m′Dm′n′,
(17)
in which Wn,n′
mask DFT coefficients. Replacing DmnbyˆDmnin the definition
of the power spectrum of a given map leads to the power spec-
trum of the masked data (a.k.a.the pseudo-powerspectrum).For
a signal T plus noise N map, the ensemble averaged of the pseu-
dospectrum tracing a statistically isotropic process, reads
m,m′ is a convolution kernel that depends only on the
?ˆP(ℓmn)? =
?
m′n′
???Wn,n′
m,m′
???2Fm′n′C(ℓm′n′) + ?ˆN(ℓmn)?,
(18)
where we have introduced the total transfer function Fm′n′ ac-
countingforthe beam,the ‘map-making’pixelisationeffects and
reprojection from curved, HEALPix maps to flat, square maps.
Thebeam transferfunctionis givenbythe beampowerspectrum
describedin Sect. 3.2. The ‘map-making’pixelisationeffects are
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Planck Collaboration: CIB anisotropies with Planck
described by the power spectrum of the pixel window function
for full-sky maps provided by the HEALPix package (the ini-
tial HEALPix maps are built with Nside = 2048 corresponding
to a pixel size of 1.7′). As explained in Planck HFI Core Team
(2011a), time domainfiltering is includedas part of the scanning
beam, such that any time domain filtering effects end up in the
estimate of the beam instead of as part of Fm′n′. Finally, each
curved map with a 1.7′resolution is reprojected on its tangent,
flat space with a pixel size of ∼ 3.5′. This induces first a repix-
elisation effect as the output map is less resolved than the input
one and second, a slight displacement of the pixel centers. The
cumulative impact of ‘image deformation’ and ‘repixelisation’
is estimated via Monte-Carlo: we first generate a set of full-sky
maps and compute the MC average of their pseudospectra. This
set of maps is then re-projectedon flat maps for which MC aver-
ageoftheirpseudospectraintheflat-skyapproximationarecom-
puted. The ratio of the flat-sky pseudospectrum divided by the
full-sky pseudospectrum gives a measurement of re-projection
effect. Note that those simulations have been performed assum-
ing different shapes for the input angular power spectra. The de-
rived reprojection transfer functions were in perfect agreement
underlying the robustness of the approach.
An unbiased estimate of Cℓ is obtained by first subtract-
ing the noise contribution and then deconvolving the mask and
beam effects encoded in the convolution kernel
For the sky coverage of the considered fields, the rapid oscil-
lations of the convolution kernels introduce strong correlations
between spatial frequencies and make its inversion numerically
intractable. (Pseudo-)Power spectra are therefore estimated in
frequency bands (labeled b hereafter). The binning operator is
???Wn,n′
m,m′
???2Fm′n′.
Rmn
b=

ℓβ
∆b
0
mn
if ℓb
otherwise
low≤ ℓmn< ℓb+1
low
,
(19)
where ∆bis the number of wave vectors ℓmnthat fall into the bin
b. The reciprocaloperatorthat relates the theoretical value of the
one-dimensional binned power spectrum Pbto its value at ℓmnis
Qb
mn=

1
ℓβ
0
mn
if ℓb
low≤ ℓmn< ℓb+1
otherwise
low
(20)
For optimal results, the spectral index β should be chosen to get
ℓβCℓas flat as possible. (In the case of the CMB, β ≃ 2 is the
equivalent of the standard ℓ(ℓ + 1) prefactor.) In the case of CIB
anisotropy Cℓscales roughly as ℓ−1, and we therefore adopt a
binning with β = 1. Nevertheless, we checked that our results
were robust against the choice of β: we simulate a power spec-
trum scaling as ℓ−1but reconstruct it assuming β = 0, 1 and 2 in
POKER. For each choice of β, the estimated power spectrum was
in perfect agreement with the input one (for a more complete
discussion, see Ponthieu et al. 2011).
The binned pseudo-power spectra is
ˆPb=
?
m,n∈b
Rmn
b
ˆP(ℓmn),
(21)
and the CIB power spectrum is related to its binned value, Cb,
via
C(ℓm′n′) ≃ Qb′
With such binned quantities, Eq. 18 can be re-written as
m′n′Cb′.
(22)
?ˆPb
?
=
?
b′
Mbb′Cb′ +
?ˆNb
?
