Planck 2013 results. XV. CMB power spectra and likelihood

A&A 571, A15 (2014) DOI: 10.1051/0004-6361/201321573 c ESO 2014 Astronomy & Astrophysics Planck 2013 results Special feature Planck 2013 results. XV. CMB power spectra and likelihood Planck Collaboration: P. A. R. Ade93, N. Aghanim65, C. Armitage-Caplan98, M. Arnaud79, M. Ashdown76,6, F. Atrio-Barandela19, J. Aumont65, C. Baccigalupi92, A. J. Banday101,10, R. B. Barreiro73, J. G. Bartlett1,74, E. Battaner102, K. Benabed66,100, A. Benoît63, A. Benoit-Lévy26,66,100, J.-P. Bernard101,10, M. Bersanelli38,55, P. Bielewicz101,10,92, J. Bobin79, J. J. Bock74,11, A. Bonaldi75, L. Bonavera73, J. R. Bond9, J. Borrill14,95, F. R. Bouchet66,100,?, F. Boulanger65, M. Bridges76,6,69, M. Bucher1, C. Burigana54,36, R. C. Butler54, E. Calabrese98, J.-F. Cardoso80,1,66, A. Catalano81,78, A. Challinor69,76,12, A. Chamballu79,16,65, H. C. Chiang29,7, L.-Y Chiang68, P. R. Christensen88,42, S. Church97, D. L. Clements61, S. Colombi66,100, L. P. L. Colombo25,74, C. Combet81, F. Couchot77, A. Coulais78, B. P. Crill74,89, A. Curto6,73, F. Cuttaia54, L. Danese92, R. D. Davies75, R. J. Davis75, P. de Bernardis37, A. de Rosa54, G. de Zotti50,92, J. Delabrouille1, J.-M. Delouis66,100, F.-X. Désert58, C. Dickinson75, J. M. Diego73, H. Dole65,64, S. Donzelli55, O. Doré74,11, M. Douspis65, J. Dunkley98, X. Dupac45, G. Efstathiou69, F. Elsner66,100, T. A. Enßlin84, H. K. Eriksen71, F. Finelli54,56, O. Forni101,10, M. Frailis52, A. A. Fraisse29, E. Franceschi54, T. C. Gaier74, S. Galeotta52, S. Galli66, K. Ganga1, M. Giard101,10, G. Giardino46, Y. Giraud-Héraud1, E. Gjerløw71, J. González-Nuevo73,92, K. M. Górski74,103, S. Gratton76,69, A. Gregorio39,52, A. Gruppuso54, J. E. Gudmundsson29, F. K. Hansen71, D. Hanson85,74,9, D. Harrison69,76, G. Helou11, S. Henrot-Versillé77, C. Hernández-Monteagudo13,84, D. Herranz73, S. R. Hildebrandt11, E. Hivon66,100, M. Hobson6, W. A. Holmes74, A. Hornstrup17, W. Hovest84, K. M. Huffenberger27, G. Hurier65,81, A. H. Jaffe61, T. R. Jaffe101,10, J. Jewell74, W. C. Jones29, M. Juvela28, E. Keihänen28, R. Keskitalo23,14, K. Kiiveri28,49, T. S. Kisner83, R. Kneissl44,8, J. Knoche84, L. Knox31, M. Kunz18,65,3, H. Kurki-Suonio28,49, G. Lagache65, A. Lähteenmäki2,49, J.-M. Lamarre78, A. Lasenby6,76, M. Lattanzi36, R. J. Laureijs46, C. R. Lawrence74, M. Le Jeune1, S. Leach92, J. P. Leahy75, R. Leonardi45, J. León-Tavares47,2, J. Lesgourgues99,91, M. Liguori34, P. B. Lilje71, M. Linden-Vørnle17, V. Lindholm28,49, M. López-Caniego73, P. M. Lubin32, J. F. Macías-Pérez81, B. Maffei75, D. Maino38,55, N. Mandolesi54,5,36, D. Marinucci41, M. Maris52, D. J. Marshall79, P. G. Martin9, E. Martínez-González73, S. Masi37, M. Massardi53, S. Matarrese34, F. Matthai84, P. Mazzotta40, P. R. Meinhold32, A. Melchiorri37,57, L. Mendes45, E. Menegoni37, A. Mennella38,55, M. Migliaccio69,76, M. Millea31, S. Mitra60,74, M.-A. Miville-Deschênes65,9, D. Molinari35,54, A. Moneti66, L. Montier101,10, G. Morgante54, D. Mortlock61, A. Moss94, D. Munshi93, J. A. Murphy87, P. Naselsky88,42, F. Nati37, P. Natoli36,4,54, C. B. Netterfield21, H. U. Nørgaard-Nielsen17, F. Noviello75, D. Novikov61, I. Novikov88, I. J. O’Dwyer74, F. Orieux66, S. Osborne97, C. A. Oxborrow17, F. Paci92, L. Pagano37,57, F. Pajot65, R. Paladini62, D. Paoletti54,56, B. Partridge48, F. Pasian52, G. Patanchon1, P. Paykari79, O. Perdereau77, L. Perotto81, F. Perrotta92, F. Piacentini37, M. Piat1, E. Pierpaoli25, D. Pietrobon74, S. Plaszczynski77, E. Pointecouteau101,10, G. Polenta4,51, N. Ponthieu65,58, L. Popa67, T. Poutanen49,28,2, G. W. Pratt79, G. Prézeau11,74, S. Prunet66,100, J.-L. Puget65, J. P. Rachen22,84, A. Rahlin29, R. Rebolo72,15,43, M. Reinecke84, M. Remazeilles75,65,1, C. Renault81, S. Ricciardi54, T. Riller84, C. Ringeval70,66,100, I. Ristorcelli101,10, G. Rocha74,11, C. Rosset1, G. Roudier1,78,74, M. Rowan-Robinson61, J. A. Rubiño-Martín72,43, B. Rusholme62, M. Sandri54, L. Sanselme81, D. Santos81, G. Savini90, D. Scott24, M. D. Seiffert74,11, E. P. S. Shellard12, L. D. Spencer93, J.-L. Starck79, V. Stolyarov6,76,96, R. Stompor1, R. Sudiwala93, F. Sureau79, D. Sutton69,76, A.-S. Suur-Uski28,49, J.-F. Sygnet66, J. A. Tauber46, D. Tavagnacco52,39, L. Terenzi54, L. Toffolatti20,73, M. Tomasi55, M. Tristram77, M. Tucci18,77, J. Tuovinen86, M. Türler59, L. Valenziano54, J. Valiviita49,28,71, B. Van Tent82, J. Varis86, P. Vielva73, F. Villa54, N. Vittorio40, L. A. Wade74, B. D. Wandelt66,100,33, I. K. Wehus74, M. White30, S. D. M. White84, D. Yvon16, A. Zacchei52, and A. Zonca32 (Affiliations can be found after the appendices) Received 26 March 2013 /Accepted 5 May 2014 ABSTRACT This paper presents the Planck 2013 likelihood, a complete statistical description of the two-point correlation function of the CMB temperature fluctuations that accounts for all known relevant uncertainties, both instrumental and astrophysical in nature. We use this likelihood to derive our best estimate of the CMB angular power spectrum from Planck over three decades in multipole moment, `, covering 2 ≤`≤2500. The main source of uncertainty at `< ∼1500 is cosmic variance. Uncertainties in small-scale foreground modelling and instrumental noise dominate the error budget at higher `s. For ` < 50, our likelihood exploits all Planck frequency channels from 30 to 353 GHz, separating the cosmological CMB signal from diffuse Galactic foregrounds through a physically motivated Bayesian component separation technique. At `≥50, we employ a correlated Gaussian likelihood approximation based on a fine-grained set of angular cross-spectra derived from multiple detector combinations between the 100, 143, and 217 GHz frequency channels, marginalising over power spectrum foreground templates. We validate our likelihood through an extensive suite of consistency tests, and assess the impact of residual foreground and instrumental uncertainties on the final cosmological parameters. We find good internal agreement among the high-`cross-spectra with residuals below a few µK2at `< ∼1000, in agreement with estimated calibration uncertainties. We compare our results with foreground-cleaned CMB maps derived from all Planck frequencies, as well as with cross-spectra derived from the 70 GHz Planck map, and find broad agreement in terms of spectrum residuals and cosmological parameters. We further show that the best-fit ΛCDM cosmology is in excellent agreement with preliminary Planck EE and T E polarisation spectra. We find that the standard ΛCDM cosmology is well constrained by Planck from the measurements at `< ∼1500. One specific example is the spectral index of scalar perturbations, for which we report a 5.4σdeviation from scale invariance, ns=1. Increasing the multipole range beyond `≃1500 does not increase our accuracy for the ΛCDM parameters, but instead allows us to study extensions beyond the standard model. We find no indication of significant departures from the ΛCDM framework. Finally, we report a tension between the Planck best-fit ΛCDM model and the low-`spectrum in the form of a power deficit of 5–10% at `< ∼40, with a statistical significance of 2.5–3σ. Without a theoretically motivated model for this power deficit, we do not elaborate further on its cosmological implications, but note that this is our most puzzling finding in an otherwise remarkably consistent data set. Key words. cosmic background radiation – cosmological parameters – cosmology: observations – methods: data analysis ?Corresponding author: F. R. Bouchet, e-mail: [email protected] Article published by EDP Sciences A15, page 1 of 60 A&A 571, A15 (2014) 1. Introduction This paper, one of a set associated with the 2013 release of data from the Planck1mission (Planck Collaboration I 2014– Planck Collaboration XXXI 2014), describes the CMB power spectra and the corresponding likelihood that we derive from the Planck data. The power spectrum of the cosmic microwave background (CMB) is a unique signature of the underlying cosmological model (e.g., Spergel et al. 2003;Hinshaw et al. 2013). It has been measured over the whole sky by COBE and WMAP, and over smaller regions by ground-based and sub-orbital experiments (e.g., Tristram et al. 2005;Jones et al. 2006;Reichardt et al. 2009;Fowler et al. 2010;Das et al. 2011;Keisler et al. 2011;Story et al. 2013;Das et al. 2014). By mapping the whole sky to scales of a few arc minutes, Planck now measures the power spectrum over an unprecedented range of scales from a single experiment. To estimate cosmological parameters from the power spectrum requires a likelihood function that propagates uncertainties. In this paper we describe the power spectra obtained from the Planck temperature data, as well as the associated likelihood function. Since the probability distribution of the power spectrum is non-Gaussian at large scales, we follow a hybrid approach to construct this likelihood (Efstathiou 2004,2006), using a Gibbs sampling based approach at low multipoles, `, and a pseudo-C`technique at high multipoles (Hivon et al. 2002) as well as an approximation to the quadratic maximum likelihood estimator (Rocha et al. 2011,2010) at high multipoles. The high-`part of the Planck likelihood (`≥50) is based on power spectra estimated from each Planck detector in the frequency range 100 to 217 GHz, allowing careful assessment of each detector’s response to the sky emission. We implement three independent likelihood methods. The first, used in the distributed likelihood code, estimates the power spectrum at every multipole, together with the associated covariance matrix. The second takes a simplified form, binning the spectra, and is used to explore the stability of the results with respect to different instrumental and astrophysical systematic effects, the third one takes a CMB map estimated from component separation techniques (hence using all Planck frequency maps) and estimates the band powers and associated covariance matrices. The methods give consistent results. Unresolved extragalactic foregrounds make a significant contribution to the power spectra at high multipoles. We develop a model for these foregrounds, designed to allow the Planck likelihood to be combined with high resolution data from the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT). We combine frequencies and model unresolved foregrounds in a physical way, as in e.g., Shirokoffet al. (2011); Dunkley et al. (2011); Reichardt et al. (2012), performing component separation at small scales at the power spectrum level. On large scales, ` < 50, Galactic contamination is more significant. We use the Planck temperature maps in the range 30 ≤ν≤353 GHz to separate Galactic foregrounds in the maps, and then estimate the full probability distribution of the CMB power spectrum. 1Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific 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 scientific consortium led and funded by Denmark. Table 1. Detectors used to make the maps for this analysis. Set name Frequency Type Detectors FWHM [GHz] [arcmin] 100-ds0 . . . . . . . . . . 100 PSB All 8 detectors 9.65 100-ds1 . . . . . . . . . . 100 PSB 1a+1b +4a+4b 100-ds2 . . . . . . . . . . 100 PSB 2a+2b +3a+3b 143-ds0 . . . . . . . . . . 143 MIX 11 detectors 7.25 143-ds1 . . . . . . . . . . 143 PSB 1a+1b +3a+3b 143-ds2 . . . . . . . . . . 143 PSB 2a+2b +4a+4b 143-ds3 . . . . . . . . . . 143 SWB 143-5 143-ds4 . . . . . . . . . . 143 SWB 143-6 143-ds5 . . . . . . . . . . 143 SWB 143-7 217-ds0 . . . . . . . . . . 217 MIX 12 detectors 4.99 217-ds1 . . . . . . . . . . 217 PSB 5a+5b +7a+7b 217-ds2 . . . . . . . . . . 217 PSB 6a+6b +8a+8b 217-ds3 . . . . . . . . . . 217 SWB 217-1 217-ds4 . . . . . . . . . . 217 SWB 217-2 217-ds5 . . . . . . . . . . 217 SWB 217-3 217-ds6 . . . . . . . . . . 217 SWB 217-4 Notes. Spider web bolometers (SWB) are used individually; polarisation sensitive bolometer pairs (PSBs, denoted a and b) are used in pairs, and we consider only the maps estimated from two pairs of PSBs. The relevant effective beams, and their uncertainties, are given in Planck Collaboration VII (2014). The channel maps, or “ds0” maps, are solely used as inputs to the component separation methods, and the resulting CMB map spectra are used in Sect. 7.5 below only as a consistency check. This paper is structured as follows. In Sect. 2we describe the pseudo-C`likelihoods, and in Sect. 3set up the foreground model. The power spectra and derived cosmological parameters are presented in Sects. 4and 5, and an assessment of their accuracy and robustness is made in Sects. 6and 7. In Sect. 8we describe the low-`likelihood, and conclude by presenting the complete Planck likelihood in Sect. 9. 2. High-`likelihoods The Planck maps consist of the order 5 ×107pixels for each detector (Górski et al. 2005), so a likelihood described directly at the pixel level would be too time consuming. A significant compression of data can be achieved with minimal information loss using pseudo-C`power spectra, even in the case of incomplete sky coverage. Here we describe the form of the likelihood function of the compressed data, given a sky signal and instrumental model. Following Hamimeche & Lewis (2008), we assume a Gaussian form of the likelihood (Bond et al. 2000) based on pseudo-spectra that have been corrected to account for partial sky masking (Hivon et al. 2002) (for an overview on high-` Likelihoods see e.g. Rocha et al. 2010). We use a “fine-grained” data description, computing spectra of maps from individual detectors or detector sets. Table 1describes the 13 maps used in the analysis, spanning 100 to 217 GHz, which are compared with the general purpose “ds0” maps used in input to the component separation methods. These maps are based on the Planck nominal mission. In the case of the polarised detector sets (PSB type in Table 1), the map-making algorithm uses the four combined detectors to produce maps of the T,Q,UStokes parameters, but only the temperature maps produced from each PSB detector set are used in the likelihood analyses. We compute the spectra at these multiple frequencies to simultaneously constrain the CMB and foreground contributions. We choose these frequencies as a A15, page 2 of 60 Planck collaboration: Planck 2013 results. XV. trade-offbetween adding further information, and adding further complexity to the foreground model, which would be needed to include the adjacent 70 and 353 GHz channels (see Sect. 3 for further discussion). In our baseline analysis the spectra are computed at each multipole, together with an estimate of the full covariance matrix with off-diagonal errors between different spectra and multipoles. As in the WMAP analysis, we use only cross-spectra between detectors, alleviating the need to accurately model the mean noise contribution. In this section, we begin with a reminder of the pseudospectrum approach, and describe our baseline likelihood distribution, hereafter referred to as the CamSpec likelihood. We then show how a compression of spectra within a given frequency can be achieved with negligible loss of information. We describe the signal and instrument model, including detector noise properties, calibration, and beam uncertainties. Next, we describe an alternative, simpler, form of the likelihood, hereafter referred to as Plik, based on binned power spectra with an inverse-Wishart distribution. This does not require the pre-computation of large covariance matrices, so changing the sky or instrument modelling is straightforward. This simpler form of the likelihood is used to assess the robustness of our likelihood methodology with respect to technical choices and astrophysical foreground modelling. In Sect. 7.5 we also compare these likelihoods to the Likelihood estimated from band power spectra computed directly from CMB maps estimated by multi-frequency component separation (Planck Collaboration XII 2014). The band powers and covariance matrices are estimated with the XFaster approach (Rocha et al. 2011,2010). 2.1. The CamSpec likelihood The temperature maps produced from individual detector sets are first weighted with apodised Galactic and point source masks that are common within a given frequency band (see Sect. 3) for details. On top of these masks crafted to reject astrophysical contamination, each detector set has a small number O(104) of unobserved pixels that are added to the corresponding detector set masks. Let us define ˜ Ti `mas the spherical harmonic coefficients of the weighted temperature map of detector set i. The pseudospectrum at multipole `, for the detector pair (i,j),is then given by ˜ Ci j `=1 2`+1X m ˜ Ti `m˜ Tj† `m,(1) where the dagger, †, denotes the Hermitian transpose. This is related to the “deconvolved” spectrum, ˆ CTi j , by a coupling matrix, ˜ CTi j =MTT i j ˆ CTi j .(2) For an isotropic signal on the sky, the ensemble average of these deconvolved spectra are equal to the spectra of the theoretical models (including CMB and isotropic unresolved foregrounds) that we wish to test. For completeness, the coupling matrices are given explicitly in Appendix A.1. In the first method, CamSpec, we form the deconvolved spectra ˆ C`without any prior smoothing of the pseudo-spectra ˜ C`. Even for the largest sky masks used in our analysis (see Sect. 3), the coupling matrices are non-singular. The deconvolution requires the evaluation of ∼N2 map coupling matrices for a data set with Nmap sky maps, which takes a moderate, but not excessive, amount of computer time. A more challenging computational task is to compute the covariances of the pseudo-spectra, i.e., Cov( ˜ CTi j ˜ CTpq ).Here we need to compute N4 map coupling matrices, and the problem rapidly becomes computationally intractable even for relatively low values of Nmap. For the moment we assume that these covariance matrices are available and describe their computation in Appendix A.4. We use the notation ˜ X=Vec( ˜ C) to denote a column vector for which the index pof a single element Xpdenotes the map combination (i,j) and multipole `. We denote the covariance matrix of this vector as ˜ M=(˜ X−h ˜ Xi)( ˜ X−h ˜ Xi)T.