1. Beam Deconvolution Map-Making
Charmaine Armitage
Benjamin Wandelt
UIUC

2. Deconvolution Map-Making Technique
Allows for arbitrary beam shapes
Main beam distortions
Planck beam has elliptic contours with  up to 1.4
Far sidelobes
Uses algorithm for fast, exact convolution on the sphere and works in spherical harmonic space (see Appendix)
Speeds up the convolution by a factor of a million (versus pixel-based methods) for nearly symmetric beams and lmax of a few thousand
Iterative scheme to solve for the maximum-likelihood estimate of the true sky
ATN-1As=ATN-1d

3. Current Status of PREBEAM
Have shown gain in map-reconstruction accuracy through deconvolution technique for temperature maps (Armitage & Wandelt, PRD )
Currently generating polarization maps: Polarized REgularized BEAm deconvolution Map-making (PREBEAM)
Uses M3 interface: allows for use of common LevelS CTP data

4. Preliminary Polarization Results
INPUT SKY: Simulated, noise-free
BEAM: LFI-27a, LFI-27b, LFI-28a, LFI-28b with FWHMmax/FWHMmin ~ 1.2
SCAN: Planck-like cycloidal scan (sampling frequency reduced)
RESOLUTION: pixel-size ~ 0.25 degrees 
FINDINGS:
Standard map-making method (binning of TOD from elliptical beam into pixel map) results in polarization maps with large signal at high frequency
Coupling of temperature to polarization
Reconstruct accurate map with PREBEAM

5. LFI 27

6. LFI 28

7. Preliminary Polarization Results
Input Q map
Standard map (sym. beam)
Standard map (ellip. beam)
PREBEAM map (ellip. beam)

8. Preliminary Polarization Results
Input Q map
Standard map (sym. beam)
Standard map (ellip. beam)
PREBEAM map (ellip. beam)
Beam-induced polarization systematic has RMS of the same order magnitude as the entire polarization signal

9. Deconvolution Goals
We have begun to demonstrate the effectiveness of PREBEAM to remove systematic effects due to beam distortions, with a particular view to polarization anisotropies
Future work will be motivated by the need to assess systematics due to beam asymmetries
We aim to develop PREBEAM to run a full Planck simulation for the 30GHz channel for a worst case assessment of beam-induced systematics, including the full beam response pattern
Near term impact: scanning strategy; assessment of polarization capabilities of Planck mission
Post launch: recovery of faint polarization signal in the presence of beam-induced systematics

10. Appendix: Convolution
General convolution on the sphere of two arbitrary functions b and s
Fourier transform:

11. Appendix: Transpose convolution
General transpose convolution on the sphere:
Fourier transform: