4 The effects of beam distortions on CMB science

The critical question is to evaluate the effect of beam distortions on PLANCK science. In particular we want to know how the distorted beams affect the determination of the anisotropy power spectrum, and hence the estimation of cosmological parameters.

Standard criteria for the performance of optical systems do not directly address the study of the impact of beam distortions on CMB science. Because of this, we have developed three independent methods to characterize beam distortions and quantify their effect on CMB science. "A priori'' they have somewhat different sensitivities to various types of beam distortions; nevertheless, they give similar answers, giving us confidence that we understand the effects of beam distortions.

We observe that the effect of main beam distortion in presence of foreground contamination should be in principle carefully considered. On the other hand, previous works confirm that the combination of these contaminations is not critical for PLANCK, at least for the LFI channels. Galaxy fluctuations combined to beam distortions produce a significant increase of the added noise (by a factor $\sim 3$) with respect to the case of a pure CMB sky only close to the galactic plane and at lowest frequencies (Burigana et al. 1998a). The combination of radiosource fluctuations and beam distortions produce in general only very small effects (Burigana et al. 1999a). Therefore, we consider here only the effect of the main beam distortions for a pure CMB fluctuation sky.

The methods presented below have been applied to the simulated beams computed by assuming an edge taper value of 30 dB (see Sect. 5 for details about the edge taper and its effects).

Figure 6: Effective angular resolution of the PHASE A telescope calculated using Method 1. Upper right and bottom panels--Comparison of calculated window functions (heavy lines) with the window functions of perfect Gaussian beams (i.e., $\exp [-\ell(\ell+1)\sigma^2]$) with FWHM of 10', 12', and 14'(top to bottom, respectively). Labels inside the boxes give the offsets of the feed corresponding (top to bottom order) to each heavy line. The faster the window function falls off the broader the beam. The less Gaussian the window function the more distorted the beam. The two most off-set beam patterns ( $Az=El=~\pm~5^\circ$) are incompletely mapped in the provided data sets, and their corresponding Fourier transforms are odd, and incomplete.     Top left panel-- rms temperature differences that would be obtained with the actual distorted beams for the eleven feed positions show in the other panels, identified by symbol, for two difference CDM models (closed above, open below). The abscissa is the FWHM of the Gaussian beam that produces the same rms. A 100GHz feed on-axis would have an effective resolution of 11' (with the assumed 30dB edge taper). In the symmetric focal assembly configuration for the PHASE A telescope, however, the 100GHz LFI feeds are at a mean distance from the optical axis of $3.3^\circ$, i.e. $Az=El=\pm$$2.3^\circ$. The effective angular resolution of the PHASE A telescope for the LFI 100GHz feeds, therefore, is expected to be between $\simeq 12'$and $\simeq 15'$. This is a serious degradation of the angular resolution

Figure 7: The same as Fig. 6 (see the corresponding caption) but for the BASELINE configuration. This design is close to meet the goal of $\simeq 10'$ resolution at 100 GHz

Figure 8: The same as Fig. 6 (see the corresponding caption) but for the ENLARGED configuration. This design fully meets the goal of $\simeq 10'$ resolution at 100 GHz

   
4.1 Method 1: The effective window function

In the first method an effective window function is calculated for the distorted beams and the loss in angular resolution from the corresponding rms temperature perturbations is then estimated.

The first step is to computing the spherical transform $w_{\ell m}$ for each of the beam response functions $w(\Omega)$:

$$w_{\ell m} = \int {\rm d}\Omega\, w(\Omega)\, Y_{\ell m} (\Omega) \, ,$$ (1)

with the beam centered at the pole of the coordinate frame. Then we simply consider the one-point variance of a distorted beam-smoothed temperature field:

$$\langle \tilde{T}^2\rangle = {1\over 4\pi} \sum_\ell (2\ell +... ...um_{m=-\ell}^\ell w_{\ell m}^2 \over (2\ell +1) } \right] \, ,$$ (2)

where the object in square brackets can be considered as a (Fourier space) symmetrized response function (labelled $W^2(\ell)$).