(3) As explained later, the deconvolved detector set cross-spectra given by Eq. (2) can be efficiently combined within a given frequency pair after a small effective recalibration, taking their respective isotropised effective beam transfer function and noise levels into account (see Appendix A.3 for the detailed procedure). Here, like in all Planck papers, the effective beam means the beam, which accounts for the combined effect of the instrument and of the complete processing chain. Covariance estimates of these combined spectra can be deduced from those of the detector set cross-spectra. The covariance matrix is computed for a fixed fiducial model, and we approximate the likelihood as a Gaussian, described in Appendix A.5. The likelihood thus takes the form p=e−Swith S=1 2ˆ X−XTˆ M−1ˆ X−X. For the current analysis we include the following (deconvolved) spectrum combinations, ˆ X=(ˆ C100 ×100 `,ˆ C143 ×143 `,ˆ C217 ×217 `,ˆ C143 ×217 `),(4) coupled to a parametric model of the CMB and foreground power spectra. The multipole ranges we select, as well as the masks, depend on frequency, as described in Sect. 5. The tailored selection of multipole ranges and masks we adopt here is designed to keep the Galactic contamination as low as possible in both real and harmonic space, while keeping a reasonably low cosmic variance. This procedure allows us to keep the Galactic emission model simple (see Sect. 3.2). We do not include the 100×143 and 100×217 spectra since these spectra carry little additional information about the primary CMB anisotropies, but would require us to solve for additional unresolved foreground parameters. This tradeoffof information versus complexity was also considered for the use of the 70 GHz and 353 GHz data, which we choose not to include except for cross-checks. The fiducial covariance matrix is composed of the blocks shown in Fig. 1. The off-diagonal blocks in this matrix accurately account for the correlations between the power spectra at different frequencies. This description would be sufficient for perfectly known calibrations and beam transfer functions of each detector sets’ cross-spectra. Planck Collaboration VII (2014) describes in detail these uncertainties, and shows that for each detector set pair, (i,j), the effective beam transfer function can be expressed as Bi j(`)=Bi j mean(`) exp        nmodes X k=1 gi j kEi j k(`)      ,(5) described further in Appendix A.6, with nmodes beam error eigenmodes Ei j k(`), and their covariance matrix. These modes are then combined into generalised beam eigenmodes corresponding to A15, page 3 of 60 A&A 571, A15 (2014) Fig. 1. Covariance matrix blocks used in the likelihood, accounting for the correlations between cross-spectra estimated from the 100, 143, and 217 GHz channels. the spectra ˆ X. The associated covariance matrix is used to construct a Gaussian posterior distribution of the eigenmodes, which allows marginalisation over the beam uncertainties. Finally, in the construction of the covariance matrix, one needs to accurately specify the contribution of the instrumental noise. Even if there is no bias on the spectra due to instrumental noise, having removed auto-spectra, the latter dominates the covariance matrix on small scales. Fortunately, the Planck scanning strategy at the ring level allows us to make estimates of the noise pseudo-spectra from half-ring difference maps (see Planck Collaboration XII 2014). These half-ring difference maps, together with the knowledge of the noise variance per pixel for each detector set, can be used to derive the noise contribution to the covariance matrix with good accuracy (see Appendix A.8 for details). 2.2. The Plik likelihood We now describe the alternative form of the likelihood, inspired by Cardoso et al. (2008), used for cross-checks and robustness tests. We start from the full-sky exact likelihood for a Gaussian signal, which for Nmap detector maps is given by p(maps|θ)∝exp −      X ` (2`+1)Kˆ C`,C`(θ)       , where θis a vector containing the parameters of the signal model, and ˆ C`are the empirical angular spectra. KA,Bdenotes the Kullback divergence between two n-variate zero-mean Gaussian distributions with covariance matrices Aand B, and is given by KA,B=1 2htrAB−1−log detAB−1−ni. As already noted, a sky cut introduces off-diagonal couplings between different multipoles. In this method we bin the power spectra in such a way that these off-diagonal terms of the covariance are negligible. This is adequate to model sources with slowly varying spectra, such as foregrounds, and the CMB anisotropies for standard cosmologies. In this case, the likelihood takes the form p(maps|θ)∝exp −L(θ),with L(θ)= Q X q=1 nqKˆ Cq,Cq,(6) where the angular spectra ˆ C`for each cross-frequency spectrum have been averaged into Qspectral bins using spectral windows wq(`) (q=1,...,Q), with ˆ Cq=X ` wq(`)`ˆ C`,Cq=X ` wq(`)C`. Here wq(`) denotes the window function for the qth bin, and the same symbol, C, is used to denote binned or unbinned spectra. The effective number of modes in the qth bin is nq=fsky ·(P`wq(`)2)2 P`wq(`)4/(2`+1)· We adopt a spectral binning defined by wq(`)=         `(`+1)(2`+1) P`q max `q min `(`+1)(2`+1) `q min ≤`≤`q max, 0 otherwise. The Plik bin width is ∆`=9 from `=100 to `=1503, then ∆`=17 to `=2013, and finally ∆`=33 to `max =2508. This ensures that correlations between any two bins are smaller than 10%. While this binned likelihood approximation does not fully capture all couplings between different multipoles, it has a notable advantage in computational speed, and it agrees well with the primary likelihood. It is therefore very well suited for performing an extensive suite of robustness tests, as many more parameters can be considered in a short time. Further, instrumental effects can be investigated quickly to assess the agreement between pairs of detectors within a frequency channel, such as individual detector calibrations and beam errors. A specific example is the impact of (effective) beam uncertainty parameters on the likelihood. This can be investigated by re-expressing the model covariance matrices as C`=B`(γ)C`(θ)B`(γ)T,(7) where C`(θ) is the model covariance including both signal and noise, and B`(γ) is a diagonal matrix encoding the beam and calibration errors with elements given by2 Bi `(γ)=exp        nmodes X k=1 δi kEii k(`)      .(8) Here, Eii k(`) are the eigenmodes of the (auto-)spectra, similar to Eq. (5). Note that Eq. (7) does not contain the mean beam transfer function, since it is already included in the empirical spectra. Thus, using Eq. (7)Plik approximates the cross-spectrum beam errors as the harmonic mean of the corresponding auto-spectrum beam errors, under the assumption that B`is diagonal between detectors. This approximate factorisation is intrinsically linked to the assumed Kullback shape of the Plik likelihood, and is later demonstrated to work well for both simulations and data. 2From Eqs. (5), (7), and (8), we have δi k=gii k/2 at first order. A15, page 4 of 60 Planck collaboration: Planck 2013 results. XV. The Plik likelihood method also provides a direct estimate of the detector noise power spectra as it can include the empirical auto-spectra, and we find that these noise estimates are in good agreement with the noise spectra used to construct the CamSpec likelihood covariance matrix. The method can also produce a binned CMB power spectrum independent of the underlying cosmological model, providing a direct quality assessment of the foreground model parametrisation. In practice, we proceed in two steps. First, we jointly estimate the noise together with all other parameters using both auto and cross-spectra. Then we fix the noise estimates, and use the fiducial Gaussian approximation to explore the remaining free parameters excluding the autospectra, optionally including only specific data combinations. 3. Foreground emission model and sky masks 3.1. Sky masks The Galactic emission varies strongly in both complexity and strength across the sky. It is therefore necessary to find a balance between maximizing the sky coverage to reduce statistical uncertainties, and establishing a simple yet efficient foreground model. In this paper, we threshold an ILC-subtracted 353 GHz temperature map to define a basic set of diffuse Galactic masks, which form a sequence of increasing sky fraction, to minimise the contribution from diffuse dust emission. We refer to them using the percentage of the sky retained: G22, G35, G45, G56, G65. We also use a point source mask, labelled PS96, which is based on the union of the point sources detected from the channels in the range 100 to 353 GHz. The point source flux cut is not critical, since the amplitudes of the Poisson contributions of unresolved sources are allowed to vary over a wide range in the likelihood analysis. Thus, we do not impose tight priors from source counts and other CMB experiments on the Poisson amplitudes. All theses masks are shown in Fig. B.1. For Planck, we need to estimate the covariance matrices to percent level precision. For temperature spectra, and in the absence of point source holes, this precision can be achieved with sharp, non-apodised Galactic masks (Efstathiou 2004). However, the inclusion of point source holes introduces nonnegligible low-`power leakage, which in turn can generate errors of several percent in the covariance matrices. In order to avoid such power leakage, we also derive a series of apodised masks. For the Galactic masks, we proceed as follows. First, we smooth each mask with a five-degree Gaussian beam, and zero any pixels below a threshold of 0.15 (chosen to approximately keep the zeroed area the same as that of the original, unapodised mask). We then subtract 0.15 from the remaining pixels, and rescale the resulting mask by 1/(1−0.15). This procedure is iterated a further two times. In order to retain sufficient sky area for the most conservative sky mask, a slightly less aggressive version of mask G22 was used to seed the apodisation process for that case. Each point source is apodised to 300FWHM, resulting in the PSA82 point source mask. The resulting set of apodised masks are shown in Fig. B.2. For all cosmology analyses, we use three of the union of one of the apodised galactic masks and of the apodised point sources mask – CL31, CL39, and CL49, which are shown in Fig. 2. Table 2summarizes the various masks and the corresponding sky fraction they retain. 3.2. Galactic emission The contamination from diffuse Galactic emission at low to intermediate multipoles can be reduced to low levels compared Fig. 2. Set of masks (CL31, CL39, CL49) used for the likelihood analyses, in Mollweide projection. Table 2. Series of masks used in this paper. Galactic mask Apodised Galactic masks Cosmology mask G22 GA21 – G35 GA34 CL31 G45 GA38 CL39 G56 GA54 CL49 G65 GA60 – Notes. As described in the text, each series in a line corresponds to a particular threshold level at 353 GHz. The Galactic and cosmology masks are labelled by their prefix G and CL. The two digits at the end of each mask name refers to the sky fractions retained by each mask. The cosmology masks are obtained by merging apodised Galactic masks (prefixed by GA) with our unique apodised point source mask, PSA82. For completeness, let us note that we also occasionally use the combinations mask0 ≡G22 ∪PSA82 and mask1 ≡G35 ∪PSA82 to study the diffuse Galactic emission. to CMB anisotropies by a suitable choice of masking. However, even with conservative masking, the remaining Galactic emission at high multipoles is non-negligible compared to other unresolved components, such as the cosmic infrared background (CIB) anisotropies at 143 and 217 GHz. A clear way of demonstrating this is by differencing the power spectra computed with different masks, thereby highlighting the differences between the isotropic and non-isotropic unresolved components. A15, page 5 of 60 A&A 571, A15 (2014) Fig. 3. Differences between power spectra evaluated from the union masks G22 ∪PSA82 and G35 ∪PSA82 (≡mask0 and mask1 respectively in the legend), showing the presence of Galactic dust. For `≤1400 the spectra show the 217−143 “double-differenced” power spectrum, rescaled to correct for dust emission at 143 GHz. For ` > 1400 the 217 mask differenced power spectrum is plotted. The blue line shows the 857 GHz mask-differenced power spectrum scaled to 217 GHz as described in the text. The green dotted line shows the best-fit of Eq. (9) scaled to 217 GHz. This is the model that we use to subtract dust emission at high multipoles from the 143 ×143, 143 ×217 and 217 ×217 spectra with suitably adjusted scaling factors. Let us define D ≡ `(`+1)C`/2π, a notation we shall use throughout this paper. Figure 3shows (up to `≤1400) the 217 GHz power spectrum difference for the union masks mask0 ≡G22 ∪PSA82 and mask1 ≡G35 ∪PSA82, minus the corresponding difference for the 143 GHz frequency channel. Any isotropic contribution to the power spectrum (CMB, unresolved extragalactic sources etc.) cancels in such a maskdifferenced power spectrum, leaving a non-isotropic signal of Galactic origin. Nevertheless, at low multipoles, a mask differenced power spectrum is still severely affected by cosmic variance of the primordial CMB; but this can be eliminated by forming a “double-difference” power spectrum between frequencies. This is what is plotted at `≤1400 in Fig. 3, rescaling to account for the small dust contribution at 143 GHz. Above `=1400, the noise in the 143 GHz power spectra becomes significant, and so we plot directly the 217 mask-differenced power spectrum. The figure gives an accurate estimate of the dust contribution within mask1 to the 217 GHz spectra over the entire multipole range plotted. In the same figure, these difference spectra are compared to the unbinned mask-differenced 857 GHz power spectrum, scaled to 217 GHz adopting a multiplicative factor3of (9.93×10−5)2; the dotted line shows a smooth fit to the unbinned spectrum. The agreement between this prediction and the actual dust emission at 217 GHz is excellent, and this demonstrates conclusively the existence of a small-scale dust emission component with an amplitude of ∼5−15 µK2at 217 GHz if mask1 is used. For cosmological parameter analysis this small-scale dust component must be taken into account, and several approaches may be considered: 1. Fit to a template shape, e.g., as shown by the dotted line in Fig. 3. 3The scaling coefficient for the 143 GHz spectrum is (3.14 ×10−5)2, derived from the 7-parameter fitting function of Eq. (A.46). Fig. 4. 857 GHz mask-differenced power spectrum (points), interpreted as Galactic dust emission. The solid line shows the best-fit model defined by Eq. (9). 2. Reduce the amplitude by further masking of the sky. 3. Attempt a component separation by using higher frequencies. The main disadvantage of the third approach is a potential signal-to-noise penalty, depending on which frequencies are used, as well as confusion with other unresolved foregrounds. This is particularly problematic with regards to the CIB, which has a spectrum very similar to that of Galactic dust. In the following we therefore adopt the two former solutions. It is important to understand the nature of the small scale dust emission, and, as far as possible, to disentangle this emission from the CIB contribution at the HFI cosmological frequencies. We use the 857 GHz power spectrum for this purpose, noting that the dust emission at 857 GHz is so intense that this particular map provides an effectively noise-free dust emission map. In Fig. 4we again show the 857 GHz mask power spectrum difference, but this time plotted on a log-log scale. The solid line shows the corresponding best-fit model defined by D`=A(100/`)α [1 +(`/`c)2]γ/2,(9) with A=5.729 ×108µK2,α=0.169, `c=90.5, and γ=0.427. At high multipoles this fit asymptotically approaches C`∝`−2.6, which is compatible with previous knowledge about diffuse Galactic emission, i.e., a power-law behaviour with an index close to −3 extending to high multipoles (see e.g., Miville-Deschênes et al. 2007). Note that throughout the paper, we use CMB thermodynamic units. The upper panel in Fig. 5shows the 857 GHz spectra for the four Galactic masks (G22, G35, G45 through to G56) with the point-source mask applied. They are compared to the 857 GHz CIB power spectrum from Planck Collaboration XVIII (2011), which for ` > 500 can be described approximately as `C`≈7× 106µK2. The best fit models to Eq. (9) are also shown (with α, `cand γfixed to the values derived above, but with the amplitude allowed to vary) fitted to `≤500, where we expect diffuse dust emission to be dominant. The lower panel of Fig. 5 shows the same power spectra after subtracting the best-fit dust model. After subtracting the Galactic dust component, the recovered power spectra are consistent with the CIB measured in A15, page 6 of 60 Planck collaboration: Planck 2013 results. XV. Fig. 5. Top: 857 GHz power spectra for the four different masks defined in the text. The dotted lines show the best-fit model defined by Eq. (9) fit to `≤500, capturing the Galactic dust. An estimate of the CIB power spectrum is shown in orange points (Planck Collaboration XVIII 2011). Bottom: power spectra after subtracting the Galactic dust model. The horizontal purple line shows a constant amplitude of 7 ×106µK2, which roughly describes the Planck CIB results. Planck Collaboration XVIII (2011) for all masks. The excess at high multipoles may be due to a combination of aliasing of large scale power through the point source masks at `> ∼3000, Galactic point sources, and uncertainties in the 857 GHz beams. The model explains by construction the “double-difference” plot shown in Fig. 3. Specifically, this emission is consistent with a cirrus-like power spectrum, C`∝`−2.6, extrapolated to high multipoles. Furthermore, the results of Fig. 5demonstrate that over a wide area of sky, we can understand the 857 GHz power spectrum in terms of a “universal” cirrus spectrum together with an isotropic CIB component. These results provide strong evidence that an extragalactic CIB component dominates over the diffuse Galactic emission at multipoles ` > 700 over the full range of HFI frequencies outside the G22 region (i.e. over the “cleanest” ∼20% of sky). Although this is demonstrated for 857 GHz in Fig. 5, it applies over the entire HFI frequency range since Galactic dust and CIB have similar spectra. We therefore use the dust template of of Eq. (9) with suitably adjusted scaling factors to remove the high-`contribution from dust from the empirical 143×143, 143×217 and 217×217 spectra, which are used in CamSpec. We take a different approach for the Galactic dust correction with the Plik likelihood. Rather than correcting the empirical spectra during a pre-processing step, the Plik likelihood implements an explicit one-parameter model that describes the dust contribution to the cross-spectrum between detectors iand j, CDust `(i,j)=ADust F(νi, ν0)F(νj,ν0) ` 500!