© ESO, 2016

1. Introduction

This paper, one of a series associated with the 2015 Planck data release, is the second of two describing the processing of the data from the High Frequency Instrument (HFI). The HFI is one of the two instruments on board Planck¹, the European Space Agency’s satellite dedicated to precision measurements of the cosmic microwave background (CMB). The HFI uses cold optics (at 4 K, 1.6 K, and 100 mK), filters, and 52 bolometers cooled to 100 mK. Coupled with the Planck telescope, it enables us to map the continuum emission of the sky in intensity and polarization at frequencies of 100, 143, 217, and 353 GHz, and in intensity at 545 and 857 GHz. Paper A (Planck Collaboration VII 2016) describes the processing of the data at the time-ordered level and the measurement of the beam. Paper B (this paper) describes the HFI photometric calibration and mapmaking.

Paper A describes how the time-ordered information (TOI) of each of the 52 bolometers is processed and flagged. Sampled at 5.544 ms, the TOI is first corrected for the analog-to-digital converter (ADC) non-linearity, then it is demodulated and converted to the absorbed power with a simple non-linear bolometric correction. Glitches are flagged and glitch tails are removed from the TOI. Thermal fluctuations are removed on the 1 min timescale. Sharp lines in the temporal power spectrum of the TOI from the influence of the 4-K cooler are removed. Finally, the bolometer time response is deconvolved and the TOI is low-pass filtered. At this point, the TOIs are cleaned but not yet calibrated. The measurement of the beam is performed using a combination of observations of planets for the main beam and GRASP² physical optics calculations for the sidelobes. The focal plane geometry, or the relative position of bolometers in the sky, is deduced from Mars observations.

This paper describes how the prepared TOIs are used to make the calibrated maps for all Planck HFI bands. After a summary of the photometric definitions (Sect. 2), this paper gives a description of the main steps of the mapmaking processing, focusing on the changes made since the previous Planck releases (Planck Collaboration VI 2014; Planck HFI Core Team 2011). The major difference concerns the calibration (Sect. 3), which is now based on the orbital CMB dipole for the lower frequency channels (100, 143, 217, and 353 GHz, also called CMB channels) while the 545 and 857 GHz channels are photometrically calibrated using the signal from Uranus and Neptune. Section 4 explains the mapmaking upgrades, including the polarization treatment. Section 5 describes the maps, the solar dipole measurement, and the derivation of far sidelobes and zodiacal maps. Section 6 presents the noise characteristics and the null tests obtained by splitting the Planck HFI dataset into different groups based on ring period, time period, or detector sets. Consistency checks are performed in order to assess the fidelity of the maps. Finally, Sect. 7 is dedicated to the description of systematic effects, in particular in polarization. The major systematic residuals in Planck HFI data are due to the leakage from temperature to polarization induced by flux mismatch between associated bolometers. This is the result of either bandpass mismatch, or zero-level uncertainty, or calibration uncertainty. We present a first attempt to correct the maps for the intensity-to-polarization leakage. At lower frequencies, even after correction, the residuals are still higher than the noise level and thus the maps cannot yet be used for cosmological analysis.

2. Photometric equations

The power absorbed by a given detector at time t can be written as the sum of three terms corresponding to the first three Stokes parameters (I_p,Q_p,U_p) at the sampled pixel p of the beam-convolved sky, $\begin{array}{ccc}{\mathit{P}}_{\mathit{t}}\mathrm{=}\mathit{G}\mathrm{[}{\mathit{I}}_{\mathit{p}}\mathrm{+}\mathit{\rho }{\mathrm{\{}{\mathit{Q}}_{\mathit{p}}\cos \mathrm{2}\mathrm{(}{\mathit{\psi }}_{\mathit{t}}\mathrm{+}\mathit{\alpha }\mathrm{)}\mathrm{+}{\mathit{U}}_{\mathit{p}}\sin {\mathrm{2}\mathrm{(}{\mathit{\psi }}_{\mathit{t}}\mathrm{+}\mathit{\alpha }\mathrm{)}}^{\mathrm{\}}}}^{\mathrm{]}}\mathrm{+}{\mathit{n}}_{\mathit{t}}\mathit{,}& & \end{array}$(1)where G encodes a photometric calibration factor, ρ is the detector polarization efficiency, ψ_t is the roll angle of the satellite, α is the detector polarization angle, and n_t represents all the noise contributions to the absorbed power (photon noise, phonon noise, glitch residuals, etc.). The polarization efficiency is derived from the cross-polarization coefficient η through ρ = (1−η) / (1 + η). It allows us to describe spider-web bolometers (SWB, ρ ≈ 0) as well as polarization-sensitive bolometers (PSB, ρ ≈ 1). According to the bolometer model and given the stability of the HFI operational conditions during the mission, the gain G is expected to be constant over the whole mission (see Sect. 1 of Paper A) once the bolometer non-linearities have been corrected.

For an axisymmetric beam response, the “smearing” and “pointing” operations commute, and one can solve directly for the pixelized beam-convolved map, $\begin{array}{ccc}{\mathit{P}}_{\mathit{t}}& \mathrm{=}& \mathit{G}{\left(\mathrm{1}\mathit{,\rho }\cos \mathrm{2}\mathrm{(}{\mathit{\psi }}_{\mathit{t}}\mathrm{+}\mathit{\alpha }\mathrm{)}\mathit{,\rho }\sin \mathrm{2}\mathrm{(}{\mathit{\psi }}_{\mathit{t}}\mathrm{+}\mathit{\alpha }\mathrm{)}\right)}^{†}\mathrm{·}\mathrm{(}{\mathit{I}}_{\mathit{p}}\mathit{,}{\mathit{Q}}_{\mathit{p}}\mathit{,}{\mathit{U}}_{\mathit{p}}\mathrm{)}\mathrm{+}{\mathit{n}}_{\mathit{t}}\\ & \mathrm{\equiv }& \end{array}$(2)where the direction of observation at the time t, (θ_t,φ_t), falls into pixel p and we define the map-pointing matrix A and the sky signal T = (I,Q,U).

3. Calibration

The bolometer signal measured through current-biasing is proportional to the small variation in the incoming power from the sky. To express the measurement in sky temperature units, one has to determine a gain per detector based on a known source in the sky. For the HFI low-frequency channels (100 to 353 GHz), we use the CMB orbital dipole as a primary calibrator. This signal fills the entire beam and is almost insensitive to the beam profile and only marginally affected by pointing errors, while its signal-to-noise is high enough thanks to the full-sky coverage. Moreover, it is a stronger signal than CMB anisotropies (by a factor of around 10), but not bright enough to cause non-linearities in the detectors, and has the same electromagnetic spectrum as the anisotropies. At higher frequencies (545 and 857 GHz), calibration is performed on planets.

3.1. CMB dipole conventions

The CMB dipole is induced by the Doppler effect of the relative motion of the satellite with respect to the CMB frame, $\begin{array}{ccc}{\mathit{T}}_{\mathrm{Doppler}}\mathrm{\left(}\mathit{t,}\hat{{u}}\mathrm{\right)}\mathrm{=}\frac{{\mathit{T}}_{\mathrm{CMB}}}{{\mathit{\gamma }}_{\mathit{t}}\mathrm{(}\mathrm{1}\mathrm{-}{{\beta }}_{\mathit{t}}\mathrm{·}\hat{{u}}\mathrm{)}}\mathit{,}& & \end{array}$(3)where β_t = v_t/c and ${\mathit{\gamma }}_{\mathit{t}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{\beta }}_{\mathit{t}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{2}}$, v_t is the satellite velocity at time t, and $\hat{{u}}$ is the unit vector along the line of sight.

The solar system motion with respect to the CMB frame, giving rise to what is referred to as the solar dipole, is the dominant component of the satellite velocity. A residual contribution (called the orbital dipole) is induced by the yearly motion of the satellite with respect to the solar system barycentre. The solar dipole can be considered as sky-stationary during the observations and is thus projected onto the sky as an ℓ = 1 component with amplitude previously measured by COBE and WMAP, 3355 ± 8 μK (Hinshaw et al. 2009). Relativistic corrections to the solar dipole produce second-order anisotropies at multipoles ℓ ≥ 1 with amplitudes proportional to β^ℓ and, more importantly, couple the two dipole components, as will be discussed below. Although the orbital dipole velocity is typically an order of magnitude lower than the solar dipole, it is time dependent and its time-variability is precisely determined by the satellite velocity, which is known at the level of 10^-4 km s^-1. Finally, to calibrate in temperature, we only rely on an external measurement of the CMB absolute temperature. We use T_CMB = 2.7255 K (Fixsen 2009).

The expansion of Eq. (3)in β gives $\begin{array}{ccc}\mathrm{\Delta }\mathrm{\left(}\hat{{u}}\mathrm{\right)}\mathrm{=}\frac{{\mathit{T}}_{\mathrm{Doppler}}}{{\mathit{T}}_{\mathrm{CMB}}}\mathrm{-}\mathrm{1}\mathrm{\approx }{\beta }\mathrm{·}\hat{{u}}\mathrm{-}\frac{{\mathit{\beta }}^{\mathrm{2}}}{\mathrm{2}}\mathrm{+}{\left({\beta }\mathrm{·}\hat{{u}}\right)}^{\mathrm{2}}\mathrm{+}\mathrm{𝒪}\mathrm{(}{\mathit{\beta }}^{\mathrm{3}}\mathrm{)}\mathit{.}& & \end{array}$(4)If we decompose the velocity into a solar boost β₁ and an orbital boost β₂ then $\begin{array}{ccc}\mathrm{\Delta }\mathrm{=}{\mathrm{\Delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{\Delta }}_{\mathrm{2}}\mathrm{-}{{\beta }}_{\mathrm{1}}\mathrm{·}{{\beta }}_{\mathrm{2}}\mathrm{+}\mathrm{2}\mathrm{(}{{\beta }}_{\mathrm{1}}\mathrm{·}\hat{{u}}\mathrm{)}\left({{\beta }}_{\mathrm{2}}\mathrm{·}\hat{{u}}\right)\mathrm{+}\mathrm{𝒪}\mathrm{(}{\mathit{\beta }}^{\mathrm{3}}\mathrm{)}\mathit{,}& & \end{array}$(5)where the first term corresponds to the solar dipole, the second term is the orbital dipole, and the third and fourth terms show the coupling between each term due to relativistic corrections.

3.2. Absolute calibration on orbital dipole

The calibration algorithm takes advantage of the orbital dipole not being fixed on the sky, unlike the solar dipole (during the length of the mission). In practice, relativistic corrections and second order in the development of the conversion from I_ν to T_cmb couple solar and orbital dipoles creating an additional non-stationary signal, which also depends on the frequency. We use the total CMB dipole computed using Eq. (3)as the calibration reference signal, assuming that it has the same scaling with frequency as the higher multipole CMB anisotropies. This approximation will leave in the HFI frequency maps the frequency dependent fraction of the kinematic quadrupole arising from the second-order term in the β expansion described in Kamionkowski & Knox (2003) and Quartin & Notari (2015). The amplitude of the kinematic quadrupole is expected to be lower than 0.5% on the CMB-calibrated Planck-HFI channels. However, it is not directly correlated to the orbital dipole on which we calibrate the data. Thus, we expect the systematic error induced on the gain estimation to be much smaller.

