4 Results of beam tests

For sake of clarity, we have first performed our simulations neglecting the galactic emission in order to obtain general results holding for low foreground regions. A flat CDM model with H₀=50 km/s/Mpc and a baryon density in terms of critical density $\Omega_{\rm b} = 0.05$, with a 24% of mass in Helium, has been adopted for the present tests, but some other models have been considered for comparison (see Sect. 4.7). For the standard Planck scanning strategy, a test on a given scan circle consists of about 680 integrations at 30 GHz and of about 1700 integrations at 100 GHz. We can estimate the numerical error introduced by interpolation and integration techniques by comparing for few cases the results obtained by using an integration grid of $48 \times 48$ points with those obtained with a grid of $96 \times 96$ points (see Col. 8 of Table 1).

  

Table 1: $\rm rms_{th}$ value of $(T_{\rm ell}-T_{\rm circ})$ for the cases presented in Sect. 4.1

4.1 Results from high resolution maps

We have firstly performed our tests using maps at a resolution of about 5' (COBE-cube resolution 11). Figures 2 and 3 show our results for a typical test at 30 GHz and at 100 GHz respectively. Even in correspondence of the largest temperature variations induced by CMB quadrupole large scale waves, the temperature differences remain practically equal to those obtained in other sky regions (see the bottom panels of Figs. 2 and 3).

  

Figure 2: Top panel: difference between the thermodynamic temperature observed by asymmetric and symmetric beams for a typical scan circle as function of the scan integration number along the circle. Bottom panel: thermodynamic temperature observed by asymmetric (triangles) and symmetric (crosses) beams and differences (circles) between the two measurements for the same scan circle. The figure refers to the test 2 in Table 1

  

Figure 3: The same as Fig. 2 but at 100 GHz (see test 5 in Table 1)

Typical results obtained by using maps at COBE-cube resolution R=11 are tabulated in Table 1 (tests 1-6) in terms of $\rm rms_{th}$ for different beam FWHM's and distortion parameters r. Column 8 in Table 1 gives the numerical error, $\Delta {\rm rms_{th}}$,in the determination of $\rm rms_{th}$ derived from the comparison of results obtained for a typical scan circle with a symmetric beam by using different integration grids (of course the reported value of $\Delta {\rm rms_{th}}$ is $\sqrt{2}$ times that derived from this test).

These results may be qualitatively interpreted from a geometrical point of view: the contribution of the different parts of the sky observed by beams with different shapes becomes more important and produces a growing effect as the FWHM and/or r increases (Burigana et al. 1997).

On the other hand we may expect that these results depend in part on the resolution of the maps used for the test. Indeed the use of maps at lower resolution tends to smear out the differences of temperature between different pixels and we expect that the value of $\rm rms_{th}$ may decrease by passing from higher to lower resolution maps, as confirmed by the results of the test 7 in Table 1, performed by using the same map at 30 GHz of the above tests but degraded at a lower resolution (see Sect. 4.4).

4.2 Searching for an analytical description of beam distortion effect

The results of the previous section indicate that $\rm rms_{th}$ mainly depends on three quantities: the beam distortion parameter, r, the beam ${\rm FWHM'}$,where the suffix "prime'' indicates that it is expressed in arcminutes, and the map resolution, R, or equivalently the typical pixel dimension, d. A simple expression for the beam distortion effect as function of these parameters can be useful for many purposes. We will search here for a formula based on the assumption of separability of the variables, i.e. of the kind:

$$\begin{eqnarray} {\rm rms_{th}}\!\!&=& \!\! {\rm const} \cdot f[{\rm FWHM'}; (d/... ...1mm} {\rm FWHM'}, r] \cdot h[r; (d/{\rm FWHM}), {\rm FWHM'}] \, ,\end{eqnarray}$$ (5)

where the function f,g,h depends only on their first argument, i.e. the quantities after ";'' are considered constant. This simple work-hypothesis allow us to obtain an analytical description of the main beam distortion effect with a relatively small number of simulations. We have later verified that it is accurate enough for the present purposes (see Sect. 4.6). In the following sections we present numerical tests that allow to derive simple but quite accurate representations of these functions.

4.3 Dependence on the beam FWHM

For understanding the dependence of $\rm rms_{th}$ on the beam FWHM we have performed four tests by keeping constant the ratio, $d/{\rm FWHM}$, between the typical pixel dimension and the beam size, and the beam distortion parameter r. We have considered a case with a beam size $\rm (FWHM=72')$ somewhat larger than those of Planck LFI radiometers in order to give results holding also for observations performed with smaller telescopes. The corresponding results are tabulated in Table 2 (tests 8-11). They can be well described by a function f (see left panel of Fig. 4):

$$\begin{eqnarray} f[{\rm FWHM'}; (d/{\rm FWHM}), r] \propto &-& {\rm FWHM'}^2 \nonumber \\ &+& 96.0 \, {\rm FWHM'} +1800 \, .\end{eqnarray}$$ (6)

We note that this function presents a maximum at ${\rm FWHM} \sim 48'$;indeed (see tests 10 and 11 in Table 2) for FWHM of about $1^\circ$the beam distortion effect is smaller than for example for FWHM of about $0.5^\circ$; this fact can be produced by the smearing out of the effect related to the different contribution of the parts of the sky observed by beams with different distortion parameters, when the beam size is large enough.