−γd gDust igDust j.(10) Here F(ν,ν0)=νβdB(Td, ν) ∂B(TCMB,ν) ∂T /νβdB(Td, ν0) ∂B(TCMB,ν0) ∂T ,(11) where the dust amplitude, ADust, is measured in units of µK2, νiis the reference frequency for map i,ν0is a reference frequency, which is taken to be 143 GHz, B(T, ν) is the emission law of a black body with temperature T, and the dust colourcorrection terms, gDust i, are computed by integrating the dust spectrum within the spectral band of each detector (set). We fix the frequency and angular scaling parameters to γd=2.6, βd=1.6 and Td=18 K. 3.3. Poisson power from unresolved point sources Unresolved galaxies contribute both shot noise and clustered power to the Planck maps. The Poisson contribution leads to a scale independent tem, C`=const.We model this power with a single amplitude parameter for each auto-spectrum (APS 100,APS 143, and APS 217) and a cross correlation coefficient for each cross spectrum (APS 143 ×217 =rPS 143 ×217 qAPS 143APS 217). These quantities are not of primary interest for cosmological results, so to avoid modelling error we do not separate the power into that sourced by “dusty" or “radio" galaxies (i.e., with increasing or decreasing brightness with frequency, respectively) as is done in the analysis of the ACT and SPT power spectra (Dunkley et al. 2011,2013; Reichardt et al. 2012). We also make no assumptions about their coherence between frequencies. The Poisson power is related to the flux density dN/dSvia C`=1 4πZdˆnZScut(ˆn) 0 dS S 2dN dS,(12) where we have explicitly introduced the Planck flux cut Scut(ˆn). Since Planck utilises a constant signal-to-noise cut, and the Planck noise varies significantly across the sky, this flux cut has a spatial dependence. Although this does not alter the shape of the Poisson term, extra care must be taken when comparing results4with models of dN/dS. In Sect. 7.3, we explore the consistency between the Poisson power recovered from the Planck power spectrum analysis and predictions from source count measurements. 4One must also account for the fact that these numbers correspond to the amplitude for a suitably averaged spectral band, which is approximately that of the map, and is described in detail in Planck Collaboration IX (2014). A15, page 7 of 60 A&A 571, A15 (2014) 3.4. Clustered power from unresolved point sources Unresolved galaxies also contribute power because they trace large-scale structures. The mean flux from the radio galaxies is much smaller than that from the dusty galaxies, so only the dusty galaxies contribute a significant clustering term (Millea et al. 2012). The CIB clustering has been studied extensively, starting with Bond et al. (1986,1991). Further theoretical investigation (Scott & White 1999;Haiman & Knox 2000) was stimulated by the detection of the infrared background in the COBE data (Puget et al. 1996;Fixsen et al. 1998), and the detection of bright “sub-millimetre” galaxies in SCUBA data (Hughes et al. 1998). Subsequently, the clustering has been detected at 160 microns (Lagache et al. 2007), at 250, 350 and 500 microns by the Balloon-borne Large Aperture Submillimeter Telescope (BLAST, Viero et al. 2009;Hajian et al. 2012) and at 217 GHz by SPT and ACT (Hall et al. 2010; Dunkley et al. 2011). Recent Planck measurements of the CIB (Planck Collaboration XVIII 2011) have extended the measurements at 217 GHz, 353 GHz, and 545 GHz to larger scales, and recent Herschel measurements (Viero et al. 2013) have improved on the BLAST measurements and extended them to smaller angular scales. Rather than attempt to establish a physical model of the CIB, we adopt in this analysis a phenomenological model that captures the CIB uncertainties for both Planck and high-`experiments. Our baseline model for the clustered CIB component is a power-law spectrum with a free spectral index, DCIB `∝`γCIB , with an amplitude at each frequency, ACIB 143 and ACIB 217 , and a crosscorrelation between frequencies, ACIB 143 ×217 =rCIB √(ACIB 143 ACIB 217 ). We assume that the CIB clustering power at 100 GHz is negligible. 3.5. Unresolved Sunyaev-Zeldovich effects Based on analysis of ACT and SPT data, the thermal SunyaevZeldovich (tSZ) contribution is expected to contribute approximately DtSZ `=3000 ∼9µK2at 100 GHz and DtSZ `=3000 ∼4µK2 at 143 GHz (Reichardt et al. 2012;Dunkley et al. 2013;Sievers et al. 2013). The kinetic Sunyaev-Zeldovich (kSZ) effect is expected to have a similar, or smaller, contribution, with DkSZ `=3000 < ∼ 7µK2. In addition, theoretical arguments (Reichardt et al. 2012; Addison et al. 2012b) suggest that there should be a tSZ × CIB correlation that should contribute about the same order of magnitude as the kSZ term at 143 GHz. For Planck, all of these SZ contributions are small in comparison to other unresolved foregrounds and are therefore poorly constrained by Planck data alone. Nevertheless, to eliminate biases in cosmological parameters (Millea et al. 2012;Zahn et al. 2005), we model their contributions, with appropriate constraints from higher resolution CMB experiments, using three templates. First, for the thermal SZ effect we adopt the family of templates described by Efstathiou & Migliaccio (2012). These are based on the Komatsu & Seljak (2002) model, but use the “universal” X-ray electron pressure profile, Pe,ofArnaud et al. (2010) extrapolated to high redshift via Pe(z)∝[(1 −ΩΛ)(1 +z)3+ ΩΛ]4/3−/2.(13) Here, describes departures from self-similar evolution, and a value of =0.5, which is adopted as the default for parameter estimation purposes, provides a good match to the results Fig. 6. SZ templates at 143 GHz computed for a normalization of σ8=0.8. The tSZ templates are from the model of Efstathiou & Migliaccio (2012) for three values of the evolution parameter ,=0 (top), =0.25 (middle) and =0.5 (lower). The kinetic SZ template is from Trac et al. (2011). The tSZ ×CIB cross correlation (143 × 143 GHz) is from the Addison et al. (2012b) template with parameters described in the text, and is negative for 143 ×143 GHz. from recent hydrodynamical numerical simulations incorporating feedback processes (Battaglia et al. 2010,2012). Figure 6 shows the tSZ templates for three values of ; the template shape is not particularly sensitive to . We treat the (dimensionless) normalization of the tSZ template at 143 GHz as an adjustable parameter, DtSZ `=AtSZ 143DtSZ,template `.(14) This parameter fixes the amplitude at 100 GHz via the frequency dependence of the tSZ effect, CtSZ ∝ xex+1 ex−1−4!2 ,x=hν kT ·(15) We neglect the tSZ at 217 GHz. Second, for the kinetic SZ effect we adopt the template described by Trac et al. (2011), and as in Eq. (14) we treat the dimensionless amplitude of the template, AkSZ, as a free parameter, DkSZ `=AkSZDkSZ,template `.(16) Third and finally, for the cross-correlation between the thermal SZ component and the CIB we adopt the template described by Addison et al. (2012a). In this case, the amplitude is parametrised in terms of a single correlation coefficient, DtSZ×CIB `=−2ξqDtSZ143 3000 DCIB143 3000 DtSZ×CIB,template `(143×143), DtSZ×CIB `=−ξqDtSZ143 3000 DCIB217 3000 DtSZ×CIB,template `(143×217).            (17) These templates are plotted in Fig. 6, normalized to σ8=0.8 and with ξ=1.0 using a fiducial CIB amplitude. Note that with these parameters, the tSZ ×CIB cross-spectrum approximately cancels the kSZ spectrum at 143 GHz. A15, page 8 of 60 Planck collaboration: Planck 2013 results. XV. As seen in Fig. 6, the SZ contributions are at the level of a few µK2, which, although small, must be taken into account to assess inter-frequency residuals. However, one can see that these templates have similar shapes at multipoles < ∼2000, and therefore they cannot be disentangled using Planck data alone. On the other hand, higher resolution experiments can break this degeneracy, and as shown in Planck Collaboration XVI (2014), the combination of Planck, ACT, and SPT, better constrains the amplitude of the thermal SZ effect. The ACT and SPT data at 150 GHz can be fitted to high, sub-µK2, accuracy without kSZ and tSZ x CIB templates, yet we expect a kSZ contribution of at least the amplitude shown in Fig. 6, and larger if we account for patchy reionisation (see, e.g., Knox 2003, and references therein) and references therein). This implies a cancellation of the kSZ and tSZ x CIB contributions at 150 GHz (Addison et al. 2012b), as discussed in greater detail in Planck Collaboration XVI (2014). 4. Consistency checks and combined cross-spectra The large number of cross spectra in a detector-by-detector power spectrum analysis allows for a number of internal consistency checks of the data. Within a frequency band, we expect to see exactly the same sky signals (primordial CMB, Galactic, and extra-galactic foregrounds), and so any intra-frequency residuals reflects instrumental systematics, for example beam errors, “gain” fluctuations, and band-pass mismatch. In contrast, interfrequency residuals are harder to analyse because the sky signals vary with frequency. An accurate model of the unresolved foregrounds is therefore required to assess inter-frequency residuals. Furthermore, as we show below, the scatter caused by chance CMB–foreground cross-correlations can dominate the inter-frequency residuals. For a precision experiment such as Planck, where the power spectra are expected to be signal dominated over a wide multipole range, intraand inter-frequency residuals provide a powerful way of assessing possible systematic errors. It is essential that contributions of systematic errors to both types of residual are small enough that they have negligible impact on cosmological parameter analysis. Figure 7shows a selection of temperature cross-spectra and estimates of the analytic covariance matrices, together with the best-fit cosmological model described in Sect. 5. Unresolved foregrounds have been subtracted using the best-fit foreground parameters of the model described in Sect. 3. The scatter varies substantially between cross-spectra, reflecting differences in the instrument noise and effective resolution of different detector combinations. The analytic error model summarized in Appendix A.2 is indicated, modified by the non-white noise correction. This model provides an excellent description of the scatter seen in the data, over the full multipole range shown in plots, with an accuracy of a few percent or better. 4.1. Intra-frequency residuals In this section we analyse the intra-frequency residuals at 143 and 217 GHz. There are Nspec =10 cross-spectra at 143 GHz and 15 cross-spectra at 217 GHz5. At each frequency, we solve for multiplicative (“effective” calibration) coefficients, yi, that minimise χ2=X `X i j,j>i (yiyjˆ Ci j `−h ˆ C`i)2,(18) 5There is only one cross-spectrum at 100 GHz. Table 3. Map calibration coefficients. Map yiMap yi 143-ds1 0.9990 217-ds1 0.9982 143-ds2 0.9994 217-ds2 0.9975 143-ds3 1.0000 217-ds3 1.0000 143-ds4 0.9988 217-ds4 0.9992 143-ds5 0.9980 217-ds5 0.9981 – – 217-ds6 0.9985 where hˆ C`i=1 Nspec X i j,j>i yiyjˆ Ci j `,(19) subject to the constraint that y1=1 (where i=1 corresponds to detector 5 at 143 GHz and detector 1 at 217 GHz). Note that the power spectra in Eq. (18) and (19) are corrected for beam transfer functions. To minimise the possible impact of beam errors and noise, we restrict the sum in Eq. (18) to the multipole range 50 ≤`≤500 where the spectra are signal dominated. Numerical values for the calibration coefficients are given in Table 3, using mask CL31. The calibration factors are insensitive to the choice of mask or multipole range. The results of Table 3show that effective calibration factors of ∼0.2% are quite typical for HFI maps, in the 100−217 GHz frequency range. These recalibrations are of the order of magnitude of the statistical errors of the calibrations on dipole (see Planck Collaboration VIII 2014, Table 2). Note that the data are corrected for individual bolometer time transfer functions (TTFs; Planck Collaboration VII 2014). For each detector, the TTF model is tuned to minimise survey differences and by construction normalized to unity at the spin frequency of the satellite (0.01666 Hz) to preserve the dipole calibration. The consistency of intra-frequency power spectrum residuals therefore provides a test of the consistency of the TTFs in addition to the beam transfer functions. Figure 8shows the remarkable consistency of the power spectra at each frequency. The upper panels show the spectra corrected for the beam and effective calibration, together with the mean cross spectra. The lower panels show the dispersion around the mean. Up to `≃700 the ∆`=31 binned crossspectra show a dispersion in their values typically less than expected from ∼0.2% calibration variations. This excess scatter (compared to the noise-induced expected scatter) has negligible impact on cosmological parameter analysis. The residuals of the cross spectra in band averages of width ∆`∼61 are shown in Fig. 9, before and after correction for the effective intra-frequency calibrations. The reduction in scatter after correction is evident at `< ∼500, and the residual scatter is consistent with instrument noise and beam errors. At 217 GHz, beam errors dominate over noise at multipoles < ∼1000. There is no evidence that the excess scatter is caused by a small number of “anomalous” detectors. 4.2. Inter-frequency residuals The results of the previous section show that the intra-frequency cross-spectra between detector/detector sets are consistent to within a few µK2at multipoles `< ∼1000. In a likelihood analysis, there is therefore little loss of information in compressing the power spectra for each distinct frequency combination, as A15, page 9 of 60 A&A 571, A15 (2014) Fig. 13. Foreground model over the full range of HFI cosmological frequency combinations. The upper panel in each plot shows the residual between the CamSpec measured power spectrum and the “best-fit” primary CMB power spectrum, i.e., the unresolved foreground residual for each frequency combination. The lower panels show the residuals after removing the best-fit foreground model. The lines in the upper panels show the various foreground components. Major foreground components are shown by the solid lines, colour coded as follows: total foreground spectrum (red); Poisson point sources (orange); CIB (blue); thermal SZ (green), Galactic dust (purple). Minor foreground components are shown by the dotted lines: kinetic SZ (green); tSZ ×CIB cross correlation (purple). The 100 ×143 and 100 ×217 GHz spectra are not used in the CamSpec likelihood. Here we have assumed rPS 100 ×143 =1 and rPS 100 ×217 =1. A15, page 16 of 60 Planck collaboration: Planck 2013 results. XV. Fig. 14. Comparison of pairs of foreground subtracted cross spectra, demonstrating consistency of the residuals with respect to the best-fit theoretical model. The red line in each of the upper panels shows the theoretical six parameter ΛCDM spectrum for the best-fit parameters listed in Table 8. The lower panels show the residuals with respect to this spectrum, together with error bars computed from the diagonal components of the covariance matrices of the band averages. The points here are band-averaged in bins of width ∆`∼31. and Plik for AkSZ are in good agreement, showing the stability of the cosmological parameters to the likelihood method. In Sect. 7, we further explore the stability of the cosmological and foreground distributions to technical choices made in the likelihood and data selections. 