6 Conclusions

We have developed optical calculation codes to compute the focal surface, optimize feedhorn positions and calculate the main beam patterns on the focal surface of a suitable set of PLANCK-like telescopes. In particular we focussed on off-axis Gregorian telescopes with three different primary mirror apertures possibly including the so-called baffle option. Main beam patterns have been found to be mainly affected by spherical aberration and coma. Our results are in good agreement with those of TICRA (1997) based on the GRASP8 code and provide an adequate base for studying the impact of optical distortions on PLANCK science.

Three different independent methods of analysis have been proposed to quantify the optical distortion impact on PLANCK observations. The first two (semi-analytical) methods provide fast and accurate evaluations of the angular resolution degradation of PLANCK beams for different locations on the focal plane. The third (fully numerical) method also allows to quote the amount of additional noise introduced by main beam distortions with respect to Gaussian beams equivalent from the point of view of angular resolution. For what concerns the effective angular resolution, the three approaches are in very good agreement, thus confirming the reliability of our methods. For PLANCK-LFI beam locations on the focal surfaces of 1.3, 1.55 and 1.75m telescopes we found averaged effective angular resolution (FWHM) of $\langle W_{\rm e} \rangle \simeq 13.1', 11.0', 9.7'$ and averaged rms increased noise $\langle \sigma_{{\rm th}} \rangle \simeq 2.2, 2.1, 2.0~\mu$K respectively.

By using the standard formula for $C_\ell$ uncertainty in CMB experiment, we provided simple estimates of the angular resolution degradation effect on the recovered CMB power spectrum at large $\ell$. The Fisher Matrix method has been implemented to compute the LFI uncertainties on the estimation of a suitable set of cosmological parameters as a function of the telescope aperture. As expected, a larger aperture and the resulting better resolution is particularly important for recovering cosmological parameters which show unambiguously imprints at large multipoles. This is particularly crucial for open models where all the relevant cosmological information encoded into the Doppler peaks is shifted towards large multipoles. As an example the error on $\Omega_\Lambda$ decreases from 4.5% to 2.7% by increasing telescope aperture from 1.3m to 1.75m. Present results have to be considered as lower limits of the impact of main beam distortions since only the resolution degradation and not the additional noise has been considered. Furthermore the straylight contamination, which has not been included here, may increase the overall uncertainty, mainly affecting the low $\ell$ part of the power spectrum, and then just those cosmological parameters practically unaffected by main beam distortions. We stress here that both LFI and HFI on-board PLANCK satellite, a third generation of CMB space missions, are going to be designed to accurately recover the CMB anisotropy pattern for whole set of cosmological models not ruled out by present data limited only by astrophysical contaminations.

From the above results, it is clear that a telescope with an aperture $\,\lower2truept\hbox{${> \atop\hbox{\raise4truept\hbox{$\sim$ }}}$ }\,1.7$m has to be regarded as a preferable choice to fully achieve PLANCK-LFI goals. On the other hand, the actual $\simeq 1.5$m baseline telescope results to be a reasonable compromise between the overall mission constraints and the scientific goals. The BASELINE telescope aperture is a significant improvement with respect to the PHASE A telescope, providing an angular resolution quite close to the LFI goal and allowing for an optimized trade-off between main beam distortion and sidelobe effects. From the encouraging studies of Villa et al. (1998a) and Mandolesi et al. (1999), it results that an Aplanatic telescope design shows significant improvements with respect to the regularity of the beam shapes (quite close to ellipses with typical axes ratios of $\sim 1.2$) as well as to the uniformity of the beam angular resolution on the focal surface. This kind of designs is actually the considered baseline for the industrial studies by ALCATEL for the optimization of the PLANCK telescope in order to fully achieve the scientific goals.

Acknowledgements

It is a pleasure to thank T. Cafferty, M. Dragovan, P. Guzzi, V. Jamnejad, J.-M. La Marre, A. Lange, C. Lawrence, M. Malaspina and L. Wade for useful discussions.

Figure 13: The same as Fig. 6 but for the BAFFLE.1550 configuration

Figure 14: The same as Fig. 6 but for the BAFFLE.1750 configuration

Figure 15: Synthetic analysis of simulated beams based on the Method 3 for all the considered telescope designs. Left panels: effective width, $FWHM_{{\rm eff}}$, of distorted beams a function of their elevation and azimuth. Right panels: rms $_{{\rm th}}$ of the temperature differences between observations performed by a distorted beam and the corresponding symmetric Gaussian one (solid line) and between observations performed by a Gaussian beam with the same FWHM of the distorted one and a Gaussian beam with the same FWHM of the central "real'' beam (dotted line). (Channels at 100 GHz)