A&A 602, A41 (2017) Astronomy DOI: 10.1051/0004-6361/201629815 & c ESO 2017 Astrophysics
Cosmology with the cosmic microwave background temperature-polarization correlation F. Couchot1, S. Henrot-Versillé1, O. Perdereau1, S. Plaszczynski1, B. Rouillé d’Orfeuil1, M. Spinelli1, 2, and M. Tristram1
1 Laboratoire de l’Accélérateur Linéaire, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France e-mail: [email protected] 2 Department of Physics and Astronomy, University of the Western Cape, Robert Sobukwe Road, Bellville 7535, South Africa Received 30 September 2016 / Accepted 14 February 2017
ABSTRACT
We demonstrate that the cosmic microwave background (CMB) temperature-polarization cross-correlation provides accurate and robust constraints on cosmological parameters. We compare them with the results from temperature or polarization and investigate the impact of foregrounds, cosmic variance, and instrumental noise. This analysis makes use of the Planck high-` HiLLiPOP likelihood based on angular power spectra, which takes into account systematics from the instrument and foreground residuals directly modelled using Planck measurements. The temperature-polarization correlation (TE) spectrum is less contaminated by astrophysical emissions than the temperature power spectrum (TT), allowing constraints that are less sensitive to foreground uncertainties to be derived. For P ΛCDM parameters, TE gives very competitive results compared to TT. For basic ΛCDM model extensions (such as AL, mν, or Neff ), it is still limited by the instrumental noise level in the polarization maps. Key words. cosmic background radiation – cosmological parameters – methods: data analysis
1. Introduction spectrum level in the likelihood. Most of the results presented in Planck Collaboration XIII(2016) are based on TT angular The results from the Planck satellite have recently demon- power spectra which present the higher signal-to-noise ratio. strated the consistency between the temperature and the polar- However, foreground residuals in temperature combine several ization data (Planck Collaboration XIII 2016). Adding the infor- different components which are difficult to model in the power mation coming from the velocity gradients of the photon–baryon spectra domain as they are both non-homogeneous and non- fluid through the polarization power spectra to the measure- Gaussian. Any mismatch between the foreground model and the ment of the temperature fluctuations improves the constraints data can thus result in a bias on the estimated cosmological pa- on cosmological parameters and helps break some degenera- rameters and, in all cases, will increase their posterior width. On cies. One of the best examples is the measurement of the the contrary, in polarization, even though the signal-to-noise ra- reionization optical depth using the large-scale signature that tio is lower, the only foreground that affects the Planck data is reionization leaves in the EE polarization power spectrum the polarized emission of the Galactic dust. As we show, this al- (Planck Collaboration Int. XLVII 2016). Moreover, as suggested lows for a precise reconstruction of the cosmological parameters in Galli et al.(2014), for a cosmic variance limited experiment, (especially with TE spectra) with less impact from foreground polarization power spectra alone can provide tighter constraints uncertainties. on cosmological parameters than the temperature power spec- trum, while for an experiment with Planck-like noise, constraints The cosmological parameters reconstructed with TT spectra should be comparable. are compared to those obtained independently with TE and EE. In this paper, we discuss in greater detail the constraints on In each case, we detail the foreground modelling and the prop- cosmological parameters obtained with the Planck 2015 polar- agation of its uncertainties. We use the High-` Likelihood on ization data (including foregrounds and systematic residuals). Polarized Power spectra (HiLLiPOP) likelihood which is based We find that the level of instrumental noise allows for an accurate on the Planck data in temperature and polarization. HiLLiPOP reconstruction of cosmological parameters using temperature- is one of the four high-` likelihoods developed within the Planck TE polarization cross-correlation C` only. Constraints from Planck consortium for the 2015 release and is briefly presented and EE polarization spectrum are dominated by instrumental noise. compared to others in Planck Collaboration XIII(2016). It is a In addition, we investigate the robustness of the cosmological full temperature+polarization likelihood based on cross-spectra interpretation with respect to astrophysical residuals. from Planck maps at 100, 143, and 217 GHz. It is based on a In the Planck analysis (Planck Collaboration XV 2014; Gaussian approximation of the C` likelihood which is well suited Planck Collaboration XI 2016), the foreground contamination for multipoles above ` = 30. In contrary to the Planck pub- is mitigated using masks which are adapted to each fre- lic likelihood (Planck Collaboration XIII 2016), the foreground quency, reducing the sky fraction to the region where the fore- description in HiLLiPOP directly relies on the Planck astro- ground emission is low. The residuals of diffuse foreground physical measurements. For the Λ cold dark matter (ΛCDM) emission are then taken into account using models at the cosmology, using a τ prior, it gives results very compatible
Article published by EDP Sciences A41, page 1 of 24 A&A 602, A41 (2017)
Planck 353 GHz map as a tracer of the thermal dust emission TT 4 in intensity. In practice, we smoothed the Planck 353 GHz map 10 EE to increase the signal-to-noise ratio before applying a thresh-
] TE old which depends on the frequency considered. Masks are then B ◦ M 3
2 apodized using a 8 Gaussian taper for power spectra estima- C 10
K tion. For polarization, Planck dust maps show that the diffuse µ [ emission is strongly related to the Galactic magnetic field at ` 2
C 10 large scales (Planck Collaboration Int. XIX 2015). However, at
π the smaller scales which matter here (` > 50), the orientation 2 /
) 1 of dust grains is driven by local turbulent magnetic fields which 1 10
+ produce a polarization intensity proportional to the total intensity `
( dust map. We thus use the same Galactic mask for polarization ` 100 as for temperature. Molecular lines from CO produce diffuse emission on star
-1 forming region. Two major CO lines at 115 GHz and 230 GHz 10 0 500 1000 1500 2000 2500 enter the Planck bandwidths at 100 and 217 GHz, respectively multipole ` (Planck Collaboration XIII 2014). We smoothed the Planck re- constructed CO map to 30 arcmin before applying a threshold at Fig. 1. Signal (solid line) versus noise (dashed line) for the Planck 2 K km s−1. The resulting masks are then apodized at 15 arcmin. cross-spectra for each mode TT, EE, and TE (in red, blue, and green, In practice, the CO masks are almost completely included in the respectively). Galactic masks, decreasing the accepted sky fraction only by a few percentage points. For point sources, the Planck 2013 and 2015 analyses mask with the Planck public likelihood, except for the (τ, As) pair which is more consistent with the low-` data. Consequently, it the sources detected with a signal-to-noise ratio above 5 in the Planck point-source catalogue (Planck Collaboration XXVI also shows a better lensing amplitude AL (see the discussion in Couchot et al. 2017). 2016) at each frequency (Planck Collaboration XVI 2014; The paper is organized as follows. In Sect.2, we describe the Planck Collaboration XI 2016). On the contrary, the masks power spectra used in this analysis. We discuss the Planck maps used in our analysis rely on a more refined procedure that and the sky region for the power spectra estimation. Section3 preserves Galactic compact structures and ensures the com- presents the likelihood functions both in temperature and in po- pleteness level at each frequency, but with a higher flux cut larization, and details the model of each associated foreground (340, 250, and 200 mJy at 100, 143, and 217 GHz, respec- emission. We then present in Sect.4 the results for the ΛCDM tively). The consequence is that these masks leave a slightly cosmological model and check the impact of priors on the as- greater number of unmasked extragalactic sources, but pre- serve the power spectra of the dust emission (as described in trophysical parameters. Section5 gives the results on the AL pa- rameter considered as an internal cross-check of the CMB like- Planck Collaboration Int. XXX 2016). For each frequency, we lihoods. Finally, in Sect.6, we demonstrate the impact of the mask a circular area around each source using a radius√ of three foreground parameters for the temperature likelihood and the TE times the effective Gaussian beam width (σ = FWHM/ ln 8) at likelihood in terms of both the bias and the precision of the cos- that frequency. We apodize these masks with a Gaussian taper of mological parameters. FWHM = 15 arcmin. Finally, we also mask strong extragalactic objects including both point sources and nearby extended galaxies. The masked 2. Data set galaxies include the LMC and SMC and also M31, M33, M81, 2.1. Maps and masks M82, M101, M51, and CenA. The combined masks used are named M80, M70, and M55 The maps used in this analysis are taken from the Planck 2015 1 (corresponding to effective fsky = 72%, 62%, 48%), associ- data release and described in detail in Planck Collaboration VIII ated with the 100, 143, and 217 GHz channels, respectively (2016). We use two maps per frequency (A and B, one for (Fig.2). Tests have been carried out using more conserva- each half-mission) at 100, 143, and 217 GHz. The beam asso- tive Galactic masks (with fsky = 65%, 55%, and 40% for ciated with each map is provided by the Planck Collaboration 100, 143, and 217 GHz, respectively) showing perfectly com- (Planck Collaboration VII 2016). Figure1 compares the signal patible results with those of the smaller masks. Compared to with the noise of the Planck maps for each mode TT, EE, and the masks used in the Planck 2015 analysis, the retained sky TE. fraction is almost identical. Indeed, the Galactic masks used Frequency-dependent apodized masks are applied to these in Planck Collaboration XI(2016) retain 70%, 60%, and 50% maps in order to limit the foreground contamination in the power respectively. spectra. We use the same masks in temperature and polarization. The masks are constructed first by thresholding the total inten- sity maps of diffuse Galactic dust to exclude strong dust emis- 2.2. Power spectra sion. In addition, we also remove regions with strong Galactic CO emission, nearby galaxies, and extragalactic point sources. We use Xpol (an extension to polarization of Tristram et al. Diffuse Galactic dust emission is the main contaminant for 2005) to compute the cross-power spectra in temperature and po- CMB measurements in both temperature and polarization at fre- larization (TT, EE, and TE). Xpol is a pseudo-C` method which quencies above 100 GHz. We build Galactic masks using the also computes an analytical approximation of the C` covari- ance matrix directly from data. Using the six maps presented 1 Planck PLA: http://pla.esac.esa.int in Sect. 2.1, we derive the 15 cross-power spectra for each CMB
A41, page 2 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation
Table 1. Multipole ranges used in the analysis and corresponding num- ber of multipoles available (n` = `max − `min + 1).
TT EE TE 100 × 100 [ 50, 1200] [100, 1000] [100, 1200] 100 × 143 [ 50, 1500] [100, 1250] [100, 1500] 100 × 217 [500, 1500] [400, 1250] [200, 1500] 143 × 143 [ 50, 2000] [100, 1500] [100, 1750] 143 × 217 [500, 2500] [400, 1750] [200, 1750] 217 × 217 [500, 2500] [400, 2000] [200, 2000] n` 9556 7256 8806
Notes. The total number of multipoles is 25 618.
The multipole ranges used in the likelihood analysis have been chosen to limit the contamination of the Galactic dust emis- sion at low-` and the noise at high-`. Table1 gives the multi- pole ranges, [`min, `max], considered for each of the six cross- frequencies in TT, TE, and EE. The spectra are cosmic-variance limited up to ` ' 1500 in TT and ` ' 700 in TE (outside the troughs of the CMB signal). The EE mode is dominated by in- strumental noise.