6.2. Comparison to simulations We now investigate the precision and accuracy with which cosmological parameters, and foreground parameters to a lesser extent, can be recovered from realistically simulated data. Here we compare the posterior distributions of cosmological and foreground parameters, together with calibration and beam error parameters, inferred using the Plik likelihood, with the input values of a set of simulations, referred to as “Full Focal Plane” (FFP6). The signals in these simulations are based on the “Planck Sky Model” (Delabrouille et al. 2013), which includes a detailed model of the astrophysical emission, both Galactic and extragalactic, at the Planck frequencies. The simulations also reproduce in detail the main instrumental systematic effects of Planck, including correlated time-line noise, instrumental pointing, flags, anisotropic detector beams, and spectral bandpasses. One thousand CMB and noise realisations were generated using the same foreground emission, and a hundred realisations were performed at the level of different detector sets. These simulations are described further in Planck Collaboration (2013). A15, page 17 of 60 A&A 571, A15 (2014) Fig. 15. Top:Planck maximum-likelihood (primary) CMB spectrum compared with the best-fit six parameter ΛCDM spectrum. Bottom: power spectrum residuals with respect to the ΛCDM model. The error bars are computed from the diagonal elements of the band-averaged covariance matrix, as given by Eq. (A.25), including contributions from foreground and beam transfer function errors. In order to test for any methodological bias, we estimate cosmological parameters from 100 FFP6 simulations using the Plik likelihood. They consist of random realisations of CMB anisotropies and noise, superimposed with a single realisation of a frequency-dependent foreground template. We assume the foreground power spectra to be known exactly, and include their additional power as a constant component in our model. We do not estimate the foreground parameters in this case. The goal of this test is to demonstrate the reliability of our analysis pipeline and explore the effects of the noise and CMB-foreground chance correlations on parameter estimation. To compare our results to the simulation inputs, we also estimate cosmological parameters from the 100 CMB realisations, without including noise or foregrounds. In the likelihood evaluations we down-weight the high-`part of the power spectrum as if there were noise at the level of the noise simulations. A direct comparison of the derived parameters from this CMB-only analysis and the full simulation (that contains noise and foreground emission) allows us to remove the scatter introduced by cosmic variance. The result of the analysis is shown in Fig. 22, where we plot the distribution of the difference of the mean estimated parameters, between the CMB-only simulation and the noisy CMB simulation with foregrounds, in units of the standard deviation of each individual distribution. An unbiased pipeline will give a difference consistent with zero. Averaged over 100 simulations, only the bias on Ωch2is statistically significant, but is <0.3σ. An interesting point to note from these histograms is that the distributions are rather wide, mostly around 0.5σ, while by construction, we only studied the effect of noise and CMB-foreground chance correlations in this test. The FFP6 simulations can also be used to assess the sensitivity to foreground modelling errors. In Fig. 23, we compare the posterior marginal distributions of cosmological parameters, estimated with different assumptions about the foreground model. The blue lines, which correspond to the analysis of CMB-only simulations (but accounting for the noise covariance in the likelihood) correspond to the idealised case where the foregrounds play no role, and where the noise-induced variance has been averaged. The red, thick lines show results obtained when marginalising over the parameters of the model of foregrounds that is applied to the Planck data (see Sect. 3), with a fixed value of the CIB spectral index γCIB. Purple and grey lines show respectively the effect of leaving γCIB free when marginalising, or fixing it to the displaced value of 0.4, more than 2σ away from the peak posterior. The green lines show the effect of leaving γDust (spectral index of the Galactic dust emission) free in the marginalisation. The distributions are all in reasonable agreement with the input parameters of the simulation. In addition, we see that varying assumptions on the parameters of the foreground model (red, green, purple, and yellow lines) have negligible impact on the recovered cosmological parameters. Finally, the broadening of the posteriors between the CMB-only exploration and the full case (including noise random realisations and foregrounds) is expected, as the latter includes all the sources of variance, including CMB-foreground and CMB-noise chance correlations. It is worth noting that the FFP6 foreground simulations, based on extrapolations of existing observations, cannot be described by the simple foreground model used in the likelihood analyses. The upper bounds on biases introduced by a possible mismatch between the simulated foreground templates and the model used in the analysis, inferred from Fig. 22, should be representative of the Planck data analysis. The negligible impact of the various assumptions made on the foreground model parameters of Fig. 23 confirms that the cosmological parameter estimations are robust to details of the foreground model. 7. Consistency checks In this section we investigate the stability of the distributions of cosmological and foreground parameters. The technical choices made in constructing the high-`likelihood fall into three broad categories. The first category covers internal parameter choices that leave the data selection unchanged. This includes choices such as the binning strategy, marginalising or not over calibration and beam errors, and the description of the noise model. The second category includes variations in the data selection, such as the multipole range used, and the choice of masks, or of spectra. The final category accounts for variation in the foreground model. We perform a suite of tests to investigate the impact of these choices on parameters. We use the Plik likelihood, and all tests are compared to the baseline Plik spectra. Most of the results can be summarized by “whisker plots”, which compare the main properties of the posterior distribution of the cosmological and foreground parameters. More detailed results are reported in Appendix D.1. In this section we also compare our estimated cosmological parameters to those derived from spectra computed from the LFI 70 GHz channel. We additionally check the consistency of parameters with results obtained using the power spectrum of CMB maps derived by component separation methods (described in Planck Collaboration XII 2014) that use Planck data at all frequencies. This battery of tests demonstrates the stability of the inferred cosmological parameters. A final test is to compare the predicted polarisation spectrum of the best fitting ΛCDM model with spectra measured from Planck. As discussed in Planck Collaboration I (2014) and Planck Collaboration VI (2014), the Planck polarisation data is not yet used in our cosmological analysis, as further tests must be performed, but the current results increase our confidence in the robustness of the high-`temperature likelihood. A15, page 18 of 60 Planck collaboration: Planck 2013 results. XV. Fig. 16. Zoom-in of regions of the Planck primary CMB power spectrum using finer bin widths (∆`=15 for ` < 1000 and ∆`=7 for 1000 ≤`≤ 2200. In the upper panels, the red lines show the best-fit ΛCDM spectrum, and the blue lines join the Planck data points. Error bars are computed as in Fig. 15. 7.1. Impact of technical choices for fixed data selection Here we consider three changes: (1) fixing the inter-frequency calibration and beam errors to the best-fit values, rather than marginalising; (2) including sub-pixel effects; and (3) including noise correlation between detectors. Figure 24 shows the corresponding impact on parameters. The effect of fixing the calibration and beam errors is negligible on most cosmological parameters within the six parameter ΛCDM model, with the exception of nswhere we see a 0.16σ shift. There is a bigger effect on the foreground parameters due to their partial degeneracy, and their sub-dominant contribution to the total power. There is only a small correlation between the cosmological parameters and the calibration coefficients, so marginalising or fixing their value has little impact on the cosmology. The calibration coefficients are, however, strongly correlated with each other, in particular at 217 GHz, since they are also significantly correlated with e.g., the CIB amplitudes in the 217 ×217 and 217 ×143 GHz spectra. This is important to keep in mind when comparing the calibration estimates obtained here with those obtained from the CMB dipole in the HFI data processing paper (Planck Collaboration VI 2014). Nevertheless, as can be seen in Fig. 26, the peak posterior values of the relative calibration coefficients are found to differ from 1 at most at the few parts per thousand for all the 13 detectors sets involved, in agreement with the estimates of the calibration accuracy of the maps (Planck Collaboration VI 2014), although a wide flat prior has been applied on these coefficients. The same test applied on simulations with no beam or calibration errors shows how well this test is passed. This confirms that the deviations found at the 0.1% are significantly detected, and it is important to show that these deviations have little impact on the cosmology. The estimated values of the beam errors do not imply that extra beam corrections are required. A comparison of the prior and posterior distributions suggests that we have quite conservatively estimated the uncertainties from the beam determination. We keep this conservative approach in CamSpec, in which we marginalise analytically over all beam eigenmodes except for the dominant 100 GHz mode, β1 1, which we sample directly. As an extended test, we investigate the effect of possible errors in the beam transfer function when the helium abundance YP is also allowed to vary freely (i.e., without imposing constraints from Big Bang Nucleosynthesis), as it has a larger effect on the small scale spectrum. Varying this parameter leads to a substantial broadening of the posterior distributions for Ωbh2,θMC, and ns, as can be seen in Fig. 27. We confirm that marginalising over calibration and beam errors has a small impact on all A15, page 19 of 60 A&A 571, A15 (2014) 0 500 1000 1500 2000 2500 −200 −100 0 100 200 Dresidual, [µK2] 100 ×100 0 500 1000 1500 2000 2500 10−1 100 101 102 103 104 D[µK2] 100 ×100 data model cmb cib and sz poisson galactic dust ksz subpixel correction cross det correction 0 500 1000 1500 2000 2500 −200 −100 0 100 200 Dresidual, [µK2] 143 ×143 0 500 1000 1500 2000 2500 10−1 100 101 102 103 104 D[µK2] 143 ×143 0 500 1000 1500 2000 2500 −200 −100 0 100 200 Dresidual, [µK2] 143 ×217 0 500 1000 1500 2000 2500 10−1 100 101 102 103 104 D[µK2] 143 ×217 0 500 1000 1500 2000 2500  −200 −100 0 100 200 Dresidual, [µK2] 217 ×217 0 500 1000 1500 2000 2500  10−1 100 101 102 103 104 D[µK2] 217 ×217 Fig. 17. Left: residuals between the Planck power spectrum derived with Plik using “validation” settings (described in the text) and the Plik best-fit model. The light grey lines show residuals for individual detector pairs within each frequency combination. The blue lines show the inverse covariance weighted averages of the individual residuals, together with their errors computed from the covariance matrix. Right: decomposition of the total best-fitting model power spectra into CMB, combined thermal SZ and CIB, unresolved point sources, kinetic SZ, and Galactic dust. cosmological parameters, including YP. On the other hand, it has a somewhat larger impact on some of the foreground parameters (see Appendix D.1). We then investigate the impact of two sub-dominant effects: the “sub-pixel effect” and the possible presence of a correlation in the noise between detectors or detector sets. The sub-pixel effect has a convolving effect on the power spectra that is similar to gravitational lensing of the CMB, but is purely a result of the Planck scanning strategy and the map-making procedure (Planck Collaboration VII 2014). The scanning strategy on rings with very low nutation levels results in the centroid of the samples being slightly shifted from the pixel centres; however, the map-making algorithm assigns the mean value of samples in the pixel to the position at the centre of the pixel. This has a nondiagonal effect on the power spectra, but the correction can be computed given the estimated power spectra for a given data selection, and recast into an additive, fixed component of the model covariance matrix. The possible noise correlation between detectors may appear due to factors such as common residual thermal fluctuations, electronic chain noise, or cosmic ray showers. To build a model of this correlated component, we compute the crosspower spectra between detectors of difference maps that are free of signal. This procedure should capture all correlations on time scales shorter than half a ring’s observation. These estimates are noisy, so we compute an average amplitude of the correlation for `≥1000. We find that the impact on cosmological parameters of both these effects is negligible, with less than a 0.1σvariation on the mean posterior values. A15, page 20 of 60 Planck collaboration: Planck 2013 results. XV. 0 1000 2000 3000 4000 5000 6000 D`,[µK2] 0 500 1000 1500 2000 2500 ` −200 −100 0 100 200 D`residual, [µK2] Fig. 18. Top: CMB power spectrum estimated using the Plik likelihood, after subtracting the best-fit foreground model. Bottom: residual difference (blue line) between the Plik spectrum and the best-fit ΛCDM model. The solid green line shows the difference between the best-fit models derived from the Plik and CamSpec likelihoods. 0.021 0.022 0.023 Ωbh2 0.112 0.120 0.128 Ωch2 1.040 1.042 100θMC 0.06 0.09 0.12 τ 0.925 0.950 0.975 ns 3.04 3.12 3.20 ln(1010As) 64 68 72 H0 8 12 zre 0.30 0.35 0.40 Ωm 13.65 13.80 13.95 Age 0.60 0.65 0.70 ΩΛ Plik reference Camspec reference Plik mask 3 Fig. 19. Comparison of cosmological parameters estimated from the CamSpec (blue) and Plik (red for mask CL39; purple for mask CL49) likelihoods. All parameters agree to better than 0.2σwhen CL49 is used for Plik, matching the sky area used by CamSpec at 100 GHz. 7.2. Impact of data selection Here we consider three changes: (1) varying the angular range used in the likelihood; (2) varying the Galactic mask; and (3) discarding individual frequency channels. These are expected to result in changes in the parameter distributions due to the fact that we are changing the input data. The corresponding impact on parameter distributions are illustrated in Fig. 24. We first vary the maximum and minimum multipole. Using `max =1008 gives parameter distributions of similar width to those obtained by WMAP. We find that all basic ΛCDM cosmological parameters have converged by `max ≃1500, since no parameters are specifically sensitive to the damping tail. The convergence of the posteriors on foreground parameters is, as expected, slower when increasing `max, as most of them are 15 30 45 ACIB 143 20 40 60 ACIB 217 0.25 0.50 0.75 rCIB 143×217 8 16 24 AtSZ 0.25 0.50 0.75 ξtSZ×CIB 0.25 0.50 0.75 γCIB 100 200 300 APS 100 30 60 90 APS 143 40 80 120 APS 217 0.25 0.50 0.75 rPS 143×217 15 30 AkSZ 0.000036 0.000042 ADust Plik reference Plik+highL Camspec Camspec+highL Fig. 20. Comparison of foreground parameters estimated with the CamSpec and Plik likelihoods. The purple (red) lines show the CamSpec (Plik) distributions using only Planck data, and the green (blue) lines show the CamSpec (Plik) results when additionally including ACT and SPT. dominant at small scales. The Galactic dust normalization decreases with `max due to its correlation with the CIB components. Similarly, changing `min from 100 to 50 has a negligible effect on cosmological parameters, and mostly affects the determination of the Galactic dust amplitude, which decreases for `min as it is better measured on large scales; its correlations with the CIB clustered and Poisson contributions explain the slight variations in the corresponding parameters ACIB 217 ,γCIB, and APS 217. We then investigate the impact of varying the Galactic mask, from the most conservative (CL20) to the least conservative (CL49), for fixed multipole range 100 ≤`≤2508. As expected, the errors decrease as the sky fraction increases. From CL20 to CL39, cosmological and foreground parameters are stable. The foreground parameters change significantly, however, when we use CL49 for all channels, showing that our foreground model is unable to properly fit the data: the clearest sign of this failure is the unphysically low value of the CIB spectral index (γCIB), indicating that our CIB component determination is getting contaminated by a dust-like component with a steeper angular power spectrum than the CIB, but shallower than that of our (single) Galactic dust component. At low Galactic latitudes, the presence of compact Galactic sources leads to a flatter angular power spectrum than that of (high-latitude) diffuse thermal dust, and this likely affects the CIB determination. This justifies the choice made in CamSpec to use a conservative masking strategy (CL31) for the 143 and 217 GHz channels. Next, we remove one frequency channel at a time. Results change by less than 0.5σexcept when removing the 217 GHz channel. Removing the 217 GHz channel retains only 21 crossspectra out of the 78. As a finer test, we consider surveydifference tests, which were performed in the HFI data processing paper (Planck Collaboration VI 2014) at each frequency, for all combinations of the input maps used in the likelihood analysis (two at 100 GHz, five at 143 GHz, and six at 217 GHz), and for two survey differences: Survey 1 – Survey 2 and Survey 1 – Survey 3. It was found there that in addition to the 217-ds1 × 217-ds2 cross spectrum, only two other cross-spectra failed A15, page 21 of 60 A&A 571, A15 (2014) Ω b h 2 Ω c h 2 100θ MC τ n s ln(10 10 A s ) A CIB 143 A CIB 217 r CIB 143×217 A tSZ ξ tSZ×CIB γ CIB A PS 100 A PS 143 r PS 143×217 A PS 217 A Dust A kSZ Ω Λ Age Ω m z re H 0 Ωbh2 Ωch2 100θMC τ ns ln(1010As) ACIB 143 ACIB 217 rCIB 143×217 AtSZ ξtSZ×CIB γCIB APS 100 APS 143 rPS 143×217 APS 217 ADust AkSZ ΩΛ Age Ωm zre H0 -44% 41%-40% 24%-22%14% 44%-76%41%26% 15% 7% 5% 94% 3% -5% 1% 5% -3% -8% -6% -6% 3% 0% -3% -8% -3% 65% -1% -1% 3% -6% -4% -9% 6% 0% -3% 0% 3% -8% -6% -11%53%16%57% -2% 5% 2% 0% -3% 1% 11%-11%13% 0% -1% 4% -4% 2% -1% 4% 2% 60%-46%-37%-11% 0% 3% 0% 5% 5% 8% 8% 24% -31%-50%-2% 20% 0% 4% 1% 6% 14%13%-22%-4% 7% -30%19% 1% 47% 1% 4% 0% 2% 11% 7% 5% 21% -6% 9% 28% 33% 2% 38% 3% 1% 1% 7% 21%12%-39%-65%1% -6% 4% -42%8% 56%21% 0% 2% -1% 0% -3% 0% 17%50%-19% -4% -4% 79%2% -6% 19%-35% -18%0% -6% -13%-34%-18% -18% -18% -8% -11%5% 1% -33%-68%-45% -55%0% 53%-98% 51%24%78%-4% 0% -3% 1% 0% -5% -4% -2% -3% -4% 0% -2% -2% -87%67%-75%-26%-62%-9% 1% 4% -1% 0% 1% 3% 0% 0% 0% -2% 1% 13%-77% -53%98% -51%-24%-78%4% 0% 3% -1% 0% 5% 4% 2% 3% 4% 0% 2% 2% -99%77% 13%-13%9% 99% 18%96% -3% -2% -7% -8% 0% 2% 5% 7% 2% 7% 0% -12%14%-15% -14% 61%-96% 57%25%78%-2% 0% -3% 1% 0% -4% -4% -2% -3% -3% 0% -2% -4% 99%-84%-99% 14% −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 21. Correlation matrix between all the cosmological (top block), foreground (middle block), and derived (bottom block) parameters, estimated using the Plik likelihood. this test, namely 217-1 ×217-ds2 and 217-1 ×217-3. We therefore consider here a data selection where only the (other) “best 217 GHz” spectra are retained in the analysis. Figures 24 and 25 show that this has no discernible influence on the determination of all basic ΛCDM cosmological parameters (but for a slight broadening of the posterior distribution), while the main foreground parameter change is a 1σshift of the point source amplitude at 217 GHz. We also note that since the public release of the data in March 2013, we have found evidence that the “`∼1800 dip” in some 217 GHz detector cross-spectra (see Fig. 17) is stronger in the first six-month survey than in subsequent surveys and that its amplitude may be reduced by additional data flagging targeted at EMI-EMC interference from the 4 K cooler drive and read-out electronics. This dip is therefore likely to be a (small) residual systematic effect in the data, but one, which has little impact on cosmological parameter determination in the standard model. Planck Collaboration XVI (2014) extends the verification to other cosmological models and data selections. 