Plots of $W^2(\ell)$ computed for the beam patterns are shown for the PHASE A telescope in Fig. 6: the derived symmetrized Fourier beam shapes are not Gaussian. It is impossible to specify a simple, single, effective FWHM for such distorted beams. Instead, we show in the top-left panel of Fig. 6 the results of computations of the actual rms temperature perturbation in such beams from the usual CDM model (upper line) and the open CDM model (both roughly COBE-DMR normalized - but only relative effects are relevant here). The curves show the rms temperature variation as a function of FWHM of a symmetric Gaussian beam. Superposed symbols, to be matched with the other panels for beam identification, correspond to the actual values of rms temperature as measured by the distorted beams. From such a plot, within the context of a CMB anisotropy model, one can identify an effective angular resolution for non-symmetric beams. Note the symmetry of this effect in the V direction and the large asymmetry in the U direction which strictly reflects the intrinsic asymmetry of the adopted Gregorian configuration; the locations of the feedhorns in the focal plane have been designed to take this effect into account (see Villa et al. 1997 and Mandolesi et al. 1997, 1998). One can see that the PHASE A telescope has angular resolution poorer than 11'; the mean distance of 100 GHz feeds from the optical axis is of $3.3^\circ$ or equivalently, for beams located along the "diagonals'' ( $\vert Az \vert = \vert El \vert$), $\vert Az \vert = \vert El \vert = 2.3^{\circ}$ a value in the range $2^{\circ} \le \vert Az \vert = \vert El \vert \le 3^{\circ}$ where the effective angular resolution of the PHASE A telescope ranges between a minimum value of $\simeq 12'$ and a maximum value of $\simeq 17'$. By avoiding the unfavourable locations at $\vert Az \vert = \vert El \vert$ with El>0, the worst effective angular resolution reduces to $\simeq 14' \div 15'$, that still remains a serious degradation of the angular resolution.

Figures 7 and 8 show equivalent results for the BASELINE and ENLARGED telescopes. The significant improvement is due to the better optical performance as well as to the well known geometrical property that larger aperture telescopes lead to beam locations closer to the optical axis direction, so automatically releasing the issue of the distortions. Indeed, for these designs the mean offset of the LFI 100GHz feeds from the optical axis is respectively 2 $\hbox{$.\!\!^\circ$ }$8 and 2 $\hbox{$.\!\!^\circ$ }$5. The BASELINE telescope is close to meet the $\simeq 10'$ goal at 100GHz for all of the HFI and LFI feeds (see also the Table 3); of course, further improvements can be reached by the ENLARGED configuration.

   
4.2 Method 2: SNR-weighted effective window functions

In the second method observations are modeled by convolving the sky with the calculated beams and then adding noise. Specifically, the noise power spectrum is added to the product of the power spectrum of the sky signal and the window function. The window function is derived using a minimum variance estimator as described below (this requires use of a cosmological model, but the results are almost independent of the model assumed). Since the window function multiplies the signal, the optimum window function is as close to unity over as large a range of $\ell$ as possible.

For simplicity we approximate the sky as flat on the scales probed by the beam (i.e., $\sin\theta\approx\theta$; a very good approximation for beam sizes less than 1$^\circ$). Then the spherical-harmonic transform becomes a Fourier transform and the 2-D window function is the (square of the) Fourier transform of the intensity of the beam. To ask what "equivalent'' azimuthally symmetric beam corresponds to the 2-D window function thus computed requires a definition of what physically we mean by equivalent.

Figure 9: Comparison of the results of Methods 1 and 2 for six feed positions. The differences are small except at very high $\ell$ values where the window functions are very small

If we are concerned primarily with parameter estimation or reconstruction of the angular power spectrum, we can define "equivalent'' to mean that which gives the same derivatives of the power spectrum with respect to the cosmological parameters (see Bond et al. 1997). Operationally this means a weighted sum of the m-modes which minimizes the variance. If $C(\ell)$ is our cosmological signal, $N(\ell)$ is the noise, and we approximate the sum over m as an integral over $\theta_\ell$, then the optimal $W(\ell)$ solves:

$$\left( 1 + \left[ {N\over CW}\right]_\ell \right)^{-1} \equiv... ...\left( 1 + {N(\ell)\over C(\ell)} W^{-1}({\bf l}) \right)^{-1}$$ (3)

where $W({\bf l})$ is the 2-D window function and ${\bf l}$ is a 2-D vector of length $\ell$. The only dependence on $\theta_\ell$comes from the window function. In the angular average most of the weight at high $\ell$ comes from those parts of the beam that are narrowest. At low $\ell$, where the beam smearing is insignificant, all samples of the sky contribute equally. The criterion for keeping or suppressing directions in Fourier space is that the signal ( $C_\ell W({\bf l})$) be measurable over the noise ($N_\ell$).