3. Likelihood function On the full-sky, the distribution of auto-spectra is a scaled-χ2 with 2` + 1 degrees of freedom. The distribution of the cross- spectra is slightly different (see Appendix A in Mangilli et al. 2015); however, above ` > 50 the number of modes is large enough so that we can safely assume that the C` are Gaussian distributed. When considering only a part of the sky, the values of C` are correlated so that for high multipoles, the resulting dis- tribution can be approximated by a multi-variate Gaussian taking into account `-by-` correlations
X X 0 0 0 0 Fig. 2. M80, M70, and M55 masks. A combination of an apodized i j h −1ii j,i j i j −2 ln L = R Σ R 0 + ln |Σ|, (3) Galactic mask and a compact object mask is used at each frequency ` ``0 ` i6 j ``0 (see text for details). i06 j0
i j i j ˆ i j mode: one each for 100 × 100, 143 × 143, and 217 × 217; four where R` = C` − C` denotes the residual of the estimated each for 100 × 143, 100 × 217, and 143 × 217 as outlined below. cross-power spectrum C` with respect to the model Cˆ ` for each From the coefficients of the spherical harmonic decomposi- polarization mode considered (TT, EE, TE) and each frequency X { T E B } D T E tion of the (I,Q,U) masked maps a˜ `m = a˜`m, a˜`m, a˜`m , we form ({i, j} ∈ [100, 143, 217]). The matrix Σ = RR is the full co- the pseudo cross-power spectra between map i and map j, variance matrix which includes the instrumental variance from i j 1 X the data as well as the cosmic variance from the model. The C˜ = a˜ i∗ a˜ j , (1) ` 2` + 1 `m `m latter is directly proportional to the model so that the matrix Σ m should, in principle, depend on the model. In practice, given our current knowledge of the cosmological parameters, the the- where the vector C˜ ` includes the four modes ˜TT ˜EE ˜TE ˜ET oretical power spectra typically differ from each other at each {C` , C` , C` , C` }. We note that the TE and ET cross- power spectra do not carry exactly the same information since ` by less than they differ from the observed C` so that we computing T from map i and E from map j is not the same can expand Σ around a reasonable fiducial model. As described as computing E from map j and T from i. They are computed in Planck Collaboration XV(2014), the additional terms in the independently and averaged afterwards using their relative expansion are small if the fiducial model is accurate and its ab- weights for each cross-frequency. The pseudo-spectra are then sence does not bias the likelihood. Using a fixed covariance ma- | | corrected from beam and sky fraction using trix Σ, we can drop the constant term ln Σ . We therefore expect the likelihood to be χ2-distributed with a mean equal to the num- i j ˜ 0 i j i j ber of degrees of freedom ndof = n` − np (n` is given in Table1 C` = (2` + 1)M``0 C`0 , (2) and np is the number of fitted parameters) and a variance equal where the coupling matrix M depends on the masks used for each to 2ndof. set of maps (Peebles 1973) and includes beam transfer functions We define several likelihood functions based on the informa- usually extracted from Monte Carlo simulations (Hivon et al. tion used: hlpT for TT cross-spectra, hlpE for EE cross-spectra, 2002). hlpX for TE cross-spectra, and hlpTXE for the combination
A41, page 3 of 24 A&A 602, A41 (2017) TT EE TE
100x100
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100x100100x143100x217143x143 143x217 217x217100x100100x143100x217143x143143x217217x217100x100100x143100x217143x143143x217217x217 Fig. 3. Full HiLLiPOP covariance matrix including all correlations in multipoles between cross-frequencies and power spectra.
TT EE TE ET of all cross-spectra. The hlpX likelihood combines information (cd), is an N-by-N matrix (where N = n` + n` + n` + n` ) and from TE and ET cross-spectra. reads The next two sections describe the computation of the co- D E D E ˜ ab,cd ˜ab ˜cd∗ ˜ab ˜cd∗ ˜ab ˜cd∗ variance matrix and the building of the model, focusing on the Σ``0 ≡ ∆C` ∆C`0 = C` C`0 − C` C`0 differences with the Planck public likelihood. D a c∗ ED b∗ d E D a d∗ ED b∗ c E X a˜`ma˜`0m0 a˜`ma˜`0m0 + a˜`ma˜`0m0 a˜`ma˜`0m0 = (2` + 1)(2`0 + 1) 3.1. Semi-analytical covariance matrix mm0
We use a semi-analytical estimation of the C` covariance matrix by expanding the four-point Gaussian correlation using Isserlis’ computed using Xpol. The matrix encloses the `-by-` correla- formula (or Wick’s theorem). tions between all the power spectra involved in the analysis. The Each two-point correlation of pseudo-a`m can be expressed computation relies directly on data estimates. It follows that con- as the convolution of C` with a kernel which depends on the tributions from noise (correlated and uncorrelated), sky emission polarization mode considered (from astrophysical and cosmological origin), and the cosmic D E X variance are implicitly taken into account in this computation Ta∗ Tb TaTb 0,Ta 0,Tb∗ a˜ a˜ 0 0 = C W W 0 0 without relying on any model or simulations. `m ` m `1 `m`1m1 ` m `1m1 ` m The covariance matrix Σ of cross-power spectra is directly 1 1 D E 1 X n o related to the covariance Σ˜ of the pseudo cross-power spectra Ea∗ Eb Ea Eb +,Ea∗ +,Eb Ba Bb −,Ea∗ −,Eb a˜ a˜ 0 0 = C W W 0 0 +C W W 0 0 through the coupling matrices: `m ` m 4 `1 `m`1m1 ` m `1m1 `1 `m`1m1 ` m `1m1 `1m1
D E −1 −1 D E 1 X +,E ab,cd ab cd∗ ab ˜ ab,cd cd∗ Ta∗ Eb Ta Eb 0,Ta∗ b Σ ≡ ∆C ∆C 0 = M Σ M 0 (4) a˜`m a˜`0m0 = C` W`m` m W`0m0` m `1`2 ` ` ``1 `1`2 ` `2 2 1 1 1 1 1 `1m1 with (a, b, c, d) ∈ {T, E} for each map A, B, C, D. We compute Σ˜ for each cross-spectra block independently where the kernels W0, W+, and W− are defined as linear com- that includes `-by-` correlation and four-spectra mode correla- bination of products of Y`m of spin 0 and ±2 (see AppendixA). tion {TT, EE, TE, ET}. The TE and ET blocks are both computed As suggested in Efstathiou(2006), neglecting the gradients of individually and finally averaged. The matrix Σ˜ , which gives the the window function and applying the completeness relation for correlations between the pseudo cross-power spectra (ab) and spherical harmonics (Varshalovich et al. 1988), we can reduce
A41, page 4 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation the products of four W into kernels similar to the coupling ma- 106 trix M defined in Eq. (2). In the end, the blocks of Σ matrices 105 4 reads 104
] 2 103 % [ ] TaTb,TcTd TaTc TbTd TaTd TbTc 4 Σ ' C 0 C 0 M + C 0 C 0 M 2 1 TT,TT TT,TT K 10 `` `` `` `` − a µ n
[ 0 2 a 1 σ Ea Eb,Ec Ed Ea Ec Eb Ed Ea Ed Eb Ec 2 10 / ' σ Σ C 0 C 0 M + C 0 C 0 M m
EE,EE EE,EE i 2 `` `` `` `` 0 s
10 σ 2 T E ,T E TaTc Eb Ed Ta Ed EbTc a b c d -1
] 2 103
power spectra (see AppendixA for details). In practice, to avoid % [ ]
4 2 1
K 10 any correlation between C` estimates and their covariance, we − a µ n
[ 0 2 a 1 σ use a smoothed version of each measured power spectrum (using 2 10 / σ m i 2 0 s
10 σ a Gaussian filter with σ` = 5) to estimate the covariance matrix. 2 -1
A41, page 5 of 24 A&A 602, A41 (2017)
The model is a function of the cosmological (Ω) and 5 M80xM80 ˆ model 10 nuisance (p) parameters: C` (Ω, p). The latter include M80xM70 instrumental parameters accounting for instrumental uncertain- ] M80xM55
B 3 ties and scaling parameters for each astrophysical foreground M 10 M70xM70 model as described in the following sections. In the end, we have C K M70xM55
µ 1 a total of 6 instrumental parameters (only calibration is consid- [ 10 M55xM55 T T ered, see Sect. 3.2.1), 9 astrophysical parameters (7 for TT, 1 for ` C TE, 1 for EE), and 6+ cosmological parameters (ΛCDM and ex- -1 10 tensions), i.e. a total of 21+ free parameters in the full likelihood function (see AppendixB). We note that the Planck public like- lihood depends on more nuisance parameters: 15 for TT (com- 3 pared to 13 for hlpT), 9 for TE (compared to 7 for hlpX), and 9 10 for EE (compared to 7 for hlpE). ] 1 B 10 M C
3.2.1. Instrumental systematics K -1
µ 10 [ E E The instrumental parameters of the HiLLiPOP likelihood are the ` -3 inter-calibration coefficients (c, which are measured relative to C 10 β the 143A map), and the amplitudes ( ) of the beam error modes -5 (µ`). In practice, we have linearized Eq. (5) for the coefficients c 10 and fit for small deviations around zero (c → 1 + c), while fix- 3 ing c143A = 0 for normalization. The uncertainty in the absolute 10 calibration is propagated through a global rescaling factor Apl. ] 1 The effective beam window functions B` account for B 10 M the scanning strategy and the weighted sum of individ- C
K -1
ual detectors performed to obtained the combined maps µ 10 [
(Planck Collaboration VII 2016). It is constructed from Monte E T ` -3
Carlo simulations of CMB convolved with the measured beam C 10 on each time-ordered data sample. The uncertainties in the de- termination of the HFI effective beams come directly from simu- 10-5 lations and is described in terms of the Monte Carlo eigenmodes 100 101 102 103 µ` (Planck Collaboration XV 2014). In the Planck 2013 analysis, it was found that, in practice, only the first beam eigenmode for multipole ` the 100 × 100 spectrum was relevant (Planck Collaboration XVI 2014). For the 2015 analysis, Planck Collaboration XI(2016) Fig. 5. Dust power spectra at 353 GHz for TT (top), TE (middle), and EE (bottom). The power spectra are computed from cross-correlation found no evidence of beam error in their multipole range thanks between half-mission maps for different sets of masks as defined in to higher accuracy in the beam estimation, which reduced the Sect. 2.1 and further corrected for CMB power spectrum (solid black amplitude of the beam uncertainty. As a consequence, in our line) and CIB power spectrum (dashed black line). analysis, we fixed their contribution to zero (β = 0). artificial knee in the residual dust power spectra around ` ∼ 200 3.2.2. Galactic dust (Sect. 3.3.1 in Planck Collaboration XI 2016). In contrast, our Galactic dust power spectra are directly comparable to those de- The TT, EE, and TE Galactic dust C` templates are obtained from the cross-power spectra between half-mission maps at rived in Planck Collaboration Int. XXX(2016). Moreover, here 353 GHz (as in Planck Collaboration Int. XXX 2016). This is re- we do not assume that the dust power spectra have the same spa- peated for each mask combination associated with the map data tial dependence across masks. set. The estimated power spectra are then accordingly rescaled For each polarization mode (TT, EE, TE), we then extrap- to each of the six cross-frequencies considered in this anal- olate the dust templates at 353 GHz for each cross-mask to the cross-frequency considered ˆ Mi M j ysis. We compute the 353 GHz cross-spectra C` for each pair of masks (M , M ) associated with the cross-spectra i × i j,dust dust dust Mi M j,dust i j C = Adust aν aν C , (6) j (Fig.5). We then subtract the Planck best-fit CMB power ` i j ` dust dust dust spectrum. For TT, we also subtract the CIB power spectrum where the aν = f (ν)/ f (353 GHz) extrapolated fac- (Planck Collaboration XXX 2014). In addition to Galactic dust, tors are estimated for intensity or polarization maps. We use unresolved point sources contribute to the TT power spectra at a greybody emission law with a mean dust temperature of Mi M j,dust T 19.6 K and spectral indices β = 1.59 and βP = 1.51 as 353 GHz. To construct the dust templates C` for our anal- ysis, we thus fit a power-law model with a free constant A`α + B measured in Planck Collaboration Int. XXII(2015). The result- dust in the range ` = [50, 2500] for TT, while a simple power-law is ing aν factors are (0.0199, 0.0387, 0.1311) for total intensity used to fit the EE, TE power spectra in the range ` = [50, 1500]. and (0.0179, 0.0384, 0.1263) for polarization at 100, 143, and Thanks to the use of the point source mask (described 217 GHz, respectively. TT in Sect. 2.1), our Galactic dust residual power spectrum is In HiLLiPOP, this results in three free parameters (Adust, EE TE much simpler than in the case of the Planck official likeli- Adust, Adust) describing the amplitude of the dust residuals in each hood. Indeed, the masks used in the Planck analysis remove mode. This model based on Planck internal measurements is some Galactic structures and bright cirrus, which induces an simpler than the one used in the Planck official likelihood, which
A41, page 6 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation allows the amplitude of each cross-frequency to vary (ending 100 with a total of 16 free parameters) and puts constraints on the 100x100 -1 dust SED through the use of strong priors. 10 100x143 10-2 100x217 3.2.3. Cosmic infrared background 143x143
] -3 B 10 143x217 The thermal radiation of dust heated by UV emission from M C -4 217x217 young stars produces an extragalactic infrared background K 10 µ whose emission law is very close to the Galactic dust emis- [ ` -5 sion. The Planck Collaboration has studied the CIB in detail in C 10 Planck Collaboration XXX(2014) and provides templates based -6 on a model that associates star forming galaxies with dark matter 10 halos and their sub-halos, using a parametrized relation between -7 the dust-processed infrared luminosity and (sub-)halo mass. This 10 model provides an accurate description of the Planck and IRAS 10-8 CIB spectra from 3000 GHz down to 217 GHz. We extrapolate 0 500 1000 1500 2000 2500 this model here, assuming it remains appropriate when describ- multipole ` ing the 143 GHz and 100 GHz data. The halo model formalism, which is also used for the tSZ Fig. 6. CIB power spectra templates. The SED and the angular de- pendence is given by Planck Collaboration XXX(2014). The CMB TT and the tSZ × CIB models (see Sects. 3.2.4 and 3.2.5), has the power spectrum is plotted in black. general expression (Planck Collaboration XXIII 2016)
AB,1h AB,2h 3.2.4. Sunyaev-Zel’dovich effect C` = C` + C` , (7) The thermal Sunyev-Zel’dovich emission (tSZ) is also parame- AB,1h where A and B stand for tSZ effect or CIB emission, C` is terized by a single amplitude and a fixed template measured in AB,2h Planck Collaboration XXI(2014) at 143 GHz, the one-halo contribution, and C` is the two-halo term. The AB,1h i j,tSZ tSZ tSZ tSZ one-halo term C` is computed as C = A a a C , (10) ` tSZ νi ν j ` Z dV Z d2N where atSZ = f tSZ(ν)/ f tSZ(143) is the thermal Sunyaev- CAB,1h = 4π dz dM W1hW1h, (8) ν ` dzdΩ dMdV A B Zel’dovich spectrum normalized at 143 GHz. We recall that, ig- noring the bandpass corrections, the tSZ spectrum is given by d2N where dMdV is the dark matter halo mass function from dV tSZ x hν Tinker et al.(2008), dzdΩ the comoving volume element, and f (ν) = x coth − 4 with x = · (11) 1h 2 kBTcmb WA,B is the window function that accounts for selection effects and total halo signal. Instead, the contribution of the two-halo After integrating over the instrumental bandpass, we obtain AB,2h tSZ term, C` , accounts for correlation in the spatial distribution f = −4.031, −2.785, and 0.187 at 100, 143, and 217 GHz, of halos over the sky. respectively (see Table 1 in Planck Collaboration XXII 2016). For the CIB, the two-halo term (i.e. the term that con- The Planck official likelihood uses the same parametrization siders galaxies belonging to two different halos) is dominant but with an empirically motivated template power spectrum at low and intermediate multipoles and is very well con- (Efstathiou & Migliaccio 2012). strained by Planck. The one-halo term is flat in C` and not The kinetic Sunyev-Zel’dovich (kSZ) is produced by the pe- well measured as it is degenerated with the shot noise. Hence, culiar velocities of the clusters containing hot electron gas. We in Planck Collaboration XXX(2014) strong priors on the shot use power spectra extracted from reionization simulations. We noises have been used to get the one-halo term. In HiLLiPOP, supposed that the kSZ follows the same SED as the CMB and we did not include any shot noise term in the CIB template to only fit a global free amplitude, AkSZ. We chose a combination of avoid degeneracies with the amplitude of infrared sources (see templates coming from homogeneous and patchy reionization.