7.3. Testing the extragalactic sources model In this section we describe various tests of the validity of our model for the extragalactic sources. Further tests are also reported in Appendix B of Planck Collaboration XVI (2014). 7.3.1. Poisson power from extragalactic sources Here we check that the Poisson power estimated in the likelihood, which comes from sources below Planck’s detection threshold, is roughly consistent with the level expected given number counts of detected galaxies. Figure 28 shows source A15, page 22 of 60 Planck collaboration: Planck 2013 results. XV. −1 0 1 ∆Ωbh2[σ]−1 0 1 ∆Ωch2[σ]−1 0 1 ∆θ[σ] −0.5 0.0 0.5 ∆τ[σ]−1 0 1 ∆nS[σ]−1.0−0.5 0.0 0.5 1.0 ∆ log 1010AS[σ] −1 0 1 ∆Ωm[σ]−1 0 1 ∆ΩΛ[σ]−1 0 1 ∆H0[σ] −2−1 0 1 2 ∆Age [σ] Fig. 22. Distribution of the difference between cosmological parameter mean posterior values, as estimated by Plik (assuming foreground spectra are known) from the (CMB +fixed foregrounds +noise) FFP6 simulations, and the corresponding posterior parameter mean values from CMB only simulations (based on the same CMB realisations as in the FFP6 simulations), in units of the FFP6 posterior standard deviations. This demonstrates the absence of methodological bias at the level of the intrinsic dispersion between realisations, while removing from the distributions the dominating scatter coming from the purely CMB induced cosmic variance. In these figures, the dominant source of scatter comes from the CMB-foreground chance correlations in the signal dominated part of the spectrum, and from the noise scatter at large multipoles. counts from Planck (Planck Collaboration Int. VII 2013) at 100, 143, and 217 GHz derived from the Planck Early Release Compact Source Catalogue (Planck Collaboration VII 2011; Planck Collaboration XIII 2011). At 143 and 217 GHz we also show the source counts from SPT as reported in Vieira et al. (2010) at 150 and 220 GHz, and from ACT (Marriage et al. 2011) at 150 GHz. The models of de Zotti et al. (2005) and Tucci et al. (2011) are also shown and are discussed in Planck Collaboration Int. VII (2013). Planck Collaboration Int. VII (2013) use spectral information to separate the sources into “synchrotron” and “dusty” sources, and show that the counts at 100–217 GHz are dominated by synchrotron sources at flux densities above ∼400 mJy. Vieira et al. (2010) performed a similar separation. The counts at 150 GHz are dominated by synchrotron sources at flux densities S>10 mJy, but dusty galaxies contribute roughly equally at 220 GHz at flux densities < ∼30 mJy (Vieira et al. 2010;Hall et al. 2010). The ACT counts have not been separated according to spectral type, but should be dominated by radio sources at these flux densities. Figure 28 show models fit to the counts using the function S5/2dN dS=AS 5/2 (S/S0)α+(S/S0)β+B 1−exp S S1!!,(20) 0.0208 0.0216 0.0224 Ωbh2 0.112 0.120 0.128 Ωch2 1.036 1.038 100θMC 0.06 0.09 0.12 τ 0.950 0.975 1.000 ns 3.06 3.12 3.18 ln(1010As) 64 68 72 H0 8 12 zre 0.28 0.32 0.36 Ωm 13.80 13.95 14.10 Age 0.64 0.68 0.72 ΩΛ Input value Standard CMB only Free γCIB Free γDust γCIB = 0.4 Fig. 23. Posterior marginal distributions of cosmological parameters obtained by Plik on a single FFP6 realisation, under different assumptions on the foreground model. Red thick lines correspond to the foreground model parametrisation described in Sect. 3with fixed γCIB. Purple lines (resp. green, grey) show the effect of leaving γCIB free (resp. letting γDust free, or fixing γCIB to 0.4, 2σaway from the peak posterior value). Finally, the blue lines correspond to the analysis made on CMBonly simulations. The input values of the parameters of the simulation are shown as dashed vertical lines. Table 7. Parameters for point source model of Eq. (20), fitting detected source counts shown in Fig. 28. Parameter 100 GHz 143 GHz 217 GHz A. . . . . . . . . . . . 18.24 8.38 8.58 S0. . . . . . . . . . . 1.58 1.65 1.48 α. . . . . . . . . . . . 1.88 1.89 1.90 β. . . . . . . . . . . . 3.35 3.78 4.10 B. . . . . . . . . . . . 14.91 8.73 8.53 S1. . . . . . . . . . . 14.91 5.17 1.78 where A,S0,α,β,B, and S1are free parameters. The best-fit values of these parameters are given in Table 7. Given this model, and given the approximate flux cut applied to the Planck maps, the expected contribution of radio sources to the Planck power spectra, at flux densities smaller than 400, 350, and 225 ±50 mJy at 100, 143, and 217 GHz, are 8.47 ±1, 6.05 ±0.8, and 3.10 ±0.7 Jy2/sr. The contribution of unresolved infrared galaxies to the power spectra is not negligible. They are expected to dominate at 217 GHz, even if they are sub-dominant in the Planck counts. Indeed, faint IR galaxies create a “bump” in the S5/2dN/dSdistribution, below the detection limit of ACT or SPT. This bump is seen at higher frequencies, e.g., with the Herschel SPIRE instrument (see Planck Collaboration Int. VII 2013, for details). This bump of infrared galaxies has not been measured at frequencies of 217 GHz and below. However, measurements with the AzTEC telescope at 1.1mm (270 GHz, Scott et al. 2012) can be used to extrapolate the counts down to 217 GHz. This leads to a predicted peak in the number counts (S5/2dN/dS) around 1.4 mJy at a level of 190 Jy1.5/sr, somewhat higher than the values in Hall et al. (2010). The corresponding contribution of infrared galaxies to the power spectrum is estimated in Planck Collaboration XVIII (2011) to be 16 Jy2/sr at 217 GHz but with significant uncertainty. A15, page 23 of 60 A&A 571, A15 (2014) Plik reference Camspec reference Beams and calib margin. No subpix No correlated noise `max = 1008 `max = 1503 `max = 2013 `min = 50 CL20 mask CL31 mask CL49 mask no 217 GHz, fixed γcib no 143 GHz no 100 GHz best 217 GHz Plik and highL Comp. sep. `max = 2000 LFI 70 GHz 0.020 0.021 0.022 0.023 0.024 0.025 Ωbh2 Plik reference Camspec reference Beams and calib margin. No subpix No correlated noise `max = 1008 `max = 1503 `max = 2013 `min = 50 CL20 mask CL31 mask CL49 mask no 217 GHz, fixed γcib no 143 GHz no 100 GHz best 217 GHz Plik and highL Comp. sep. `max = 2000 LFI 70 GHz 0.10 0.11 0.12 0.13 0.14 Ωch2 1.036 1.038 1.040 1.042 1.044 1.046 1.048 100θMC 0.06 0.08 0.10 0.12 τ 0.90 0.95 1.00 1.05 ns 2.95 3.00 3.05 3.10 3.15 ln(1010As) 60 65 70 75 80 H0 7 8 9 10 11 12 13 14 15 zre 0.20 0.25 0.30 0.35 0.40 0.45 Ωm 13.2 13.4 13.6 13.8 14.0 14.2 Age 0.55 0.60 0.65 0.70 0.75 0.80 ΩΛ Fig. 24. Comparison of the distributions of cosmological parameters obtained with Plik in the reference case (left) with a set of validation test cases. The red line indicates the median and blue the mean, computed from the posterior histograms. The box shows the 68% confidence interval; the outer line the 95% interval. Summing the expected contributions from radio and IR galaxies, we estimate the following values for D3000 for Planck: 200, 75, and 120 µK2at 100, 143, and 217 GHz respectively. These predictions are much less certain at 217 GHz due to the absence of infrared galaxy counts at this frequency. 7.3.2. Clustered power in CIB fluctuations Here we check the consistency of the estimated clustered power in CIB fluctuations. As already noted, from the CamSpec and Plik likelihoods we find only an upper limit on the clustered CIB power at 143 GHz. With CamSpec we detect clustered power at 217 GHz with ACIB 217 =32 ±10 µK2, and Poisson power with APS 217 =92 ±22 µK2. This Poisson power is dominated by the CIB fluctuations. For the Plik likelihood, we have ACIB 217 =49 ±7µK2and APS 217 =58 ±19 µK2. The sum of the CIB power at 217 GHz, and at pivot scale `=3000, is in the range ≈105–125 µK28. 8Note that Planck does not measure directly the CIB power at the pivot scale `=3000, hence these extrapolated values are sensitive to possible shape mismatch of the clustered CIB fluctuation power spectra at lower multipoles. We compare this level to the measurements by the ACT and SPT experiments, which probe higher angular resolution. Fitting a common model to the ACT power spectra from Das et al. (2014), and the SPT spectra from Keisler et al. (2011); Reichardt et al. (2012), the analysis in Dunkley et al. (2013) finds ACIB 219.6=54 ±16 µK2for the CIB clustered component (and APS,CIB 219.6=78 ±12 µK2for the CIB Poisson component) at effective frequency 219.6 GHz. For SPT, the clustered level is ACIB 219.6=59 ±12 µK2(and Poisson APS,CIB 219.6=69 ±10 µK2), also at an effective frequency of 219.6 GHz. This is consistent with the SPT analysis in Reichardt et al. (2012). These are estimated assuming γCIB =0.8 and rCIB 143 ×217 =1, and that the CIB emission can be modelled with frequency as a modified blackbody, following Addison et al. (2012a). The total CIB signal seen by Planck, extrapolated to `= 3000 scales, is therefore consistent with the ACT and SPT observations, but given the limited angular range of Planck, the clustered and Poisson part are degenerate. This motivates us to include the ACT and SPT data in many of our cosmological analyses. When combining Planck, ACT and SPT data together, using the same foreground model (except for Poisson power, which A15, page 24 of 60 Planck collaboration: Planck 2013 results. XV. Plik reference Camspec reference Beams and calib margin. No subpix No correlated noise `max = 1008 `max = 1503 `max = 2013 `min = 50 CL20 mask CL31 mask CL49 mask no 217 GHz, fixed γcib no 143 GHz no 100 GHz best 217 GHz Plik and highL 0 10 20 30 40 ACIB 143 Plik reference Camspec reference Beams and calib margin. No subpix No correlated noise `max = 1008 `max = 1503 `max = 2013 `min = 50 CL20 mask CL31 mask CL49 mask no 217 GHz, fixed γcib no 143 GHz no 100 GHz best 217 GHz Plik and highL 20 40 60 80 100 ACIB 217 0.0 0.2 0.4 0.6 0.8 1.0 rCIB 143×217 0 5 10 15 20 25 AtSZ 0.0 0.2 0.4 0.6 0.8 1.0 ξtSZ×CIB −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 γCIB 0 50 100 150 200 250 300 350 APS 100 0 50 100 150 200 APS 143 0 50 100 150 200 250 300 350 APS 217 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 rPS 143×217 0 10 20 30 40 50 AkSZ 0.00000 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 ADust Fig. 25. Comparison of the distributions of foreground model parameters obtained with Plik, as in Fig. 24. Fig. 26. Estimated calibration and beam eigenmode parameters, compared to the priors, using (left) 100 to 217 GHz data, and (right) a single “FFP6” simulation where there was no calibration or beam errors. The shaded area shows the width of the prior imposed on the first beam eigenvalue. The individual SWB detectors (those denoted in the legend by their number rather than by a detset number) are not single out by this comparison. A15, page 25 of 60 A&A 571, A15 (2014) 0.8 0.9 1.0 Spectrum amplitude (q) −0.1 0.0 0.1 0.2 Spectrum tilt (n) WMAP Planck Fig. 36. Probability distribution for a two-parameter (amplitude and tilt) model derived from the Planck (blue) and WMAP (red) low-`likelihoods, including angular scales between `=2 and 30. while Fig. 36 shows the corresponding constraints on the twoparameter amplitude-tilt model employed in Sect. 8.1.2, including multipoles between `=2 and 30. Neglecting the minor differences in the masks adopted by the two codes, these power spectra and parameter constraints are largely dominated by cosmic variance, and one should therefore expect the two distributions to be almost identical. Instead, from Fig. 36 we see that the WMAP low-`spectrum is 2.5–3% higher than the Planck spectrum. For a detailed discussion of this discrepancy, including a comparison at higher `, see Planck Collaboration XXXI (2014). Here we only note that the effect is robust with respect to foreground removal and power spectrum evaluation algorithms, and also point out that the effect at low multipoles is too large to be explained by uncertainties in the Planck transfer functions (Planck Collaboration II 2014;Planck Collaboration VII 2014) or calibration (Planck Collaboration V 2014;Planck Collaboration VIII 2014). Also note that the amplitude of the low-`spectrum relative to the Planck best-fit model, (q,n)= (1,0), derived including the full multipole range between 2 ≤ `≤2500, is somewhat low in Fig. 36, with a best-fit amplitude of q∼0.9. This observation is discussed and quantified in greater detail in Sect. 9.3. 9. The Planck CMB spectrum and likelihood 9.1. Hybridisation of lowand high-`likelihoods The high-`and low-`likelihoods introduced in Sects. 2and 8 each describe only a part of the full Planck data set. To estimate cosmological parameters from all the angular scales probed by Planck, they must be combined into a single likelihood function that describes all multipoles from `=2 to 2500. In principle, it is desirable to include as many multipoles as possible in the low-`likelihood, since it captures the full nonGaussian structure of the likelihood. The Gaussian approximation for the likelihood using pseudo-spectra also improves at higher multipole due to the increasing number of degrees of freedom (Efstathiou 2004). For Planck we adopt a transition multipole of `trans =50, a compromise between obtaining robust convergence properties for the low-`likelihood, and ensuring that the Gaussian approximation holds for the high-`likelihood (Hamimeche & Lewis 2009). To combine the likelihoods, we must account for the weak correlations between the lowand high-`components. We consider three options: 1. Sharp transition: the low-`likelihood ends at `max =49; the high-`likelihood starts at `min =50; no correlations are accounted for. 2. Gap: the low-`likelihood ends at `max =32; the high-`likelihood starts at `min =50; no correlations are accounted for, but the gap is sufficiently wide that any correlations are negligible. 3. Overlap with correction: the low-`likelihood ends at `max =70; the high-`likelihood starts at `max =50; the double-counting of the overlap region is accounted for by subtracting from the log-likelihood a contribution only including 50 ≤`≤70 as evaluated by the Commander estimator. Under the assumption that no correlations extend from `≤50 to `≥70, this approach is exact). We estimate cosmological parameters using all three methods, and find that the posterior means typically vary by <0.1σ. The largest variation is seen including the running of the spectral index of scalar perturbations, in which the posterior mean changes by 0.2σ. Further, all deviations at the 0.1–0.2σlevel are seen for case 2 above, which excludes data compared to the other two; case 1 and 3 give nearly indistinguishable results. Since case 1 is implementationally simpler, and can be estimated more efficiently (see Sect. 8), we select this method, adopting a sharp transition at `max =50. 9.2. The Planck power spectrum and ΛCDM constraints Using the full Planck likelihood, we now present the final 2013 Planck CMB power spectrum. For this, we fix all nuisance parameters to their maximum-likelihood values. The resulting spectrum is shown in Fig. 37 together with the corresponding best-fit six-parameter ΛCDM model. The agreement between the observations and the model is excellent over most of the multipole range. Only at low `s is it possible to see a systematic offset in the form of a slight power deficit; this is addressed separately in the next section. Table 8provides a summary of the ΛCDM parameters derived using the methodology described in Planck Collaboration XVI (2014) from the Planck likelihood. Here we use the same prior ranges on all parameters as in Planck Collaboration XVI (2014). These are as in Table 5and 6, except for ACIB 143 ,AtSZ, and AkSZ, which are modified to [0,20], [0,10], and [0,10] respectively. Results are given for Planck alone, and in combination with the low-`WMAP polarisation likelihood (Planck+WP). For each case, we report both posterior maximum and mean values. Uncertainties denote 68% confidence limits. A detailed discussion of these results is presented in Planck Collaboration XVI (2014), including an analysis of extended cosmological models, and their compatibility with other astrophysical data sets. The bounds derived from Planck alone are significantly tighter than those from the 9 year WMAP data alone, and comparable or better than those inferred from WMAP combined with SPT and ACT observations. These new constraints provide a precision test of the ΛCDM model. In general, we find good agreement with results derived from other astrophysical data sets, although there are a few exceptions that are in moderate tension with Planck (Planck Collaboration XVI 2014). Considering each of the six ΛCDM parameters in turn, we first note that Planck constrains the physical baryon density to Ωbh2=0.02207 ±0.00033, which is in remarkable agreement A15, page 32 of 60 Planck collaboration: Planck 2013 results. XV. 2 10 50 0 1000 2000 3000 4000 5000 6000 D[µK2] 90◦18◦ 500 1000 1500 2000 2500 Multipole moment,  1◦0.2◦0.1◦0.07 ◦ Angular scale Fig. 37. The 2013 Planck CMB temperature angular power spectrum. The error bars include cosmic variance, whose magnitude is indicated by the green shaded area around the best fit model. The low-`values are plotted at 2, 3, 4, 5, 6, 7, 8, 9.5, 11.5, 13.5, 16, 19, 22.5, 27, 34.5, and 44.5. Table 8. Constraints on the basic six-parameter ΛCDM model using Planck data. Planck Planck+WP Parameter Best fit 68% limits Best fit 68% limits Ωbh2. . . . . . . . . 