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

A&A 520, A8 (2010)inflation, giving us a unique window on physics at ∼10 16 GeVenergies.The strategic role of LFI polarimetry within the Planck missionis: (i) to constrain the steep-spectrum polarised foregrounds,dominated by Galactic synchrotron emission; and (ii) to mapthe sky close to the minimum of foreground contamination at70 GHz, albeit with less sensitivity to the CMB than availablefrom Planck’s High Frequency Instrument (HFI, Lamarre et al.2010). This will provide an independent check on the HFI resultswith different systematic uncertainties, and a much lower levelof contamination by polarised thermally-emitting dust.Mandolesi et al. (2010) demonstratethatCMBpolarisationcan be detected in the power spectrum with a signal-to-noise ofup to 100:1. Since the power spectrum is proportional to the skysignal squared, this sets the following overall requirements onpolarisation calibration:– global multiplicative artefacts ≪0.5%;– errors in the instrumental polarisation angles ≪0.05 rad =3 ◦ ;– artefacts uncorrelated with the CMB polarisation ≪10% ofpolarised intensity.The constraint on angles arises as follows: a global angle errorof δ rotates each E, B harmonic component vector (alm E , aB lm )byan angle of 2δ. HencefortheCMBwhereE-modes stronglydominate, ClEE = 〈|alm E |2 〉 is reduced by cos 2 2δ, i.e.anerrorof4δ 2 ,tolowestorder.Randomangleerrorswillhaveasmallerimpact, so this is a safe upper limit.We will demonstrate that the first two requirements are easilymet by the LFI. The worst instrumental artefacts are expected tobe due to various forms of leakage into the polarisation of thestrong total intensity signal from our Galaxy, but over much ofthe sky this will not be a serious contaminant.Stronger requirements on calibration precision are placed bythe desire to produce accurate maps of foreground polarisation,especially along the Galactic plane, since we know from WMAPthat this is the dominant signal at LFI frequencies and resolution.While we do not expect to recover maps which are noise-limitedat all pixels, we show that measurement of polarisation to 1% oftotal intensity or better appears achievable, although some potentialhurdles remain to be overcome.In this paper we present a system-level overview of the LFIas a polarimeter. Section 2 reviews the standard notation ofStokes parameters and discusses the several coordinate systemsused to express them in this paper. Section 3 describes the overallarchitecture of the system, while Sect. 4 connects this to theJones and Mueller matrix formalisms, to allow us to build upthe system-level performance from component-level measurementsand models. The LFI is most generally characterised byapolarisationresponseStokesvector(whichdependsonbothfrequency and sky position) for each detector. In principle thisformalism provides a complete description of all multiplicativeinstrumental effects, and hence of all multiplicative systematicerrors, which can be defined as differences between the true responseand the (relatively) idealised response assumed in thedata reduction.Analyses of polarisation systematics frequently specialisethis general approach to capitalise on simplifying features of theinstrument: for instance, Mueller matrices may be independentof direction, in which case a perturbation analysis may be appliedto isolate the dominant departures from the ideal identitymatrix: for example see O’Dea et al. (2007)forthecaseofarotatingwave-plate. Similarly, Hu et al. (2003) giveafirst-orderperturbation analysis of the impact on polarisation of departuresof the beamshape from an ideal circular Gaussian. Partly becausePlanck is not primarily a polarimetric mission, we cannot makemuch use of such simplifications, although the dominant beamdependentpolarisation residuals do indeed correspond to someof the patterns discussed by Hu et al.Section 5, therefore,presentsquantitativedetailsofthesystemparameters that affect the polarisation response vectors, asknown prior to launch. Since LFI detectors are highly linearover the range of sky signal strengths expected on-orbit, theonly other class of systematic errors are additive effects suchas 1/ f noise; in fact the suppression of such terms is the drivingfactor in the design of both the LFI instrument and its dataanalysis pipeline. Such terms are addressed in Sects. 6 and 7:Section 6 discusses additive terms due to residual instrumentaltemperature fluctuations, based on the cryogenic tests for LFIand Planck,whileSect.7 addresses the impact of 1/ f noise.The effective polarisation response varies from sky pixel tosky pixel under the control of the scanning strategy, so the onlyway to assess the impact of residual instrumental effects on angularpower spectra is through simulations of a complete skysurvey. This is also done in Sect. 7, whichalsoallowsustodiscussthe possibility of checking the polarisation calibration usingastronomical sources. Section 8 summarises our results.2. Stokes parameters and coordinatesIt is convenient to express the polarisation state of electromagneticradiation either via Stokes parameters {I, Q, U, V} or, morenaturally, via the linearly polarised intensity p and orientationangle Θ. Weusetheterm“orientation”ratherthandirectionforΘ to signify that a rotation of 180 ◦ has no physical significance,which is to say that linear polarisation is a spin-2 quantity inthe sense of Zaldarriaga & Seljak (1997). The Stokes parameterscan be defined in terms of the complex amplitudes E x , E yof the wave in the ˆx and ŷ directions (ẑ being the propagationdirection) via:I = 〈 |E x | 2 + |E y | 2〉 (1)Q = 〈 |E x | 2 −|E y | 2〉 = p cos 2Θ (2)U = 2 〈 R(E ∗ xE y ) 〉 = p sin 2Θ (3)V = 2 〈 I(E ∗ xE y ) 〉 (4)(e.g. Kraus 1966). Stokes I is the total intensity, irrespective ofpolarisation; Q and U represent linear, and V circular, polarisation.Stokes parameters (and p)mayrepresenteitherfluxdensityor intensity (brightness). In CMB analysis I is often referred toas “temperature” while Q and U are termed “polarisation”, butthis is misleading inasmuch as in this context all Stokes parametersare measured in temperature units (cf. Berkhuijsen 1975).In the following we often use the Stokes vector S =(I, Q, U, V) T (we use calligraphic script for Stokes vectors andthe matrices that act on them to distinguish them from real-spacevectors). For I and V this is just a notational convenience as theytransform as scalars under real-space rotation; but the projectionof S into the (Q, U) planehasavectornature,inthatitscomponentsdepends on the chosen coordinate system: an angle 2Θin (Q, U) correspondstoanorientationofΘ on the sky. To definethe zero-point of Θ,weneedtorelatethelocalx and y usedabove, defined only for one line of sight, to a global coordinatePage 2 of 26