The orbital dipole signal is modulated on a one-year period. To take into account the time variation of this signal, we need to add a term in Eq. (2), $\begin{array}{ccc}{P}\mathrm{=}\mathit{G}\mathrm{\times }\mathrm{(}\mathrm{A}{T}\mathrm{+}{{t}}_{\mathrm{orb}}\mathrm{)}\mathrm{+}{n}\mathit{,}& & \end{array}$(6)where t_orb is the time-dependent orbital dipole signal, while in this formula the solar dipole is part of the sky signal T. We note that we can arbitrarily set all or part of the solar dipole signal either in the calibration template or in the sky without changing the resulting gain estimation.

Since t_orb is known, we can solve Eq. (6)for each bolometer independently, rewriting the system as $\begin{array}{ccc}{P}\mathrm{=}\mathrm{A}\begin{array}{c}\mathrm{˜}\\ {T}\end{array}\mathrm{+}\mathit{G}{{t}}_{\mathrm{orb}}\mathrm{+}{n}\mathit{,}& & \end{array}$(7)where the unknowns are the sky-signal in absorbed power units $\begin{array}{c}{\mathrm{˜}}_{}\\ {T}\end{array}$ and the gain G. The calibration problem is thus linear and can be solved directly. The maximum-likelihood estimate of the gain G is obtained by combining all available samples using the noise covariance matrix N = ⟨ n^†n ⟩ and marginalizing over $\begin{array}{c}{\mathrm{˜}}_{}\\ {T}\end{array}$, solving the equation $\begin{array}{ccc}\mathrm{(}{{t}}_{\mathrm{orb}}^{\mathrm{T}}{\mathrm{N}}^{-1}\mathrm{Z}{{t}}_{\mathrm{orb}}\mathrm{)}\mathrm{\times }\mathit{G}\mathrm{=}{{t}}_{\mathrm{orb}}^{\mathrm{T}}{\mathrm{N}}^{-1}\mathrm{Z}{d}\mathit{,}& & \end{array}$(8)where d is the vector of input data P_t and we define $\begin{array}{ccc}\mathrm{Z}\mathrm{=}\mathrm{I}\mathrm{-}\mathrm{A}{\mathrm{(}{\mathrm{A}}^{\mathrm{T}}{\mathrm{N}}^{-1}{\mathrm{A}}^{\mathrm{)}}}^{-1}{\mathrm{A}}^{\mathrm{T}}{\mathrm{N}}^{-1}\mathit{.}& & \end{array}$(9)In practice, the noise is treated by assuming that the destriping reduces the matrix N to a diagonal one once the data d have been corrected for a constant offset (see Sect. 4.2). For the HFI, the mapmaking problem is degenerate for the reconstruction of polarization if we solve for a single detector at a time. Neglecting polarization in Eq. (8)for polarization-sensitive detectors biases the calibration solution. Moreover, we need a very accurate relative calibration between detectors that are combined to reconstruct polarization in order to minimize leakage from intensity to polarization (see Sect. 7.3). For this reason, for the Planck 2015 release we have extended the algorithm described in Tristram et al. (2011) to perform a multi-detector gain estimation for all bolometers at a given frequency together with the offsets (see Sect. 4.2).

3.3. Long time-constant residuals

The observation strategy of Planck results in the path across a particular part of the sky being almost reversed six months later. As described in Paper A, we take advantage of this to derive the time transfer function below one second. Nevertheless, longer time responses (larger than the second), even with low amplitudes, may bias the calibration estimation by distorting the dipole signal and causing some leakage of the solar into the orbital dipole. To take into account the systematic residuals after time-constant deconvolution into account, we built a simplified model describing a pure single-mode sinusoidal dipole signal (including solar and orbital dipole) convolved with an exponential decay in the time domain. In the frequency domain, this reads $\begin{array}{ccc}{\mathrm{𝒯}}_{\mathrm{dip}}\mathrm{=}\mathrm{ℱ}\mathrm{(}{\mathit{t}}_{\mathrm{dip}}\mathrm{\ast }\mathit{B}{\mathrm{e}}^{\mathrm{-}\mathit{t}\mathit{/}\mathit{\tau }}\mathrm{)}\mathrm{=}\frac{\mathit{B}}{\mathrm{1}\mathit{/}\mathit{\tau }\mathrm{+}\mathrm{2}\mathrm{i}\mathit{\pi \nu }}{\mathit{\delta }}_{\mathit{\nu ,}{\mathit{\nu }}_{\mathrm{spin}}}\mathit{,}& & \end{array}$(10)where B and τ are the amplitude and the time constant, and ν_spin is the spin frequency; is a complex coefficient, the real part of which corresponds to the relative change in the gain G, while the imaginary part corresponds to the amplitude of the dipole mode shifted by 90°.

In practice, prior to the absolute calibration, we solve for the amplitude of a shifted-dipole template using single detector calibration, $\begin{array}{ccc}{P}\mathrm{=}\mathrm{A}\begin{array}{c}\mathrm{˜}\\ {T}\end{array}\mathrm{+}\mathit{G}{{t}}_{\mathrm{dip}}\mathrm{+}\mathit{C}{{t}}_{\mathrm{dip}}^{{\mathrm{90}}^{\mathrm{◦}}}\mathrm{+}{n}\mathit{,}& & \end{array}$(11)where C gives the amplitude of the dipole mode shifted by 90°. With this toy model, we cannot reconstruct the amplitude and the time constant because we only fit for one coefficient C, but we trace the systematic effect on the dipoles due to the very long time response. Once the coefficient C has been determined, we correct the data for $\mathit{C}{{t}}_{\mathrm{dip}}^{{\mathrm{90}}^{\mathrm{◦}}}$ to account for this additional shifted mode coming from the residual time constant.

The effect on the gain G depends on the unknown value of the time constant τ. Figure 1 shows the impact on the gains for each bolometer when including the shifted-dipole correction. At higher frequencies (353 GHz and above), the signal is no longer dominated by a dipole and cannot be approximated by the model described above. For the 2015 release, we do not include any correction for time constant residuals in the mapmaking for those channels.

3.4. Submillimetre calibration

Mars, Jupiter, Saturn, Uranus, and Neptune are all observed several times during the full length of the mission. The submillimetre channels of HFI (545 and 857 GHz) are calibrated on models of Uranus and Neptune. We do not use Jupiter and Saturn observations for calibration since both planets have strong absorption features at those frequencies, which complicate comparison with broadband measurements. The flux from Jupiter also leads to detector saturation at the highest HFI frequencies. Similarly, we choose not to use Mars as a calibrator because strong seasonal variations complicate the modelling.

Various methods are used to derive planet flux densities, including aperture photometry and point-spread function (PSF) fits. Planet measurements with Planck are studied fully in Planck Collaboration XXXIV (in prep.). We focus here on calibration using aperture photometry in the time-ordered data from the submillimetre channels. The simulation pipeline used for the main beam reconstruction computes the reconstruction bias and the error budget (the mean and the variance) for each planet observation. The comparison of flux measurements of Neptune and Uranus with up-to-date models provides the calibration factors at 545 and 857 GHz.

3.4.1. Planet models

We use the ESA 2 model for Uranus and the ESA 3 model for Neptune (Moreno 2010). Both models quote absolute uncertainties of 5%. Planet model spectra are produced from their modelled brightness temperatures using the planet solid angles at the time of observation and integrated under the individual 545 and 857 GHz bolometer bandpasses. Flux densities are colour-corrected to a reference spectrum defined by a constant νI_ν law so as to be directly comparable to HFI flux density measurements.

Planck Collaboration XXXIV (in prep.) gives a detailed account of the ratio between the expected planet fluxes and the measured values at all HFI frequencies and for the five observed planets (Mars, Jupiter, Saturn, Uranus, and Neptune). They all fall within the 5% model uncertainty range. This is a validation of the models at 100–353 GHz. Hence the models can be used with some confidence to calibrate the 545 and 857 GHz channels.

3.4.2. Aperture photometry in the timelines

We select all samples in a 2° × 2° box around the planet positions and build time-ordered vector objects (hereafter timelines) for each bolometer and each planet scan. We use the first four scans of Neptune and Uranus (season 1 to 4). We build corresponding background timelines using all the samples in a 2° × 2° box around the planet position when the planet is not there. The resulting background has a much higher spatial density than the planet timelines. We use this to build a background mini-map with 2′ × 2′ pixels that can then be interpolated at each sample position in the planet timelines in order to remove a local background estimate (see Appendix B in Paper A for details).

The aperture photometry measurement procedure applied to our planet timelines is an extension of the usual aperture photometry approach to irregularly gridded data. Flux is integrated in an aperture of radius 3 times the effective FWHM. Typically, aperture photometry is applied to maps of fixed-size pixels, which means integrating the flux in the aperture is equivalent to summing the pixel values. In our case, we have to take into account the inhomogenous spatial distribution of the samples. To do this we assume that the beam products perfectly describe the spatial light profile of Neptune and Uranus, and compute a spatial sampling correction factor as the ratio of the integrated flux in a highly spatially oversampled beam and the integrated flux in a beam sampled on the planet timelines. This sampling correction has to be computed for each bolometer and planet crossing, because the spatial sampling varies between planet observations. We estimate the statistical uncertainty of the measurements as the standard deviation of the samples in an annulus of radius 3 to 5 times the effective FWHM of the beam.

We find large variations between the individual planet measurements in each detector and at each season of the full mission survey (Fig. 2) which we attribute to the underestimation of the measurement uncertainty. The signal from Neptune and Uranus is not expected to vary in time apart from the differences in solid angle which are very small and already taken into account. While accurately corrected by the timeline aperture photometry algorithm presented here, the limited available spatial sampling of the planet signal at these frequencies could explain part of the variations. We therefore decided to include the seasonal root mean square (rms) of the measurements (we have four observations per bolometer per planet) in the measurement uncertainty.

The averaged calibration factors for each detector for each planet are in very good agreement. The final calibration factors are the means of both planet estimates. We compare them to the 2013 Planck release in Fig. 3: the calibration factors changed by 1.9 and 4.1% at 545 and 857 GHz, respectively, which is within the planet modelling uncertainty. Combined with other pipeline changes (such as the analog-to-digital converter non-linearity corrections), the 2015 frequency maps have decreased in brightness by 1.8 and 3.3% compared to 2013.

Calibration uncertainties are given in Table 1. In order to produce the frequency maps, detectors are weighted by their inverse noise variance. We use the same weights to compute the corresponding calibration errors. We estimate combined statistical errors of 1.1% and 1.4% at 545 and 857 GHz, respectively, to which one should add linearly (as should be done for systematics) the 5% uncertainty arising from the planet models. Errors on absolute calibration are therefore 6.1 and 6.4% at 545 and 857 GHz, respectively. Since the reported relative uncertainty of the models is of the order of 2%, we find the relative calibration between the two HFI highest frequency channels to be better than 3%.

4. Mapmaking

4.1. Summary

Each data sample is calibrated in K_CMB for the 100, 143, 217, and 353 GHz channels, or  MJy sr^-1 (assuming a constant νI_ν law) for the 545 and 857 GHz channels, using the calibration scheme described above. The bolometer gains are assumed to be constant throughout the mission, which was not the case for the 2013 release. The Planck total dipole (solar and orbital) is computed and subtracted from the data.