Sect. 3.2.6). i j,kSZ hKSZ pKSZ The power spectra template for each cross-frequency in C` = AkSZ C` + C` . (12) 2 −1 Jy sr (with the IRAS convention νI(ν) = const.) are then For the homogeneous kSZ, we use a template power spectrum converted in µK2 using a slightly revised version of Table 6 CMB given by Shaw et al.(2012) calibrated with a “cooling and star in Planck Collaboration IX(2014): aconv = 1/244.06, aconv = 100 143 formation” simulation. For the patchy reionization kSZ we use conv −1 1/371.66, and a217 = 1/483.48 KCMB/MJy sr at 100, 143, and the fiducial model of Battaglia et al.(2013). Both templates are 217 GHz, respectively. Those coefficients account for the inte- shown in Fig.7. The Planck official likelihood considers a tem- gration of the CIB emission law in the Planck bandwidth. plate from homogeneous reionization only, but the impact on the The CIB templates used in HiLLiPOP (Fig.6) are then cosmology is completely negligible. rescaled with a free single parameter ACIB:
Ci j,CIB = A aconvaconvCνiν j,temp. (9) 3.2.5. tSZxCIB correlation ` CIB νi ν j ` The halo model can naturally account for the correlation be- The same parametrization was finally adopted in the Planck of- tween two different source populations, each tracing the under- ficial analysis for the 2015 release. lying dark matter, but having different dependence on host halo
A41, page 7 of 24 A&A 602, A41 (2017)
100 100 100x100 100x100 -1 -1 10 100x143 10 100x143 10-2 100x217 10-2 100x217 143x143 143x143
] -3 ] -3 B 10 143x217 B 10 143x217 M M C -4 217x217 C -4 217x217 K 10 K 10 µ µ [ [ ` -5 ` -5 C 10 C 10
10-6 10-6
10-7 10-7
-8 -8 10 0 500 1000 1500 2000 2500 10 0 500 1000 1500 2000 2500 multipole ` multipole `
0 10 Fig. 8. tSZxCIB power spectra templates. The SED and the angular dependence are fixed. Dashed lines are negative. 10-1
10-2 3.2.6. Unresolved PS
] -3
B 10 M
C At Planck frequencies, the unresolved point sources signal in- -4 K 10 corporates the contribution from extragalactic radio and infrared µ [ ` -5 dusty galaxies (Tucci et al. 2005). We use a specific mask for C 10 each frequency to mitigate the impact of strong sources (see Sect. 2.1). Planck Collaboration XI(2016) gives the expected 10-6 amplitudes for the Poisson shot noise from theoretical models 10-7 that predict number counts dN/dS for each frequency. Their analyses take into account the details of the construction for the -8 point source masks, such as the fact that the flux cut varies across 10 0 500 1000 1500 2000 2500 multipole ` the sky or the “incompleteness” of the catalogue from which the masks are built at each frequency. We computed the expectation aradio Fig. 7. Top: tSZ power spectra templates at each cross-frequency. at each cross-frequency for the point source amplitudes ( νi,ν j Dashed lines are negative. SED are fixed and we fit the overall am- IR for the radio sources and aνi,ν j for the infrared sources) based plitude AtSZ. Bottom: frequency independent kSZ template. The black on the flux-cut considered for our own point sources masks (see line is the CMB power spectrum. Sect. 2.1) using the model from Tucci et al.(2011) for the ra- dio sources and from Béthermin et al.(2012) for dusty galax- ies (see Table4). We note that we found di fferent prediction properties (Addison et al. 2012). An angular power spectrum can numbers for radio galaxies than those reported in Table 17 of thus be extracted for the correlation between unresolved clusters Planck Collaboration XI(2016). contributing to the tSZ effect and the dusty sources that make We consider a flat Poisson-like power spectrum for each up the CIB. While the latter has a peak in redshift distribution component and rescale by two free amplitudes Aradio and AIR : between z ' 1 and z ' 2, and is produced by galaxies in dark PS PS 11 13 matter halos of 10 –10 M , tSZ is mainly produced by local 14 (z < 1) and massive dark matter halos (above 10 M ). This Ci j,PS = Aradio aradio + AIR aIR . (14) implies that the CIB and tSZ distributions present a very small ` PS νi,ν j PS νi,ν j overlap for the angular scales probed by Planck, and their corre- lation is thus hard to detect (Planck Collaboration XXIII 2016). In polarization, we neglect the point source contribution from We use the templates shown in Fig.8, computed using a both components (Tucci et al. 2004). tSZ power spectrum template based on Efstathiou & Migliaccio It is important to notice that building a reliable multi- (2012) and a CIB template as described in Sect. 3.2.3. The power frequency model for the unresolved sources is difficult. Indeed, 2 −1 spectra templates in Jy sr (with the convention νI(ν) = const.) it depends on the flux-cut used to construct each mask, but 2 are then converted to µKCMB using the same coefficients as for also on the procedure used to identify spurious detections of the CIB (Sect. 3.2.3). high-latitude Galactic cirrus as point sources in the catalogue. As for the other foregrounds, we then allow for a global free The uncertainty on the flux-cut estimation is particularly im- amplitude, AtSZxCIB, and write portant in the case of radio sources as the flux-cuts considered for CMB analysis (typically around 200 mJy) are close to the peak of the number count. That is the main reason why the
i j,tSZxCIB conv conv νiν j,temp Planck public likelihood analysis considers one amplitude for C = AtSZxCIB a a C . (13) ` νi ν j ` point sources per cross-spectrum.
A41, page 8 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation
0.13 hlpTXE
2 hlpT 0.12 h c hlpE Ω 0.11 hlpX
1.043 s
θ 1.042 0
0 1.041 1 1.040
0.10 τ 0.05
1.00 s n 0.95
3.15 ) s
A 3.10 0 1
0 3.05 1 (
g 3.00 o l
0.0200.0220.0240.026 0.11 0.12 0.13 1.0401.0411.0421.043 0.05 0.10 0.95 1.00 3.00 3.05 3.10 3.15 2 2 100θ τ ns 10 Ωbh Ωch s log(10 As)
Fig. 9. Posterior distribution for the six cosmological ΛCDM parameters for HiLLiPOP and a prior on τ (0.058 ± 0.012).
3.3. Additional priors the constraints from the high-resolution experiments ACT and SPT (see Planck Collaboration XI 2016). As demonstrated in The various parameters considered in the model described in this Couchot et al.(2017), this is not strictly equivalent, in particu- section are not all well constrained by the CMB data themselves. lar for results on A . We choose to leave the correlation free. We complement our model with additional priors coming from L external knowledge. For the instrumental nuisances, Gaussian priors are applied on the calibration coefficients based on the uncertainty estimated 4. HiLLiPOP results in Planck Collaboration VIII(2016): c0 = c1 = c3 = 0 ± 0.002, c4 = c5 = 0.002 ± 0.004 (Table 5), and Apl = 1 ± 0.0025. This section is dedicated to the results derived with the Given its angular resolution, Planck is not equally able to HiLLiPOP likelihood functions (hlpTXE, hlpT, hlpE, and hlpX). constrain the different astrophysical emissions. We choose to We discuss the cosmological parameters as well as the astro- apply Gaussian priors on the dominant ones, including galac- physical foregrounds and instrumental nuisance. We pay partic- tic dust, CIB, thermal-SZ, and point sources. The width of the ular attention to the difference between the results obtained with priors is driven by the uncertainty of the foreground modelling. TT spectra (hlpT) and those obtained with TE spectra (hlpX). We recall that this model tries to capture residuals from highly We choose not to use any low-` information and pre- non-Gaussian and non-isotropic emission using the template in fer to apply a simple prior on the optical reionization depth C` with fixed spectral energy densities (SED). As a consequence, (τ = 0.058 ± 0.012) as given by the lollipop likelihood in it is difficult to derive an accurate estimation of the expected am- Planck Collaboration Int. XLVII(2016). We have checked that, plitudes. We used a Gaussian centred on one with a 20% width for the ΛCDM model, the parameters are undistinguishable (1.0 ± 0.2) as priors for the rescaling amplitudes of the five fore- when using the corresponding Planck low-` likelihood. We radio IR grounds (Adust, ACIB, AtSZ, APS , and APS). use the Gaussian priors on the inter-calibration coefficients and The Planck Collaboration suggests the addition of a 2D on astrophysical rescaling factors (dust, CIB, tSZ, and point prior on both amplitudes of tSZ and kSZ in order to mimic sources) as discussed in Sect. 3.3.