0.022068 0.02207 ±0.00033 0.022032 0.02205 ±0.00028 Ωch2. . . . . . . . . 0.12029 0.1196 ±0.0031 0.12038 0.1199 ±0.0027 100θMC . . . . . . . 1.04122 1.04132 ±0.00068 1.04119 1.04131 ±0.00063 τ. . . . . . . . . . . . 0.0925 0.097 ±0.038 0.0925 0.089+0.012 −0.014 ns. . . . . . . . . . . 0.9624 0.9616 ±0.0094 0.9619 0.9603 ±0.0073 ln(1010As) . . . . . 3.098 3.103 ±0.072 3.0980 3.089+0.024 −0.027 ΩΛ. . . . . . . . . . 0.6825 0.686 ±0.020 0.6817 0.685+0.018 −0.016 Ωm. . . . . . . . . . 0.3175 0.314 ±0.020 0.3183 0.315+0.016 −0.018 σ8. . . . . . . . . . . 0.8344 0.834 ±0.027 0.8347 0.829 ±0.012 zre . . . . . . . . . . . 11.35 11.4+4.0 −2.811.37 11.1±1.1 H0. . . . . . . . . . 67.11 67.4±1.4 67.04 67.3±1.2 109As. . . . . . . . 2.215 2.23 ±0.16 2.215 2.196+0.051 −0.060 Ωmh2. . . . . . . . . 0.14300 0.1423 ±0.0029 0.14305 0.1426 ±0.0025 Age/Gyr . . . . . . 13.819 13.813 ±0.058 13.8242 13.817 ±0.048 z∗. . . . . . . . . . . 1090.43 1090.37 ±0.65 1090.48 1090.43 ±0.54 100θ∗. . . . . . . . 1.04139 1.04148 ±0.00066 1.04136 1.04147 ±0.00062 zeq . . . . . . . . . . . 3402 3386 ±69 3403 3391 ±60 Notes. The top section contains constraints on the six primary parameters included directly in the estimation process, and the bottom section contains constraints on derived parameters. with standard BBN predictions based on a determination of the primordial abundance of deuterium, Ωbh2=0.021±0.001 (Iocco et al. 2009), but with a fractional uncertainty of 1.5 %, three times smaller than the BBN uncertainty. The physical density of dark matter is measured with a fractional uncertainty of 2.6%, providing new constraints on specific dark matter production scenarios. The single most precise parameter, however, is the angular size of the sound horizon at the last-scattering surface, θMC, A15, page 33 of 60 A&A 571, A15 (2014) which is measured with a fractional uncertainty of 0.065 % by Planck, improving on the combined WMAP, ACT, SPT, and SNLS constraint by a factor of two. Next, given that no polarisation data are included in the current data release, it is remarkable that Planck alone constrains the optical depth to reionisation, τ, with a fractional error of 40%. This is made possible by Planck’s high angular resolution and sensitivity, which allows a high signal-to-noise measurement of lensing in the small-scale CMB power spectrum. This in turn breaks the well-known e−2τAsdegeneracy between τ and the amplitude of scalar perturbations, As. The fractional uncertainty on Asfrom Planck alone is 7%. Having sufficient power to measure τfrom small angular scale temperature data, Planck naturally also provides very strong constraints on the spectral index of scalar perturbations, ns, leading to a fractional uncertainty of 0.97 %. The scaleinvariant Harrison-Zeldovich spectrum, ns=1, is ruled out at a significance of 4.1σfrom the Planck temperature spectrum alone. The significance of this exclusion rises to 5.4σ when the low-`WMAP polarisation is included. The analyses presented in Planck Collaboration XVI (2014) and Planck Collaboration XXII (2014) show that the preference for a (red) tilted primordial spectrum remains very strong also within most extensions beyond the minimal ΛCDM model. The implications of this results for inflationary models are discussed in Planck Collaboration XXII (2014). With our choice of cosmological parameters, the Hubble parameter, H0, and the fractional density of the cosmological constant, ΩΛ=1−Ωm, are derived parameters. They are probed by CMB observations mainly through their impact on θMC, and, to lesser extent, by the impact of ΩΛon the late-time integrated Sachs-Wolfe effect. Since θMC is accurately measured, a particular combination of H0and ΩΛis very well constrained by Planck, although in a model-dependent way; θMC depend on other cosmological parameters, such as the spatial curvature radius, neutrino masses, the number of relativistic degrees of freedom, or a possible dark energy equation of state parameter. The results reported in Table 8rely on the assumption of a flat ΛCDM cosmology with three neutrino species, two of which are assumed to massless and one featuring a small mass mν= 0.06 eV, reflecting the lower bound on neutrino masses imposed by neutrino oscillation experiments. Under these assumptions, Planck finds preferred ranges for H0and ΩΛthat are lower than previous CMB experiments. For instance, Planck+WP gives H0=67.3±1.2 km s−1Mpc−1, to be compared with 70.5± 1.6 km s−1Mpc−1for the combined WMAP9+eCMB data set presented by Hinshaw et al. (2013). The underlying cosmology in the two analyses is the same, excepted for the small neutrino mass introduced in our default ΛCDM model. However, if we assume all three neutrino species to be massless, our best-fit and mean values for H0increase only by 0.6 km s−1Mpc−1. Thus, the tension is clearly driven by the data rather than by theoretical assumptions. Planck Collaboration XVI (2014) shows that our results for H0and ΩΛare in very good agreement with Baryon Acoustic Oscillation data, but in moderate tension with other cosmological probes. For instance, our Planck+WP bounds on H0disagree at the 2.5σlevel with direct determinations of the Hubble parameter using Cepheid and supernovae (Riess et al. 1998) or quasar time delays (Suyu et al. 2013), as well as with the results of the Carnegie Hubble Program (Freedman et al. 2012). Our bounds on ΩΛare in a slight 2σtension with the results of the SNLS supernovae collaboration (Conley et al. 2011;Sullivan et al. 2011), although in better agreement with the Union2.1 compilation (Suzuki et al. 2012). Our combined Table 9. Results of the Hausman test applied to the temperature power spectrum for 2 ≤`≤32. Data set sobs 1P(s1<sobs 1) [%] Commander . . . . . -0.647 0.73 NILC . . . . . . . . . –0.649 0.73 SEVEM . . . . . . . . –0.804 0.50 SMICA . . . . . . . . –0.589 1.33 WMAP9 ILC . . . . –0.234 7.18 determination of σ8and Ωmshows larger tension with recent data based on cosmic shear or cluster count techniques. On the other hand the Planck best-fit ΛCDM model is in good agreement with the halo power spectrum derived from the luminous red galaxy catalogue of the Sloan Digital Sky Survey (Reid et al. 2010), especially when the analysis is restricted to linear scales. 9.3. Significance of the low-`tension with ΛCDM models From the above discussion, it is clear that the ΛCDM framework provides an excellent model for most of the Planck data. However, as noted in Sect. 8.3 and seen in Fig. 37, the low-` Planck temperature power spectrum appears to be in some tension with the best-fit Planck ΛCDM model, which for Planck is almost exclusively determined by the small-scale spectrum. In this section we assess the significance and impact of this tension between low and high `s using three different statistical tests. We start by applying a modified Hausman test (Polenta et al. 2005;Planck Collaboration II 2014) to the low-`spectra derived from the four foreground-cleaned Planck maps (Planck Collaboration XII 2014) and the 9-year WMAP ILC map, using multipoles up to `max =32. This test uses the statistic s1=suprB(`max,r), where B(`max,r)=1 √`max int(`maxr) X `=2 H`,r∈[0,1](29) H`=ˆ C`−C` pVar ˆ C` ,(30) and ˆ C`and C`denote the observed and model power spectra, respectively. Intuitively, this statistic measures the relative bias between the observed spectrum and model, measured in units of standard deviations, while taking the so-called “look-elsewhere effect” by maximizing s1over multipole ranges into account. We use realistic Planck “FFP6” simulations (Planck Collaboration I 2014) to derive the empirical distribution of s1under the null hypothesis. Figure 38 compares the results obtained from the data with the simulation distribution, and Table 9lists significances. As measured by this statistic, we see that a negative bias is found in the low-`Planck power spectrum relative to the ΛCDM model at the 99% confidence level. For the WMAP ILC map the significance of the negative bias nominally decreases to 93%. This is consistent with the findings in Sect. 8.3, where it was shown that the WMAP temperature power spectrum is 2.5–3 % higher than the Planck spectrum at low `’s. However, as discussed in Planck Collaboration XXXI (2014), a similar amplitude difference between the two experiments is also seen at smaller scales. Since the current test compares the observed WMAP data with the best-fit Planck ΛCDM model, the present test is not optimal for assessing internal consistency between low and high `s within the WMAP data. A15, page 34 of 60 Planck collaboration: Planck 2013 results. XV. 0.00 0.02 0.04 0.06 0.08 -1 0 1 2 3 Commander Sevem Smica Nilc WMAP9 s1 P(s1) Fig. 38. Results of the Hausman test applied to the temperature power spectrum for 2 ≤`≤32. The black histogram shows the expected distribution, estimated with simulations, of the s1test statistic. The vertical bars represent Planck CMB maps and the 9 year WMAP ILC map. Note that the statistic is indistinguishable for the NILC and Commander maps. Next, to obtain a quantitative measure of the relative power discrepancy between low and high `s, we fit the two-parameter amplitude–tilt power spectrum model (see Sect. 8.1.2) to the Planck data using the low-`likelihood restricted to various multipole ranges defined by 2 ≤`≤`max, where `max is allowed to vary. Thus, this measures the amplitude of the low-`spectrum relative to the best-fit Planck ΛCDM spectrum, which is driven by the smaller angular scales. Figure 39 shows the resulting constraints on the power spectrum amplitude, q, as a function of `max, after marginalising over the tilt, n. For comparison, we also show similar constraints derived using the low-` WMAP temperature likelihood up to `=30. The best-fit amplitude is q∼0.9 for `max =20–35, different from unity at a statistical significance of 2–2.5σby this measure. The WMAP spectrum shows a consistent behaviour, up to the same overall scaling factor of 2.5–3% between Planck and WMAP discussed above. We have verified that these results are insensitive to the (well-known) low quadrupole moment by excluding C2from the analysis; the large cosmic variance of this particular mode results in a low overall statistical weight in the fit. Finally, we assess the impact of the low-`power deficit on the ΛCDM model estimated using the Planck likelihood10 (augmented with the WMAP polarisation likelihood). We fit a low-` rescaling amplitude, Alow for ` < `low jointly with the ΛCDM parameters, i.e., C`=AlowCΛCDM `for ` < `low and C`=CΛCDM ` for `≥`low. Figure 40 shows the resulting posterior distributions for Alow for `low =32 (green) and `low =49 (blue). The purple line shows the same when replacing the Planck low-` likelihood with the WMAP low-`likelihood (`low =32). The corresponding best-fit values are Alow =0.899 ±0.046 (Planck; `low =32), Alow =0.953 ±0.033 (Planck;`low =49) and Alow =0.953 ±0.048 (WMAP; `low =32), respectively. As already noted in Sect. 8.3, the observed deviations from unity are too large to be explained by the <1% uncertainties in the Planck transfer functions (Planck Collaboration II 2014;Planck Collaboration VI 2014). In Fig. 41 we show the posterior distributions for Ωch2,ns and H0after marginalising over Alow for `low =49. (Adopting `low =32 results in negligible differences for all parameters 10 We have verified that the following results are insensitive to whether Plik or CamSpec are used for the high-`likelihood. 20 30 40 50 Maximum multipole moment, max 0.8 0.9 1.0 1.1 Best-fit amplitude, A Planck WMAP Fig. 39. Power spectrum amplitude, q, relative to the best-fit Planck model as a function of `max, as measured by the low-`Planck and WMAP temperature likelihoods, respectively. Error bars indicate 68 and 95% confidence regions. Fig. 40. Posterior distributions for the low-`spectrum amplitude, Alow, estimated using the Planck likelihood, with `low =32 (green) and `low =49 (blue). The purple line show the distribution derived using the WMAP temperature likelihood with `low =32. except Alow). Shifts of 0.6–1σare observed compared to the reference model, Alow =1. We note that H0, which already has a “low” value (for a detailed discussion, see Planck Collaboration XVI 2014), prefers an even lower value when allowing a rescaling of the low-`spectrum. As a final test, we replace the entire low-`likelihood, both temperature and polarisation, with a Gaussian prior on the optical depth of reionisation, τ=0.089±0.014, matching the WMAP measurement (Hinshaw et al. 2013). The resulting posteriors are shown as purple lines in Fig. 41, and agree well with the case including a low-`scaling factor, but are, in fact, slightly further away from the reference model. Although not very significant in an absolute sense, these results do indicate that the high-`likelihood is challenged in finding models that also fit the low-`power spectrum. A15, page 35 of 60 A&A 571, A15 (2014) 0.113 0.122 0.130 Ωc h 2 0.930 0.955 0.981 ns 62.9 66.8 70.7 H 0 0.8 0.9 1.0 1.1 Alow Fig. 41. Comparison of the posterior distributions for Ωch2,ns, and H0for the default six-parameter ΛCDM model constrained by Planck (red); compared to the case when we allow a variable low-`power spectrum amplitude at `≤49 (blue); and when replacing the low-`temperature likelihood with a Gaussian prior on τ, the optical depth of reionisation (purple). The lower right panel shows the posterior distribution for the low-`amplitude, Alow. To summarize, we have phenomenologically quantified a tension between the low-`CMB power spectrum at `< ∼40 and the Planck best-fit ΛCDM model. Its significance varies between 2.5 and 3σdepending on the estimator used. The effect is seen in all four Planck foreground-cleaned CMB maps with little variation. It is also present in the 9-year WMAP data, although an overall amplitude difference of 2.5–3% between the data sets complicate a direct comparison. To make further progress, one would seek to establish a physical model that predicts a low-`power deficit compared to high `’s, and that may also predict other observable effects, which may be tested with cosmological data. Such a model may be related to the tentative detections of violations of statistical isotropy discussed in Planck Collaboration XXIII (2014), e.g., the low CMB temperature variance, the hemispherical power asymmetry, or the alignment between the quadrupole and octopole moments. 10. Discussion and conclusions We have presented the Planck likelihood, which provides a detailed and accurate characterisation of the two-point statistics of the CMB temperature field, accounting for all significant sources of uncertainty; statistical, instrumental, and astrophysical. This likelihood function allows us to present an estimate of the CMB temperature power spectrum that spans more than three decades in `with unprecedented precision; a spectrum that saturates the cosmic variance limit at all scales > ∼0.1◦, nearly exhausting the information content of the temperature anisotropies, and, in fact, becoming limited by uncertainties due to astrophysical foreground modelling. This is precisely what was originally promised at the time when Planck was selected by ESA in March 1996. On large angular scales, ` < 50, the Planck likelihood is based on a Gibbs sampling approach that allows joint CMB power spectrum and component separation analysis, while accurately marginalising over a physically motivated foreground model constrained by the 30–353 GHz Planck frequencies. On intermediate and small scales, the Planck likelihood employs a fine-grained set of cross-spectrum combinations among the 100, 143, and 217 GHz detector maps to constrain the high-` CMB power spectrum, ensuring that no noise bias can compromise the results, while at the same time allowing for physical foreground modelling in terms of power spectrum templates. This emphasis on physical foreground modelling has made it possible to combine the full power of the Planck data with observations from higher-`CMB experiments. We have validated our results through an extensive suite of consistency and robustness analyses, propagating both instrumental and astrophysical uncertainties to final parameter estimates. Further, we have studied in detail the well-known degeneracies that exist between the foreground and cosmological parameters at high `s when only including Planck observations, and shown that they have only a weak impact on cosmological conclusions. On a more detailed level, we draw the following conclusions: –The consistency between power spectra measured within the different frequency channels is remarkable. In the signaldominated regime for single detectors, at `< ∼1000, the cross-spectra show an RMS dispersion of a few µK2in multipole bands of δ` =31. This confirms the relative calibration of the 100, 143, and 217 GHz detectors to ∼0.2%. –The differences, ∆D`, between the 143 ×143, 143 ×217, and 217 ×217 cross-spectra averaged over multipole bands of δ` ≈31 have a dispersion over 800 ≤`≤1500 of 9, 5, and 5 µK2, respectively, after subtracting the best-fit foreground model. This dispersion is not primarily of instrumental origin, but can be predicted from a model of the chance correlations between foregrounds and CMB fluctuations. –At high `s, the power spectrum of the four foregroundcleaned CMB maps derived through component separation are consistent within their uncertainties. The cosmological parameters derived from these maps are consistent with those estimated by the Planck likelihood for `< ∼2000, despite very different foreground models. –At low `s, the power spectrum differences among the four foreground-cleaned CMB maps are below 50 µK2for nearly every single multipole. Residuals with respect to the 9 year WMAP ILC map are slightly larger, typically 100 µK2or more. A detailed comparison between Planck and WMAP reveals a systematic power spectrum amplitude difference at the 2–3% level that cannot be accounted for within the Planck instrumental error budget. This is consistent with the findings presented in Planck Collaboration XXXI (2014). –Parameters derived from the 70 GHz Planck frequency map