As in the Planck 2013 release, we average the measurements for each detector in each pixel visited during a stable pointing period (hereafter called a ring) while keeping track of the bolometer orientation on the sky. The subsequent calibration and mapmaking operations use this intermediate product as input. The calibrated TOIs are only modified by a single offset value per ring and per detector. The offsets are determined with the destriping method described in Tristram et al. (2011). Here, the size of the pixels where the consistency of different rings is tested is 8′ (N_side = 512). Maps in intensity and polarization are used to assess the consistency of the destriper solution. The offsets are simultaneously determined for all bolometers at a given frequency using the full mission data. For all the maps produced using a given bolometer, the same offset per ring is used (except in the case of half-rings, see below).

The products of the HFI mapmaking pipelines are maps of I, Q, U together with their covariances (II, IQ, IU, QQ, QU, UU), pixelized according to the HEALPix scheme (Górski et al. 2005). The map resolution is N_side = 2048, and the pixel size is 1.́7. The mapmaking method is a simple projection of each unflagged sample to the nearest grid pixel. In the case of polarization data with several detectors solved simultaneously, the polarization equation is inverted on a per-pixel basis (see Sect. 4.2). For each sky map, a hit count map is computed (number of samples per pixel; one sample has a duration of 5.544 ms).

The zodiacal light component, which varies in time, is estimated using templates fitted on the survey-difference maps (see Sect. 5.4). We provide maps without zodiacal light for which these templates are systematically subtracted from the data of each bolometer prior to the mapmaking.

Unlike in the 2013 release, the far sidelobes are not removed from the maps. At most, this leaves residuals of the order of 0.5–1.5 μK in the 100–353 GHz maps with uncertainties on the residuals of roughly 100% (Planck Collaboration VII 2016). At higher frequencies, Galactic pick-up from the far sidelobes produces significant residuals in about half of the detectors of the order of 0.03 MJy sr^-1 at 545 and 0.3 MJy sr^-1 at 857 GHz (see Planck Collaboration X 2016). The total solid angle in the spillover is a better known quantity, and we describe the effect of the far sidelobes on the calibration in Sect. 5.3.

4.2. Mapmaking method

In the same way as in Planck Collaboration VIII (2014), we use a destriping algorithm to deal with the HFI low-frequency noise. In this approach, the noise in a ring r is represented by an offset, denoted o_r, and a white noise part n, which is uncorrelated with the low-frequency noise. For a given bolometer, we can write Eq. (6)as $\begin{array}{ccc}{\mathit{P}}_{\mathit{t}}\mathrm{=}\mathit{G}\mathrm{\times }\mathrm{(}{\mathit{A}}_{\mathit{tp}}\hspace{0.17em}{\mathit{T}}_{\mathit{p}}\mathrm{+}{\mathit{t}}_{\mathrm{orb}}\mathrm{)}\mathrm{+}{\mathrm{\Gamma }}_{\mathit{tr}}\hspace{0.17em}{\mathit{o}}_{\mathit{r}}\mathrm{+}{\mathit{n}}_{\mathit{t}}\mathit{,}& & \end{array}$(12)where Γ_tr is the ring-pointing matrix (which associates the data sample t with the ring number r). The unknowns are the gain G, the offsets for each ring o_r, and the sky signal for each pixel T_p = (I_p,Q_p,U_p). As there is a degeneracy between the average of the offsets and the zero level of the maps, we impose the constraint ⟨ o_r ⟩ = 0. The absolute zero level of the maps is determined as described in Sect. 4.4.

For the production of the maps for the 2015 HFI data release, we first build the rings for all detectors. We apply the following frequency-dependent processing to these compressed data sets.

For CMB frequencies (100, 143, and 217 GHz) channels we proceed as follows:

• 1.

  we estimate a first approximation of the orbital dipole gaintogether with the offsets and the amplitude of thelong-time-constant residuals for each bolometer independently,neglecting the polarization signal;

• 2.

  we then derive the gains and offsets for a fixed amplitude of the long-time-constant residuals using the multi-bolometer algorithm.

At 353 GHz, the long-time-constant residuals are more difficult to constrain. They are driven more by Galactic emission drifts than by the dipole, which dominates at lower frequencies, so that the model described in Sect. 3.3 is not relevant. Hence for this frequency we use a simpler pipeline without a long-time-constant residuals template:

• 1.

  we estimate the orbital dipole gain together with the offsets foreach bolometer independently, neglecting the polarizationsignal;

• 2.

  we then estimate the final offsets using a destriping procedure for all bolometers at this frequency.

For the two highest frequencies, at 545 and 857 GHz, the pipeline is considerably different because we use the planets (Uranus and Neptune) as calibration sources:

• 1.

  we estimate a first approximation of the offsets for eachbolometer independently;

• 2.

  we derive the gains from the planet flux comparison;

• 3.

  we then estimate the final offsets with a destriping procedure for all bolometers at a given frequency.

Finally, using the pre-computed gains and offsets, we project the ring data onto maps for each data set. For polarization, we invert the 3 × 3 system derived from Eq. (12)for each pixel independently with the criterion that the condition number be lower than 10³, $\begin{array}{ccc}{T}\mathrm{=}{\mathrm{(}{\mathrm{A}}^{\mathrm{T}}{\mathrm{N}}^{-1}{\mathrm{A}}^{\mathrm{)}}}^{-1}{\mathrm{A}}^{\mathrm{T}}{\mathrm{N}}^{-1}{d}\mathit{,}& & \end{array}$(13)where d are the calibrated, cleaned ring data (offsets and orbital dipole removed) d = ⁽P−Γ o⁾/G−t_orb. We note that we use HEALPix (not IAU) conventions for the sign and normalization of the Q and U Stokes parameters.

For destriping, we use the same tool as before, polkapix, which was thoroughly validated in Tristram et al. (2011). Maps are built by simple co-addition in each pixel of the destriped, calibrated, and time-varying component-subtracted signal. We subtract the CMB dipole as measured by Planck (see Sect. 5.1).

We introduced the following modifications with respect to Planck Collaboration VIII (2014):

• we included polarization in the destriping for the channels thatinclude polarization-sensitive bolometers;

• we enlarged the masked fraction of the sky from 10 to 15%, based on Galactic emission, to avoid signal gradients leaking into offsets;

• to improve the offset accuracy, we computed one set of offsets combining all detectors, using full-mission data, and used them for all derived maps involving these detectors.

This last change induces a small noise correlation between detector-set maps (see Sect. 6.5). In 2013 we computed independent offsets for each detector or detector set, including the full mission.

4.3. Map products

The principal HFI final product consists of six maps that cover the six frequencies (100–857 GHz) for the full mission in intensity at high resolution (N_side = 2048). However, many more maps are needed to assess the noise and the consistency of the data. Figure 4 summarizes the various splits produced. The branches that are delivered in the 2015 release are described in Appendix A.

Maps from different halves of each ring period (first and last) are computed independently of each other, including the offset per half-ring. Thus, half-ring half-difference maps can give a quick account of the noise level in the maps.

For each frequency, we also produce temperature and polarization maps using detector sets (each set including four polarization-sensitive bolometers). In addition, we produce a temperature map for each spider-web bolometer.

The Planck scanning strategy samples almost all the sky pixels every six months, with alternating scan directions in successive six-month periods. The full cold HFI mission encompasses five surveys, each covering a large fraction of the sky. Surveys 1–2 and 3–4 are paired to produce Year 1 and Year 2 maps (see Table A.2). Maps are produced for the full-mission dataset together with the survey, year, and half-mission maps. Each map is associated with a hit-count map and variance maps (II; and QQ, UU, IQ, IU, and QU when polarization is reconstructed). Overall, a total of about 6500 sky maps have been produced. We have used this data set to evaluate the performance of the photometric calibration by examining difference maps (see Sect. 6.2).

4.4. Zero levels

Planck-HFI cannot measure the absolute sky background. The mapmaking procedure does not change the mean value of the input TOI. We therefore adjust the monopole on the maps a posteriori in a similar manner to the method used in Planck Collaboration VIII (2014), which relies on external datasets. To achieve this, we need to take into account two major components of the monopole:

• 1.

  Galactic dust emission: we estimate the brightness in theHFI single-detector maps that corresponds to zerogas column-density (i.e. zero Galactic dust emission) by using theH i column density, which isassumed to be a reliable tracer of the Galactic gas column-densityin very diffuse areas (see Planck CollaborationVIII 2014, Sect. 5.1). The offsetsderived are then subtracted from each detector’s data in theprocessing.

• 2.

  Extragalactic emission: the cosmic infrared background (CIB) monopole is taken into account by adding the levels from Béthermin et al. (2012) to the maps (see Table 6).

The sum of the two offsets is appropriate for total emission analysis. For Galactic studies, only the Galactic zero level has to be set which can be achieved by subtracting the CIB levels (Table 6) from the released maps. Unlike the previous release, in the 2015 maps the zero-level correction (both CIB and Galactic) has been applied.

Zodiacal light has not been accounted for in this procedure. The correction to be applied at each frequency to set the Galactic zero level using zodiacal-light-corrected maps are given in Table 6.

4.5. Polarization efficiency and orientation

The Planck-HFI bolometers were characterized on the ground before launch. Rosset et al. (2010) reported pre-flight measurements of the polarization efficiency of the HFI polarization-sensitive bolometers with an accuracy of 0.3%. The absolute orientation of the focal plane was measured at a level better than . The relative orientation between bolometers is known with an accuracy better than .

Spider-Web Bolometers are much less sensitive to polarization. Nonetheless, we take into account their polarization efficiency, which is between 1 and 9%, although their orientations have been less accurately determined (errors can be up to a few percent), as described in Rosset et al. (2010).

5. HFI temperature and polarization maps

5.1. Solar dipole measurement

The ℓ = 1 mode of CMB anisotropy is unique in that its amplitude is dominated by a large component associated with our motion with respect to the CMB rest frame. In this section, we present the CMB solar dipole results based on Planck-HFI maps at the two lowest frequencies, 100 and 143 GHz. Low-frequency maps are dominated by CMB over a large fraction of the sky. Nevertheless, the inhomogeneous nature of the dust emission can bias CMB solar dipole estimates.

We cleaned the Galactic emission from the HFI maps using a local correlation with the 857 GHz map. We model each HFI map I_ν as $\begin{array}{ccc}{\mathit{I}}_{\mathit{\nu }}\mathrm{-}\mathit{C}\mathrm{=}\mathit{q}{\mathit{I}}_{\mathrm{857}}\mathrm{+}{\mathit{D}}_{\mathrm{res}}\mathit{,}& & \end{array}$(14)where C is the CMB anisotropy (here we use the SMICA map, Planck Collaboration IX 2016, from which we remove any residual dipole component) and I₈₅₇ is the Planck 857 GHz map that is assumed to have a negligible contribution from the solar dipole³. The term D_res includes the dipole and any systematic effects from both I₈₅₇ and I_ν. In bright regions of the sky, D_res also contains extra emission that is uncorrelated or only partially correlated with I₈₅₇, for instance free-free emission or CO.

In order to capture any spatial variations of the dust spectral energy distribution (SED), we estimated q and D_res on an N_side = 64 grid. For each N_side = 64 pixel, we performed a linear regression of the N_side = 2048 pixels of I_ν vs. I₈₅₇, assuming a constant dust SED over a 55′ area. We then fit for the dipole amplitude and direction in D_res using sky pixels where I₈₅₇ < 2, 3, or 4  MJy sr^-1 (corresponding to 37, 50, or 58% of the sky respectively) to limit the effect of Galactic-emission residuals (CO, free-free emission, and small-scale dust SED variations).

The results are given in Table 2. We measure a solar system peculiar velocity of 370.06 ± 0.09 km s^-1 with respect to the CMB rest frame. We use the CMB temperature from Fixsen (2009; 2.7255 ± 0.0006 K) to convert that measurement into a CMB dipole measurement.

The error bars here only include statistical uncertainties, which are very low thanks to the Planck-HFI signal-to-noise ratio. We evaluate the additional systematic uncertainties from the variation of the results between independent bolometer maps. For the amplitude, the peak-to-peak variation between bolometers and combined maps is ±0.8 μK at 100 and 143 GHz. Variations with sky coverage are of the same order. We note that the uncertainty from the FIRAS temperature should be added to the budget (±0.74 μK). For the coordinates, we found variations of ± in longitude and ± in latitude. These differences are observed when comparing results at different frequencies, and are likely to result from uncertainties in the foreground subtraction. This is also consistent with the magnitude of the direction shifts we observe when changing the sky fraction.

As an independent check, we also produce a cleaned CMB map using an internal linear combination (ILC) method. We used the HFI maps at 100, 143, and 217 GHz smoothed with a 1° FWHM Gaussian kernel. We note that smoothing the data with a 1° kernel reduces the solar dipole in the maps by 0.005%, i.e. 0.2 μK, which we corrected for afterwards. We then estimate the solar dipole amplitude and direction using a Galactic mask that removes less than 15% of the sky to avoid the inner Galactic plane where the residuals are most intense. The measurement is compatible with the results in Table 2.

In the end, the amplitude (d) and direction (Galactic longitude, latitude) of the solar dipole measured by Planck-HFI is

$\begin{array}{ccc}& & \mathit{d}\mathrm{=}\mathrm{3364.29}\mathrm{\pm }\mathrm{0.02}\mathrm{\left(}\mathrm{stat}\mathrm{\right)}\mathrm{\pm }\mathrm{0.8}\mathrm{\left(}\mathrm{sys}\mathrm{\right)}\mathrm{\pm }\mathrm{0.74}\mathrm{\left(}\mathrm{FIRAS}\mathrm{\right)}\mathit{\mu }\mathrm{K}\\ & & \left(\begin{array}{c}\\ \\ \end{array}\right)\mathrm{=}\left(\begin{array}{c}\\ & & \\ & & \end{array}\right)\mathit{.}\end{array}$This is to be compared to the official Planck solar dipole measurement obtained in combination with the Planck-LFI:

$\begin{array}{ccc}& & \mathit{d}\mathrm{=}\mathrm{3364.5}\mathrm{\pm }\mathrm{2.0}\mathit{\mu }\mathrm{K}\\ & & \left(\begin{array}{c}\\ \\ \end{array}\right)\mathrm{=}\left(\begin{array}{c}\\ \\ \end{array}\right)\mathit{.}\end{array}$

Compared to the WMAP five-year results (; Hinshaw et al. 2009), this is 9.3 μK (0.28%) higher in amplitude while shifted by in longitude and latitude. Part of the difference (0.6 μK) is due to the revised CMB monopole temperature compared to Mather et al. (1999; 2.725 K). This total dipole (solar, orbital, relativistic, and interactions thereof) is removed from the calibrated TOI before final mapmaking.

5.2. Planck-HFI maps

Frequency maps have been produced using inverse noise weighting. In Figs. 5 and 6 we show the six intensity frequency maps from 100 to 857 GHz at full resolution (N_side = 2048). Figure 7 presents polarization maps at the first four frequencies (100, 143, 217, and 353 GHz), degraded to lower resolution (N_side = 256) in order to enhance the signal-to-noise ratio. These maps have been corrected from bandpass leakage as will be discussed in Sect. 7.3.1. In both intensity and polarization, we clearly see the emission from the Galactic dust increasing with frequency. In intensity, CMB anisotropies are visible at high latitude in the low-frequency channels (between 100 and 217 GHz). In polarization, the 100 GHz maps are contaminated in the Galactic plane by residual CO leakage coming from bandpass mismatch between bolometers.

5.3. Far sidelobes

As noted in Planck Collaboration VII (2016), far sidelobes affect the response of the instrument to large-scale structures. In addition, the far sidelobes also affect the HFI calibration.

At low frequencies, HFI calibrates by fitting to the sinusoidal signal created by the dipole modulated by the Planck circular scanning strategy. As outlined in Appendix B of Planck Collaboration XXXI (2014), this effectively weights different parts of the beam in general, and the sidelobes in particular, by their angle from the spin-axis. For example, far sidelobe contributions close to the spin-axis actually affect the calibration very little. Similarly, since we are calibrating with signals that are “in phase” with the known phase of the main beam as it scans, the further a sidelobe contribution is in angle around the spin-axis from the main lobe, the less it contributes to the calibration. So, a sidelobe contribution that is 90° in scan phase from the main lobe, for example, would not contribute to the HFI calibration, while something close to the main beam would potentially have a large effect. The change of the gain due to the far sidelobes is calculated by fitting the dipole to full timeline simulations of the dipole convolved by the far sidelobes. The factors are 0.09% at 100 GHz, 0.05% at 143 GHz, 0.04% at 217 GHz, and negligible at 353 GHz. The delivered 100–217 GHz maps have been scaled by these gain changes. It should be noted that these numbers are uncertain at the 20–30% level, depending on a multitude of details, such as how the telescope is modelled.

For the planet photometry, some level of knowledge of the amplitude of the far sidelobes is needed to correctly compare the reconstructed flux with the planet brightness. However, the relative far sidelobe power is lower than 0.3% (Tauber et al. 2010) for all HFI frequencies, which is well below the systematic uncertainties of the planet emission models we are using, which are around 5% (see Sect. 3.4). Therefore far sidelobes can safely be ignored in the 545 and 857 GHz calibration.

5.4. Zodiacal emission

Zodiacal emission is reconstructed and subtracted in the same fashion as that used for the 2013 Planck results (Planck Collaboration XIV 2014). The basic procedure for characterizing and removing zodiacal emission from the Planck maps is to

• make frequency maps for each horn and survey as described inprevious sections;

• make survey difference maps for each horn and year;

• find the date ranges over which each N_side = 256 pixel was observed, and veto those pixels that were observed over a time-span of more than one week;

• use the COBE model (Kelsall et al. 1998) to recreate the different zodiacal emission components, assuming black-body emissivities;

• fit the components to the survey difference maps for each horn and year to extract the actual emissivities;

• use the average of the fitted emissivities to reconstruct the implied zodiacal emission seen during each pointing period, for each horn, and remove these from each detector.

Figure 8 shows the profiles for each HFI band for survey 3, made by differencing maps made without zodiacal emission removal from those made with zodiacal emission removal and binning them in ecliptic latitude. The diffuse cloud and bands are each marked only once, but exist at the same latitudes at each frequency in differing relative amounts. These profiles also show some small offset effects near the ecliptic poles, which arise because the destriping is redone when zodiacal emission is removed.

The emissivities for each zodiacal component at each of the HFI frequencies are given in Table 3 and are plotted in Fig. 9. As noted in Planck Collaboration XIV (2014), there seems to be a jump between the emissivities for the bands at DIRBE wavelengths and the emissivities of Bands 1 and 3 at Planck wavelengths. This is being investigated, but is assumed to be a consequence of the assumption in the DIRBE analysis that all three bands have the same emissivities, while the Planck analysis allows them to be different. For the Planck cosmological studies this should be irrelevant, since the zodiacal analysis is being used only to remove the interplanetary dust contamination – the overall amplitudes of the emissivities, which are completely degenerate with the assumed particle density in the bands, are not being interpreted physically.

6. Noise description and subset differences

6.1. Map variance

As demonstrated in Paper A, the noise spectra for the Planck-HFI bolometers show significant deviation from white noise, resulting in correlations between pixels after map projection. At large scales, the correlations are dominated by low-frequency noise, while at high resolution neighbouring pixels are correlated as a result of time-response deconvolution and filtering. The Planck 2015 release does not provide a pixel-pixel correlation matrix; only the variance per pixel is given for each delivered map. At first order, the variance maps are proportional to 1 /N_hit, where N_hit is the number of samples per pixel.

The half-difference half-ring maps, projected using the same gain but destriped independently, are a good representation of the noise variance in the HFI maps. In the difference between the first and the second half of a ring, the sky signal vanishes almost completely. Moreover, most of the HFI systematics are scan-synchronous and thus also vanish in the difference.

Table 4 compares the noise per sample from three estimators: (a) the mean value of the pre-whitened variance map (i.e. scaled using the hit counts); (b) the variance of the pre-whitened half-ring half-difference map; and (c) the average of the half-ring map power spectra in the ℓ range 100–5000 (see Sect. 6.2). For polarization, the numbers are averages over Q and U for the maps and E and B for the spectra. The different estimators are sensitive to different kinds of systematic effects, such as time-response residuals and signal gradient in pixels. Nevertheless, the three noise estimators give very consistent results.

The variance maps (I–I, Q–Q, and U–U) are inhomogeneous owing to the Planck scanning (which have negligible wobbling), the relative position of the detectors in the focal plane, and the rejection of some rings or groups of rings (see Paper A). Moreover, the typical HEALPix pixel size is about at N_side = 2048 resolution while Planck scans the sky on roughly (but not exactly) ecliptic meridians separated by 2.́5 (near the ecliptic equator). As a consequence, for single survey maps, lines of empty pixels appear between the scanning trajectories around (l,b) = (0°, ± 45°). Even when surveys are combined, inhomogeneities arise because the HEALPix pixels are elongated parallelograms. The axis of their elongation changes at the boundaries between the 12 primary HEALPix pixels. In the same regions of the sky and in Galactic coordinates, these elongations are parallel with the scanning trajectories, which induces moiré patterns in the coverage maps.

The degree of correlation between the Stokes parameters within each pixel reflects the distribution of the detector orientations, which results from the scanning strategy. The I–Q and I–U correlations are about 14, 9, 6, and 12% at 100, 143, 217, and 353 GHz. The Q–U correlation is about 11, 2, 3, and 8% at 100,143, 217, and 353 GHz. In Appendix A.2 we show the sky distribution of these correlations.

6.2. Map differences

The redundancy of the Planck scanning history and focal plane layout provides numerous ways to check data consistency. We can create differences between maps built using data splits, as described in Sect. 4.3. In the limit that the signal is the same in each data subset, the difference map should contain only noise. The TOI processing includes several operations that introduce correlations on various time scales; these are discussed below.

In the dipole-calibrated frequency channels (100–353 GHz), the signal differences are small enough that the data-split map differences can be evaluated at a spectral level, giving insight into the residual systematic errors. For high-frequency channels, we discuss the residuals in the map domain.

Given maps of two subsets of the data, M_A and M_B, we construct the half-difference as $\mathrm{\Delta }\mathit{M}\mathrm{=}{}^{\mathrm{(}}\mathit{M}_{\mathit{A}}\mathrm{-}{\mathit{M}}_{\mathit{B}}{}^{\mathrm{)}}\mathit{/}\mathrm{2}$. We compute the power spectrum in temperature and polarization of the half-difference, masking the sky with the Planck point source mask and the galaxy masks used in Planck Collaboration XI (2016), i.e. leaving 65, 59, 48, and 32% of the sky unmasked at 100, 143, 217, and 353 GHz, respectively.