  

Table 2: $\rm rms_{th}$ value of $(T_{\rm ell}-T_{\rm circ})$ for the cases presented in Sects. 4.3 and 4.4

  

Figure 4: From the left to the right: the function f versus the beam FWHM as derived from the tests 8-11; the function g versus $d/{\rm FWHM}$ as derived from the tests 12-14; the function h versus r as derived form the tests 1-6 (crosses: tests at 30 GHz; diamonds: tests at 100 GHz; see Sect. 4.5)

4.4 Dependence on the map resolution

For understanding the dependence of $\rm rms_{th}$ on the ratio $d/{\rm FWHM}$ we have carried out three tests by keeping constant the beam size and the beam distortion parameter r. We have choosen a value ${\rm FWHM}=18'$ which is intermediate between those relevant for Planck LFI radiometers. The corresponding results are tabulated in Table 2 (tests 12-14). They can be well described by a function g (see middle panel of Fig. 4):

$$g[(d/{\rm FWHM}); {\rm FWHM'}, r] \propto - (d/{\rm FWHM}) +1.72 \, .$$ (7)

4.5 Dependence on the beam distortion parameter

The results of the tests 1-6 in Table 1 cannot be directly used to estimate the dependence of the beam distortion effect on the beam distortion parameter r, being based on maps with different ratios $d/{\rm FWHM}$ and different FWHM. On the other hand from Eqs. (6) and (7) we know how these two parameters may affect the test. Then we fit the values of $\rm rms_{th}$ of the tests 1-6 in Table 1 divided by the functions f and g. By including the obvious condition of null distortion effect in the case r=1, for the function h we obtain the simple expression (see right panel of Fig. 4):

$$h[r; (d/{\rm FWHM}), {\rm FWHM'}] \propto -(r-1)^2+2.26 (r-1)\, .$$ (8)

4.6 Normalization of the analytical approximation

Given the function f,g,h we can easily find the normalization constant of the Eq. (5) by exploiting the results of Tables 1 and 2 together. Finally we have the formula:

$$\begin{eqnarray} {\rm rms_{th}} &\simeq & 6.43\; 10^{-4}\, \mu{\rm K} [ -(r-1)^2... ...mber \\ &\times& [-{\rm FWHM'}^2 + 96.0 \, {\rm FWHM'} +1800] .\end{eqnarray}$$ (9)

This approximation works quite well for FWHM , as we have verified by comparing it with the results of Tables 1 and 2 and a few number of other different tests. Typical errors are about 0.25 $\mu {\rm K}$ and always less than 0.7 $\mu {\rm K}$ and it works even better for (typical errors less than 0.1 $\mu {\rm K}$). We note that tests based on sky maps directly simulated at low resolution, and not obtained from degradation of high resolution maps, tend to show a somewhat larger effect (for example about 30% greater for R=9). In the asymptotic case of maps at very high resolution ($d \rightarrow 0$)we can neglect the dependence on d, by setting g=1, and replace the proportionality constant with $\simeq 1.11 \; 10^{-3}\,\mu{\rm K}$ in Eq. (9).

4.7 Check for other cosmological models

The above results are obtained by exploiting a simulated sky map in the case of pure CDM model for CMB fluctuations. We have performed other tests with different cosmological scenarios. We have considered the case of a beam with a FWHM of 18' and a beam distortion parameter r=1.3 for two maps directly generated at resolution R=10 obtained for a flat MDM model ($\Omega_{\rm Hot}=0.3$,$\Omega_{\rm Cold=0.65}$, $\Omega_{\rm b} = 0.05$, H₀=50 km/s/Mps) and for a flat $\Lambda$CDM model ($\Omega_{\Lambda}=0.7$, $\Omega_{\rm Cold}=0.25$, $\Omega_{\rm b} = 0.05$, H₀=50 km/s/Mps). We have ${\rm rms_{th}} = 1.56$ and 1.74 $\mu {\rm K}$ respectively, in good agreement with the results of Sect. 4.6.

4.8 Averaging over three samplings

Although the accurate study of the effect of beam distortions when the telescope rotation around the satellite spin axis is taken into account is far from the purposes of the present work, we consider here that the Planck sampling scheme assumes 3 samplings per FWHM. During this time the telescope continuously changes its pointing direction; we have carried out a simulation neglecting the telescope motion within a single sampling time, but assigning to each of the 3 samplings a different telescope direction, so roughly considering the telescope motion during the integration time. We simply compare the average of three samplings obtained by a symmetric beam with that obtained by an elliptical one. When we compute the differences between these two data streams for a typical scan on a pure CMB fluctuation map for a distortion parameter r=1.3 (the case of the test 2) we obtain a $\rm rms_{th}$ value of $1.85\,\mu{\rm K}$,just a little lower than that found for the test 2. This test indicates that the oversampling of the sky introduced by the telescope motion cannot be used in simple way to significantly reduce the effect of main beam distortions on anisotropy measurements, as intuitively expected from the fact that during an integration time does not change significantly the orientation of a distorted beam respect to the observed sky region.