Thus the "effective window function'' depends on the signal-to-noise ratio, and so will be theory specific, but this dependence is very weak. In practice $W_\ell$ is well approximated by the quadrature mean of the two 1-D window functions obtained by slicing $W({\bf l})$ in two orthogonal directions (i.e., $\sigma_{\rm eff}^2 \approx 1/2\left[\sigma_x^2 + \sigma_y^2\right]$).

Although this calculation differs in detail from that of Method 1, the results are quite similar, as shown in Fig. 9.

   
4.3 Method 3: Simulated observations

In the third method a sky model (we assume a standard CDM model for the present tests) is numerically convolved with the calculated beams truncated at 1^ $^{\circ}\;$ from the beam center, as well as with Gaussian beams of FWHM from 6' to 17' in steps of 1' (the integration uses a 2-dimensional Gaussian quadrature with a typical grid of $48 \times 48$ points; see Burigana et al. 1998a for further details). All the beams are "artificially'' relocated along the telescope optical axis and their centers observe the same set of positions in the sky; the different beam orientations in the sky related to the PLANCK scanning strategy is also taken into account.

Figure 10: Results of Method 3. Left panel: contours of effective angular resolution, given as the FWHM of the symmetric Gaussian beam whose convolution with a CDM sky has the smallest rms difference with the convolution of the actual distorted beams of the PHASE A telescope. Central panel: the same for the BASELINE telescope. This design is close to meet the goals for angular resolution for both HFI and LFI feeds. Right panel: the same for the ENLARGED telescope. This design meets the goals for angular resolution for both HFI and LFI feeds

We then calculate the rms of the difference between the convolutions obtained by using each simulated beam and Gaussian beams with increasing FWHM: the Gaussian beam which minimizes the rms difference is considered as the "equivalent'' Gaussian beam, which FWHM defines the effective angular resolution of the considered distorted beam. The values obtained with this method are in excellent agreement with those obtained with Methods 1 and 2. Figure 10 summarizes the results as a contour plot of effective angular resolution over the entire focal surface. Also, Table 3 gives the results of this method for the LFI feeds; we compute also the averaged rms differences (in terms of thermodynamic temperature), $<\sigma_{{\rm th}}>$, between the convolutions from the simulated feeds and their corresponding equivalent Gaussian beams, that can be seen as an estimate of the additional error introduced by the beam distortion in absence of appropriate deconvolution techniques able to properly take into account the beam shape: we find $<\sigma_{{\rm th}}>$ = $2.2, 2.1, 2.0~\mu$K for the PHASE A, BASELINE and ENLARGED design respectively.

   

Table 2: Beam properties at 100 GHz for telescopes of different aperture. We give the effective FWHM (in arcmin), $W_{\rm e}$: Results from pairs of beams located at the same U and $\mid V \mid$ have been averaged. By averaging over all the feeds we have: $< W_{\rm e} >\ =\ 13.1',\, 11.0',\, 9.7'$respectively for the 1.3, 1.55, 1.75 m aperture telescopes

D $\rightarrow$	1.3 m	1.55 m	1.75 m
Feed	$W_{\rm e}$	$W_{\rm e}$	$W_{\rm e}$
1	12.00	10.00	8.90
2 & 17	12.15	10.25	9.00
3 & 16	12.35	10.45	9.20
4 & 15	13.00	11.00	9.80
5 & 14	12.90	10.90	9.55
6 & 13	13.15	11.00	9.70
7 & 12	13.40	11.20	9.85
8 & 11	13.90	11.65	10.25
9 & 10	14.20	12.05	10.70

As shown by Fig. 10 and Table 2, a larger telescope allows to reach the nominal 10' resolution; this is due both to the better resolution of the best LFI feed and to the partial reduction of the (absolute) angular resolution degradation between the best and the worst LFI feed in the LFI ring region; unfortunately, increasing the primary mirror has a small impact on $<\sigma_{{\rm th}}>$. This is due to the optical aberrations which are not reduced by scaling the telescope parameters.

This method gives also a particularly convenient way of quantifying one of the most important effects of distorted beams, namely, that data at the same position on the sky taken from multiple feeds at a given frequency cannot simply be averaged together, being each beam differently distorted, possibly with a different orientation (Burigana et al. 1998b).

