Planck Pre-Launch Status Papers - APC - Université Paris Diderot ...

A&A 520, A13 (2010)In the case of orientation errors only, to first order,∆ α IQ = −1 2 (ω 1 − ω 2 ) 〈sin 2ψ〉− 1 2 (ω 3 − ω 4 ) 〈cos 2ψ〉 (31)∆ α IU = 1 2 (ω 1 − ω 2 ) 〈cos 2ψ〉− 1 2 (ω 3 − ω 4 ) 〈sin 2ψ〉 (32)∆ α QQ = − [(ω 1 + ω 2 ) − (ω 3 + ω 4 )] 〈cos 2ψ sin 2ψ〉 (33)∆ α QU = (ω 1 + ω 2 ) 〈 cos 2 2ψ 〉 + (ω 3 + ω 4 ) 〈 sin 2 2ψ 〉 (34)∆ α UQ = −(ω 1 + ω 2 ) 〈 sin 2 2ψ 〉 − (ω 3 + ω 4 ) 〈 cos 2 2ψ 〉 (35)∆ α UU = [(ω 1 + ω 2 ) − (ω 3 + ω 4 )] 〈cos 2ψ sin 2ψ〉 . (36)In these Eqs. (22)–(36), the average is over the samples fallinginto a given pixel. It depends only on the scanning strategy.Figure 2 shows the angle distribution on the sky for a realisticPlanck scanning strategy. Planck shows large inhomogeneitiesthat induce additional terms with respect to the case of a singlebolometer.Leakage from intensity to polarization. Erroron gain onlyproduces leakage from intensity to polarization (seeEq. (A.8)). This leakage is driven by the relative errors insidea given horn which indicates that an absolute error onthe gain (same for all detectors) will not produce any leakage.Neither polarization efficiency nor detector orientationerrors induce any leakage from I into polarization Q and U(see Eqs. (A.9)−(A.14)).Leakage from polarization to intensity. Bothpolarizationefficiencyand orientation error produce leakage from polarizationto intensity. It is driven by the difference of errors withinone horn and the relative weight of each horn depends on thedistribution of ψ (see Fig. 2).Polarization mixing. Polarizationcalibration parameters mixboth Q and U. ThismeansthattheyinduceleakagefromQto U through the term ∆ ρ QU(and from U to Q through theterm ∆ ρ UQ )butalsoaltertheamplitudeofpolarization(∆ρ QQand ∆ ρ UU 0). If we consider identical errors for eachdetector, we are in the limiting case where orientation errorinduces only leakage (Eqs. (34), (35)) and polarizationefficiency only changes the amplitude of polarization(Eqs. (27), (30)) as described by Eq. (16). In the case ofPlanck-HFI, and considering independent errors, none ofthese simplifications apply. In particular, different parameteraverages from one horn to the other induce both Q and Umixing and amplitude modification.6.2. Results for E and B-mode power spectraThe semi-analytical method described in Sect. 5 is able to propagateinstrumental errors up to the six CMB power spectra: TT,EE, BB, TE, TB and EB.Inthissection,wewillfocusonthe E and B-mode power spectra and discuss results obtainedfor Planck-HFI in case of absolute (Sect. 6.2.1)andrelativeuncertainties(Sect. 6.2.2). Other spectra (like TBand EB)thatarepredicted to be null for CMB signal, can be very useful in revealing“leakage” due to systematics. However, many systematic effectscan produce such leakage, which will make their separateidentification very complicated when using only these modes.6.2.1. Global error over the focal plane/calibrationon the skyAbsolute calibration of total power is done using the orbitaldipole that has the same electromagnetic spectrum as the CMBFig. 2. Amplitude of the various terms in Eqs. (22)−(36) describingthefocal plane angle distribution on the sky for a mock but realistic Planckscanning coverage (HEALPix maps at nside = 1024, Galactic coordinates).From top to bottom: |〈cos 2ψ〉|, |〈sin 2ψ〉|, |〈sin 4ψ〉|/2, 〈cos 2 2ψ〉.and is not degenerate with the underlying sky signal as its signchanges after 6 months of observation. From Eqs. (23)and(24),absolute error on the gain g will not produce any leakage in polarizationsignals:⎛ ⎞γ 00∆ g s = ⎜⎝000000⎟⎠ s for gain. (37)As far as polarization is concerned, we need a polarized sourceon the sky. The Crab nebulae, a supernova remnant, is a goodcandidate as it shows a large polarization emission in the Planck-HFI frequency bands. It has been observed in a wide rangePage 6 of 12