In order to use this half-difference map to assess noise in the full maps, we account for widely varying integration time in the two subsets using a pixel-by-pixel weight map, which is multiplied by the half-difference map ΔM prior to computing the angular power spectrum. The weight is constructed as W = 2/$\sqrt{{\mathrm{(}}^{}\mathrm{1}\mathit{/}{\mathit{n}}_{\mathit{A}}\mathrm{+}\mathrm{1}\mathit{/}{\mathit{n}}_{\mathit{B}}{\mathrm{)}}^{}{}^{\mathrm{(}}\mathit{n}_{\mathit{A}}\mathrm{+}{\mathit{n}}_{\mathit{B}}{\mathrm{)}}^{}}$ , where n_A is the hit count map for M_A and n_B is the hit count map for M_B. In the limit that the half-difference map consists entirely of white noise, this exactly accounts for the differences in the hit counts. The TT and EE spectra of the difference maps are plotted in Fig. 10 and are described in the sections below. The BB spectra are nearly the same as the EE spectra and are not shown.

6.2.1. Half-ring map differences

The half-ring difference is sensitive to high-frequency noise, since most low-frequency modes (on time scales longer than 1 h) are common to both data sets and thus vanish. In the harmonic domain, the noise is nearly white with an amplitude compatible with the noise estimated in the map domain (see Sect. 6.1). At large multipoles, the noise blows up because of the time transfer function deconvolution, before being cut off by the low-pass filter. At lower multipoles, half-ring differences show low-frequency noise residuals due to the destriping. Indeed, the destriping is performed independently for each half, or essentially half the data are used to solve the offsets for the full ring maps. The residuals from the offset determination are therefore expected to be twice as large as in the full-mission map.

In addition, the deglitching operation performed during the TOI processing uses the full data set to estimate the signal in each ring, thereby introducing some correlation between the two halves of each ring. Taking the difference between the two half-rings removes the correlated portion of the noise at the few percent level.

6.2.2. Half-mission map differences

With half-mission differences, we can check for long-time-scale variations and for apparent gain variation with time due to the analog-to-digital converter non-linearities. Moreover, because of the slightly shifted pointing between the first and second halves of the mission, the effect of a signal gradient within a pixel (especially on the Galactic plane where the signal is strong) is larger than for the half-ring map differences.

Because the number of observations in a given pixel can be very different between the two half-mission maps, using a weighting as described above is essential. Including the weighting, the half-mission differences give a power that is 10–20% higher than the corresponding half-ring difference. This fraction of additional power is nearly the same in all the channels 100–353 GHz, and is the same in both temperature and polarization. We understand this small additional power to be due to effects from the TOI processing that introduce correlations in the noise between the subsets. The half-ring maps, as stated above, have correlations introduced by the deglitcher that are subsequently removed by the differencing. These correlations are not present between the half-mission data sets, so their difference shows a higher noise power.

6.2.3. Detector-set map differences

This difference probes systematic effects that are bolometer-dependent. In the case of 143, 217, and 353 GHz, we note that the detector-set split excludes the unpolarized detectors, and the noise in TT is correspondingly higher than in the half-mission and half-ring split. The 100 GHz channel has only polarization-sensitive bolometers and the TT spectrum of the difference is much closer to the spectrum seen for the other data splits.

There are several other effects that make the power spectrum of the detector-set difference stand apart. A unique time response function is deconvolved from each bolometer. In the half-ring and half-mission data splits, the deconvolved function is identical in the two halves. With the detector-sets, however, the time response is in general slightly different in the two halves. This effect leads to a tilt in the spectrum of the detector-set difference maps relative to the half-ring or half-mission split. Moreover, at 353 GHz, signal residuals are larger owing to relative calibration uncertainties between detectors.

6.2.4. Map differences at low-ℓ

At low multipoles, despite the huge progress in the control of the systematics, data are still contaminated by systematic residuals. Figure 11 shows the EE power spectra from the half-difference maps at 100, 143, and 217 GHz, and compared to the noise power spectrum from simulations (FFP8, described in Planck Collaboration XII (2016). While the half-ring differences are compatible with noise, at multipoles typically lower than 50 the detector-set and half-mission differences are dominated by excess power that is larger than the EE CMB signal. The origin of the excess power will be explored in a forthcoming publication.

6.2.5. High-frequency channels

For the highest frequency channels (545 and 857 GHz) the data-split map differences are dominated by residual signals. Figure 12 shows the rms of the differences of intensity maps at 545 and 857 GHz for half-ring, half-mission, and year data splits compared to the same data split performed on a simulated noise map. At low signal, the difference is consistent with instrumental noise. At high signal levels, an additional residual appears in the difference map that is roughly proportional to the signal level. Part of this is due to pointing errors. For year and half-mission, the effect is enhanced by the combination of residual gain variations and the relative difference of pointing between the two splits. Over most of the sky, the signal is reproducible to better than 1% for these frequencies. Bolometer map differences (not shown here) are, in addition, sensitive to the relative calibration error.

6.3. Noise cross-correlation

Here we check for correlations in the noise by computing cross-spectra between the difference maps described earlier. We look at 100 (Fig. 13), 143 (Fig. 14), and 217 GHz (Fig. 15) in comparison with the expectations from projecting noise realizations on the sky (using the FFP8 noise realizations described in Planck Collaboration XII (2016) and the end-to-end simulations described in Paper A).

When the half-mission cross-spectra of half-ring differences are computed, the results are roughly consistent with the FFP8 noise simulations. At 143 GHz in temperature, the end-to-end simulation produces a slight rise in power at low multipoles that is not seen in the data.

Large correlations are seen in the half-ring cross-spectra of half-mission differences. These are at least partially induced by our processing since the end-to-end simulations also show correlations that are not as large in amplitude as those seen in the data, but show a similar spectral shape. These correlations are mainly due to the deglitcher, as described above.

6.4. Temperature-polarization cross-variance

In absence of spatial correlations, noise correlations between temperature I and polarized Q and U modes vanish in the harmonic domain, thanks to the orthogonality of the spherical harmonic decomposition. Consequently the TE and TB auto-spectra are not biased by noise in the way that the TT, EE, and BB spectra are. In practice, transfer function deconvolution, filtering, and pixelization effects can produce spatial correlations at high multipoles, resulting in a noise bias that is observed in the TE and TB angular power spectra. In Fig. 16 we compare the auto and cross-spectra for the half-ring, half-mission, and detector set splits. These pseudo-spectra have been built by masking Galactic emission and point sources (approximately 40% of the sky). The auto-spectra are biased at high multipoles (starting at ℓ ≈ 1500). The amplitude of this bias and its sign depend on the frequency and on the mode considered. Nevertheless, none of the cross-spectra show significant departures from the null expectation.

We observe that the amplitude of the noise bias in auto-spectra is mitigated when adding more independent data sets, such as detectors or surveys (survey maps show larger amplitude than half-mission and full-mission). These results are fully reproduced in the FFP8 simulations.

6.5. Subset map cross-covariance

When mapping subsets of the available data (selecting detectors and/or time spans) we have a choice between solving for independent baseline offsets for the subset in question or reusing full-mission, full-frequency baselines (as in the 2015 HFI map release). Full-mission baselines are more accurate, leaving less large-scale noise in the maps, but introduce noise correlation between detector-set maps.

We can measure the resulting bias in cross-spectra through noise simulations. Comparison between noise spectra from a Monte Carlo analysis at 100 GHz using both methods of destriping is shown in Fig. 17 for the case of detector sets. Noise spectra for each detector set show more large-scale power when using independent baselines than in the full mission case. On the other hand, the cross-power spectrum is biased by up to a few times 10^-3μK² below ℓ = 10; the same is true for the half-mission subset.

7. Systematic effects

We now describe the major systematic effects that could potentially affect the maps: the gain variations; errors in the absolute gain determination; errors in the polarization efficiency and orientation; and, most of all, the detector-to-detector gain mismatch. The latter includes bandpass mismatch (which affects the response to the foregrounds of the detectors at the same frequency) and relative gain uncertainties, both of which create intensity-to-polarization leakage. All these effects are constrained using tests involving the combination of maps, residuals in maps, cross-power spectra, and dedicated simulations.

7.1. Gain stability

Gain stability has been significantly improved with respect to the Planck 2013 release. This is mainly due to the analog-to-digital converter non-linearity correction, combined with the new determination of the time transfer function (see Paper A). The amplitude of the apparent gain variation has been improved from 1–2% to less than 0.5% for all cases. Residual gain variations are compatible with zero when including the correction for the long-time-constant residuals, as discussed in Sect. 3.3.

We check the stability of the gain over time using the same tool as in Planck Collaboration VIII (2014) called bogopix. For each bolometer, the code fits simultaneously for the gain g_r and the offsets o_r for each ring, marginalizing over the sky signal T: $\begin{array}{ccc}{\mathit{P}}_{\mathit{t}}\mathrm{=}{\mathit{g}}_{\mathit{r}}\mathrm{\times }\mathrm{(}{\mathit{A}}_{\mathit{tp}}\mathrm{·}{\mathit{T}}_{\mathit{p}}\mathrm{+}{\mathit{t}}_{\mathrm{orb}}\mathrm{)}\mathrm{+}{\mathit{o}}_{\mathit{r}}\mathrm{+}{\mathit{n}}_{\mathit{t}}\mathit{.}& & \end{array}$(15)Given the low amplitude of the observed gain variations (less than 0.5%), we linearize Eq. (15)and solve by iteration (see Planck Collaboration VIII 2014); one or two iterations are sufficient to ensure convergence. To initialize the iterations, we start from the constant gain solution G (see Sect. 3.2).

We compute the gain variations from single-bolometer data (neglecting polarization). Polarized signals will affect the gain determination. To reduce this bias, we ignore sky regions where the polarized emission is the strongest, which lie mostly in the Galactic plane.

Figure 18 shows the results of bogopix for bolometers at 100, 143, and 217 GHz, smoothed over a four-day period. At higher frequencies (353 GHz and above), the gain variations are much lower than the gain uncertainty. Owing to the Planck scanning strategy, the Galactic foreground is larger for some rings, while the orbital dipole amplitude is almost constant. This increases the dispersion around those regions and potentially induces some bias in the gain determination. In the end, we find gain variations with amplitudes lower than 0.3, 0.4, and 0.5% at 100, 143, and 217 GHz, respectively.

The residual apparent gain variations are essentially coming from the uncertainty in the current analog-to-digital converter non-linearity correction, the uncertainty in the long-time-constant estimation, and the effect of long-term thermal variations on the bolometer and electronics response.

7.2. Calibration accuracy

7.2.1. Inter-frequency accuracy

The precision of the calibration can be assessed by looking for residual dipoles in the maps. If the calibration of a map is slightly incorrect, the removal of the solar dipole in the mapmaking process leaves a residual dipole. However, identifying such a residual dipole is difficult because of the presence of other sources of power at ℓ = 1, mostly due to Galactic emission and zodiacal light, but also to imperfect correction of systematic effects such as far sidelobes.