,
(23)
with
Mbb′ =
?
m,n∈b
?
m′,n′∈b′
Rmn
b
???Wn,n′
m,m′
???2Fm′n′Qb′
m′n′.
(24)
An unbiased estimate of the binned angular power spectrum of
the signal is thus given by
Pb=
?
b′
M−1
bb′
?ˆPb′ − ?ˆNb′?
?
.
(25)
It is easily checked that ?Pb? = Cb.
The complete recovery of the CIB anisotropy angular power
spectra is therefore done in three step :
A. Given the mask, W, associated to our CIB map and the
transfer function, Fm,n, compute and invert Mbb′ as given in
Eq.24.Thedifferentfields(withasizeatmost5.1◦×5.1◦)are
systematically embedded in a 10◦×10◦square map. We use
binary masks, i.e., W = 1 for observed (or kept-in-the anal-
ysis) pixels and W = 0 for unobserved pixels. The estimated
power spectra are binned with a bandwidth of ∆b= 200 and
the first bin starting at ℓ = 804.
B. Derivethe noise bias ?ˆNb?, givenby first the instrumentnoise
described in Sect. 3.3 and second the additional corrections
given in Sect. 3.4.
C. Compute the final estimate ofCbfrom Eq. 25 andCb= ?Pb?.
Uncertainties on Pb come from sampling variance, noise
variance, astrophysical contaminants and systematic effects. In
the following section, we present how we estimate each of these
contributions, from Monte-Carlo simulations.
4.2. Error bar estimation
4.2.1. Statistical uncertainties
As presented in Sect. 4.1, the uncertainties on Pb come from
signal sampling variance, noise and uncertainties on the sub-
traction of CMB and Galactic dust. The first two are described
by stochastic processes with known power spectra, the last two
come from uncertainties in the weights applied to templates in
the subtraction process at the map level.
We have developed the tools necessary to simulate maps
given any input angular power spectrum for each field and fre-
quency. The simulation pipeline consists of simulating maps
given an input power spectrum (in the case of CIB, noise and
CMB residual) and maps of template residuals (conservative
Gaussian random fractions of the templates). These maps are
then combined and analyzed by the power spectrum estimator.
Each realization provides an estimated power spectrum with the
same statistical properties as our estimate on the data and alto-
gether, these simulations provide the uncertainties on our esti-
mate. The covariance matrix of Pbis
Cbb′ = ?(Pb− ?Pb?MC)(Pb′ − ?Pb′?MC)?MC,
with ?·?MCstanding for Monte-Carlo averaging.The error bar on
each Pbis
σPb=
and the bin-bin correlation matrix is given by its standard defini-
tion
(26)
?
Cbb,
(27)
Ξbb′ =
Cbb′
√CbbCb′b′.
(28)
4ℓ = 80 corresponds to the inverse of the largest angular scale con-
tained in the considered fields.
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Planck Collaboration: CIB anisotropies with Planck
Simulation pipeline –The simulated maps are 10◦× 10◦. They
contain six components, accounting for the different ingredients
supposedly present in the actual data maps:
1. A CIB anisotropy component obtained from a random,
Gaussian realization of the CIB anisotropy power spectrum.
As a model for such a spectrum, we use a fit to PCIB, esti-
mated from the data further multiplied by the power spec-
trum of the beam, pixel and reprojection transfer function.
2. A residual CMB component derived from a random,
Gaussian realization of the power spectrum given in Eq. 12
using the known Wiener filter, beam differences between
143GHz and other channels, and the WMAP best fit CMB
temperature power spectrum.
3. The instrument noise as derived in Sect. 3.3. Since the noise
is slightly colored, we simulate it using a fit of its measured
power spectrum.
4. Extra-instrument noise incurred by CMB removal using
Eq. 11 as a model of its power spectrum.
Inadditiontothosefouringredientsstandingforsignalandnoise
(a CMB residualbeingviewedas an extra-sourceof‘noise’from
the viewpoint of CIB), we add the two foreground templates
which are removed, with some uncertainties:
5. A CMB map with a Gaussian uncertaintydistributed with 2%
and 3% standard deviation at 217 and 353GHz respectively
(the CMB is negligible at higher frequencies compared to
CIB and dust). The 2% and 3% are justified in Sect. 4.3.
