# Beam deconvolution in noisy CMB maps.pdf

arXiv:astro-ph/0304326v1 17 Apr 2003Astronomy however, as we are going to show along thepaper, method II works.Let us assume a certain pixelisation and an asymmetric beam which smoothes maps of theCMB sky. We first consider that only one observation per pixel is performed. The beam couldhave either the same orientation for all the pixels or different orientations for distinct pixels; inboth cases, the beam smoothes the sky temperature T to give Ts according to the relation:T is =Msummationdisplayi=1Bi jT j , (2)4 C. Burigana nevertheless, rotatingasymmetric beams lead to a Bi j matrix which is more complicated than that corresponding to acircular beam (studied in Arnau by this reason, we are going to verify that Jacobimethod works for any beam, first in the absence of noise and, then, when there is an admissiblenoise level. In matrix form, Eq. (2) can be written as follows: Ts = BT.If we now consider that each pixel is observed N times either with an unique beam anddifferent orientations per pixel or with various non-circular rotating beams (as it occurs in projectsas P where there are various beams for each frequencies), then, we can write N matrixequations (one for each measure) of the form T (α)s = B(α)T, where index α ranges from 1 to N.The average temperature assigned to pixel i is T ia = (1/N)summationtextNα=1 T (α)is and the above system of Nmatrix equations leads toTa = BaT , (3)where matrix Ba describes the average beam, which can be calculated as follows:Bi ja = 1NNsummationdisplayα=1B(α)i j ; (4)hence, for a given experiment involving various measurements per pixel, the average beam (4)might be calculated at each pixel and, then, we can try to use the Jacobi method to solve thesystem of linear equations (3).We see that, in the absence of noise, method II could work in the most general case, in whichvarious beams move according to the most appropriate observational strategy. For a multi-beamexperiment one should in principle simulate the effective scanning strategy and the convolutionwith the sky signal for each beam and then apply the formalism described above by taking intoaccount the various resolutions and shapes of the beams. On the other hand, the power of thismethod is that it works independently of the small differences between the resolutions and shapesof the various beams at the same frequency in a given experiment. Therefore, considering thedata from a single average beam, instead of the data from the whole set of beams, but with thesensitivity per pixel obtained by considering the whole set of receivers at the given frequency inthe case in which the noise is taken into account, allows to reduce the amount of data storage andsimplify the analysis without introducing a significant loss of information about the accuracy ofthe method.Instrumental uncorrelated Gaussian noise makes beam deconvolution more problematic; nev-ertheless, as we demonstrate in this work (Sects. 3.2 and 3.4), method II works for experimentsC. Burigana thus, our pixel sizeis ∆ = 3.43′ (∆ = 6.86′). These sizes are allowed by HEALPIX and, consequently, this choicewill facilitate some comparisons. We use seventy five of these regions covering about the forty percent of the sky. With this coverage and ∆ = 3.43′ (∆ = 6.86′), the angular power spectrum can beestimated (from simulated maps) with good accuracy fromℓmin = 100 toℓmax = 10800/∆ ≃ 3100(ℓmax ≃ 1550). See S´aez the deviations grow beyond the sixth (fifth) acoustic peak.3.2. Case B: noisy, small patchesIn this case, beam, rotation, coverage and pixelizations are identical to those of case A; however,there is instrumental uncorrelated Gaussian noise with σN = 9 µK (σN = 4.5µK), in antennatemperature, for ∆ = 3.43 (∆ = 6.86), just the noise expected by combining all the beams ofP at 100 GHz. A joint treatment of the impact of main beam distortions and of correlated1/fα type noise (see e.g. Seiffert et al. 2002) and other kind of instrumental systematics (seee.g. Mennella et al. 2002) is out of the scope of this paper. On the other hand, this does notrepresents a crucial limitation, since blind destriping algorithms can strongly reduce the impactof these effects (see e.g. Delabrouille 1998, Maino et al. 1999, Mennella et al. 2002) also in thepresence of optical distortions (Burigana et al. 2001) and, possibly, of non negligible foregroundfluctuations (Maino et al. 2002). The system to be solved has the form:T is =Msummationdisplayi=1Bi jT j + Ni , (8)where Ni is the noise at pixel i. Using matrices, this equation can be written as follows:Ts = B(T + B−1N). (9)This last equation is