Following the method presented in Sect. 5.1 used to estimate the solar dipole direction and amplitude, we cleaned the Galactic emission from the HFI maps using a local correlation with the 857 GHz map. Adding the solar dipole that was removed in the mapmaking process (Sect. 5.1) to D_res produces a map that contains the true solar dipole. We then fit for its amplitude, fixing its direction to the official Planck value (lon, lat ) to limit the effect of other residuals still present in D_res (Galactic, systematic effects). The fit is done using sky pixels where I₈₅₇< 2 (faintest 37% of the sky) to limit the effect of Galactic emission residuals (CO, free-free emission, and small scale dust spectral energy distribution variations). Table 5 gives the ratio of fitted amplitude to the removed dipole found for the HFI maps at frequencies from 100 to 545 GHz. Because effects other than a miscalibration can contribute to a residual dipole in the maps, these ratios provide upper limits on the calibration accuracy at each frequency. The results indicate that the calibration at 100 and 143 GHz is precise at a level of few 10^-4. At 217 GHz, the fit is compatible with a residual dipole at the 0.2% level. At higher frequencies, the fits indicates residuals at 0.52% and 1.23% at 353 and 545 GHz, respectively.

These results are in agreement with those obtained while performing component separation, as shown in Table 4 from Planck Collaboration IX (2016) and in Planck Collaboration X (2016). They are also in agreement with the results from the cosmological parameter determination, where intercalibration coefficients are also fitted for (see Planck Collaboration XI 2016). The agreement between these measurements computed over different multipole ranges highlights the quality of the Planck-HFI calibration, together with the accuracy of the transfer function reconstruction.

7.2.2. Intra-frequency accuracy

For polarization reconstruction with Planck-HFI data, we have to combine data from several detectors. Any relative calibration error will induce an intensity-to-polarization leakage (see Sect. 7.3). For the CMB channels, we have assessed the relative calibration accuracy for each detector at a given frequency using two complementary methods.

As in Sect. 6.2 (Fig. 14) of Planck Collaboration VIII (2014), we derive relative inter-calibration factors for each detector (for 100 to 353 GHz), rescaling their cross-pseudo-power spectra, estimated over 40% of the sky (30% for 353 GHz) in the ℓ range 25–300, which encompasses the first acoustic peak. We used colour-correction factors at 353 GHz, because even at high latitude the dust emission is large. As in 2013, we keep the maximum of these factors as a conservative estimate of the relative calibration accuracy. In 2015, we find 0.09, 0.07, 0.16, and 0.78% for 100, 143, 217, and 353 GHz, respectively (compared to 0.39, 0.28, 0.21, and 1.35% in 2013). Since single-detector maps are built ignoring polarization, these values should be considered as conservative upper limits on the relative detector-to-detector calibration accuracy.

We complemented these estimations by analysing the solar dipole residual on the differences of single detector maps. We fit the dipole amplitude fixing its direction while masking 30% of the sky in the Galactic plane to avoid regions affected by band-pass differences. We find maximum amplitudes 0.5, 0.6, and 3.0 μK for 100, 143, and 217 GHz, respectively, which – relative to the solar dipole amplitude (3364.5 μK, see Sect. 5.1) – gives accuracies of the same order as the aforementioned spectra analysis. As in this previous method, the main limitation comes from polarization which is ignored in the single detector maps.

While significantly better than for the 2013 release, calibration mismatch between bolometers at a given frequency is one of the main systematic residuals contaminating the HFI large angular scales in polarization, as explained in the next section.

7.3. Intensity-to-polarization leakage

Any gain mismatch between the measurements of detectors belonging to the same frequency channel will result in intensity-to-polarization and cross-polarization leakage in the channel maps. In Planck, the dominant leakage effect has three main origins:

• monopole mismatch from the uncertainty in the mean offsetdetermination;

• gain mismatch that produces leakage from the whole intensity signal into polarization;

• bandpass mismatch that mainly generates intensity-to-polarization leakage from foreground emission (with a non-CMB spectrum). In the case of HFI, the leakage effect is dominated by CO and thermal dust emission.

All these leakage sources are especially important for the large angular scales. Beam-mismatch polarization leakage occurs at small angular scales and is discussed in Paper A. Although the first two mismatches can be minimized by obtaining more accurate measurements of offsets and gains, respectively, the bandpass mismatch cannot be removed in the mapmaking process if we want to project CMB and foregrounds at the same time.

The power absorbed by a given bolometer b at time t is expressed using the Stokes parameters (I_p,Q_p,U_p), which characterize the emission in intensity and polarization in the corresponding sky pixel p. The polarized HFI channels are calibrated using the CMB orbital dipole and the total calibrated power absorbed by the bolometer b can be written as $\begin{array}{ccc}{\mathit{m}}_{\mathit{t}}^{\mathit{b}}& \mathrm{=}& \mathrm{(}\mathrm{1}\mathrm{+}{\mathit{ϵ}}_{\mathrm{gain}}^{\mathit{b}}\mathrm{)}\\ & & \mathrm{\times }\left\{\sum _{\mathit{k}}{\mathit{C}}_{\mathit{k}}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}\mathrm{)}\mathrm{[}{\mathit{I}}_{\mathit{p}}^{\mathit{k}}\mathrm{+}{\mathit{\rho }}^{\mathit{b}}{\mathrm{(}{\mathit{Q}}_{\mathit{p}}^{\mathit{k}}\cos {\mathit{\phi }}_{\mathit{t}}^{\mathit{b}}\mathrm{+}{\mathit{U}}_{\mathit{p}}^{\mathit{k}}\sin {{\mathit{\phi }}_{\mathit{t}}^{\mathit{b}}}^{\mathrm{)}}}^{\mathrm{]}}\right\}\\ & & \end{array}$(16)where the polarization efficiency ρ^b and the polarization angle ${\mathit{\phi }}_{\mathit{t}}^{\mathit{b}}\mathrm{=}\mathrm{2}\mathrm{(}{\mathit{\psi }}_{\mathit{t}}\mathrm{+}{\mathit{\alpha }}^{\mathit{b}}\mathrm{)}$ are explicitly dependent on the bolometer b, with the index k ranging over the different sky components. Additionally, we have the following definitions:

• ${\mathit{ϵ}}_{\mathrm{gain}}^{\mathit{b}}$encodes the gain mismatch of bolometer b withrespect to the mean calibration of the channel;

• ${\mathit{ϵ}}_{\mathrm{offset}}^{\mathit{b}}$corresponds to the overall offset of bolometer b, which is small but not vanishing;

• ${\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}$is the bandpass mismatch specific to bolometer b, which affects all sky components except the CMB (see description below);

• C_kis the average transmission of sky component k in a given channel.

Each of these ϵ terms is responsible for leakage from intensity to polarization in a manner that can in principle be quantified and corrected for, as described hereafter. We note that we only consider first-order terms in ϵ, as any higher-order contribution is negligible.

7.3.1. Bandpass mismatch

Each emission component k (where k = CMB, dust, synchrotron, etc.) is integrated over the bandpass of the detector according to a given spectrum f_k(ν). Since the polarized HFI channels are calibrated using the CMB orbital dipole, we define the transmission coefficients $\begin{array}{ccc}{\mathit{C}}_{\mathit{k}}^{\mathit{b}}& \mathrm{=}& \frac{\mathrm{\int }{\mathit{f}}_{\mathit{k}}\mathrm{\left(}\mathit{\nu }\mathrm{\right)}{\mathit{H}}_{\mathit{\nu }}^{\mathit{b}}\mathrm{d}\mathit{\nu }}{\mathrm{\int }{\mathit{f}}_{\mathrm{CMB}}\mathrm{\left(}\mathit{\nu }\mathrm{\right)}{\mathit{H}}_{\mathit{\nu }}^{\mathit{b}}\mathrm{d}\mathit{\nu }}\\ & & \mathrm{\equiv }{\mathit{C}}_{\mathit{k}}\mathrm{(}\mathrm{1}\mathrm{+}{{\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}}^{\mathrm{)}}\mathit{,}\end{array}$(17)where ${\mathit{H}}_{\mathit{\nu }}^{\mathit{b}}$ is the spectral response of bolometer b and ${\mathit{C}}_{\mathit{k}}\mathrm{=}{\sum }_{\mathit{b}}{\mathit{C}}_{\mathit{k}}^{\mathit{b}}\mathit{/}{\mathit{N}}_{\mathrm{bolo}}$ is the average value of the ${\mathit{C}}_{\mathit{k}}^{\mathit{b}}$ in a given channel. These transmission coefficients express the k-component emission in CMB units. If all bolometers had the same spectral responses then ${\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}$ would be equal to zero, i.e. ${\mathit{C}}_{\mathit{k}}^{\mathit{b}}\mathrm{=}{\mathit{C}}_{\mathit{k}}$, in which case no leakage related to bandpass-mismatch would be produced.