By considering the set of data observed by Gaussian symmetric beams we have computed the differences in observed sky thermodynamic temperature, and the relative rms $_{{\rm th}}$ values, between one beam and all the others and we have calculated that two Gaussian beams differing by 1' yield rms $_{{\rm th}}$ value of about $1.25~\mu$K. Then the relation rms $_{{\rm th}} (\mu{\rm K}) \sim 1.25 \Delta FWHM$ (arcmin) (solid line in Fig. 11) sets a lower limit to the rms of the temperature difference observed by distorted beams with a given difference $\Delta FWHM$ in effective resolution; the expected rms differences are larger due to asymmetries in the beam shapes, as shown in Fig. 11 for the PHASE A telescope.

Figure 11: Distribution of rms differences versus the difference of effective angular resolution for the PHASE A telescope; for the telecope designs in Table 1, this distribution is essentially independent of the telescope aperture (see the text for further details)

Figure 12 gives a histogram of the differences between the signal measured at the same sky positions by the various LFI 100GHz feeds (solid lines) of the PHASE A and ENLARGED telescope for a suitable set of pointing directions. The mode of the distributions is $\simeq 3~\mu$K, with some values exceeding $\simeq 5~\mu$K. These values should be compared to the noise per beam $\sim 1~\mu$K that will be achieved by all PLANCK feeds up to 350GHz near the ecliptic poles. Note that this distribution is the superposition of the distributions of temperature differences obtained by comparing feeds with different spread in effective angular resolution; the central (peaked) part of this distribution (dotted lines) is dominated by the distributions concerning feeds with similar effective resolution whereas the distribution wings are dominated by the distributions concerning feeds with quite different effective resolutions (dashed lines).

Figure 12: Histograms of differences between the signal measured at the same sky positions by the 17 LFI 100GHz feeds for a suitable set of pointing directions on a CDM sky, for the PHASE A telescope (left panel) and the ENLARGED telescope (right panel) (see the text for further details)

Unfortunately, this dispersion of the sky temperature measurements cannot be significantly reduced by increasing the telescope aperture. It can be reduced only with a different FPU configuration which allows a location of all the 100 GHz feeds closer to the centre or with a different telescope design able to redistribute the impact of optical aberrations in a more uniform way on the sky field of view (Villa et al. 1998a; Mandolesi et al. 1999) or, finally, in the data analysis.

No doubt some of these large effects can be removed in the data analysis, and it is certain that thousands of work-years will be devoted to the analysis of PLANCK data. Nevertheless, all previous experience with CMB data from ground, balloon, and space experiments shows that the more and the larger the systematic effects that must be scrubbed from the data in analysis, the more uncertain the result.

   
4.4 Beam distortions and science

The effects of degraded angular resolution on the scientific return from PLANCK have been estimated by calculating the uncertainties in cosmological parameters extracted from the angular power spectrum that would be obtained from the two telescope designs.

The fractional error on the CMB fluctuation angular power spectrum C_l's can be written summing in quadrature the cosmic variance and the instrumental noise (Knox 1995)

$${\delta C_\ell \over C_\ell} = \left({4\pi \over A}\right)^{1... ...}} \left[ { 1 + {A\sigma^2 \over N C_\ell W_\ell} } \right] \,$$ (4)

where A is the surveyed area of the sky, $\sigma$ is the rms pixel noise, N is the number of pixel in the sky map, $W_l \simeq {\rm exp}(-l^2 FWHM_{{\rm eff}}^2 / 8 {\rm ln} 2)$ is a Gaussian approximation of the window function.

Of course, a detailed study, which is far from the purposes of the present work, requires the combined analysis of all the frequency channels and to take also into account the foreground contamination and an accurate quantification of the efficiency in the separation of the different components (e.g. Tegmark et al. 1999 and references therein) through Wiener filtering (Bouchet et al. 1995; Tegmark & Efstathiou 1996), MEM (Hobson et al. 1999), wavelets (Sanz et al. 1999a,b; Tenorio et al. 1999) and independent component analysis (Bedini et al. 1999). On the other hand, we know that the "cosmologically clean'' channels from 70 to 143-217 GHz (according to the adopted evolutionary scheme for the galaxies in the far infrared) are minimally affected by foreground contamination at small scales and we consider their performance in absence of foreground contamination as a guideline for the discussion of impact of beam distortion on CMB science; moreover, from simple optical scaling laws and provided that we limit to the "cosmological'' channels quite close to the 100 GHz frequency, we expect that the effect of beam distortions in these channels is quite similar to that presented here at 100 GHz.