are in excellent agreement with the reference results derived using the Planck likelihood; when the latter is limited to `≤1000, the agreement is even more striking. This confirms the strong internal consistency between the LFI and HFI instruments. –The best-fit ΛCDM model derived from the Planck likelihood predicts T E and EE spectra in good agreement with the measured polarisation signature over a broad range of frequencies (70 to 217 GHz) and multipoles (`< ∼1000). At 100, 143, and 217 GHz, the instrumental noise in the EE spectrum is at the µK2level for `< ∼1000, and the visible (small) differences between the spectra suggests different levels of foreground contribution. –We report a tension between the Planck best-fit ΛCDM model and the low-`spectrum in the form of a power deficit A15, page 36 of 60 Planck collaboration: Planck 2013 results. XV. of 5–10% at `< ∼40, with a statistical significance of 2.5–3σ. Thus, while the minimal ΛCDM model provides an outstanding fit for intermediate and small angular scales, this tension may suggest that the model is incomplete. In this respect, it is worth noting that other, but possibly related, anomalies have been reported in a companion paper studying statistical isotropy in the Planck sky maps at statistically significant levels. In summary, we find that the majority of the Planck data can be described by a minimal six-parameter ΛCDM model with a very high degree of accuracy. Within this model the statistical uncertainties are dominated by astrophysical foreground modelling by scales of `≃1500. At lower `s, the unprecedented quality of the Planck data is such that the only fundamental limit is that we can only observe one CMB sky. In other words, Planck is cosmic variance dominated at `< ∼1500, extragalactic foreground dominated at `> ∼1500, and dominated nowhere by instrumental noise or systematic errors. Using only Planck temperature data, we report a detection of ns<1 at more than 4σconfidence, which by itself is already a bit stronger than the limit derived from WMAP, SPT, ACT, and SNLS combined. Complementing the Planck observations with the 9 year WMAP polarisation data increases the significance further to 5.4σ. The multipole range above ` > 1500 is crucial for constraining possible extensions to the minimal ΛCDM model; for a detailed exploration of a wide range of such models, see Planck Collaboration XVI (2014) and Planck Collaboration XXII (2014). There we report some tensions among the CMB damping tail parameters, including ΩK, nrun, and YP. However, none of these indicate significant departures from the ΛCDM framework. In the near future, we will extend our analysis to produce a cosmic variance limited likelihood and power spectrum reaching to higher multipoles. To some extent, this will be achieved through more sophisticated astrophysical foreground modelling, and by exploiting additional frequency information. However, the two major steps forward will be, first, to include the Planck polarisation observations in the likelihood analysis, and, second, to exploit the full data set generated by the two Planck instruments. The amount of HFI data available for analysis is nearly double that, which is presented here, and the LFI instrument is still observing at the time of writing. Acknowledgements. The development of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members with the technical or scientific activities they have been involved into, can be found at http://www.sciops.esa.int/index.php?project= planck&page=Planck_Collaboration. We acknowledge the use of the CLASS Boltzmann code (Lesgourgues 2011) and the Monte Python package (Audren et al. 2013) in earlier stages of this work. 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D, 70, 083511 Xia, J.-Q., Negrello, M., Lapi, A., et al. 2012, MNRAS, 422, 1324 Zahn, O., Zaldarriaga, M., Hernquist, L., & McQuinn, M. 2005, ApJ, 630, 657 Appendix A: High-`likelihood details A.1. Power spectra and the coupling matrix We denote the pixel weight function for temperature by wT i. The pseudo-spectra of Eq. (1) are constructed using the following: ˜ T`m=X s ∆TswT sΩsY∗ `m(θs),(A.1) where the sum is over the pixels in the map. For all of the methods presented in the paper the weighting per pixel is uniform, except for the apodisation of the different Galactic and point source masks referred to in the main text. The coupling matrix appearing in Eq. (2) is given by Hivon et al. (2002): MTT `1`2=(2`2+1) 4πX `3 (2`3+1) ˜ W`3 `1`2`3 000!2 , ≡(2`2+1)ΞTT (`1, `2,˜ W) (A.2) where for the cross spectrum (i,j), ˜ W`is the power spectrum of the window function ˜ Wi j `=1 (2`+1) X m ˜wi `m˜wj∗ `m.(A.3) A.2. Pseudo-C`covariance matrices For the case of narrow window functions and uncorrelated pixel noise (σT i)2, the covariance matrices can be approximated as h∆˜ CTi j `∆˜ CTpq `0i ≈ CT `CT `0hΞTT (`, `0,˜ W(ip)( jq))+ΞTT (`, `0,˜ W(iq)( jp))i +(CT `CT `0)1/2 ×hΞTT (`, `0,˜ W2T(ip)( jq)) + ΞTT (`, `0,˜ W2T(iq)( jp)) (A.4) + ΞTT (`, `0,˜ W2T(jp)(iq)) + ΞTT (`, `0,˜ W2T(jq)(ip))i + ΞTT (`, `0,˜ WTT(ip)( jq)+ ΞTT (`, `0,˜ WTT(iq)( jp)), where Ξis the matrix defined in Eq. (A.2). The window functions are given by: ˜ W(i j)(pq) `=1 (2`+1) X m ˜w(i j) `m˜w(pq)∗ `m,(A.5) ˜ WTT(i j)(pq) `=1 (2`+1) X m ˜wT(i j) `m˜wT(pq)∗ `m,(A.6) ˜ W2T(i j)(pq) `=1 (2`+1) X m ˜w(i j) `m˜wT(pq)∗ `m,(A.7) where ˜w(i j) `m=X s wi swj sΩsY∗ `m(θs),(A.8) ˜wT(i j) `m=X s (σT s)2wi swj sΩ2 sY∗ `m(θs).(A.9) To avoid cumbersome notation, we have omitted indices from the theoretical spectra appearing in Eq. (A.4). In practice, these spectra include unresolved foreground contributions and are smoothed by the appropriate “beam” transfer functions bi j, which actually describe the overall transfer function from the sky to the maps11. In addition, these covariance matrices are cor11 These effective beam transfer function represents the combined effect of the instrument and the processing chain. They are derived from the scanning beam, derived from the data obtained while scanning planets, which therefore describe the combined effect of the instrument and of the temporal flow processing (sometimes referred to the F(`) term as in the original “master” paper by Hivon et al. 2002). We then derive the effective beam, which further accounts for the effect of map-making, which combines samples with scanning beams of several orientation in each pixel, according to the scanning strategy. This is described in detail in the beam and processing papers (Planck Collaboration IV 2014; Planck Collaboration VI 2014;Planck Collaboration VII 2014). A15, page 38 of 60 Planck collaboration: Planck 2013 results. XV. rected for the pixel window functions p`(i.e., covariance matrices such as h∆˜ CTi j `∆˜ CTpq `0iare divided by p2 `p2 `0). Finally, the Planck maps have correlated pixel noise. We adopt an approximate, heuristic modification of the formulae above to include this effect in the covariance matrices, based on the power spectra of half-ring difference maps. This procedure is detailed in Appendix A.8. A.3. Combining intra-frequency cross-spectra For Planck, the vector containing the power spectra, and its associated covariance matrix, are both large and so require substantial compression to make the computation of a high-`likelihood fast enough for parameter estimation. As described in Appendix A.7, after we correct for the “effective” calibration factors for each individual detector set, the power spectra at each frequency are consistent to extremely high accuracy. Any remaining residuals have a negligible impact on the cosmological analysis. Thus, we combine the cross-spectra from different detectors within a given frequency combination into a single power spectrum. We do not average across frequency combinations since the unresolved foregrounds depend on frequency. Further compression can be accomplished, if desired, only after unresolved foreground parameters have been determined. We form the linear combination of individual cross-spectra, for each multipole, in the following way: ˆ CTk `=X i j⊂k,i,j αTTi j `yiyjˆ CTi j ` (bTi j `)2(pT `)2·(A.10) Here the index kdenotes the particular frequency cross-spectrum combination (e.g., 100 ×100, 143 ×217), the coefficients yi denote the multiplicative factors for each map, bTi j `is the (isotropised) beam transfer function for the map combination i j, and p`is the isotropised pixel window function12. The coefficients αi j are normalized so that X i j⊂k,i,j αTTi j `=1, αTTii `=0.(A.11) How can we determine the coefficients αi j? A near optimal combination, ˆ Xk `, is given by solving X pq ˆ M−1 pq ˆ Xk `=X pq ˆ M−1 pq ˆ Xpq `,(A.12) where ˆ M−1 pq is the block of the inverse covariance matrix appropriate to the spectrum combination k. If the covariance matrix ˆ M accurately describes the data, the solution of Eq. (A.12) properly accounts for the correlations between the cross-spectra. Solving Eq. (A.12) requires the inversion of a large matrix, so we adopt a simpler solution by weighting each estimate by the diagonal component of the relevant covariance matrix, e.g., αTTi j `∝1/Cov( ˆ CTi j `ˆ CTi j `).(A.13) This has the effect of assigning each cross-spectrum equal weight in the signal dominated regime and an inverse variance weighting in the noise dominated regime. This is the correct 12 Note that for the masks used here, the isotropised pixel window function provided by HEALPIX is sufficiently accurate. solution in the noise dominated regime. The analysis of intrafrequency residuals presented in Appendix A.7 shows that in the signal dominated regime we see excess variance (with no obvious dependence on the detector/detector set combination) compared to what we expect from instrument noise alone. This excess variance is small compared to the signal and is caused by residual beam errors, consistent with the beam eigenmode amplitudes discussed in Appendix A.7, that are not included in the covariance matrices. This is our justification for assigning roughly equal weight to the spectra in the signal dominated regime. When we construct a likelihood from the combined estimates we construct the full covariance matrix including crosscorrelations between the various spectra. In this matrix, the cross-correlations in the signal dominated regime are dominated by cosmic variance if different masks are used for different frequencies. If identical masks are used for all frequencies, the cross-correlations in the signal-dominated regime are dominated by the cross correlations between the CMB and unresolved foregrounds, which are included in the analytic covariance matrices and act as a regularizing contribution (see Appendix C). A.4. Covariance matrix of combined spectra The estimates of the Planck cross-spectra are linear combinations of the pseudo-C`estimates, so their covariance matrices are given by e.g., Cov( ˆ CTk `ˆ CTk0 `0)=XαTTi j `αTTpq `0Cov( ˆ CTi j `ˆ CTpq `0).(A.14) Analytic expressions for these covariance matrices have been given in Efstathiou (2004,2006); Hamimeche & Lewis (2008), and are described in Appendix A.2. The covariance matrices are computed assuming a fixed fiducial theoretical model including an unresolved foreground model for each frequency combination. Typically, the unresolved foregrounds introduce corrections to the covariance matrices of a few percent in the transition region between signal and noise domination. In addition, we compute the fiducial model by applying appropriate beam functions bTi j for each detector combination. As discussed above, the number of coupling matrices required to compute expressions such as Eq. (A.14) scales as N4 map and so becomes prohibitively expensive as the number of crossspectra becomes large. However, most of these coupling matrices are similar, differing primarily in the amplitude of the noise levels and in minor respects such as a small number of missing pixels. We can therefore adopt the same masks and weightings for groups of cross-spectra and compute coupling matrices only for distinct combinations. This dramatically reduces the computational burden. A similar approach was adopted by Lewis (2008) to analyse the WMAP 5-year temperature maps. It is also straightforward to calculate covariance matrices for differences between different averages. If we form two spectra averaging over different detector combinations ˆ CTa `=X i,j,i,j αi j ˆ CTi j `,(A.15) ˆ CTb `=X i,j,i,j βi j ˆ CTi j `,(A.16) A15, page 39 of 60 A&A 571, A15 (2014) Fig. A.1. Toy power spectra drawn from a Gaussian distribution with mean given by a ΛCDM power spectrum and a covariance given by the bandaveraged C`covariance matrix, (top), and the corresponding differences with respect to the input ΛCDM model. Note the apparent presence of “coherent oscillatory features” in the difference spectra. These are fully described by the power spectrum covariance matrix. To assess the statistical significance of apparently “unexpected features” in the power spectrum, it is critical to include all sources of correlations among different C`’s. then the covariance matrix of the difference ˆ CTa `−ˆ CTb `is simply Cov( ˆ CTa `−ˆ CTb `)=X i,j,p,q (αi jαpq +βi jβpq −αi jβpq −βi jαpq) ×h∆ˆ CTi j `∆ˆ CTpq `i.(A.17) As a pedagogical illustration of the importance of these correlations, we show in Fig. A.1 two toy power spectra drawn from a Gaussian distribution with mean given by a ΛCDM spectrum and covariance given by Cov( ˆ CTa `−ˆ CTb `). That is, these spectra are not computed from a real sky map, but simply drawn directly from the C`error model, and therefore exclusively contain correlations modelled by the covariance matrix. The apparent “coherent oscillatory features” seen in the difference spectra (bottom panels) are therefore fully described by the CamSpec covariance matrix, accounting for correlated beam and foreground uncertainties, mask-induced coupling etc. The left panel shows a typical realisation, while the case in the right panel is selected as, visually speaking, one of the most “peculiar” within a relatively small set of simulatoins. When assessing the statistical significance of “unexpected features” in the real CMB spectrum, e.g., similar to those seen in Figs. 15 and 16, it is critical to account for these correlations. A.5. The “fiducial Gaussian” approximation We use a likelihood based on the so-called “fiducial Gaussian” approximation. Here we present our justification for this choice, based on an “expansion in covariance” of the exact likelihood in the exact full-sky, isotropic noise case. Assuming the CMB, noise, and foregrounds are Gaussian, then the probability distribution for the alm coefficients of a collection of maps, given a model, is p=e−S, where S=X ` (`+1/2) tr hC`−1ˆ C`i+ln |C`|,(A.18) up to a model-independent normalization.Here ˆ C`is the matrix of empirical spectra at a given multipole, and C`are their expectation values for the model in question. Now, a key point to note is that theoretical power spectra typically differ from each other at each `by less than they differ from the observed ˆ C`, because of cosmic variance and noise in the latter. So we are justified in expanding Eq. (A.18) about a reasonable fiducial model. Considering a single value of `for simplicity, writing C=Cf+∆,(A.19) we obtain S=Sf+(l+1/2)tr −Cf−1∆Cf−1ˆ ClCf−1∆ +Cf−1∆Cf−1∆Cf−1ˆ Cl −1 2Cf−1∆Cf−1∆ + . . .!(A.20) to second order in ∆. We may now complete the square in ∆after extracting a term that is small if the fiducial model is accurate. Up to terms independent of ∆, we have, to second order in ∆: S2=(`+1/2)tr C−1 f∆Cf−1∆(C−1 fˆ C−1) +1 2(`+1/2)tr (C−ˆ C)C−1 f(C−ˆ C)C−1 f.(A.21) Here we have recombined the perturbation and the fiducial model back together, using Eq. (A.19), in the second term to obtain exactly the “fiducial Gaussian” likelihood. The first term is a correction to the fiducial Gaussian likelihood that is typically small if the fiducial model is accurate. One can motivate neglecting this term by noticing that in its absence the approximate likelihood is unbiased (as the exact one is). One trades getting second derivatives exactly right in the vicinity of the fiducial model with getting the position, though not the depth, of the minimum right. Vectorizing the distinct elements of C−ˆ C(following Appendix A of Hamimeche & Lewis 2009), and recognizing A15, page 40 of 60 Planck collaboration: Planck 2013 results. XV. the coefficients as the inverse covariance matrix elements of the spectra under the fiducial model, we obtain Sfid =1 2(ˆ C1T−C1T,ˆ C2T−C2T, . . .) ׈ M−1(ˆ C1T−C1T,ˆ C2T−C2T, . . .)T,(A.22) where ˆ Mis the fiducial covariance matrix of the spectra, and the upper indices run on the different pairs of frequencies. This suggests an easy generalisation to the coupled cut sky pseudo-spectra, given our calculation of their covariances in Appendix A.2. We now replace the power spectra above with corresponding appropriate averages of detector cross-spectra. With bold face now denoting spectra laid out as vectors, and ˆ M the grand fiducial covariance, our final action is: Sfid =1 2(ˆ C1T−C1T,ˆ C2T−C2T, . . .) ׈ M−1(ˆ C1T−C1T,ˆ C2T−C2T, . . .)T.(A.23) Another advantage of the “fiducial Gaussian” approximation is that instrumental uncertainties (calibration errors, beam errors, etc.) do not appear in the inverse covariance, but only in the expression of the theoretical spectra CTin Eq. (A.23) above. Note that if we fix the foreground model CFk for each spectrum k, together with the calibration coefficients and beam parameters, we can minimise the likelihood (Eq. (A.23)) with respect to a “best-fit” primary CMB spectrum. This “best-fit” spectrum is given by the solution of X kk0`0 (ˆ M−1 ``0)kk0ˆ CCMB `0=X kk0`0 (ˆ M−1 ``0)kk0(ck0ˆ Ck0 `0−ˆ CFk0 `0),(A.24) where the ckare spectrum effective calibration factors (see Appendix A.7). The covariance matrix of the estimates ˆ CCMB ` is given by the inverse of the Fisher matrix: h∆ˆ CCMB `∆ˆ CCMB `0i=      X kk0 (ˆ M−1 ``0)kk0       −1 .