Considering only bandpass mismatch corresponds to setting ${\mathit{ϵ}}_{\mathrm{gain}}^{\mathit{b}}\mathrm{=}\mathrm{0}$ and ${\mathit{ϵ}}_{\mathrm{offset}}^{\mathit{b}}\mathrm{=}\mathrm{0}$ in Eq. (16). Then, ordering all the data samples, ${\mathit{m}}_{\mathit{t}}^{\mathit{b}}$, for a bolometer observing a position p on the sky into a single vector D^b, defining A to be the pointing matrix in temperature and polarization, and n the noise vector, Eq. (16)reads $\begin{array}{ccc}{{D}}^{\mathit{b}}\mathrm{=}\sum _{\mathit{k}}{\mathit{C}}_{\mathit{k}}\mathrm{A}\left(\begin{array}{c}\\ \\ \\ \end{array}\right)\mathrm{+}\sum _{\mathit{k}}{\mathit{C}}_{\mathit{k}}{\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}\mathrm{A}\left(\begin{array}{c}\\ \\ \\ \end{array}\right)\mathrm{+}{n}\mathit{.}& & \end{array}$(18)Using all bolometers b within a channel, the mapmaking procedure solves for the total signal Stokes parameters $\mathrm{(}{\mathit{I}}_{\mathit{p}}^{\mathrm{tot}}\hspace{0.17em}\mathit{,}{\mathit{Q}}_{\mathit{p}}^{\mathrm{tot}}\hspace{0.17em}\mathit{,}{\mathit{U}}_{\mathit{p}}^{\mathrm{tot}}\mathrm{)}$ in pixel p, formally computing, $\begin{array}{ccc}\left(\begin{array}{c}\\ \\ \\ \end{array}\right)\mathrm{=}{\mathrm{(}{\mathrm{A}}^{\mathrm{T}}{\mathrm{N}}^{-1}{\mathrm{A}}^{\mathrm{)}}}^{-1}{\mathrm{A}}^{\mathit{T}}{\mathrm{N}}^{-1}{D}\mathit{,}& & \end{array}$(19)which becomes $\begin{array}{ccc}\left(\begin{array}{c}\\ \\ \\ \end{array}\right)\mathrm{=}\sum _{\mathit{k}}{\mathit{C}}_{\mathit{k}}\left(\begin{array}{c}\\ \\ \\ \end{array}\right)\mathrm{+}\sum _{\mathit{k}}{\mathit{C}}_{\mathit{k}}\sum _{\mathit{b}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{b}}\mathrm{-}\mathrm{1}}{\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}{\mathrm{\Gamma }}_{\mathit{p}}^{\mathit{b}}\left(\begin{array}{c}\\ \\ \\ \end{array}\right)\mathit{,}& & \end{array}$(20)where ${\mathrm{\Gamma }}_{\mathit{p}}^{\mathit{b}}\mathrm{\equiv }{{}^{\mathrm{(}}\mathrm{A}^{\mathrm{T}}{\mathrm{N}}^{-1}{\mathrm{A}}^{\mathrm{)}}}^{-1}{\mathrm{A}}^{\mathrm{T}}{\mathrm{N}}^{-1}{\mathrm{\Delta }}^{\mathit{b}}\mathrm{A}$. We have introduced the matrix Δ^b, the elements of which are equal to zero except for the diagonal elements relevant to bolometer b, which are set to 1. The last term of Eq. (20)is the leakage term in pixel p, where intensity will leak into Q and U, Q into I and U, and U into I and Q, according to the mismatch coefficients ${\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}$ and the values of the 3 × 3 matrix $\begin{array}{ccc}{\mathrm{\Gamma }}_{\mathit{p}}^{\mathit{b}}\mathrm{=}\left(\begin{array}{c}\\ & & \\ & & \\ & & \end{array}\right)\begin{array}{c}\mathit{b}\\ \mathit{p}\end{array}\mathit{.}& & \end{array}$(21)Considering all pixels, the quantities ${\mathrm{\Gamma }}_{\mathit{XX}}^{\mathit{b}}$ correspond to nine sky maps for bolometer b. These maps can be fully determined from the mapmaking solution and may be understood as patterns of the mismatch leakage. In practice, cross-polarization leakage and polarization-to-intensity leakage are negligible compared to the intensity-to-polarization contribution and we therefore consider the latter only. The Γ_IQ and Γ_IU maps have been systematically produced by the mapmaking pipeline⁴. With these assumptions, the leakage induced by bandpass-mismatch in Q and U for the sky component k reads $\begin{array}{ccc}{\mathit{L}}_{\mathit{IQ,IU}}^{\mathrm{BP}\mathit{,k}}\mathrm{=}{\mathit{C}}_{\mathit{k}}{\mathit{I}}^{\mathit{k}}\sum _{\mathit{b}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{bolo}}\mathrm{-}\mathrm{1}}{\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}\hspace{0.17em}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathit{.}& & \end{array}$(22)In consequence, for a given calibrated intensity template of the sky component k (i.e. ${\mathit{I}}_{\mathrm{template}}^{\mathit{k}}\mathrm{=}{\mathit{C}}_{\mathit{k}}{\mathit{I}}^{\mathit{k}}$) we can compute leakage correction maps as $\begin{array}{ccc}{\mathit{L}}_{\mathit{IQ,IU}}^{\mathrm{corr}\mathit{,k}}\mathrm{=}{\mathit{I}}_{\mathrm{template}}^{\mathit{k}}\sum _{\mathit{b}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{bolo}}\mathrm{-}\mathrm{1}}{\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}\hspace{0.17em}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{=}{\mathit{I}}_{\mathrm{template}}^{\mathit{k}}\sum _{\mathit{b}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{bolo}}\mathrm{-}\mathrm{1}}\frac{{\mathit{C}}_{\mathit{k}}^{\mathit{b}}}{{\mathit{C}}_{\mathit{k}}}\hspace{0.17em}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathit{,}& & \end{array}$(23)where the last equality assumes ${\sum }_{\mathit{b}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{bolo}}\mathrm{-}\mathrm{1}}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{=}\mathrm{0}$ by construction.

Leakage correction maps have been produced for all polarized HFI channels. The relevant foregrounds at these frequencies are dust (all channels) and CO (all channels except 143 GHz). To do so, the coefficients ${\mathit{C}}_{\mathrm{dust}}^{\mathit{b}}$ have been computed from Eq. (17), where the spectral responses of the bolometers H^b(ν) are those obtained from pre-launch ground-based measurements of the bandpasses (Planck Collaboration IX 2014). The dust spectrum is taken as a grey body with spectral index β = 1.62 and temperature T = 19.7 K, which are the all-sky average values found in Planck Collaboration XI (2014). For the intensity template required in Eq. (23), we use the thermal dust intensity maps at 353 GHz obtained from the Planck thermal dust model (Planck Collaboration XI 2014). Combining all these ingredients, the dust correction maps ${\mathit{L}}_{\mathit{IQ,IU}}^{\mathrm{corr}\mathrm{\left(}\mathrm{dust}\mathrm{\right)}}$ are produced according to Eq. (23)and delivered in the 2015 HFI data release (Fig. 19).

We note, however, that the reliability of these corrections is limited by uncertainties both in the physical nature of the foreground components and in the determination of the bolometer spectral responses. For the sake of simplicity, a constant spectral index and a constant temperature across the sky have been assumed for the thermal dust emission. Furthermore, the calibrated thermal dust intensity templates are those derived from the 2013 Planck thermal dust model; although they are close, these templates do not strictly correspond to the calibration of the 2015 maps. Also, the leakage corrections are particularly sensitive to the differences in transmission between bolometers (i.e. the ${\mathit{C}}_{\mathit{k}}^{\mathit{b}}$ coefficients); small uncertainties on those will yield large uncertainties in the final correction maps. In conclusion, the bandpass leakage corrections should not be taken at face value, but should be thought of as order-of-magnitude estimates only. We only advocate the use of these correction maps to test the stability and estimate uncertainties of any further results using the HFI polarization maps. A result solely obtained by applying the corrections will not be reliable.

7.3.2. Calibration and monopole mismatches

Using the same formalism as above, calibration mismatch is computed by setting ${\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}\mathrm{=}\mathrm{0}$ and ${\mathit{ϵ}}_{\mathrm{offset}}^{\mathit{b}}\mathrm{=}\mathrm{0}$ in Eq. (16). Following closely Sect. 7.3.1, one finds that the total intensity-to-polarization leakage due to calibration mismatch is $\begin{array}{ccc}{\mathit{L}}_{\mathit{IQ,IU}}^{\mathrm{gain}}& \mathrm{=}& \sum _{\mathit{k}}{\mathit{C}}_{\mathit{k}}{\mathit{I}}^{\mathit{k}}\mathrm{\times }\sum _{\mathit{b}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{bolo}}\mathrm{-}\mathrm{1}}{\mathit{ϵ}}_{\mathrm{gain}}^{\mathit{b}}\hspace{0.17em}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\\ & \mathrm{\approx }& \end{array}$(24)where at first order, for low-frequency maps, the solar dipole signal (k = dipole, C_dipole = 1 by construction) provides the dominant contribution to the calibration mismatch leakage effect.

Setting ${\mathit{ϵ}}_{\mathrm{BP}\mathit{,k}}^{\mathit{b}}\mathrm{=}\mathrm{0}$ and ${\mathit{ϵ}}_{\mathrm{gain}}^{\mathit{b}}\mathrm{=}\mathrm{0}$ in Eq. (16), one shows in a similar fashion that the monopole intensity-to-polarization leakage is simply $\begin{array}{ccc}{\mathit{L}}_{\mathit{IQ,IU}}^{\mathrm{mono}}\mathrm{=}\sum _{\mathit{b}\mathrm{=}\mathrm{0}}^{{\mathit{N}}_{\mathrm{bolo}}\mathrm{-}\mathrm{1}}{\mathit{ϵ}}_{\mathrm{offset}}^{\mathit{b}}\hspace{0.17em}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathit{,}& & \end{array}$(25)where the monopole mismatch is modelled using a constant sky template I^monopole = 1, while the amplitude of the mismatch is encoded in ${\mathit{ϵ}}_{\mathrm{offset}}^{\mathit{b}}$.

Although the bandpass mismatch coefficients ${\mathit{ϵ}}_{\mathrm{BP}\mathit{,}\mathrm{dust}}^{\mathit{b}}$ can be evaluated directly from foreground modelling (assuming a given spectrum of the dust and using the spectral responses of the detectors), this is not the case for ${\mathit{ϵ}}_{\mathrm{gain}}^{\mathit{b}}$ and ${\mathit{ϵ}}_{\mathrm{offset}}^{\mathit{b}}$. It is therefore not possible to provide correction maps for these leakage effects by computing Eqs. (24)and (25)directly. However, one can consider the possibility of fitting these quantities from the maps themselves by using the ${\mathit{I}}^{\mathrm{dipole}}\mathrm{\times }{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}$ and ${\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}$ as templates of the gain and monopole leakages, respectively. Such a method, known as the generalized global fit (GGF), has been implemented and is further described in Appendix A.3.2. The leakage maps produced with this method for the 353 GHz channel are delivered in the 2015 release and have been corrected simultaneously for leakages induced by bandpass, calibration, and monopole mismatches.

7.4. In-flight validation of the polarimeter efficiency and orientation

As discussed in Sect. 4.5, the polarimeter efficiency and orientation used in this release are taken from ground measurements (Rosset et al. 2010). In order to validate these numbers in flight, we used the Crab nebula maps obtained with the IRAM 30 m telescope and the 90 GHz XPOL polarimeter (Aumont et al. 2010). These maps consist of I, Q, and U measurements with an angular resolution of 27′′ of a 10′-wide region around the Crab nebula (Tau A, M1, or NGC 1952, at J2000 coordinates RA = 5^h34^m32^s and Dec = 22°00′52′′). The same region was observed by Planck once per survey, with different scan directions for odd and even surveys. We compared single survey, single bolometer maps of the Crab region with a model obtained from the IRAM maps, and solved for the best values of polarimeter angle and efficiency.

From single survey, single bolometer data we can only solve for an intensity map, which projects on the sky the total power P_t described in Eq. (1). This power depends on the true value of the polarization angle α specific to the detector.

We compared the single bolometer, single survey maps with a model obtained with the following procedure:

• we pixelized the IRAM observations on aHEALPix grid with N_side = 2048, rotating toGalactic coordinates;

• we convolved these maps with the single bolometer, single survey effective beams using FEBeCoP (Mitra et al. 2011);

• using the Crab IRAM map as a template, and the polarization angles α, we modelled the intensity map described above in the Crab region as a function of an angular offset Δα.

We then fitted for the values of the angular offset Δα. To do that, we first removed the background from the single bolometer maps. We built a noise model combining the single detector pixel variance with the noise of the IRAM observation, taking into account the smoothing applied. We used the Rosset et al. (2010) values as a prior.

The resulting angular offsets are presented in Fig. 20. Corrections are compatible with zero, and this analysis does not favour an update of the ground-based parameters. We used the same procedure to fit the polarization efficiency ρ, but the result is completely dominated by the ground-based calibration prior.

If we assume that the CMB anisotropies have vanishing TB and EB power spectra, i.e. that there are no parity-violating physical mechanisms in the early Universe, we can also check whether the overall polarizer angle of Planck-HFI is compatible with zero. Planck Collaboration XXXI (in prep.) show that the CMB TB and EB spectra measured by HFI are consistent with zero. Their analysis gives a polarizer angle within 0.3° of zero, which is identical to the systematic error of the ground-based measurements. This is a factor of five improvement over the WMAP final results (Hinshaw et al. 2013) and is comparable with ACT (Naess et al. 2014; see also the review by Gubitosi & Paci 2013).