For high resolution and practically full sky experiments like PLANCK, the uncertainty on $C_\ell$ at large $\ell$ (larger than about 800-1000) is dominated by the instrumental noise, which at any $\ell$ gives the error on our estimate of the observable realization of the $C_\ell$'s. In this limit, given two experiment with angular resolution FWHM₁ and FWHM₂respectively and the same sensitivity per pixel, the ratio between the uncertainty of the power spectrum recovered by the two experiments is given by

$${ (\delta C_\ell)_1 \over (\delta C_\ell)_2 } \simeq {\rm exp} [-\ell^2(FWHM_2^2 - FWHM_1^2)/8{\rm ln}2] \, .$$ (5)

For the case of the PHASE A, BASELINE and ENLARGED telescopes the average (FWHM) angular resolutions are $\simeq 13', 11'$ and 9.7' respectively; at multipoles $\ell \simeq 800, 1000, 1200$ and 1500 we find then $(\delta C_\ell)_1 / (\delta C_\ell)_2$ $\simeq 1.6, 2.1, 2.9$ and 5.2 or $\simeq 1.3, 1.5, 1.9$ and 2.7 by comparing PHASE A and BASELINE telescopes or BASELINE and ENLARGED telescopes, respectively.

It is then clear as the accurate determination of those cosmological parameters based on the accurate knowledge on CMB angular power spectrum at large $\ell$ is significantly affected by the quoted degradation of angular resolution introduced by the PHASE A telescope with respect to the PLANCK nominal goal of 10' at 100 GHz. We consider here, as an example, an open (but not extreme) cold dark matter model like target model and use the standard Fisher matrix method for quantifying the impact of angular resolution degradation on PLANCK LFI observations. For simplicity, we use only the 70 and 100 GHz channels together, neglecting foreground contamination. For the 70 GHz beams we have not carried out detailed optical simulations, for the present purposes we simply estimate their averaged effective angular resolution by scaling our results at 100 GHz according to two heuristic considerations: i) in absence of optical distortions, as for example very close to the optical axis, the beam FWHMscales with $1/\nu$; ii) for similar beam positions in the sky field of view (in the present FPU configuration the LFI feeds at 70 and 100 GHz are located on a ring at approximately the same distance from the optical axis) the difference between the effective angular resolution and the nominal resolution along the optical axis quantifies the impact of optical aberrations and scales with $\nu$. At 70 GHz, we use then an effective FWHM $\simeq 17.7', 14.8', 13.1'$ respectively for the PHASE A, BASELINE and ENLARGED telescopes. The results are given in Table 3. From the optical simulations we know that aberration effects are not fully described by the degradation in effective angular resolution alone; they introduce also an additional, systematic noise which could not be simplistically added in quadrature to the white noise; in particular, its angular power spectrum is not flat in $\ell - C_\ell$ space, as discussed in Burigana et al. (1999b). For the present purposes, we neglect here these additional effects focussing only on the impact of angular resolution: Table 3 should be then regarded as an optimistic assessment of the effects of beam distortions.

   

Table 3: Uncertainties ($1-\sigma$) in Extracted Cosmological Parameters versus Telescope Design. An OCDM model with h=0.65, $\Omega _{\rm m}=0.4$, $\Omega _{\rm b}=0.06$, $\Omega _\Lambda =0$, $\Omega _{\rm K} = 0.6$ (curvature), $\tau =0.05$, $n_{\rm S}=1$ has been considered here as the fiducial model. We take $f_{{\rm sky}}=0.65$

Quantity	PHASE A	BASELINE	ENLARGED
ln h	0.0330	0.0248	0.0217
ln $\Omega_{\rm K}$	0.0377	0.0277	0.0232
$\Omega_\Lambda$	0.0448	0.0327	0.0271
ln ${\Omega_{\rm b}h^2}$	0.0134	0.0125	0.0122
$\Omega_{\rm m}$	0.0674	0.0493	0.0410
ln $n_{\rm S}$	0.0095	0.0089	0.0087
ln $\tau$	0.8074	0.7928	0.7870
ln C2	0.0832	0.0815	0.0809

As known, the result partially depends on the choice of the set of parameters used in the analysis; on the other hand, this test confirms the results expected from a qualitatively point of view, i.e. the effect of degradation in angular resolution is particularly relevant for the determination of those parameters, as h, $\Omega_{\rm m}$, $\Omega_\Lambda$ and $\Omega_{\rm K}$, that show unambiguous signatures at large multipoles. Although for flat models this effect results to be less critical, we stress here that PLANCK is designed to be a third generation of CMB space mission and has to have the capability to disentangle between different cosmological models and to accurately determine the cosmological parameters for wide sets of cosmological scenarios and not only for some target models, by using separately each of the two instruments, for security and redundancy.