A41, page 9 of 24 A&A 602, A41 (2017)
Table 2. χ2 values compared to the number of degree of freedom 1.0 1.0 1.0 (nd.o.f. = n` − np) for each of the HiLLiPOP likelihoods. 0.8 0.8 0.8
2 2 0.6 0.6 0.6 Likelihood χ nd.o.f. χ /nd.o.f. hlpTXE 27 888.3 25 597 1.090 0.4 0.4 0.4
hlpT 9995.9 9543 1.047 0.2 0.2 0.2 hlpX 9319.9 8799 1.059 0.0 0.0 0.0 hlpE 7304.5 7249 1.008 0.005 0.000 0.005 0.005 0.000 0.005 0.000 0.005 c100A c100B c143B 1.0 1.0
The results described here were obtained using the 0.8 0.8 adaptative-MCMC algorithm implemented in the CAMEL tool- 0.6 0.6 box2. We use the CLASS3 software to compute spectra models 0.4 0.4 for a given cosmology. hlpTXE 2 hlpT The χ values of the best fit for each HiLLiPOP likelihood 0.2 0.2 hlpE are given in Table2. Using our simple foreground model, we hlpX 0.0 0.0 2 0.005 0.000 0.005 0.005 0.000 0.005 are able to fit the Planck data with reasonable χ values and c217A c217B reduced-χ2 comparable to the Planck public likelihood (the ab- solute values are not directly comparable since the Planck public Fig. 10. Posterior distribution of the five inter-calibration parameters for likelihood uses binned cross-power spectra and different fore- each of the HiLLiPOP likelihoods (hlpTXE, hlpT, hlpE, and hlpX). ground modelling). We note that hlpT and hlpX show compara- ble χ2 with a similar number of degrees of freedom. comparable sensitivity for ns; instead, the Planck instrumental noise on TE spectra increases the posterior width by a factor of 4.1. ΛCDM cosmological results almost 2 (Galli et al. 2014). As expected, the results based on the hlpE likelihood are even less accurate. Figure9 shows the posterior distributions of the six ΛCDM pa- rameters reconstructed from each likelihood and their combina- tions, which are summarized in Table3. We find very consistent 4.2. Instrumental nuisances results for cosmology between all the likelihoods. For hlpE, we In the likelihood function, the calibration uncertainties are mod- find a ∼2σ tension on both ns and Ωb, which is not related to the foregrounds or to the multipole range or the sky fraction. elled using an absolute rescaling Apl and inter-calibration fac- Almost all parameters are compatible with the Planck re- tors c. The parameter Apl allows the propagation of an over- sults (Planck Collaboration XIII 2016) within 0.5σ when con- all calibration error at the cross-spectra level (which principally sidering the temperature data only or the full likelihood. Error translates into a larger error on the amplitude of the primordial bars from the Planck public likelihood and HiLLiPOP as pre- power spectrum As). We apply the same calibration factors for sented in this paper are nearly identical. As discussed in detail temperature and polarization. in Couchot et al.(2017), the di fference in τ and As can be under- The constraints on inter-calibration coefficients from the stood as a preference of the HiLLiPOP likelihood for a lower AL Planck CMB data are much weaker than the external priors. (Sect.5). In both cases, the shifted value for AL comes from Without priors, we found that the coefficients are recovered with- a tension between the high-` and the τ constraint (either from out any bias in all cases with posterior width of typically 1.5%, lowTEB or from the prior), the likelihood for HiLLiPOP alone 2%, 7%, and 5% for hlpTXE, hlpT, hlpE, and hlpX, respectively. showing almost no constraint on τ when AL is free. Figure 10 shows the posterior distributions for the inter- The results are compatible with those presented calibration factors (including external priors described in in Couchot et al.(2017), where we used low- ` data from Sect 3.3). We found a slight tension (less than 2σ) between the Planck-LFI (instead of a tighter prior on τ from the last results calibration factors recovered from temperature and for polariza- 2 of Planck-HFI). We also now impose a model for the point tion. The relatively bad χmin value of the full likelihood config- source frequency spectrum (radio sources and infrared sources) uration (Table2) is certainly partially due to this disagreement which increased the sensitivity in ns by ∼15%. between calibrations. We tried to take into account the differ- The hlpX likelihood is almost as sensitive as hlpT to ΛCDM ence between temperature and polarization calibration. To this parameters, although the signal-to-noise ratio is lower in the TE end we added, in the polarization case, additional new parame- spectra. As we discuss in Sect.6, this comes from the uncertain- ters (corresponding to the polarization efficiency) through the ties on the foreground parameters which increase the width of redefinition c → c(1 + ) for the polarization maps. We checked the hlpT posteriors. This is also the case for the Hubble parame- the results with the hlpX and hlpTXE likelihoods. The calibra- ter H0 for which we find tions in temperature are kept fixed and the is are left free in the 2 analysis. We did not see any improvement of the χmin for the full H0 = 67.09 ± 0.86 (hlpT) (15a) likelihood. The level of the calibration shifts is of the order of H0 = 67.16 ± 0.89 (hlpX), (15b) one per mil. We have checked that it has a negligible impact on both the cosmology and the astrophysical parameters. compatible with the low value reported by the Planck Collab- oration (Planck Collaboration XIII 2016). The only parameter which is significantly less constrained by the TE data is ns. In- 4.3. Astrophysical results deed, for a cosmic-variance limited experiment, TT and TE show We recall that the foregrounds in the HiLLiPOP likelihoods are 2 Available at camel.in2p3.fr modelled using fixed spectral energy densities (SED) and that, 3 http://class-code.net for each emission, the only free parameter is an overall rescaling
A41, page 10 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation
Table 3. Central value and 68% confidence limit for the base ΛCDM model with HiLLiPOP likelihoods with a prior on τ (0.058 ± 0.012).
Parameters hlpT hlpX hlpE hlpTXE 2 Ωbh 0.02212 ± 0.00021 0.02210 ± 0.00024 0.02440 ± 0.00106 0.02227 ± 0.00014 2 Ωch 0.1209 ± 0.0021 0.1204 ± 0.0020 0.1130 ± 0.0043 0.1191 ± 0.0012 100θs 1.04164 ± 0.00043 1.04184 ± 0.00047 1.04101 ± 0.00074 1.04179 ± 0.00028 τ 0.062 ± 0.011 0.059 ± 0.012 0.059 ± 0.012 0.067 ± 0.011 ns 0.9649 ± 0.0058 0.9631 ± 0.0108 0.9939 ± 0.0158 0.9672 ± 0.0037 10 log(10 As) 3.058 ± 0.022 3.046 ± 0.027 3.061 ± 0.027 3.065 ± 0.022
1.0 1.0 1.0 and hlpTXE are, respectively, 0.8 0.8 0.8
0.6 0.6 0.6 ACIB = 0.84 ± 0.15 (1.01 ± 0.13), (17)
0.4 0.4 0.4 which is perfectly compatible with the astrophysical measure- 0.2 0.2 0.2 ment from Planck for hlpTEX and at 1σ for hlpT. 0.0 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 A TT A TE A EE 1.0 dust 1.0 dust 1.0 dust
0.8 0.8 0.8 SZ: Planck data are only mildly sensitive to SZ components. In particular, we have no constraint at all on the amplitude of the 0.6 0.6 0.6 kSZ effect (AkSZ) and the correlation coefficient between SZ and 0.4 0.4 0.4 CIB (AtSZxCIB). When using astrophysical foreground informa- 0.2 0.2 0.2 tion, the external prior on AtSZ drives the final posterior: 0.0 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0 1 2 3 4 ACIB ASZ AkSZ 1.0 1.0 1.0 ASZ = 1.00 ± 0.20 (0.94 ± 0.19). (18)
0.8 0.8 0.8 0.6 0.6 0.6 Point sources: We find more power in Planck power spectra for 0.4 0.4 0.4 the radio sources than expected and a bit less for IR sources: 0.2 0.2 0.2 radio 0.0 0.0 0.0 A = 1.61 ± 0.09 (1.62 ± 0.09) (19a) 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 PS ASZxCIB Aradio AIR IR APS = 0.78 ± 0.07 (0.71 ± 0.07), (19b) Fig. 11. Astrophysical foreground amplitude posterior distributions for the HiLLiPOP likelihoods: hlpTXE (black), hlpT (red), hlpE (blue) and with no impact on cosmology (see Sect.6). We have identi- hlpX (green). Priors are also plotted (grey dashed line). fied that the tension comes essentially from the 100 GHz map which dominates the constraints for the radio source amplitude. Table4 shows the results when we fit one amplitude for each amplitude (which should be one if the correct SED is used). The cross-spectra and when compared to the model expectation. The compatibility with one for all foreground amplitudes is thus a distribution of the posteriors for the point sources amplitudes are good test for the consistency of the internal Planck templates. plotted in Fig. 12. We find relatively good agreement between Figure 11 shows the posterior distributions for the astrophysical the predictions from source counts and the HiLLiPOP results, foreground amplitudes. We discuss the results in detail in the with the exception of the 100 × 100 where the measurement dif- following sections. We check the stability of the cosmological fer by up to 4σ with the prediction. This is coherent with the results with respect to foreground parameters in Sect.6. results from the Planck Collaboration (discussed in Sect. 4.3 of Planck Collaboration XI 2016). It could be a sign for resid- Dust: The emission of galactic dust is the dominant residual ual systematics in the data but we recall that an accurate point foreground in the power spectra considered in this analysis. source modelling is very hard to obtain for a large sky coverage The recovered amplitudes for each case (and, in parentheses with inhomogeneous noise as such of Planck. This is particu- for the full likelihood) are larly important for the estimation of the radio sources amplitudes which are sensitive to both catalogue completeness and flux cut TT ± ± Adust = 0.97 0.09 (0.99 0.08) (16a) estimation. TE Adust = 0.86 ± 0.12 (0.80 ± 0.11) (16b) EE Adust = 1.14 ± 0.13 (1.20 ± 0.11). (16c) 5. AL as a robustness test The dust amplitude in temperature is recovered perfectly. The As discussed in Couchot et al.(2017), the measurement of amplitude for the EE polarization mode is found to be slightly the lensing effect in the angular power spectra of the CMB high at 1.5σ, while the TE polarization mode is low at about anisotropies provides a good internal consistency check for high- σ 1.5 . When using the full HiLLiPOP likelihood, the tension on ` likelihoods. The Planck public likelihood shows an AL differ- the dust polarization modes EE and TE reaches 2σ which is di- ent from one by up to 2.6σ. rectly related to the small tension on calibration discussed in With HiLLiPOP and the τ-prior, the best fits for AL (Fig. 13) Sect. 4.2. are A = 1.12 ± 0.09 (hlpT + τ prior) (20a) CIB: The second emission to which Planck TT CMB power L A = 1.07 ± 0.21 (hlpX + τ prior), (20b) spectra are sensitive is the CIB. The ACIB recovered for hlpT L
A41, page 11 of 24 A&A 602, A41 (2017)
Table 4. Poisson amplitudes for radio galaxies (model from Tucci et al. 2011) and dusty galaxies (model from Béthermin et al. 2012) compared 2 −1 to HiLLiPOP results. Units: Jy sr (νIν = const.).
Radio IR Total hlpT hlpTXE 100 × 100 7.8 ± 1.6 0.2 ± 0.0 7.9 ± 1.6 15.5 ± 1.4 15.8 ± 0.9 100 × 143 5.4 ± 1.1 0.5 ± 0.1 5.8 ± 1.1 10.4 ± 1.5 10.5 ± 1.0 100 × 217 4.3 ± 0.9 1.9 ± 0.4 6.2 ± 1.0 10.1 ± 1.7 10.0 ± 1.4 143 × 143 4.8 ± 1.0 1.2 ± 0.2 6.1 ± 1.0 6.3 ± 1.7 5.9 ± 1.2 143 × 217 3.6 ± 0.8 5.1 ± 1.0 8.7 ± 1.3 6.2 ± 1.8 5.3 ± 1.5 217 × 217 3.2 ± 0.8 21.0 ± 3.8 24.2 ± 3.8 16.7 ± 2.2 15.0 ± 2.1
1.0 1.0 1.0 1.0 hlpT 0.8 0.8 0.8 hlpX 0.6 0.6 0.6 0.8
0.4 0.4 0.4
0.2 0.2 0.2 0.6
0.0 0.0 0.0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 100x100 100x143 100x217 APS APS APS 1.0 1.0 1.0 0.4
0.8 0.8 0.8
0.6 0.6 0.6 0.2
0.4 0.4 0.4 0.0 0.2 0.2 0.2 0.00 0.05 0.10 0.15 0.20 0.25 τ 0.0 0.0 0.0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 143x143 143x217 217x217 APS APS APS Fig. 14. Posterior distribution for the reionization optical depth τ for hlpT (red line) and hlpX (green line) compared to the prior from Fig. 12. Posterior distributions for the six point sources amplitudes for Planck Collaboration Int. XLVII(2016) used throughout this analysis hlpTXE (black line) and hlpT (red line) compared to model prediction (dashed dark blue line). 2 −1 (dashed line). Units: Jy sr (νIν = const.).