formally identical to the matrix form of Eq. (2) and it can be solved in thesame way – using Jacobi method – to find the map T + B−1N. After applying this method, somenumerical error E is expected and, consequently, the numerical solution of system (9) is of theformT∗ = T + B−1N + E . (10)In general, T∗ is different from T (sky temperature before smoothing); hence, the angular powerspectrum extracted from the map T∗ is different from that of the unconvolved sky, which is to beextracted from map T. Results are shown in Fig. 4, which has the same structure as Fig. 3. We seethat the spectra before smoothing and after deconvolution (which are obtained from maps T andT∗, respectively) separate at middle ℓ values. In the right panels, we can verify that the deviation8 C. Burigana in this way, the effect of the noise variance in the esti-mate of the angular power spectrum of N∗ is strongly reduced (forty noise realizations suffice).When we subtract this spectrum from that of T∗, namely, when we correct the T∗ spectrum tak-ing into account noise effects, results are much better than those showed in Fig. 4 4. These newresults are presented in Fig. 5. The structure of this figure is identical to that of Figs. 3 and 4. Asin Fig. 3, the range of ℓ values – in the left panels – has been appropriately chosen to include theregion where the displayed curves separate significantly. The deconvolution is better than that ofFig. 4, where no correction for the noise has been considered. According to the right panels ofFig. 5, the relative deviation produced by deconvolution plus correction is smaller than five percent for ℓ ≤ 1500 (ℓ ≤ 1300) in the case ∆ = 3.43 (∆ = 6.86). For ∆ = 3.43 (∆ = 6.86), decon-volution works very well up to the end of the fifth (fourth) acoustic peak. For equivalent levelsof noise in different pixels, we see that – as expected – deconvolution has recovered more Cℓcoefficients for ∆ = 3.43; hence, we can say that results corresponding to ∆ = 3.43 are sensiblybetter than those of ∆ = 6.86. On account of this fact and also for the sake of briefness, pixelswith ∆ = 6.86 are not considered in cases C and D below.3.3. Case C: application to P in the absence of noiseThe selected orbit for P is a Lissajous orbit around the Lagrangian point L2 of the Sun-Earth system (see e.g. Bersanelli et al. 1996). The spacecraft spins at 1 r.p.m. and the field ofview of the two instruments – LFI and HFI (High Frequency Instrument, Puget et al 1998) –is about 10◦ × 10◦ centered at the telescope optical axis (the so-called telescope line of sight,LOS) at a given angle α from the spin-axis direction, given by a unit vector, s, chosen to be4 We observe that an analogous approach can be pursued also in the presence of correlated noise, providedthat the noise properties can be known from laboratory measures and/or directly reconstructed from the data(Natoli et al. 2002). Of course, in this context, destriping (or, possibly, map-making, see e.g. Natoli et al.2001) should be previously applied both to the data and to the simulated pure noise data.C. Burigana & D. S´aez: Beam deconvolution in noisy CMB maps 9pointed in the opposite direction with respect to the Sun. In this work we consider values ofα ∼ 85◦, as adopted for the baseline scanning strategy. The spin axis will be kept parallel to theSun–spacecraft direction and repointed by ≃ 2.5′ every ≃ 1 hour (baseline scanning strategy).Hence P will trace large circles in the sky and we assume here, for simplicity, 60 exactrepetitions of the set of the pointing directions of each scan circle. A precession of the spin-axiswith a period, P, of ≃ 6 months at a given angle β ∼ 10◦ about an axis, f, parallel to the Sun–spacecraft direction (and outward the Sun) and shifted of ≃ 2.5′ every ≃ 1 hour, may be includedin the scanning strategy, possibly with a modulation of the speed of the precession in order tooptimize data transmission (Bernard et al. 2002). The quality of our deconvolution code is ofcourse almost independent of the details of these proposed scanning strategies, and we assumehere the baseline scanning strategy for sake of simplicity.The code implemented for simulating P observations for a wide set of scanning strate-gies is described in detail in Burigana et al. (1997, 1998) and in Maino et al. (1999). In this studywe do not include the effects introduced by the P orbit, to be currently optimized, by simplyassuming P located in L2, because they are fully negligible in this context.We compute the convolutions between the antenna pattern response and the sky signal asdescribed in Burigana et al. (2001) by working at ∼ 3.43′ resolution and by considering spin-axis shifts of ∼ 2.5′ every hour and 7200 samplings per scan circle. We simulate 11000 hoursof observations (about 15 months) necessary to complete two sky surveys with the all Pbeams.With respect to the reference frames described in Burigana et al. (2001), following the recentdevelopments in optimizing the polarization properties of LFI main beams (see e.g. Sandri et al.2003), the conversion between the standard Cartesian telescope frame x,y,z and the beam framexb f,yb f,zb f actually requires a further angle ψB other than the standard polar coordinates θB andφB defining the colatitude and the longitude of the main beam centre direction in the telescopeframe. Appendix A provides the transformation rules between the telescope frame and the beamframe, as well as the definition of the reference frames adopted in this work.The orientation of these frames as the satellite moves is implemented in the code. For eachintegration time, we determine the orientations in the sky of the telescope frame and of the beamframe, thus performing a direct convolution with the sky signal by exploiting the detailed mainbeam response in each considered sky direction. The detailed main beam shape and position onthe telescope field of view adopted in this application is that computed in the past year for the feedLFI9 (Sandri et al. 2002) which shows an effective FWHM resolution of 10.68′ and deviationsfrom the symmetry producing a typical ellipticity ratio of 1.25. Such values of resolution andasymmetry parameter are in the range of them that it is possible to reach with a 1.5 m telescopelike that of P by optimizing the optical design (see e.g. Sandri et al. 2003). Although ourdeconvolution method is largely independent of the details of the considered beam shape, it isinteresting to exploit its reliability under quite realistic conditions.10 C. Burigana & D. S´aez: Beam deconvolution in noisy CMB mapsThe CMB anisotropy map has been projected into the HEALPix scheme (G`orski et al. 1999)starting from the angular power spectrum of the assumed ΛCDM model (see Sect. 1).To make the application of the deconvolution code easier and the system solution possiblewithout large RAM requirement and in a reasonable computational time 5 we implemented acode that identify in the simulated time ordered data (TOD) all and only the beam centre pointingdirections in an equatorial patch (in ecliptic coordinates) of 1024 × 1024 pixels with a ∼ 3.43′side (nside = 1024). We keep the exact information on the beam centre pointing direction andthe beam orientation (defined for instance by an angle between the axis xb f and the parallel inthe beam centre pointing direction) as computed by our flight simulator. All the samples of theTOD within the same pixel are identified and restored in contiguous positions. At this aim, wetake advantage from the nested, hierarchical ordering of the HEALPix. This is quite simple in thecurrent simplified simulation, but it will require the development of efficient and versatile toolsto manage the more general case in which the all samples from the experiment multi-beam arrayare considered, particularly for the ecliptic polar patches, which pixels are observed many andmany times because of the P scanning strategy. In the context of the P project, thiseffort will be pursued by taking advantage from the development of P Data Model (see e.g.Lama et al. 2003).¿From the simulated TOD, possibly restored as described above, we extract a map of a patchof simply coadded data and a map of a patch deconvolved by applying method II. The lattermap can be then symmetrically smoothed with a beam FWHM of 10.68′ by using the HEALPixtools for comparison with the former one, obtained from the convolution with the simulatedasymmetric beam and taking into account the scanning strategy. Of course, from the input mapwe can extract the same sky patch.We consider four different patches covering an equatorial region of ≃ 28.3 % of the sky(analogously to the case of small patches, see Sect. 3.1, avoiding the boundary regions of thefour patches slightly reduces the originally considered, ≃ 33.3 %, sky coverage).All the above maps are inverted with the anafast code of HEALPix to extract the corre-ponding angular power spectra. The result is shown in Fig. 6. Of course, all the angular powerspectra are in strict agreement at multipoles ∼ 1000 and viceversa: therefore, even allowing forchanges in the assumed effective angular resolution, the simple symmetric beam approximationcan not improve the power spectrum recovery simultaneously in the two above ranges of ℓ (seealso the text).