(A.25) A.6. Uncertainties on individual detector sets beams and calibrations Let us consider two detectors (or detector sets) Xand Y. Neglecting instrumental noise, the cross-spectrum CXY,obs `is related to the true one, CXY,sky `, through CXY,obs `=CXY,sky `WXY,eff,true `,(A.26) where WXY,eff,true `is the effective beam window function. Note that because of the optical beam non-circularity and the Planck scanning strategy, WXY `,WXX `WYY `1/2when X,Y, while WXY =WYX for any Xand Y. In the `range of interest, WXY `≥0, so we denote WXY =BXY 2, following the usual prescription for simple (circular) beam models. In what follows, we drop the XY pair superscript except when they are required for clarity. Our analyses use the best estimated Cest `of the sky power spectrum, where the measured Cobs `is corrected by a nominal effective window Weff,nom `: Cest `=Cobs `/Weff,nom `, =Csky `Weff,true `/Weff,nom `, =Csky `Beff,true `/Beff,nom `2.(A.27) The ratio Beff,true `/Beff,nom `, which determines the uncertainty on the angular power spectrum due to the beam, is estimated using Monte-Carlo simulations of planet transits. We estimate Bmean `and Wmean `from the Monte Carlo simulations as Bmean `= nMC X i=1 (Wi `)1/2/nMC,(A.28) Wmean `= nMC X i=1 Wi `/nMC,(A.29) and compute the deviations around the mean ∆i `=ln Bi `/Bmean `.(A.30) Since the relative dispersion of the simulated Wi `is small (less than 1%), the deviations are well approximated by ∆i `≃1 2ln Wi `/Wmean `.(A.31) The matrix ∆then has nMC rows and `max +1 columns. Its Singular Value Decomposition (SVD) is given by ∆=MDVT(A.32) where Mis an orthogonal nMC ×nMC matrix (i.e., MTM= MMT=InMC ), Dis a diagonal matrix with nMC non-negative eigenvalues, and Vis a matrix with `max +1 rows whose nMC columns are orthonormal vectors (i.e., VTV=InMC ). Here InMC is the identity matrix. The covariance matrix of the beam deviations is defined as C≡∆T∆/(nMC −1) =VD2VT/(nMC −1),(A.33) from which we compute the eigenmode matrix E≡DVT/(nMC −1)1/2(A.34) using the SVD of ∆. Most of the statistical content of ∆or Cis limited to the first few modes nmodes with the largest eigenvalues. We therefore keep only the largest nmodes =5 of the Ematrix. The beam uncertainty for a given spectrum is then given by B`=Bmean `exp gTE` =Bmean `exp        nmodes X k=1 gkEk `      (A.35) where gis a vector of independant Gaussian variates of unit variance with nmodes elements, and Ek(`) is the kth row of E. This can be generalised to a set of spectra. Taking three pairs of detector sets a={UV},b={XY}and c={ZT}, one can write ∆a∆b∆c=MaMbMc.         Ea0 0 0Eb0 0 0 Ec          .(A.36) and the covariance matrix is given by Cabc =∆a∆b∆cT.∆a∆b∆c =         Ea0 0 0Eb0 0 0 Ec          T .         I MaT MbMaT Mc MbT MaI MbT Mc McT MaMcT MbI         .         Ea0 0 0Eb0 0 0 Ec          , (A.37) The beam errors can therefore be correlated (and in fact are strongly so, see Planck Collaboration VII 2014). In the next Appendix, this general covariance matrix is used to derive the beam error eigenmodes of combined spectra for the CamSpec likelihood. A15, page 41 of 60 A&A 571, A15 (2014) 0.021 0.022 0.023 Ωbh2 0.12 0.13 Ωch2 1.040 1.042 1.044 100θMC 0.04 0.08 0.12 τ 0.90 0.93 0.96 ns 3.04 3.12 3.20 ln(1010As) 10 20 30 ACIB 143 40 60 ACIB 217 0.4 0.6 0.8 rCIB 143×217 5 10 15 AtSZ 0.25 0.50 0.75 ξtSZ×CIB 0.4 0.6 0.8 γCIB 80 160 240 APS 100 30 60 90 APS 143 0.25 0.50 0.75 rPS 143×217 40 80 120 APS 217 0.000032 0.000040 ADust 15 30 45 AkSZ 0.60 0.65 0.70 ΩΛ 13.65 13.80 13.95 Age 0.30 0.35 0.40 Ωm 4 8 12 zre 64 68 72 H0 Plik reference No subpixel effect Uncorrelated noise Fig. D.2. Impact of removing the sub-pixel effect (blue) or the correlated noise between detector sets (purple), compared to the Plik reference case (red). In Fig. D.11, we show 70 GHz parameters for three choices of the maximum `considered: `max =800, 1000, and 1200. The latter two are consistent. Minor discrepancies are displayed at `max =800, which can be explained since there is, in this range, no detection of the point source component. In the same figure, we show results using the Plik likelihood for `max =1008, which is consistent with Planck 70 GHz over the same `range. D.3. Consistency of the Planck low resolution CMB maps Here we extend the discussion presented in Sect. 8.3. In Fig. D.12 we show the power spectrum of the residual maps, relative to Commander, for NILC,SEVEM, and SMICA. The maximum discrepancy in the range `< ∼40 (the multipole where noise begins to become non-negligible) is localised at the quadrupole and is less than 20 µK2, whereas for the range 3 ≤`< ∼40 the differences are of order ≈5µK2. Overall, the Planck maps are more in agreement among themselves than with WMAP, except perhaps at the quadrupole. The residual map between WMAP and Planck shows power spectrum residuals from ≈10 µK2up to ≈40 µK2to `< ∼40. These figures should be compared to the residual estimated from simulated foreground maps, shown to be < ∼10 µK2at `< ∼70 in Planck Collaboration XII (2014). A15, page 48 of 60 Planck collaboration: Planck 2013 results. XV. 0.020 0.022 0.024 Ωbh2 0.10 0.12 0.14 Ωch2 1.035 1.040 1.045 100θMC 0.04 0.08 0.12 τ 0.90 0.96 1.02 ns 3.0 3.1 3.2 ln(1010As) 15 30 45 ACIB 143 50 75 100 ACIB 217 0.25 0.50 0.75 rCIB 143×217 8 16 24 AtSZ 0.25 0.50 0.75 ξtSZ×CIB 0.4 0.8 1.2 γCIB 100 200 300 APS 100 100 200 300 APS 143 0.25 0.50 0.75 rPS 143×217 100 200 300 APS 217 0.00004 ADust 15 30 45 AkSZ 0.45 0.60 0.75 ΩΛ 13.2 13.6 14.0 Age 0.30 0.45 Ωm 4 8 12 zre 60 70 80 H0 Plik reference `max = 1008 `max = 1503 `max = 2013 Fig. D.3. Impact on the cosmological and foreground parameters of varying the maximum multipole, `max, retained in the Plik likelihood. We consider `max =1008 (blue), 1503 (purple), and 2013 (green) in addition to the reference `max =2508 (red). We complement our results with those obtained from the “FFP6” simulations described in Planck Collaboration XII (2014). They consist of 1000 signal plus noise maps processed through each of the four component separation pipelines. For each Monte Carlo realisation, we follow the same procedure as in the previous section, i.e. smoothing (FWHM = 329.810) and binning the maps to Nside =32. We apply this procedure to both the CMB input maps and the output maps derived by the four component separation algorithms. Again, a Gaussian white noise with a variance of 4 µK2is added, and the noise covariance matrix is corrected accordingly. Note that the additional white noise is taken into account not only for numerical regularization (to this extent its amplitude may well be lower), but principally because the output instrumental noise processed through component separation and downgraded to low resolution is far from being white. The additional white noise makes the detailed knowledge of the full noise covariance matrix unimportant. For each realisation and for each component separation code, we compute the power spectrum of the processed map and of the A15, page 49 of 60 A&A 571, A15 (2014) 0.021 0.022 0.023 Ωbh2 0.112 0.120 0.128 Ωch2 1.040 1.042 100θMC 0.04 0.08 0.12 τ 0.925 0.950 0.975 ns 3.04 3.12 ln(1010As) 10 20 30 ACIB 143 40 60 ACIB 217 0.4 0.6 0.8 rCIB 143×217 5 10 15 AtSZ 0.25 0.50 0.75 ξtSZ×CIB 0.25 0.50 0.75 γCIB 80 160 240 APS 100 25 50 75 APS 143 0.25 0.50 0.75 rPS 143×217 30 60 90 APS 217 0.000032 0.000040 ADust 15 30 45 AkSZ 0.60 0.65 0.70 ΩΛ 13.65 13.80 13.95 Age 0.30 0.35 0.40 Ωm 8 12 zre 64 68 72 H0 Plik reference `min = 50 Fig. D.4. Impact of changing in the Plik likelihood the minimum multipole from `min =100 (red, as in reference case) to `min =50 (blue). input signal map, using a common mask. When estimating the power spectrum, the input CMB maps are regularised by adding a negligible Gaussian white noise with 0.1µK2variance. We have checked explicitly that when the white noise is added to the component separated maps, we are able to recover the input power spectrum without bias up to `∼60 for all the four component separation methods. Moreover, we compute the power spectrum of the difference maps (output processed map minus input CMB) for each realisation, in order to evaluate the total amount of residual noise. Note that this is not only given by the added regularization noise, but also from the intrinsic noise, albeit small, that is present in the maps. Figure D.13 shows the average and the 1σlevels of such noise residuals for each of the component separation solutions. We thus see, that the level of such total noise residuals is well below the difference plotted in Fig. D.12. Therefore, we argue that the existing differences between the codes are due to genuine foreground separation residuals. A15, page 50 of 60 Planck collaboration: Planck 2013 results. XV. 0.021 0.022 0.023 Ωbh2 0.11 0.12 0.13 0.14 Ωch2 1.040 1.042 1.044 100θMC 0.04 0.08 0.12 τ 0.92 0.96 ns 3.04 3.12 3.20 ln(1010As) 15 30 ACIB 143 40 60 80 ACIB 217 0.25 0.50 0.75 rCIB 143×217 6 12 18 AtSZ 0.25 0.50 0.75 ξtSZ×CIB 0.25 0.50 0.75 γCIB 80 160 240 APS 100 30 60 90 APS 143 0.25 0.50 0.75 rPS 143×217 40 80 120 APS 217 0.00004 0.00008 ADust 15 30 45 AkSZ 0.60 0.66 0.72 ΩΛ 13.65 13.80 13.95 Age 0.24 0.32 0.40 Ωm 8 12 zre 64 68 72 H0 Plik reference CL20 mask CL31 mask CL49 mask Fig. D.5. Impact of changing the Plik Galactic mask, increasing the sky area used from GA21 (blue), GA34 (purple), GA38 (red, reference), to the least conservative GA54 (green). All results use the 100 ≤`≤2508 range. Note that the CamSpec likelihood uses the G35 mask for the 143 and 217 GHz channels, and G56 at 100 GHz, but with a restricted, composite multipole range. Appendix E: Dust cleaning using Planck 353 GHz The WMAP polarisation products are weighted combinations of Ka, Q, and Vband 9 year maps. The WMAP analysis mitigates polarised foreground emission using template fitting for the synchrotron and dust emissions. As a template for synchrotron emission, the WMAP Kband channel is used, and for dust a polarisation model is used to create a template map (Page et al. 2007). Here we assess the impact on the WMAP polarisation signal when this dust template is replaced by the Planck 353 GHz map, which provides a more direct tracer of the polarisation. Note that this is only for comparison; in all other analyses, we continue to use the WMAP polarisation products as released by the WMAP team (except using the Planck aTT `mmap as discussed in Sect. 8). Foreground cleaned maps can be written as mclean =mi−αimsynch −βimdust A15, page 51 of 60 A&A 571, A15 (2014) 0.021 0.022 0.023 Ωbh2 0.11 0.12 0.13 Ωch2 1.040 1.042 1.044 100θMC 0.04 0.08 0.12 τ 0.92 0.96 1.00 ns 2.96 3.04 3.12 ln(1010As) 15 30 45 ACIB 143 25 50 75 ACIB 217 0.4 0.6 0.8 rCIB 143×217 8 16 24 AtSZ 0.25 0.50 0.75 ξtSZ×CIB 0.0 0.6 1.2 γCIB 100 200 300 APS 100 25 50 75 APS 143 0.25 0.50 0.75 rPS 143×217 40 80 120 APS 217 0.00006 0.00012 ADust 15 30 45 AkSZ 0.60 0.66 0.72 ΩΛ 13.6 13.8 14.0 Age 0.24 0.32 0.40 Ωm 8 12 zre 65 70 75 H0 Plik reference no 217 GHz no 143 GHz no 100 GHz Fig. D.6. Impact on parameters of removing one single frequency channel (i.e., all spectra with at least one frequency in the removed channel). Results are shown removing the 100 GHz (green), 143 GHz (purple), or 217 GHz (blue) channels, compared to the reference case (red). Where the 217 GHz channel is removed, the CIB spectral index is held fixed at γCIB =0.6. where m=(Q,U) are linear polarisation Stokes parameter maps, and the index iis for Ka, Q, and Vbands. Here msynch is the WMAP 9-year Kband map and for mdust we use either the WMAP dust template or the Planck 353 GHz maps. For each frequency band, the scaling coefficients αiand βiare estimated by minimizing the χ2: χ2(α, β)=mt cleanC−1mclean (E.1) where Cis the covariance matrix. Following the WMAP analysis, we do not include the signal contribution in Cand we only use the diagonal elements of the noise covariance matrix to estimate χ2. Scaling coefficients are computed using two different WMAP masks: the “processing mask” that masks a narrow region in the plane of the Galaxy, and the more conservative “P06” mask used for power spectrum estimation and cosmological analysis. A15, page 52 of 60 Planck collaboration: Planck 2013 results. XV. Ω b h 2 Ω c h 2 100θ MC τ n s ln(10 10 A s ) calib100 − calib143 − calib143 − calib143 − calib143 − calib217 − calib217 − calib217 − calib217 − calib217 − calib217 − H 0 z re Ω m Age Ω Λ Ωbh2 Ωch2 100θMC τ ns ln(1010As) calib100 −ds1 calib143 −ds1 calib143 −ds2 calib143 −6 calib143 −7 calib217 −1 calib217 −2 calib217 −3 calib217 −4 calib217 −ds1 calib217 −ds2 H0 zre Ωm Age ΩΛ -43% 35%-39% 27%-25%10% 47%-76%39%25% 19% 3% 1% 94% 3% -4% 7% -2% -4% -5% -5% -3% 5% -1% 0% -2% -1% 70% 0% 2% 0% 1% 0% 0% 64%68% -1% 5% -8% 0% -2% 0% 52%57%52% -1% 3% 0% 1% 0% 0% 57%61%56%48% 1% 0% -3% 5% 1% 4% 21% 38% 32% 25% 33% -2% 0% -4% 1% 0% 0% 19%34% 28% 26% 32% 54% 4% 0% -2% 2% 2% 0% 19%35% 31% 27% 32% 57%57% 1% 1% -1% 1% 0% 0% 17%32% 28% 24% 27% 56%55%58% 1% 1% -1% 3% 1% 2% 24%42%36% 32% 36% 63%62%67%62% 0% 1% -5% 2% 2% 1% 24%44%37% 32% 35% 66%62%66%64%75% 60%-96% 56%27%78%0% -7% -5% -2% -6% -3% -1% -1% 0% -1% 0% -2% 16%-15%5% 99% 17%96% -4% 0% 1% 0% 1% 5% 1% 1% 1% 3% 2% 16% -52%98% -50%-26%-78%1% 7% 4% 2% 6% 3% 0% 1% 0% 1% 0% 2% -99% -16% -87%67%-71%-27%-64%-11%5% 4% 0% 5% 1% 0% 2% -1% 0% 0% 1% -83% -16%77% 52%-98% 50%26%78%-1% -7% -4% -2% -6% -3% 0% -1% 0% -1% 0% -2% 99% 16%-99% -77% −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Fig. D.7. Covariance matrix between cosmological parameters and detector cross-calibration coefficients (the calibration of the 143-5 detector is set to 1 to avoid an overall degeneracy with the total signal amplitude). The template coefficients at each channel are shown in Fig. E.1, estimated using the two different Galactic masks. At Q and Vband the estimated coefficients are consistent for the two masks; at Ka band the 353 GHz map gives more stable results than the WMAP dust template. The coefficients using the P06 mask are more uncertain, however, as the residual dust signal outside the mask is low, especially for Ka band. We find that the preferred synchrotron coefficient, α, is slightly lower using the 353 GHz map, and the overall χ2, shown in Table E.1, is slightly improved using the Planck dust map. We now test the effect on cosmological parameters, in particular the optical depth to reionisation, using these two different templates. Using the Planck 353 GHz channel as the dust template, with coefficients estimated using the processing mask, lowers the best fit value of τby about 1σ(see Fig. E.2). we find τ=0.075 ±0.013, compared with τ=0.089 ±0.013 using the WMAP dust model. This in turn has the effect of lowering As, from 3.088±0.025 to 3.061±0.025, but other ΛCDM parameters are not affected. We note though that using template coefficients estimated outside the P06 Galactic mask, the optical depth using the Planck template is lowered by only 0.5σcompared A15, page 53 of 60 A&A 571, A15 (2014) A CIB 143 A CIB 217 r CIB 143×217 A tSZ ξ tSZ×CIB γ CIB A PS 100 A PS 143 A PS 217 r PS 143×217 A kSZ A Dust calib100 −ds1 calib143 −ds1 calib143 −ds2 calib143 −6 calib143 −7 calib217 −1 calib217 −2 calib217 −3 calib217 −4 calib217 −ds1 calib217 −ds2 ACIB 143 ACIB 217 rCIB 143×217 AtSZ ξtSZ×CIB γCIB APS 100 APS 143 APS 217 rPS 143×217 AkSZ ADust calib100 −ds1 calib143 −ds1 calib143 −ds2 calib143 −6 calib143 −7 calib217 −1 calib217 −2 calib217 −3 calib217 −4 calib217 −ds1 calib217 −ds2 55% 19%44% 28%15%64% 10%-8% 0% -12% 10%52%-20% -27%-1% 12%15%-27%-49%6% 18% -21%3% 0% -22%23%7% 32% -28%-70%-25%3% 4% -41%1% 43% 0% 0% 0% 22%26%32%-9% 18%22% -22%-19%-20% -21%0% 0% -18%-65%-48%-29% 20%45%9% 15% 2% 76%-6% -3% -31%26% -4% 17% 6% 0% 0% 0% -1% 8% -4% -1% -3% -3% 0% 1% -2% 1% 3% -3% -2% -2% -2% 4% 2% -1% 0% 70% -2% -5% 0% 2% -4% -3% -7% -4% 4% 0% 0% -1% 64%68% 0% -1% 0% 1% 0% 2% -5% 3% 4% 5% -3% 3% 52%57%52% 2% -2% 2% 2% 0% 0% -2% 0% 3% 0% -1% 2% 57%61%56%48% 0% -22%-19% -2% 3% -6% 0% -3% 21% 9% 0% -11%21% 38% 32% 25% 33% -1% -21%-18% -4% -1% -6% 1% -4% 19% 5% 2% -11%19%34% 28% 26% 32% 54% 1% -19%-20%-4% 2% -1% 0% -4% 17% 8% 0% -8% 19%35% 31% 27% 32% 57%57% -1% -23%-18% -1% 1% -6% 0% -6% 19% 8% 2% -10%17%32% 28% 24% 27% 56%55%58% 1% -24% -20%0% -2% -7% -2% -7% 22% 8% 0% -12%24% 42%36% 32% 36% 63%62%67%62% -3% -27% -20%-3% 0% -5% -1% -6% 23% 10% 2% -12%24% 44%37% 32% 35% 66%62%66%64%75% −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Fig. D.8. Covariance matrix between foreground parameters and detector cross-calibration coefficients, as in Fig. D.7. 0 2000 4000 6000 0 200 400 600 800 1000 Auto Cross PLC Bestfit + PS 0 2000 4000 6000 0 200 400 600 800 1000 ℓ(ℓ+ 1)Cℓ/2πµK2 ℓ(ℓ+ 1)Cℓ/2πµK2 ℓℓ Fig. D.9. Power spectrum from the Planck 70 GHz channel, before removal of unresolved sources. Both the auto-spectrum from the 70 GHz maps, and the weighted cross spectra from maps of the three 70 GHz horn pairs are shown. The best-fitting cosmological model from CamSpec, shown for comparison with the best-fitting source power added, gives a good fit to the data. A15, page 54 of 60 Planck collaboration: Planck 2013 results. XV. Fig. D.10. Cosmological parameters derived from the 70 GHz maps (solid black) are compared to CamSpec results (red dashed). to the WMAP template, indicating some spatial dependence. We conclude that the impact on cosmological parameters from the choice of dust template is small, but not insignificant. Still, the low-`polarisation from Planck has not been qualified for cosmological analyses yet, and we only used it to reject the hypothesis of a large effect. A15, page 55 of 60 A&A 571, A15 (2014) Fig. D.11. Cosmological parameters derived from the 70 GHz maps for different values of the maximum multipole – `max =800 (green dotted), `max =1000 (blue dashed), and `max =1200 (solid black) – are compared to Plik at `max =1008 (pink dot-dashed). A15, page 56 of 60 Planck collaboration: Planck 2013 results. XV. 10 20 30 40 50 60 l 0 10 20 30 40 50 NILC SEVEM SMICA WMAP-9 Fig. D.12. Power spectrum of the residual CMB maps, relative to Commander, for the NILC,SEVEM, and SMICA methods. The 1σexpected noise level is shown in black. Fig. D.13. Estimated total residual noise (intrinsic and regularizing white noise) levels for each of the four Planck CMB maps: Commander,Nilc, Sevem, and Smica. Solid lines show the average and the hatched regions show the 68% CL. A15, page 57 of 60