likelihoods give 1.0 hlpT hlpX τ = 0.122 ± 0.036 (hlpT) (21a) 0.8 τ = 0.103 ± 0.081 (hlpX), (21b)
which is, for hlpT, at 1.7σ from the HFI low-` analysis τ = 0.6 0.058 ± 0.012 (Planck Collaboration Int. XLVII 2016). The dif- ference with the τ estimation derived in Couchot et al.(2017) 0.4 comes directly from the additional constraints in the point source sector. For hlpX, the τ distribution is compatible with the Planck low-` constraint, but the constraint is weaker. 0.2 We note that when adding the information from the measurement of the power spectrum of the lensing po- tential (using the Planck lensing likelihood described in 0.0 0.6 0.8 1.0 1.2 1.4 Planck Collaboration XV 2016) the constraints on τ from hlpT A L and hlpX become comparable
Fig. 13. Posterior distribution for the AL parameter for the temperature likelihood hlpT (red line) and the temperature-polarization likelihood τ = 0.077 ± 0.028 (hlpT + lensing) (22a) hlpX (green line). τ = 0.056 ± 0.027 (hlpX + lensing), (22b)
and compatible with low-`-only results from both Planck-HFI compatible with the standard expectation. While the relative (τ = 0.058 ± 0.012, Planck Collaboration Int. XLVII 2016) and ± variation of the theoretical power spectra with AL is more im- Planck-LFI (τ = 0.067 0.023, Planck Collaboration XI 2016). portant for TE than for TT, we find a weaker constraint for TE. This illustrates the fact that the noise level in the TE power spec- trum from Planck is unable to capture the information from the 6. Foreground robustness: TT vs. TE lensing of the CMB TE at high multipoles. In this section, we investigate the impact of foregrounds on the In Couchot et al.(2017), we have shown that the Planck ten- recovery of the ΛCDM cosmological parameters. We focus on sion on AL is directly related to the constraint on τ. Indeed, the τ the results from hlpT and hlpX. constraints from the HiLLiPOP likelihoods (Fig. 14) are less in First, we show in Fig. 15, the posterior for the parame- tension with the Planck low-` likelihoods. The HiLLiPOP only ters with and without external foreground priors. These results
A41, page 12 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation 2
1.0 1.0 1.0 h 2 b 0.5 2 Ωbh Ω h c
2 s Ω θ 0.8 0.8 0.8 Ωch ) s 0 0.4 0 A
100θ 1 s 0 1
0.6 0.6 0.6 0 1
τ τ ( 0.3 10 g o 0.4 0.4 0.4 log(10 As) l k s c
n n
s n a
l 0.2 1 p k c
0.2 0.2 0.2 m
A n A planck h a 0 l 2 0 k P 1 Planck c m n c c100hm1 h a 0.1 0 l
0.0 0.0 0.0 2 0 k P 1 0.0215 0.0220 0.0225 0.0230 0.115 0.120 0.125 0.001 0.002 0.003 0.004 Planck c m n c 2 2 c100hm2 h a
+1.038 3
100θ l
Ω h Ω h s 1 4 b c k P 1 Planck c m n 1.0 1.0 1.0 c c143hm2 h 0.0 a 7 l 2 1 k P 2 Planck c m n c c217hm1 h a 7 l
0.8 0.8 0.8 1 P 2 Planck o i
c c 0.1 217hm2 d a r
0.6 0.6 0.6 Aradio A R I 0.2 AIR A Z 0.4 0.4 0.4 S
A A SZ B I C
t 0.3 A A s
CIB T 0.2 0.2 0.2 u T TT d A Adust Z S
k 0.4
0.0 0.0 0.0 A A 0.02 0.04 0.06 0.08 0.10 0.92 0.94 0.96 0.98 1.00 2.95 3.00 3.05 3.10 kSZ τ ns 10 log(10 As) ASZxCIB 0.5 2
h 0.5 Fig. 15. Posterior distributions for the six ΛCDM parameters with (solid Ω h2 b b Ω lines) and without (dashed lines) astrophysical foregrounds priors in the 2 h Ω h2 c 0.4 c Ω case of hlpT (red) and hlpX (green). s θ 0
100θ 0 )
s 1
s 0.3 A 0
τ 1 τ 0 demonstrate no impact of the priors on the final results, and sug- 1 0.2 (
10 g
log(10 A ) o gest a low level of correlation between foreground parameters s l 0.1 n s and cosmological parameters in the likelihood. Indeed, the statis- s k n c n a l tics reconstructed from the MCMC samples (Fig. 16) exhibit less p
A 1
planck k A 0.0 c m n h a than 15% correlation between the two sets of parameters. In the 0 l 0
Planck P 1
c 2 k 100hm1 c c case of temperature, we see strong correlations between the in- m n 0.1 h a 0 l 0
Planck P 1
c 2 k strumental parameters on the one hand, and between the astro- 100hm2 c c m n h a 3 l 0.2 4
Planck P physical parameters on the other hand. This is not the case for 1 c 1 k 143hm2 c c m n h a 7 hlpX, which, apart from the cosmological sector, exhibits less l 1
Planck P 2
c 2 0.3 k 217hm1 c c m n than 10% correlation. h a 7 l 1
Planck P c 2 In a second step, we have estimated the contribution of the 217hm2 c 0.4 TE foreground parameters to the error budget of the cosmological Adust parameters (Table5). This analysis assesses the degree to which 0.5 our uncertainties on the nuisance parameters impact the cosmo- Fig. 16. Correlation matrix of the likelihood parameters including logical error budget. A parameter estimation is performed to as- ΛCDM and nuisance parameters for hlpT(top) and hlpX(bottom). The sess the full error for each parameter. Then another parameter colour scale is saturated at 50%. estimation is performed with the foreground parameters fixed to their best-fit values. The confidence intervals recovered in this In temperature, almost all parameters are shifted when changing last case give the statistical uncertainties which are essentially 2 2 the nuisance values, the strongest effect being for Ωbh , Ωch , driven by noise and cosmic variance (and which correspond to TE the errors on parameters if we knew the nuisance parameters per- and ns. On the contrary, we cannot see any impact of the Adust fectly). Finally the foreground error is deduced by quadratically parameter shift even if its best-fit value is at 0.86 compared to 1. subtracting the statistical uncertainty from the total error follow- ing what was done in Planck Collaboration XI(2016). 7. Discussion In the temperature case, we see a strong impact of the nui- 2 sances on the error of Ωbh and ns. The posterior width of the With the currently available CMB measurements, the sensitivity reionization optical depth τ is strongly dominated by the prior to ΛCDM cosmological parameters is dominated by the Planck so it is marginally affected by foreground uncertainties. Finally, data in the `-range typically below ` = 2000 both in TT and even if the statistical uncertainty is larger in the case of TE, TE. For TE, adding higher multipoles coming from the mea- foreground uncertainties are negligible in the total error budget, surements of the South Pole Telescope (Crites et al. 2015) or the which makes them competitive with TT (except for ns). Atacama Cosmology Telescope (Naess et al. 2014), we find al- More important than increasing the error budget, nui- most identical results on ΛCDM cosmological parameters with- sance uncertainties can also bias the cosmological parameters. out any reduction of parameter uncertainties. This is different Figure 17 shows the results on the ΛCDM parameters for hlpT from the temperature data for which high-resolution experiments and hlpX when nuisances are fixed either to their best fit or to help to reduce the uncertainty on foreground parameters, which the value expected by the astrophysical constraints (i.e. scaling indirectly reduces the posterior width for cosmological parame- parameters fixed to 1). This corresponds to the extreme case for ters through their correlation (Couchot et al. 2017). On the low-` the potential bias, where we supposed an exact knowledge of side, measurements of TE at ` < 20 give information about the the characteristics of the complex spatial distribution of fore- reionization optical depth τ (although not equivalent to the low-` grounds and their spectra. The attempt here is to give an idea of from EE) and a longer lever arm for ns. the impact of foreground uncertainties on cosmological parame- We checked the results of the temperature-polarization cross- ters. Once again, we see a stronger impact on hlpT than on hlpX. correlation likelihood on some basic extensions to the ΛCDM
A41, page 13 of 24 A&A 602, A41 (2017)
Table 5. Errors on cosmological parameters within the ΛCDM model 8. Conclusion for hlpT and hlpX. Building a coherent likelihood for CMB data given Planck sen- Parameter Estimate Error sitivity is difficult owing to the complexity of the foreground Full Statistical Foreground emissions modelling. In this paper, we have presented a full tem- hlpT parameters perature and polarization likelihood based on cross-spectra (in- cluding TT, TE, and EE) over a wide range of multipoles (from 2 Ωbh 0.02212 0.00020 0.00018 0.00009 (27%) ` = 50 to 2500). We have described in detail the foreground 2 Ωch 0.1210 0.0021 0.0021 0.0003 (3%) parametrization which relies on the Planck measurements for as- 100θs 1.04164 0.00043 0.00044 0.00000 (0%) trophysical modelling. τ 0.062 0.011 0.011 0.002 (5%) We found results on the ΛCDM cosmological parame- ns 0.9649 0.0058 0.0052 0.0025 (24%) ters consistent between the different likelihoods (hlpT, hlpX, 10 log(10 As) 3.058 0.022 0.022 0.003 (2%) hlpE). The cosmological constraints from this work are di- hlpX parameters rectly comparable to the Planck 2015 cosmological analysis 2 Ωbh 0.02209 0.00024 0.00024 0.00004 (3%) (Planck Collaboration XIII 2016) despite the differences in the 2 foreground modelling adopted in HiLLiPOP. Both instrumental Ωch 0.1204 0.0020 0.0020 0.0005 (6%) and astrophysical nuisance parameters are compatible with ex- 100θs 1.04184 0.00047 0.00047 0.00003 (0%) τ 0.058 0.012 0.012 0.000 (0%) pectations, with the exception of the point source amplitudes in temperature for which we found a small tension with the astro- ns 0.9630 0.0111 0.0107 0.0026 (6%) 10 physical expectations. This tension may be the sign of poten- log(10 As) 3.046 0.026 0.027 0.000 (0%) tial systematic residuals in Planck data and/or uncertainty in the Notes. The full error is split between statistical and foreground errors. foreground model in temperature (especially on the dust SED or Errors are given at 68% confidence level. the various `-shape of the foreground templates). We investigated the robustness of the results with respect to the foreground and nuisance parameters. In particular, we demonstrated the impact of foreground uncertainties on the tem- 1.0 1.0 1.0 perature power spectrum likelihood. We compared these data to
0.8 0.8 0.8 the results from the likelihood based on temperature-polarization cross-correlation which involves fewer foreground components, 0.6 0.6 0.6 but is statistically less sensitive. We found that foreground un- 0.4 0.4 0.4 certainties have a stronger impact on TT than on TE with com- parable final errors (except for n ). Moreover, the hlpX like- 0.2 0.2 0.2 s lihood function does include fewer nuisance parameters (only 0.0 0.0 0.0 0.0215 0.0220 0.0225 0.0230 0.115 0.120 0.125 0.001 0.002 0.003 0.004 7 compared to 13 for hlpT) and shows less correlation in the nui- 2 2 100θ +1.039 Ωbh Ωch s 1.0 1.0 1.0 sance/foregrounds sectors which, in practice, allows much faster sampling. 0.8 0.8 0.8 This work illustrates the fact that TE spectra provide an esti- 0.6 0.6 0.6 mation of the cosmological parameters that are as accurate as TT
0.4 0.4 0.4 while being more robust with respect to foreground contamina- tions. The results from Planck in polarization are still limited by 0.2 0.2 0.2 instrumental noise in TE, but as suggested in Galli et al.(2014),
0.0 0.0 0.0 0.02 0.04 0.06 0.08 0.10 0.94 0.96 0.98 1.00 2.95 3.00 3.05 3.10 future experiments only limited by cosmic variance over a wider τ ns 10 log(10 As) range of multipoles will be able to constrain cosmology with TE even better than with TT. Fig. 17. Posterior distribution for the cosmological parameters when foregrounds are fixed either to their best-fit value (solid lines) or to the expected astrophysical value (dashed lines) for hlpT (red) and hlpX Acknowledgements. The authors thank G. Lagache for the work on the point (green). source mask and the estimation of the infrared sources amplitude, M. Tucci for the estimation of the radio point source amplitude, and P. Serra for the work on the CIBxtSZ power spectra model.
