1 Introduction

Theoretical studies of the CMB have shown that the accurate measurement of the CMB anisotropy spectrum $C^T_\ell$ with future space missions such as PLANCK will allow for tests of cosmological scenarios and the determination of cosmological parameters with unprecedented accuracy. Nevertheless, some near degeneracies between sets of cosmological parameters yield very similar CMB temperature anisotropy spectra. The measurement of the CMB polarization and the computation of its power spectrum [Seljak1996,Zaldarriaga1998] may lift to some extent some of these degeneracies. It will also provide additional information on the reionization epoch and on the presence of tensor perturbations, and may also help in the identification and removal of polarized astrophysical foregrounds [Kinney1998,Kamionkowski1998,Prunet & Lazarian1999].

A successful measurement of the CMB polarization stands as an observational challenge; the expected polarization level is of the order of $10\%$ of the level of temperature fluctuations ( $\Delta T/T \simeq 10^{-5}$). Efforts have thus gone into developing techniques to reduce or eliminate spurious non-astronomical signals and instrumental noise which could otherwise easily wipe out real polarization signals. In a previous paper [Couchot et al.1999], we have shown how to configure the polarimeters in the focal plane in order to minimize the errors on the measurement of the Stokes parameters. In this paper, we address the problem of low frequency noise.

Low frequency noise in the data streams can arise due to a wide range of physical processes connected to the detection of radiation. 1/fnoise in the electronics, gain instabilities, and temperature fluctuations of instrument parts radiatively coupled to the detectors, all produce low frequency drifts of the detector outputs. The spectrum of the total noise can be modeled as a superposition of white noise and components behaving like $1/f^{\alpha}$ where $\alpha \ge 1$, as shown in Fig. 1.

Figure 1: The power spectrum of the K34 bolometer from Caltech, the same type of bolometer planned to be used on the PLANCK mission. The measurement was performed at the SYMBOL test bench at I.A.S., Orsay (supplied by Michel Piat). The knee frequency of this spectrum is $\sim 0.014$ Hz and the planned spin frequency for PLANCK is 0.016 Hz. We can model the spectrum as the function ${S(f)=1+\left ( \frac {1.43 \ 10^{-2}}{f} \right ) ^2}$ (dashed line)

This noise generates stripes after reprojection on maps, whose exact form depends on the scanning strategy. If not properly subtracted, the effect of such stripes is to degrade considerably the sensitivity of an experiment. The elimination of this "striping'' may be achieved using redundancies in the measurement, which are essentially of two types for the case of PLANCK:

• each individual detector's field of view scans the sky on large circles, each of which is covered consecutively many times ($\sim 60$) at a rate of about ${f_{\rm {spin}}\sim 1}$ rpm. This permits a filtering out of non scan-synchronous fluctuations in the circle constructed from averaging the consecutive scans.
• a survey of the whole sky (or a part of it) involves many such circles that intersect each other (see Fig. 2); the exact number of intersections depends on the scanning strategy but is of the order of 10⁸ for the PLANCK mission: this will allow to constrain the noise at the intersection points.

Figure 2: The Mollweide projection of 3 intersecting circles. For clarity, the scan angle between the spin axis and the main beam axis is set to $60^\circ$ for this figure

One of us [Delabrouille1998] has proposed to remove low frequency drifts for unpolarized data in the framework of the PLANCK mission by requiring that all measurements of a single point, from all the circles intersecting that point, share a common sky temperature signal. The problem is more complicated in the case of polarized measurements since the orientation of a polarimeter with respect to the sky depends on the scanning circle. Thus, a given polarimeter crossing a given point in the sky along two different circles will not measure the same signal, as illustrated in Fig. 3.

Figure 3: The orientation of polarimeters at an intersection point. This point is seen by two different circles corresponding to two different orientations of the polarimeters in the focal plane. For clarity, we have just represented one polarimeter

The rest of the paper is organized as follows: in Sect. 2, we explain how we model the noise and how low frequency drifts transform into offsets when considering the circles instead of individual scans. In Sect. 3, we explain how polarization is measured. The details of the algorithm for removing low-frequency drifts are given in Sect. 4. We present the results of our simulations in Sect. 5 and give our conclusions in Sect. 6.