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

C. Rosset et al.: Planck-HFI: polarization calibrationThis estimator neglects the E − B mixing due to incomplete skycoverage (Lewis et al. 2002) andassumesacross-powerspectrumfor which noise bias is null (or if auto-spectra are used, thatthe noise bias has been previously removed) because their interactionwith the systematic effects introduced here are of secondorder.Using the previous relations, straightforward algebra leadsfrom Eq. (15)toitscounterpartinharmonicspace:∑ ∂ ˜C l∑ ∂ ˜C l∑ ∂ ˜C l∆ ˜C l = γ d + ɛ d + ω d∂γd d ∂ɛd d ∂ωd d∑+ 1 2+[ ∂2 ˜C l∂γd,d ′ d ∂γ d ′∂ 2 ˜C lω d ω d ′∂ω d ∂ω d ′∂2 ˜C lwhere, for e = γ, ɛ or ω,∂ClXY 1l∑⎡=⎢⎣ ∂aX∗ lm∂e 2l + 1 ∂e aY lm + aX∗ lmm=−l∂ 2 ClXY 1l∑⎡∂e∂e ′ =⎢⎣ ∂2 almX∗2l + 1 ∂e∂e ′ aY lm + ∂aX∗ lm∂em=−l+ ∂aX∗ lm∂a Y lm∂e ′ ∂eγ d γ d ′ + ɛ d ɛ d ′∂ɛ d ∂ɛ d ′], (18)+ aX∗ lm∂a Y ⎤lm⎥⎦ (19)∂e∂a Y lm∂e ′∂ 2 a Y lm⎤⎥⎦∂e∂e ′ · (20)We ignore cross-terms between different systematic parametersso the previous expressions are only applicable when all but oneof the parameters are set to zero. The cross-terms have beenchecked to be one order of magnitude below the direct terms.Note that we push the perturbative expansion to second order,since E-modes are much larger than B-modes and a second ordereffect on E-modes has an impact comparable to a first ordereffect on B-modes.5.3. Monte-Carlo simulationsWe have now everything in hand toperformthesemi-analyticalestimate of the polarization calibration systematic effects. Themethod can be described in 5 main steps:1. From the scanning strategy of the instrument, for each detectord, projectintoamap:cos2ψ, sin2ψ, cos2ψ sin 2ψ,and cos 2 2ψ.2. With these quantities, compute for each pixel of the map thefollowing 3 × 3matrices: [ ∑d ÃT d Ãd] −1,Λd ,anditsfirstandsecond derivatives.3. Use a simulated CMB sky s and Eq. (15)tocomputepartialderivatives ∂s/∂e (up to second order).4. Compute all cross-power spectra between s and its derivatives.5. Combine these results using gaussian random distributionsof γ d , ɛ d and ω d (with various rms σ) inEq.(18) toobtainthe final error on the angular power spectrum.The power spectra estimator used is a pseudo-C l estimator basedon the cross-power spectra algorithm (Tristram et al. 2005), extendedto polarization (Kogut et al. 2003; Grain et al. 2009). Thesemi-analytic method described in this section has been comparedto full Monte-Carlo simulations and gives results compatiblewith statistical expectations for the number of simulationsperformed.6. Application to Planck -HFI focal planeWe apply the method described in the previous section to thePlanck-HFI to set requirements on gain, polarization efficiencyand orientation. We simulated HEALPix (Górski et al. 2005)full-sky maps at a resolution of ∼3.5 arcmin(nside = 1024) sothat all pixels are seen and each pixel is uniformly sampled. Thisavoids the complications of estimating power spectra on a cutsky when allowing for the same conclusions, as our power spectrumestimator is not biased in the mean. The scanning strategythat we use is a realistic simulation of what Planck will actuallydo in a 14-month mission. The sky signal is pure CMB simulatedfrom the best ΛCDM fit to WMAP 5 years data (Dunkleyet al. 2009) withr = 0.05, supposing the CMB signal to bedominant over foregrounds residuals (at least for intensity andE-mode CMB signals).As described in Sect. 2,thePlanck scanning strategy and focalplane design do not allow the data from a single PSB pairto provide independent maps of the Stokes parameters. Here,we will use two PSB pairs calibrated in intensity and considersmall variations around their gain g d = 1, nominal angles α d ={0 ◦ , 90 ◦ , 45 ◦ , 135 ◦ } and nominal polarization efficiency ρ ′ d = 1(corresponding to perfect PSB).6.1. Error on Stokes parameter for HFIWe refer to Appendix A for the explicit form of the derivativeterms of the Stokes parameters. Here, we emphasizethe issues specific to HFI. In this case, Eq. (15) reads(see Eqs. (A.9)−(A.14))⎛∆ II ∆ IQ ∆ IU∆s = ⎜⎝∆ QI ∆ QQ ∆ QU⎞⎟⎠ s. (21)∆ UI ∆ UQ ∆ UUFor gain variations only, non-zero elements of the matrix aregiven for each pixel, to first order, by∆ g II = 1 4 (γ 1 + γ 2 + γ 3 + γ 4 ) (22)∆ g QI = 1 4 (γ 1 − γ 2 ) 〈cos 2ψ〉− 1 4 (γ 3 − γ 4 ) 〈sin 2ψ〉 (23)∆ g UI = 1 4 (γ 1 − γ 2 ) 〈sin 2ψ〉 + 1 4 (γ 3 − γ 4 ) 〈cos 2ψ〉 . (24)For polarization efficiency only, elements of the matrix are givenfor each pixel, to first order, by∆ ρ IQ = 1 4 (ɛ 1 − ɛ 2 ) 〈cos 2ψ〉− 1 4 (ɛ 3 − ɛ 4 ) 〈sin 2ψ〉 (25)∆ ρ IU = 1 4 (ɛ 1 − ɛ 2 ) 〈sin 2ψ〉 + 1 4 (ɛ 3 − ɛ 4 ) 〈cos 2ψ〉 (26)∆ ρ QQ = 1 2 (ɛ 1 + ɛ 2 ) 〈 cos 2 2ψ 〉 + 1 2 (ɛ 3 + ɛ 4 ) 〈 sin 2 2ψ 〉 (27)∆ ρ QU = 1 2 [(ɛ 1 + ɛ 2 ) − (ɛ 3 + ɛ 4 )] 〈cos 2ψ sin 2ψ〉 (28)∆ ρ UQ = 1 2 [(ɛ 1 + ɛ 2 ) − (ɛ 3 + ɛ 4 )] 〈cos 2ψ sin 2ψ〉 (29)∆ ρ UU = 1 2 (ɛ 1 + ɛ 2 ) 〈 sin 2 2ψ 〉 + 1 2 (ɛ 3 + ɛ 4 ) 〈 cos 2 2ψ 〉 . (30)Page 5 of 12