6. An Hi map with a 5%, 10% and 10% standard deviation
for its emissivity (local, IVC and HVC components re-
spectively), consistent with both the inter-calibration er-
rors (see Sect. 4.3) and the emissivity errors computed by
Planck Collaboration (2011t) using Monte Carlo simula-
tions.
The analysis pipeline –The analysis pipeline works in four
steps:
1. A first set of 1,000 MC simulations of CMB residual and
noise only is performed to assess first, the pseudo-power
spectrum of the instrument noise and CMB residual used to
debias the simulated data pseudospectrum and, second, the
noise variance given by
σNb=
?
Cnoise
bb.
2. Asecondsetof1,000MC simulations,includingallthecom-
ponents, is performed.For a given simulated map, CMB and
dust templates are removedassuming the estimated emissiv-
ities of Sect. 2.5 for dust.
3. The POKER algorithm is then applied to these ‘foreground-
cleaned’ maps to get a final estimate of the CIB angular
power spectrum. In this step, the bias involved in Eq. 25
contains the pseudo-power spectrum of the instrument noise
model and of the CMB residual model as described in the
simulation pipeline.
4. The total error bars and bin-bin correlation matrix on Pbare
obtained as the RMS of 1,000 Monte-Carlo realizations of
the simulation pipeline, as described in the previous para-
graph and using Eqs. 27 and 28.
The statistical uncertainties contain:
A. sampling variance from CIB anisotropies and residual CMB
as modeled in Eq. 12,
B. noise variance due to instrument noise and extra-noise given
by Eq 11,
C. Uncertainties on the CMB and Hi template subtraction.
In this set of simulations and analysis, we assume the beam pro-
file as described in Sect. 3.2 and ignore potential beam uncer-
tainties (see the next section for a discussion of such a system-
aticeffect).Inthefollowingwepresentresultsobtainedusingthe
FEBeCoP-derived beam profiles, but the estimated power spec-
tra using either the FICSBell-derived or the FEBeCoP-derived
beam are in very good agreement. Fig. 12 shows the results for
all fields and frequencies. We also display on Fig. 13 the bin-bin
correlation matrix showing that two bins are not correlated by
more than 10%.
4.2.2. Systematic Errors
Ourestimate of eachpowerspectrumis affectedbydifferentsys-
tematicerrorsthatmustbe accountedforseparatelyfromthe sta-
tistical errors derived in the previous section. These systematic
uncertainties may introduce a bias in the final estimate and/or
biasourMonte-Carloestimate ofthestatistical uncertaintiespre-
sented above. We review here the different sources of such sys-
tematics and evaluate their level.
Mask impact–Our power spectrum estimation is performed on
a limited sky patch. This induces power aliasing from angular
scales larger than the size of the patch, an effect which increases
as the signal power spectrum steepens. POKER is designed to ac-
count for this effect (as well as the extra aliasing induced by
holesin the map,if present)as the convolutionkernel, Mbb′, con-
tains the information on mode coupling to the larger scales. We
run POKER on data maps which have been embedded in larger
regions which are zero-padded. There is no general prescription
as to the size of the zero-padding that one should use, fortu-
nately the results are very insensitive to the particular choice
andwhetheror not the mask is apodized.By comparingdifferent
choices we found uncertainties at the 2% level, well below the
statistical uncertainties.
Template subtraction impact–Imperfect template subtraction
will also lead to the presence of ‘foreground’ residuals slightly
biasing our final estimate, PCIB
b
, of the CIB anisotropy angular
power spectrum. Such a residual level is given at first order by
δPCIB
b
≃ (δα)2?
b′
M−1
bb′ˆPtemp
b′
(29)
withˆPtemp
on the global amplitude of this template. Figure 11 illustrates,
for the particular case of the SP field, the level of these residuals.
Although negligible compared to the statistical errors, they are
accounted for in the error budget.
b1
the pseudospectrum of the template and δα the error
Beam uncertainties–Uncertainty in the beam will also bias the
estimate of the power spectrum since:
1. The beam window function enters the computation of the
convolutionkernel Mbb′. Any beam error biases our estimate
of Mbb′ and thus our final result.
2. Beam uncertainties will translate into slight misestimation
of NCMB
ℓ
(ν) and CCMBres
ℓ
(ν), potentially biasing our final es-
timate.
15