1 Introduction

Future space missions, like Planck and MAP, represent a powerful opportunity for Cosmology through the imaging of the cosmic microwave background (CMB) with great sensitivity and angular resolution. The instrument design for these space missions, as well as for a number other on-going sub-orbital CMB projects (e.g. BEAST, Boomerang, TopHat), calls for multi-frequency focal plane arrays placed at the focus of off-axis optical systems, in order to achieve proper angular resolution, sensitivity, and spectral coverage. As a consequence, not all the feedhorns can be located very close to the centre of the focal plane, where optical distortions are minimum. Optical distortions make the main beam response somewhat different from a pure gaussian, centrally symmetric shape. In addition, sidelobe effects may become more prominent and affect the measurements, an effect that will be studied in a forthcoming work. Main beam distortions may introduce two kinds of effects on anisotropy measurements: a degradation of the effective angular resolution achievable by a given experiment and a sensitivity degradation in the evaluation of the temperature anisotropy in any resolution element. Roughly speaking, these two effects can be seen as orthogonal one each other in the space $\theta - \Delta T$ of angular scales and temperature anisotropy or equivalently in the space of l - C_l of multipoles and angular power spectrum. The former shifts toward low multipoles the effective angular sensitivity of an experiment (see White 1997 and Mandolesi et al. 1997, 1998 for a discussion of this effect for Planck surveyor mission); the latter increases the final error in the determination of angular power spectrum. Here we focus on the relevance in anisotropy measurements of this second effect generated by main beam distortions.

The generation of high resolution full sky maps, consistent with the COBE-DMR normalization of the CMB anisotropies and based on reasonable angular and frequency extrapolations of the available galactic emission maps, is a basic step for simulating the mission performance.

In Sect. 2 we outlined the basic framework for generating full sky maps including CMB anisotropy and Galaxy diffuse emission. In Sect. 3 we present the method adopted to convolve the simulated map with a beam of general shape, taking into account the mission observational strategy (we refer here to the standard Planck scanning strategy, but the method can be applied to other observational schemes). The main results of our beam tests and empirical formulas that accurately quantify the effect of beam distortion on temperature measurements as functions of the relevant parameters are presented in Sect. 4. In Sect. 5 we discuss the effect of main beam distortions in presence of a relevant galactic signal. Finally, in Sect. 6 we draw out the main implications of this analysis focussing on their impact for the optimization of the feedhorns locations on the Planck focal plane.