P model: essentially AL, Neff, and mν. Given Planck sensitivity, we do not find any competitive constraints compared to the tem- References perature likelihood. For example, we find an effective number Addison, G. E., Dunkley, J., & Spergel, D. N. 2012, MNRAS, 427, 1741 of relativistic species Neff = 2.45 ± 0.45 for hlpX compared to Alam, S., Ata, M., Bailey, S., et al. 2016, MNRAS, submitted [arXiv:1607.03155] Neff = 2.95 ± 0.32 for hlpT. Adding data from high-resolution experiments, we find N = 2.84 ± 0.43, which does not help Battaglia, N., Natarajan, A., Trac, H., Cen, R., & Loeb, A. 2013, ApJ, 776, 83 eff Béthermin, M., Daddi, E., Magdis, G., et al. 2012, ApJ, 757, L23 reduce the error to the level of temperature data. Betoule, M., Kessler, R., Guy, J., et al. 2014, A&A, 568, A22 Couchot, F., Henrot-Versillé, S., Perdereau, O., et al. 2017, A&A, 597, A126 We combine CMB TE data with complementary informa- Crites, A. T., Henning, J. W., Ade, P. A. R., et al. 2015, ApJ, 805, 36 tion from the late time evolution of the Universe geometry, Efstathiou, G. P. 2006, MNRAS, 370, 343 coming from the Baryon Acoustic Oscillations scale evolu- Efstathiou, G., & Migliaccio, M. 2012, MNRAS, 423, 2492 Galli, S., Benabed, K., Bouchet, F., et al. 2014, Phys. Rev. D, 90, 063504 tion (Alam et al. 2016) and the SNIa magnitude-redshift mea- Hivon, E., Górski, K. M., Netterfield, C. B., et al. 2002, ApJ, 567, 2 surements (Betoule et al. 2014). We find very compatible results Kogut, A., Spergel, D. N., Barnes, C., et al. 2003, ApJS, 148, 161 2 with a significantly better accuracy only on Ωch . Mangilli, A., Plaszczynski, S., & Tristram, M. 2015, MNRAS, 453, 3174
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A41, page 15 of 24 A&A 602, A41 (2017)
Appendix A: Xpol computations A.1. Harmonic decomposition Following notations from Hivon et al.(2002), the decomposition into spherical harmonics reads
X T T(nˆ) = a`mY`m(nˆ) (A.1) `m X Q(nˆ) ± iU(nˆ) = ∓2a`m ∓2Y`m(nˆ) (A.2) `m ( E B 2a`m = a`m + ia`m E B (A.3) −2a`m = a`m − ia`m with 1 Z n o aE ≡ Q(nˆ) − iU(nˆ) Y∗ (nˆ) + Q(nˆ) + iU(nˆ) Y∗ (nˆ) dnˆ (A.4) `m 2 2 `m −2 `m i Z n o aB ≡ − Q(nˆ) − iU(nˆ) Y∗ (nˆ) − Q(nˆ) + iU(nˆ) Y∗ (nˆ) dnˆ. (A.5) `m 2 2 `m −2 `m When including a sky window function (or mask) w, then X X Z T = aT wT Y∗ (nˆ) Y (nˆ) Y (nˆ)dnˆ (A.6) `m `1m1 `2m2 0 `m 0 `1m1 0 `2m2 `1m1 `2m2 Z 1 X E B X E ∗ E B E`m = (a + ia ) w 2Y (nˆ) 2Y (nˆ) 0Y (nˆ)dnˆ + (a − ia ) 2 `1m1 `1m1 `2m2 `m `1m1 `2m2 `1m1 `1m1 `1m1 `2m2 X Z × wE Y∗ (nˆ) Y (nˆ) Y (nˆ) (A.7) `2m2 −2 `m −2 `1m1 0 `2m2 `2m2 Z i X E B X B ∗ E B B`m = − (a + ia ) w 2Y (nˆ) 2Y (nˆ) 0Y (nˆ)dnˆ − (a − ia ) 2 `1m1 `1m1 `2m2 `m `1m1 `2m2 `1m1 `1m1 `1m1 `2m2 X Z × wB Y∗ (nˆ) Y (nˆ) Y (nˆ), (A.8) `2m2 −2 `m −2 `1m1 0 `2m2 `2m2
P W where the window function is decomposed into spin-0 spherical harmonics (w = `m a`m 0Y`m). X We define sK with s ∈ {−2, 0, 2} (X = T, E, B), `1m1`2m2 Z X Z KX ≡ Y∗ (nˆ) Y (nˆ)wX(nˆ)dnˆ = wX Y∗ (nˆ) Y (nˆ) Y (nˆ)dnˆ; (A.9) s `1m1`2m2 s `1m1 s `2m2 `3m3 s `1m1 s `2m2 0 `3m3 `3m3 we can then rewrite E`m and B`m as X T = aT KT (A.10) `m `1m1 0 `m`1m1 `1m1 ( ) 1 X E h E E i B h E E i E`m = a 2K + −2K + ia 2K − −2K (A.11) 2 `1m1 `m`1m1 `m`1m1 `1m1 `m`1m1 `m`1m1 `1m1 ( i X E h B B i B h B B i B`m = − a 2K − −2K + ia 2K + −2K (A.12) 2 `1m1 `m`1m1 `m`1m1 `1m1 `m`1m1 `m`1m1 `1m1
We define W`1m1`2m2 , ( W 0,X ≡ KX `1m1`2m2 0 `1m1`2m2 ±,X X X . (A.13) W ≡ K ± − K `1m1`2m2 2 `1m1`2m2 2 `1m1`2m2 Thus, finally
0,T T T W`m` m 0 0 a `m X 1 1 `1m1 1 +,E i −,E E E`m = 0 W`m` m W`m` m a . (A.14) 2 1 1 2 1 1 `1m1 i −,B 1 +,B B B`m `1m1 0 − W`m` m W`m` m a 2 1 1 2 1 1 `1m1
A41, page 16 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation
A.2. Power spectra (first order a`m correlation)
We suppose independent data sets (i, j) for which we have (I, Q, U) maps and compute the spherical transform to obtain X`m (for X ∈ {T, E, B}) coefficients. The cross-power spectra are thus defined as
1 X` D E C˜XaYb = Xa Yb∗ . (A.15) ` 2` + 1 `m `m m=−`
D a b∗ E We have to compute X`mY`0m0 for each set of (X, Y) ∈ T, E, B using Eq. (A.14). In this section, we will neglect the terms in EB and TB with respect to other mode correlation.
A.2.1. Temperature D E X X D E a b∗ Ta Tb∗ 0,Ta 0,Tb∗ T T 0 0 = a a W W 0 0 `m ` m `1m1 `2m2 `m`1m1 ` m `2m2 `1m1 `2m2 with
D Xa Xb∗ E Xa Xb a a = δ` ` δm m C (x = T, E, B) (A.16) `1m1 `2m2 1 2 1 2 `1 Thus, D E X a b∗ TaTb 0,Ta 0,Tb∗ T T 0 0 = C W W 0 0 (A.17) `m ` m `1 `m`1m1 ` m `1m1 `1m1
A.2.2. Polarization D E X X * + a∗ b 1 Ea∗ +,Ea∗ Ba∗ −,Ea∗ Eb +,Eb Bb −,Eb E E 0 0 = a W − ia W × a W 0 0 + ia W 0 0 `m ` m 4 `1m1 `m`1m1 `1m1 `m`1m1 `2m2 ` m `2m2 `2m2 ` m `2m2 `1m1 `2m2 using (A.16): D E X a∗ b 1 Ea Eb +,Ea∗ +,Eb Ea Bb +,Ea∗ −,Eb E E 0 0 = C W W 0 0 + iC W W 0 0 `m ` m 4 `1 `m`1m1 ` m `1m1 `1 `m`1m1 ` m `1m1 `1m1 Ba Eb −,Ea∗ +,Eb Ba Bb −,Eb∗ −,Eb − iC W W 0 0 + C W W 0 0 `1 `m`1m1 ` m `1m1 `1 `m`1m1 ` m `1m1
EB D a∗ b E We neglect C` , so that E`mE`0m0 is real and reads
D E X n o a∗ b 1 Ea Eb +,Ea∗ +,Eb Ba Bb −,Ea∗ −,Eb E E 0 0 = C W W 0 0 + C W W 0 0 (A.18) `m ` m 4 `1 `m`1m1 ` m `1m1 `1 `m`1m1 ` m `1m1 `1m1
Analogously, D E X X Dn o n oE a∗ b 1 Ba∗ +,Ba∗ Ea∗ −,Ba∗ Bb +,Bb Eb −,Bb B B 0 0 = a W + ia W × a W 0 0 − ia W 0 0 `m ` m 4 `1m1 `m`1m1 `1m1 `m`1m1 `2m2 ` m `2m2 `2m2 ` m `2m2 `1m1 `2m2
D E X n o a∗ b 1 Ba Bb +,Ba∗ +,Bb Ea Eb −,Ba∗ −,Bb B B 0 0 = C W W 0 0 + C W W 0 0 (A.19) `m ` m 4 `1 `m`1m1 ` m `1m1 `1 `m`1m1 ` m `1m1 `1m1
A.2.3. Cross modes D E X X Dn o n oE a∗ b 1 Ta∗ 0,Ta∗ Eb +,Eb Bb −,Eb T E 0 0 = a W × a W 0 0 + ia W 0 0 `m ` m 2 `1m1 `m`1m1 `2m2 ` m `2m2 `2m2 ` m `2m2 `1m1 `2m2
TB D a∗ b E We neglect C` null, so that T`mE`0m0 is real and reads
D E D E X n o a∗ b b∗ a 1 Ta Eb 0,Ta∗ +,Eb T E 0 0 = E T 0 0 = C W W 0 0 (A.20) `m ` m `m ` m 2 `1 `m`1m1 ` m `1m1 `1m1
A41, page 17 of 24 A&A 602, A41 (2017) and D E X X Dn o n oE a∗ b 1 Ta∗ 0,Ta∗ Bb +,Bb Eb −,Bb T B 0 0 = a W × a W 0 0 − ia W 0 0 `m ` m 2 `1m1 `m`1m1 `2m2 ` m `2m2 `2m2 ` m `2m2 `1m1 `2m2
D E D E X n o a∗ b b∗ a 1 Ta Bb 0,Ta∗ +,Bb Ta Eb 0,Ta∗ −,Eb T B 0 0 = B T 0 0 = C W W 0 0 − iC W W 0 0 (A.21) `m ` m `m ` m 2 `1 `m`1m1 ` m `1m1 `1 `m`1m1 ` m `1m1 `1m1
D E X X Dn o n oE a∗ b 1 Ea∗ +,Ea∗ Ba∗ −,Ea∗ Bb +,Bb Eb −,Bb E B 0 0 = a W − ia W × a W 0 0 − ia W 0 0 `m ` m 4 `1m1 `m`1m1 `1m1 `m`1m1 `2m2 ` m `2m2 `2m2 ` m `2m2 `1m1 `2m2
D E D E n o a∗ b b∗ a 1 P Ba Eb −,Ea∗ −,Bb Ea Bb +,Ea∗ +,Bb E B 0 0 = B E 0 0 = ` m C W`m` m W 0 0 − C W`m` m W 0 0 `m ` m `m ` m 4 1 1 `1 1 1 ` m `1m1 `1 1 1 ` m `1m1 (A.22) n Ea Eb +,Ea∗ −,Bb Ba Bb −,Ea∗ +,Bb o −i C W W 0 0 + C W W 0 0 `1 `m`1m1 ` m `1m1 `1 `m`1m1 ` m `1m1
A.2.4. Application to C` ⊕,X For Eq. (A.15), we need to compute the product of 2 W`m`0m0 as in Eqs. (A.17)–(A.22). Here, we wrote integrals as 3j-Wigner symbols using
Z " 0 00 #1/2 0 00 ! 0 00 ! ∗ s+m (2` + 1)(2` + 1)(2` + 1) ` ` ` ` ` ` dˆn Y (ˆn) 0 Y 0 0 (ˆn) 00 Y 00 00 (ˆn) = (−1) (A.23) s lm s ` m s ` m 4π −s s0 s00 −m m0 m00 and make use of the orthogonality for the spinned-harmonics: ! ! P `1 `2 ` `1 `2 ` (2` + 1) 0 0 = δm m0 δm m0 `m 1 1 2 2 m1 m2 m m1 m2 m ! 0 ! . (A.24) P `1 `2 ` `1 `2 ` m m (2` + 1) 0 = δ``0 δmm0 1 2 m1 m2 m m1 m2 m
We then define
!2 X 1 X ` ` ` ≡ 0,a 0,b∗ Wa,Wb 1 2 3 MTT,TT(`1, `2; a, b) L`1`2 W` m ` m W` m ` m = (2`3 + 1)C` 1 1 2 2 1 1 2 2 4π 3 0 0 0 m1m2 `3 ! ! L` ` X 1 X 1 2 0,a +,b∗ Wa,Wb L `1 `2 `3 `1 `2 `3 MTE,TE(`1, `2; a, b) ≡ W` m ` m W` m ` m = (2`3 + 1)C (1 + (−1) ) 2 1 1 2 2 1 1 2 2 8π `3 0 0 0 −2 2 0 m1m2 `3 !2 L X 1 X `1`2 +,a +,b∗ Wa,Wb L 2 `1 `2 `3 MEE,EE(`1, `2; a, b) ≡ W` m ` m W` m ` m = (2`3 + 1)C (1 + (−1) ) 4 1 1 2 2 1 1 2 2 16π `3 −2 2 0 m1m2 `3 !2 L X 1 X `1`2 −,a −,b∗ Wa,Wb L 2 `1 `2 `3 MEE,BB(`1, `2; a, b) ≡ W W = (2`3 + 1)C (1 − (−1) ) 4 `1m1`2m2 `1m1`2m2 16π `3 −2 2 0 m1m2 `3 (A.25)
1 Wa,Wb 0 with L`` ≡ (2`+1)(2`0+1) , L = `1 + `2 + `3, which depend only on the scalar cross-power spectrum of the masks C` = P a b∗ m w w /(2` + 1). `m `m ab TaTb Ea Eb Ba Bb Ta Eb ˜ Finally, for C` = C` , C` , C` , C` , the relation between pseudo-C` (C`) and C` is given using the general coupling matrix
˜ ab 0 a×b ab C` = (2` + 1)M``0 C`0 , (A.26) and the coupling matrix M that translates pseudo-spectra to power spectra reads (see Kogut et al. 2003) MTT,TT 0 0 0 a×b 0 MEE,EE MEE,BB 0 0 M 0 = (`, ` ; wa, wb). (A.27) `` 0 MEE,BB MEE,EE 0 0 0 0 MTE,TE
A41, page 18 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation
A.3. Covariance matrix (second-order a`m correlation) ab,cd We want to write the correlation matrix Σ``0 that gives the correlation between cross-spectra (ab) and (cd) and between multipoles ` and `0 ab,cd D ab cd∗E ab −1 ab cd∗ cd∗ −1 0 ≡ ∆C ∆C 0 = M ∆C˜ ∆C˜ M 0 (A.28) Σ`` ` ` ``1 `1 `2 ` `2 with the pseudo-covariance matrix Σ˜ : ˜ ab,cd ˜ ab ˜ cd∗ ˜ ab ˜ cd∗ ab cd∗ Σ``0 = ∆C` ∆C`0 = C` C`0 − C` C`0 . (A.29)
We write the 4-a`m correlations : Xa Xb Xc Xd∗ 1 X D a b∗ c∗ d E C˜ C˜ 0 = X X X 0 0 X 0 0 (X = T , E , B ). (A.30) ` ` (2` + 1)(2`0 + 1) `m `m ` m ` m `m `m `m `m mm0 We use Isserlis’ formula (or Wick’s theorem), which gives for Gaussian variables D E D E D E D E xi x j xk xl = xi x j hxk xli + hxi xki x j xl + hxi xli x j xk . (A.31)
Thus (A.30) reads X D Xa Xb Xc Xd∗E D a b∗ ED c∗ d E D a c∗ ED b∗ d E D a d ED b∗ c∗ E ˜ ˜ 0 C` C`0 = L`` X`mX`m X`0m0 X`0m0 + X`mX`0m0 X`mX`0m0 + X`mX`0m0 X`mX`0m0 mm0 X Xa Xb Xc Xd∗ D a c∗ ED b∗ d E D a d ED b∗ c∗ E 0 = C` C`0 + L`` X`mX`0m0 X`mX`0m0 + X`mX`0m0 X`mX`0m0 (A.32) mm0 X D Xa Xb Xc Xd∗E D a c∗ ED b∗ d E D a d ED b∗ c∗ E ˜ ˜ 0 ∆C` ∆C`0 = L`` X`mX`0m0 X`mX`0m0 + X`mX`0m0 X`mX`0m0 . (A.33) mm0 We know that
∗ m X`m = (−1) X`−m. (A.34) We can thus replace m0 by −m0 in the sum of the right-hand side of (A.33) and use (A.34)
D a d ED b∗ c∗ E −2m0 D a d∗ ED b∗ c E X`mX`0m0 X`mX`0m0 = (−1) X`mX`0m0 X`mX`0m0 .
And finally, elements of the pseudo-covariance matrix Σ˜ reads
∗ X D ED E D ED E ˜ Xa Xb ˜ Xc Xd a c∗ b∗ d a d∗ b∗ c ∆C` ∆C`0 = L``0 X`mX`0m0 X`mX`0m0 + X`mX`0m0 X`mX`0m0 . (A.35) mm0
A.3.1. Basic properties We recall some basic properties:
s+m ∗ sY`m = (−1) −sY`−m, (A.36) which leads to Z ∗ Z Y∗ (nˆ) Y (nˆ) Y (nˆ)dnˆ = Y (nˆ) Y∗ (nˆ) Y∗ (nˆ)dnˆ s `1m1 s `2m2 0 `3m3 s `1m1 s `2m2 0 `3m3 Z = (−1)(s+s+0) Y∗ (nˆ) Y (nˆ) Y (nˆ)dnˆ −s `1−m1 −s `2−m2 0 `3−m3 Z = Y∗ (nˆ) Y (nˆ) Y (nˆ)dnˆ. −s `1−m1 −s `2−m2 0 `3−m3
From this we deduce that ∗ KX = KX . ±2 `1m1`2m2 ∓2 `1−m1`2−m2 A41, page 19 of 24 A&A 602, A41 (2017)
∗ ∗ ∗ Thus for W ⊕,X we have W 0,X = W 0,X , W +,X = W +,X , and W −,X = −W −,X . We also recall that the `1m1`2m2 `1m1`2m2 `1−m1`2−m2 `1m1`2m2 `1−m1`2−m2 `1m1`2m2 `1−m1`2−m2 spin-lowering and spin-raising derivative reads
s (` − s)! Y = ðsY , 0 ≤ s ≤ ` (A.37) s `m (` + s)! `m s (` + s)! Y = (−1)sð¯−sY , −` ≤ s ≤ 0 (A.38) s `m (` − s)! `m and
(ðs)∗ = ð¯ s. (A.39)
Other important properties include the following:
p ð (sY`m) = + (` − s)(` + s + 1) s+1Y`m (A.40) p ð¯ (sY`m) = − (` + s)(` − s + 1) s−1Y`m. (A.41)
Using spin-raising (resp. spin-lowering) operators (A.37) and (A.38) on Y`m(ˆn j) and then integrating twice by part, we notice that
s Z Z 0 ∗ (` − 2)! 2 ∗ −2Y 0 0 (ˆn j) −2Y (ˆn j)w jdˆn j = ð¯ Y 0 0 (ˆn j) −2Y (ˆn j) w j dˆn j ` m `1m1 (`0 + 2)! ` m `1m1 s Z 0 (` − 2)! 2 ∗ = ð Y 0 0 (ˆn j) −2Y (ˆn j) w j dˆn j (`0 + 2)! ` m `1m1 s Z 0 (` − 2)! ∗ 2 = Y 0 0 (ˆn j) ð −2Y (ˆn j) w j dˆn j (`0 + 2)! ` m `1m1 s Z 0 (` − 2)! ∗ 2 ' Y 0 0 (ˆn j) ð −2Y (ˆn j) w j dˆn j (`0 + 2)! ` m `1m1 s s Z 0 (` − 2)! (`1 + 2)! ∗ = Y 0 0 (ˆn ) Y (ˆn ) w dˆn 0 ` m j `1m1 j j j (` + 2)! (`1 − 2)! s Z 0 (` − 2)! ∗ 2 = Y 0 0 (ˆn j)ð¯ 2Y (ˆn j) w j dˆn j (`0 + 2)! ` m `1m1 s Z 0 (` − 2)! 2 ∗ ' ð¯ Y 0 0 (ˆn j) w j +2Y (ˆn j) dˆn j (`0 + 2)! ` m `1m1 s Z 0 (` − 2)! 2 ∗ = ð Y 0 0 (ˆn j) w j +2Y (ˆn j) dˆn j (`0 + 2)! ` m `1m1 Z ∗ ' Y 0 0 (ˆn ) Y (ˆn ) w dˆn , +2 ` m j +2 `1m1 j j j
where we neglected gradients of the window function w j = w(ˆn j). Finally, we also have the completeness relation for spherical harmonics (Varshalovich et al. 1988):
X ∗ sY`m(ˆni)sY`m(ˆn j) = δ(ˆni − nˆ j). (A.42) `m
A41, page 20 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation
A.3.2. Product of 2 W ⊕,X `1m1`2m2
We neglect gradients of the window function and apply the completeness relation for spherical harmonics (Eq. (A.42))
Z X 0,X∗ 0,Y X X Y ∗ ∗ W W 0 0 = w w dˆnidˆn jY`m(ˆni)Y (ˆni)Y 0 0 (ˆn j)Y` m (ˆn j) `m`1m1 ` m `1m1 i j `1m1 ` m 1 1 i j `1m1 `1m1 Z X Y ∗ = wi wi dˆniY`m(ˆni)Y`0m0 (ˆni) i 0 X Y 0,XY = W`m`0m0 (w w ) ≡ W`m`0m0 (A.43) Z X +,X∗ +,Y X X Y ∗ ∗ W W 0 0 = w w dˆnidˆn j +2Y (ˆni) +2Y (ˆni) +2Y 0 0 (ˆn j) +2Y (ˆn j) `m`1m1 ` m `1m1 i j `m `1m1 ` m `1m1 i j `1m1 `1m1 ∗ ∗ ∗ ∗ + Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) + Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) +2 `m i +2 `1m1 i −2 ` m j −2 `1m1 j −2 `m i −2 `1m1 i +2 ` m j +2 `1m1 j ∗ ∗ + Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) −2 `m i −2 `1m1 i −2 ` m j −2 `1m1 j Z X X Y ∗ ∗ ' 2 w w dˆn dˆn Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) i j i j +2 `m i +2 `1m1 i +2 ` m j +2 `1m1 j i j `1m1 ∗ ∗ + Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) −2 `m i −2 `1m1 i −2 ` m j −2 `1m1 j Z X Y ∗ ∗ ' 2 wi wi dˆni 2Y`m(ˆni) 2Y`0m0 (ˆni) + −2Y`m(ˆni) −2Y`0m0 (ˆni) i +,XY ' 2 W`m`0m0 (A.44) Z X −,X∗ −,Y X X Y ∗ ∗ W W 0 0 = w w dˆnidˆn j +2Y (ˆni) +2Y (ˆni) +2Y 0 0 (ˆn j) +2Y (ˆn j) `m`1m1 ` m `1m1 i j `m `1m1 ` m `1m1 i j `1m1 `1m1 ∗ ∗ ∗ ∗ − Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) − Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) +2 `m i +2 `1m1 i −2 ` m j −2 `1m1 j −2 `m i −2 `1m1 i +2 ` m j +2 `1m1 j ∗ ∗ + Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) −2 `m i −2 `1m1 i −2 ` m j −2 `1m1 j ' 0 (A.45) Z X 0,X∗ +,Y X X Y ∗ ∗ ∗ W W 0 0 = w w dˆnidˆn jY (ˆni)Y (ˆni) 2Y 0 0 (ˆn j) 2Y (ˆn j) + −2Y 0 0 (ˆn j) −2Y (ˆn j) `m`1m1 ` m `1m1 i j `m `1m1 ` m `1m1 ` m `1m1 i j `1m1 `1m1 Z X X Y ∗ ∗ ' 2 w w dˆn dˆn Y (ˆn )Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) i j i j `m i `1m1 i ` m j `1m1 j i j `1m1 Z X Y ∗ ' 2 (wi wi )dˆniY`m(ˆni) Y`0m0 (ˆni) i 0,XY ' 2 W`m`0m0 (A.46) Z X 0,X∗ −,Y X X Y ∗ h ∗ ∗ i W W 0 0 = w w dˆnidˆn jY (ˆni)Y (ˆni) 2Y 0 0 (ˆn j) 2Y (ˆn j) − −2Y 0 0 (ˆn j) −2Y (ˆn j) `m`1m1 ` m `1m1 i j `m `1m1 ` m `1m1 ` m `1m1 i j `1m1 `1m1 ' 0 (A.47) Z X +,X∗ −,Y X X Y ∗ ∗ W W 0 0 = w w dˆnidˆn j +2Y (ˆni) +2Y (ˆni) +2Y 0 0 (ˆn j) +2Y (ˆn j) `m`1m1 ` m `1m1 i j `m `1m1 ` m `1m1 i j `1m1 `1m1 ∗ ∗ ∗ ∗ − Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) + Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) +2 `m i +2 `1m1 i −2 ` m j −2 `1m1 j −2 `m i −2 `1m1 i +2 ` m j +2 `1m1 j ∗ ∗ − Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) −2 `m i −2 `1m1 i −2 ` m j −2 `1m1 j Z X X Y ∗ ∗ ' 2 w w dˆn dˆn Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) i j i j +2 `m i +2 `1m1 i +2 ` m j +2 `1m1 j i j `1m1 ∗ ∗ − Y (ˆn ) Y (ˆn ) Y 0 0 (ˆn ) Y (ˆn ) −2 `m i −2 `1m1 i −2 ` m j −2 `1m1 j Z X Y ∗ ∗ ' 2 wi wi dˆni 2Y`m(ˆni) 2Y`0m0 (ˆni) − −2Y`m(ˆni) −2Y`0m0 (ˆni) i −,XY ' 2 W`m`0m0 (A.48)
A41, page 21 of 24 A&A 602, A41 (2017)
A.3.3. Variance of pseudo-C` We now write elements of the pseudo-covariance matrix Σ˜ (Eq. (A.35)). We consider approximation of high multipoles (greater than X Y the width of the window function W`) for which, if the Galactic cut is sufficiently narrow, we can the replace the product C C by q `1 `2 X Y X X Y Y 0,±,X C``0 C``0 = C` C`0 C` C`0 allowing the matrix to be symmetric. Then we apply the relation to the product of W`m`0m0 (Section A.3.2) 0 on (`1, m1) and (`2, m2) successively. Finally, we identify the kernels M(`, ` , a, b) as defined in Eqs. (A.25). X D TaTb TcTd?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` T`mT`0m0 T`mT`0m0 + T`mT`0m0 T`mT`0m0 (A.49) mm0 X X X TaTc TbTd 0,Ta 0,Tc? 0,Tb? 0,Td TaTd TbTc 0,Ta 0,Td? 0,Tb? 0,Tc = L``0 C C W W 0 0 W W 0 0 + C C W W 0 0 W 0 0 W `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 `1 `2 `m`1m1 ` m `1m1 ` m `2m2 `m`2m2 0 mm `1m1 `2m2 X X TaTc TbTd 0,TaTc 0,TbTd? TaTd TbTc 0,TaTd 0,TbTc? 0 0 ' L`` C``0 C``0 W`m`0m0 W`m`0m0 + L`` C``0 C``0 W`m`0m0 W`m`0m0 mm0 mm0 D E ˜TaTb ˜TcTd? TaTc TbTd 0 T T T T TaTd TbTc 0 T T T T ∆C` ∆C`0 ' C``0 C``0 MTT,TT(`, ` ; wa wc , wb wd ) + C``0 C``0 MTT,TT(`, ` ; wa wd , wb wc ) (A.50)