8. Conclusions

This paper has described the processing applied to construct the Planck-HFI maps delivered in the 2015 release. It has also assessed the main characteristics of the maps in terms of noise and systematics, in particular resulting from the analog-to-digital converter non-linearity correction and bolometer long time constants. Since the last release, the calibration has been upgraded and is now significantly more accurate. At low frequency, it is now independent and based on the orbital dipole signal, while the planets Uranus and Neptune are used to calibrate the high end of HFI, achieving 6.1 and 6.4% absolute photometric calibration at 545 and 857 GHz, respectively. This has allowed us to measure a consistent CMB solar dipole with an unprecedented accuracy better than 10^-3 and in agreement with the independent determination by LFI.

Table 6 gives a quantitative assessment of the main characteristics of the Planck HFI maps from the 2015 release. They now cover the entire Planck HFI cold mission (885 days). The HFI aggregated sensitivity (referring to a weighted average of the 100, 143, and 217 GHz channel maps) is 26 μK_CMBarcmin in temperature and 52 μK_CMBarcmin in polarization.

The noise in the maps shows some small low-frequency excess in addition to white noise prior to time constant deconvolution. The latter then naturally raises the higher part of the noise spectra in the multipole domain. We have identified a low-level noise correlation in particular between half-ring and detector subsets that is not directly reproduced by simulations, although the level is low compared to the CMB signal.

The raw sensitivity must be matched by a long list of constraints on any possible systematic effects. This list includes: an absolute calibration at a level from 0.1% to 1.4% depending on the frequency; a resulting apparent gain variation of less than 0.5%; and a knowledge of the polarization angle and polarization absolute value respectively at the degree level and the 1% level. The instrumental beam has been measured at the percent level by using multiple planet crossings.

Despite the huge progress made in the understanding of all the aforementioned systematic effects, Planck-HFI polarization maps are still dominated by systematic residuals at large scales. These are essentially coming from the temperature-to-polarization leakage resulting from the mismatch between the bolometers that are combined to reconstruct linear polarization maps. The origins of the leakage effects include mismatch of the zero level from uncertainty in the offset determination, mismatch from gain uncertainty (even at the 10^-3 level), and bandpass mismatch. Corresponding first-order corrections for monopole, dipole, and bandpass mismatch are provided (as described in Appendix A.3.2), but residuals are still found to be larger than noise at very large scales. As a consequence, the Planck-HFI polarization maps at large scales cannot yet be directly used for cosmological studies.

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Acknowledgments

The Planck Collaboration acknowledges the support of: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, 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); ERC and PRACE (EU). A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at http://www.cosmos.esa.int/web/planck/planck-collaboration

References

Appendix A: HFI map product description

Here we summarize the HFI map products that are part of the Planck 2015 data release.

Appendix A.1: Map products

The 2015 release contains many different maps whose details are described below. All maps are given as HEALPix vectors with NESTED ordering in Galactic coordinates at the resolution corresponding to N_side = 2048 (for high resolution maps). Depending on the type of product, these vectors are packaged into a binary table and written into a FITS file. The table contains in most cases 50 331 648 rows (the length of the HEALPix vector for N_side = 2048) and either three columns (I, II, H for intensity, noise (co)variance, and hits, respectively) for the temperature-only cases or ten columns (I, Q, and U signals plus the six various I, Q, and U noise (co)variances and hits) for the polarized cases. Pixels with a condition number larger than 10³ see their hit number brought down to zero. A pixel with zero hits has an intensity value of −1.6375 × 10³⁰ (the HEALPix conventional null value).

The general matrix of products is pictured in Fig. 4. The main products are the maps of the full channels covering the full mission. For characterization and analysis purposes, the channels have been split into independent detector sets. The detector sets for the polarized channels are groups of four polarization-sensitive bolometers that can be used to build a sky map in temperature and polarization. Bolometers insensitive to polarization (Spider-Web Bolometers) of the same frequency channels are not used in the detector sets. The detector set names and the bolometers they use are listed in Table A.1. For completeness, full mission maps built using each of the unpolarized bolometers are also provided.

The mission duration has been split into single surveys, years, and the two halves of the mission duration. The surveys are defined as the observations within a contiguous rotation range of 180° for the spin axis, and as a consequence each survey does not cover the full sky. We note that the fifth survey had not been completed when HFI stopped observing, and was also interrupted by various end-of-mission tests. That last survey was also performed with a different scanning strategy than the first four surveys (Planck Collaboration I 2016). The date and ring number corresponding to the beginning and end of each time split are summarized in Table A.2. Single survey maps are provided for the full channels only. Yearly maps are provided for Year 1 and Year 2 for the full channels, the detector sets and each of the Spider-Web Bolometers, where the years span Surveys 1–2 and 3–4. Half-mission maps are also provided, where each half contains one half of the valid rings (or stable pointing periods). There are 347 discarded rings (and 26 419 valid ones), most of which occurred during the partial last survey when various end-of-life tests were performed.

Half-ring maps are produced by splitting each ring into two parts of equal duration. The difference of the two half-ring maps provides a useful estimate of the high-frequency noise and possibly of other systematics. We note that this is the only case where the destriping offsets are different from the offsets in the standard case. Half-ring maps are provided for the full mission only. They are given for the full channels, the detector sets, and the Spider-Web Bolometer maps.

Maps are corrected for zodiacal light emission. Correction maps are also provided for the frequency maps and the various time splits.

Units are K_CMB for frequencies up to 353 GHz and  MJy sr^-1assuming a constant νI_ν law above.

Appendix A.2: Stokes parameter correlations

The Planck HFI delivery includes pixelized maps of Stokes covariances (II, IQ, IU, QQ, QU, UU) solved during the mapmaking process for each pixel independently.

In Fig. A.1 we present the distribution of the I, Q, and U correlations in each pixel for the HFI frequencies where polarization is reconstructed.

Appendix A.3: Leakage correction maps

Section 7.3 describes the origin and the formalism of the intensity-to-polarization leakage in HFI and how it is, in principle, possible to quantify and correct for these systematic effects. This is particularly important for any studies of the large angular scales in polarization where dipole, monopole, and dust leakages are the main limiting factors.

Although dust and CO bandpass leakage effects can be estimated from given intensity templates and electromagnetic spectra (Sect. A.3.1), this is not the case for the calibration and monopole leakage levels, which have to be estimated directly from the maps (Sect. A.3.2).

Appendix A.3.1: Bandpass leakage correction maps

The small differences in the bandpasses of the different bolometers combined to produce polarization maps give rise to some leakage from intensity to polarization for the CO and the (Galactic) dust emission; this is estimated using the known band profiles and templates of the emission (see Sect. 7.3.1).

Bandpass leakage Q and U maps are provided for dust for all channels, using ground-based measurement of the bandpass integrated over the dust emission law. Each FITS file contains a single extension, with two columns containing the Q and U leakage maps in K_CMB units.

Appendix A.3.2: The generalized global fit (GGF) approach for leakage correction

The generalized global fit (GGF) method is a template fitting approach that has been developed to consistently solve for calibration, monopole, and bandpass mismatch leakage effects at the map level. Each Q and U map at 100, 143, 217, and 353 GHz is modelled as $\begin{array}{ccc}{\left[\mathit{Q,U}\right]}^{\mathit{\nu }}& \mathrm{=}& {\left[\mathit{Q,U}\right]}_{\mathrm{CMB}}\mathrm{+}\left[\mathit{Q,U}\right]\begin{array}{c}\mathit{\nu }\\ \mathrm{dust}\end{array}\\ & & \mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\alpha }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\beta }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{dipole}}\\ & & \mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\gamma }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{dust}}\mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\delta }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{CO}}\mathit{,}\end{array}$(A.1)where

• the first line corresponds to the physical polarization signalcoming from the CMB and the dust (synchrotron is assumednegligible at these frequencies);

• the second line corresponds to the monopole and calibration leakage terms, the templates of which make use of the leakage pattern maps Γ described in Sect. 7.3;

• the dust and CO bandpass-missmatch leakage terms are gathered in the third line, again using the leakage patterns maps Γ;

• each summation is performed over the polarized bolometers of the frequency channel ν.

We use the Planck 353 GHz Q and U maps as polarized dust templates and perform the fitting procedure at 1° resolution and N_side = 64. Focusing on polarized dust and leakage effects only, Eq. (A.1)becomes $\begin{array}{ccc}{\left[\mathit{Q,U}\right]}^{\mathit{\nu }}& \mathrm{=}& \mathrm{(}\mathrm{1}\mathrm{-}{\mathit{ϵ}}^{\mathit{\nu }}\mathrm{)}{\left[\mathit{Q,U}\right]}_{\mathrm{CMB}}\mathrm{+}{\mathit{ϵ}}^{\mathit{\nu }}{\left[\mathit{Q,U}\right]}^{\mathrm{353}}\\ & & \mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\alpha }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\beta }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{dipole}}\\ & & \mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\gamma }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{dust}}\mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathit{\nu }}{\mathit{\delta }}_{\mathit{b}}^{\mathit{\nu }}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{CO}}\\ & & \mathrm{-}{\mathit{ϵ}}^{\mathit{\nu }}\mathrm{\times }[\sum _{\mathit{b}\mathrm{\in }\mathrm{353}}{\mathit{\alpha }}_{\mathit{b}}^{\mathrm{353}}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathrm{353}}{\mathit{\beta }}_{\mathit{b}}^{\mathrm{353}}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{dipole}}\\ & & \mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathrm{353}}{\mathit{\gamma }}_{\mathit{b}}^{\mathrm{353}}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{dust}}\mathrm{+}\sum _{\mathit{b}\mathrm{\in }\mathrm{353}}{\mathit{\delta }}_{\mathit{b}}^{\mathrm{353}}{\mathrm{\Gamma }}_{\mathit{IQ,IU}}^{\mathit{b}}\mathrm{\times }{\mathit{I}}^{\mathrm{CO}}]\mathit{,}\end{array}$(A.2)where ϵ^ν is the overall factor scaling the dust from 353 GHz to the frequency channel ν and where α, β, γ, and δ define the amplitude of the monopole, dipole, dust, and CO leakage effects, respectively. The last two lines of Eq. (A.2)correct for the total leakage added when using [Q,U]³⁵³ as dust templates. This equation is at the core of the GGF method, which is then implemented as follows:

• 1.

  A first fit is performed to solve for the coefficients ofEq. (A.2), for each channel (ν = 100, 143, and217 GHz) independently. There are strongdegeneracies between the leakage templates at frequency ν and at353 GHz, so that this first fit does not providereliable α, β, γ, and δ coefficients that can be used tocompute leakage correction maps. However, it enables anaccurate determination of the overall scaling factor of the dust ϵ^νbetween the channel under scrutiny and353 GHz.

• 2.

  These ϵ^ν values are used as inputs to solve Eq. (A.2), simultaneously for all channels, in a consistent manner. At this stage, we add some extra constraints (such as minimizing detector-set and survey differences) in order to reduce the leakage degeneracies between the four channels. This generalized global fit allows us to extract the set of α, β, γ, and δ for all HFI polarized channels, including 353 GHz.

Solving for survey differences in the second step is crucial. Although the leakage coefficients are independent of scan angle, the Γ maps change because pixels are scanned at different angles in different surveys. Total leakage maps, allowing a correction for leakages induced by monopole, calibration, and bandpass mismatches, are then computed using the coefficients extracted in the second step of the procedure. Figure A.2 shows the total correction in Q (top) and U (bottom) for the 353 GHz channel. Dust and CO bandpass-mismatch leakage effects are clearly dominant near the Galactic plane, while the large patterns at high latitude are mainly due to the dipole leakage.

All Tables

All Figures