X D Ea Eb Ec Ed?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` E`mE`0m0 E`mE`0m0 + E`mE`0m0 E`mE`0m0 (A.51) mm0 XXX 1 Ea Ec Eb Ed +,Ea +,Ec? +,Eb? +,Ed Ea Ed Eb Ec +,Ea +,Ed? +,Eb? +,Ec = L``0 C C W W 0 0 W W 0 0 +C C W W 0 0 W 0 0 W `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 `1 `2 `m`1m1 ` m `1m1 ` m `2m2 `m`2m2 16 0 mm `1m1`2m2 X X 1 Ea Ec Eb Ed +,Ea Ec +,Eb Ed? 1 Ea Ed Eb Ec +,Ea Ed +,Eb Ec? ' L 0 C 0 C 0 W 0 0 W 0 0 + L 0 C 0 C 0 W 0 0 W 0 0 4 `` `` `` `m` m `m` m 4 `` `` `` `m` m `m` m mm0 mm0 D E ˜Ea Eb ˜Ec Ed? Ea Ec Eb Ed 0 P P P P Ea Ed Eb Ec 0 P P P P ∆C` ∆C`0 ' C``0 C``0 MEE,EE(`, ` ; wa wc , wb wd ) + C``0 C``0 MEE,EE(`, ` ; wa wd , wb wc ) (A.52)
X D Ba Bb Bc Bd?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` B`mB`0m0 B`mB`0m0 + B`mB`0m0 B`mB`0m0 (A.53) mm0 XXX 1 Ba Bc Bb Bd +,Ba +,Bc? +,Bb? +,Bd Ba Bd Bb Bc +,Ba +,Bd? +,Bb? +,Bc = L``0 C C W W 0 0 W W 0 0 +C C W W 0 0 W 0 0 W `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 `1 `2 `m`1m1 ` m `1m1 ` m `2m2 `m`2m2 16 0 mm `1m1`2m2 X X 1 Ba Bc Bb Bd +,Ba Bc +,Bb Bd? 1 Ba Bd Bb Bc +,Ba Bd +,Bb Bc? ' L 0 C 0 C 0 W 0 0 W 0 0 + L 0 C 0 C 0 W 0 0 W 0 0 4 `` `` `` `m` m `m` m 4 `` `` `` `m` m `m` m mm0 mm0 D E ˜ Ba Bb ˜ Bc Bd? Ba Bc Bb Bd 0 P P P P Ba Bd Bb Bc 0 P P P P ∆C` ∆C`0 ' C``0 C``0 MEE,EE(`, ` ; wa wc , wb wd ) + C``0 C``0 MEE,EE(`, ` ; wa wd , wb wc ) (A.54)
X D Ea Eb Bc Bd?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` E`mB`0m0 E`mB`0m0 + E`mB`0m0 E`mB`0m0 (A.55) mm0 1 P P P Ea Ec Eb Ed +,Ea −,Bc? +,Eb? −,Bd L``0 0 C C W W 0 0 W W 0 0 16 mm `1m1 `2m2 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 Ea Ec Bb Bd +,Ea −,Bc? −,Eb? +,Bd + C C W W 0 0 W W 0 0 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 Ba Bc Eb Ed −,Ea +,Bc? +,Eb? −,Bd + C C W W 0 0 W W 0 0 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 Ba Bc Bb Bd −,Ea +,Bc? −,Eb? +,Bd + C C W W 0 0 W W 0 0 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 = Ea Ed Eb Ec +,Ea −,Bd? +,Eb? −,Bc + C C W W 0 0 W W 0 0 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 Ea Ed Bb Bc +,Ea −,Bd? −,Eb? +,Bc + C C W W 0 0 W W 0 0 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 Ba Bd Eb Ec −,Ea +,Bd? +,Eb? −,Bc + C C W W 0 0 W W 0 0 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 Ba Bd Bb Bc −,Ea +,Bd? −,Eb? +,Bc + C C W W 0 0 W W 0 0 `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2
1 Ea Ec Eb Ed Ea Ec Bb Bd Ba Bc Eb Ed Ba Bc Bb Bd P −,Ea Bc −,Eb Bd? L 0 (C 0 C 0 + C 0 C 0 + C 0 C 0 + C 0 C 0 ) 0 W 0 0 W 0 0 ' 4 `` `` `` `` `` `` `` `` `` mm `m` m `m` m 1 Ea Ed Eb Ec Ea Ed Bb Bc Ba Bd Eb Ec Ba Bd Bb Bc P −,Ea Bd −,Eb Bc? 0 + 4 L`` (C``0 C``0 + C``0 C``0 + C``0 C``0 + C``0 C``0 ) mm0 W`m`0m0 W`m`0m0
D E Ea Eb Bc Bd? Ea Ec Eb Ed Ea Ec Bb Bd Ba Bc Eb Ed Ba Bc Bb Bd 0 P P P P ∆C˜ ∆C˜ 0 ' (C 0 C 0 + C 0 C 0 + C 0 C 0 + C 0 C 0 )M (`, ` ; w w , w w )+ ` ` `` `` `` `` `` `` `` `` EE,BB a c b d (A.56) Ea Ed Eb Ec Ea Ed Bb Bc Ba Bd Eb Ec Ba Bd Bb Bc 0 P P P P (C``0 C``0 + C``0 C``0 + C``0 C``0 + C``0 C``0 )MEE,BB(`, ` ; wa wd , wb wc )
A41, page 22 of 24 F. Couchot et al.: Cosmology with the cosmic microwave background temperature-polarization correlation
X D Ta Eb Tc Ed?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` T`mT`0m0 E`mE`0m0 + T`mE`0m0 E`mT`0m0 (A.57) mm0 X X X 1 TaTc Eb Ed 0,Ta 0,Tc? +,Eb? +,Ed Ta Ed EbTc 0,Ta? +,Ed +,Eb 0,Tc? = L``0 C C W W 0 0 W W 0 0 +C C W W 0 0 W 0 0 W `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 `1 `2 `m`1m1 ` m `1m1 ` m `2m2 `m`2m2 4 0 mm `1m1 `2m2 X X 1 TaTc Eb Ed 0,TaTc +,Eb Ed? Ta Ed EbTc 0,Ta Ed 0,EbTc? ' L 0 C 0 C 0 W 0 0 W 0 0 + L 0 C 0 C 0 W 0 0 W 0 0 2 `` `` `` `m` m `m` m `` `` `` `m` m `m` m mm0 mm0 D E ˜Ta Eb ˜Tc Ed? TaTc Eb Ed 0 T T P P Ta Ed EbTc 0 T P P T ∆C` ∆C`0 ' C``0 C``0 MTP,TP(`, ` ; wa wc , wb wd ) + C``0 C``0 MTT,TT(`, ` ; wa wd , wb wc ) (A.58)
X D TaTb Tc Ed?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` T`mT`0m0 T`mE`0m0 + T`mE`0m0 T`mT`0m0 (A.59) mm0 X X X 1 TaTc Tb Ed 0,Ta 0,Tc? 0,Tb? +,Ed Ta Ed TbTc 0,Ta? +,Ed 0,Tb 0,Tc? = L``0 C C W W 0 0 W W 0 0 + C C W W 0 0 W 0 0 W `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 `1 `2 `m`1m1 ` m `1m1 ` m `2m2 `m`2m2 2 0 mm `1m1 `2m2 X X TaTc Tb Ed 0,TaTc 0,Tb Ed? Ta Ed TbTc 0,Ta Ed 0,TbTc? 0 0 ' L`` C``0 C``0 W`m`0m0 W`m`0m0 + L`` C``0 C``0 W`m`0m0 W`m`0m0 mm0 mm0 D E ˜TaTb ˜Tc Ed? TaTc Tb Ed 0 T T T P Ta Ed TbTc 0 T P T T ∆C` ∆C`0 ' C``0 C``0 MTT,TT(`, ` ; wa wc , wb wd ) + C``0 C``0 MTT,TT(`, ` ; wa wd , wb wc ) (A.60)
X D TaTb Ec Ed?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` T`mE`0m0 T`mE`0m0 + T`mE`0m0 T`mE`0m0 (A.61) mm0 X X X 1 Ta Ec Tb Ed 0,Ta +,Ec? 0,Tb? +,Ed Ta Ed Tb Ec 0,Ta? +,Ed 0,Tb +,Ec? = L``0 C C W W 0 0 W W 0 0 + C C W W 0 0 W 0 0 W `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 `1 `2 `m`1m1 ` m `1m1 ` m `2m2 `m`2m2 4 0 mm `1m1 `2m2 X X Ta Ec Tb Ed 0,Ta Ec 0,Tb Ed? Ta Ed Tb Ec 0,Ta Ed 0,Tb Ec? 0 0 ' L`` C``0 C``0 W`m`0m0 W`m`0m0 + L`` C``0 C``0 W`m`0m0 W`m`0m0 mm0 mm0 D E ˜TaTb ˜Ec Ed? Ta Ec Tb Ed 0 T P T P Ta Ed Tb Ec 0 T P T P ∆C` ∆C`0 ' C``0 C``0 MTT,TT(`, ` ; wa wc , wb wd ) + C``0 C``0 MTT,TT(`, ` ; wa wd , wb wc ) (A.62)
X D Ea Eb Tc Ed?E D a c∗ ED b∗ d E D a d∗ ED b∗ c E ˜ ˜ 0 ∆C` ∆C`0 = L`` E`mT`0m0 E`mE`0m0 + E`mE`0m0 E`mT`0m0 (A.63) mm0 X X X 1 EaTc Eb Ed +,Ea 0,Tc? +,Eb? +,Ed Ea Ed EbTc +,Ea +,Ed? +,Eb? 0,Tc = L``0 C C W W 0 0 W W 0 0 +C C W W 0 0 W 0 0 W `1 `2 `m`1m1 ` m `1m1 `m`2m2 ` m `2m2 `1 `2 `m`1m1 ` m `1m1 ` m `2m2 `m`2m2 8 0 mm `1m1 `2m2 X X 1 EaTc Eb Ed 0,EaTc +,Eb Ed? 1 Ea Ed EbTc +,Ea Ed 0,EbTc? ' L 0 C 0 C 0 W 0 0 W 0 0 + L 0 C 0 C 0 W 0 0 W 0 0 2 `` `` `` `m` m `m` m 2 `` `` `` `m` m `m` m mm0 mm0 D E ˜Ea Eb ˜Tc Ed? EaTc Eb Ed 0 P T P P Ea Ed EbTc 0 P P P T ∆C` ∆C`0 ' C``0 C``0 MTP,TP(`, ` ; wa wc , wb wd ) + C``0 C``0 MTP,TP(`, ` ; wa wd , wb wc ) (A.64)
A41, page 23 of 24 A&A 602, A41 (2017)
Appendix B: HiLLiPOP parameter list Table B.1. Nuisance parameters for the HiLLiPOP likelihood.
Name Definition Prior (if any) Instrumental
c0 map calibration (100-A) 0.000 ± 0.002 c1 map calibration (100-B) 0.000 ± 0.002 c2 map calibration (143-A) fixed c3 map calibration (143-B) 0.000 ± 0.002 c4 map calibration (217-A) 0.002 ± 0.002 c5 map calibration (217-B) 0.002 ± 0.002 Apl absolute calibration 1 ± 0.0025 Foreground modelling radio APS scaling parameter for radio sources in TT IR APS scaling parameter for IR sources in TT ASZ scaling parameter for the tSZ in TT ACIB scaling parameter for the CIB in TT 1.00 ± 0.20 TT Adust scaling parameter for the dust in TT 1.00 ± 0.20 EE Adust scaling parameter for the dust in EE 1.00 ± 0.20 TE Adust scaling parameter for the dust in TE 1.00 ± 0.20 AkSZ scaling parameter for the kSZ effect ASZxCIB scaling parameter for cross correlation SZ and CIB
A41, page 24 of 24