Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/ppeuc/astronomy/papers/burigana.ps Äàòà èçìåíåíèÿ: Fri Jul 25 14:36:08 1997 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 10:27:53 2012 Êîäèðîâêà:

From simulated sky maps to simulated observations
C. Burigana,y D. Maino,z N. Mandolesi,y E. Pierpaoli,z
M. Bersanelli,x L. Danese,z and M.R. Attoliniy
yIstituto TESRE, Consiglio Nazionale delle Ricerche, Via Gobetti 101, 40129 Bologna, Italy
zSISSA, International School for Advanced Studies, Via Beirut 2­4, 34014 Trieste, Italy
xIFCTR, Consiglio Nazionale delle Ricerche, Via Bassini 15, 20133 Milano, Italy
We present the basic issues of our simulation program, focusing on some particular as­
pects. We have generated high resolution maps by using standard spherical harmonics
expansion for the CMB fluctuations and by extrapolating available full sky maps to pre­
dict the Galaxy emission at frequencies and angular scales appropriate to the PLANCK
space mission. These maps have been used to simulate the PLANCK observations and to
test its performance. Beam responses are typically not symmetric when they are coupled to
a telescope to reach high resolutions, whereas theoretical predictions and standard decon­
volution methods generally assume pure symmetric beams. We have simulated PLANCK
observations at 30 and 100 GHz for typical scan circles, by convolving simulated maps
of the microwave sky with asymmetric beams, to estimate the effect of beam asymmetry
on temperature fluctuations measurements. We find that the typical difference between
temperature measurements performed by symmetric and asymmetric beams is of few ¯K,
depending on the eccentricity of the beam response shape and on the beam FWHM. Al­
though this effect is not very high, it may be appreciable and must be reduced by optimising
the focal plane.
1 Introduction
The PLANCK mission will offer a powerful possibility to observe the cosmic microwave background
(CMB) fluctuations with great sensitivity and high resolution. The generation of high resolution
full sky maps, consistent with the COBE­DMR normalisation of the CMB anisotropies and based
on reasonable angular and frequency extrapolations of available maps with angular resolution of ¸ 1
deg, is a basic step for simulating the mission performance. The goal of high resolution observations
of CMB anisotropy requires the use of large aperture telescopes. A large frequency range coverage
is necessary to efficiently separate the CMB fluctuations from different astrophysical components,
galactic emission, extragalactic foregrounds, SZ effects, which are themselves very informative. 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 became more prominent and
affect the measurements, an effect that will be studied in a forthcoming work. Here we describe a first
estimate of the effect of main beam distortions on anisotropy measurements.
In section 2 we outlined the basic framework for generating full sky maps including CMB anisotropy
and Galaxy emission. In section 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 are presented in section 4. Finally in section 5 we
discuss the main implications of this analysis.
2 Generation of simulated maps
In this section we briefly present the basic issues for generating high resolution full sky maps which
include CMB fluctuations and the Galaxy emission.
1

2.1 Generation of the CMB fluctuation maps
The CMB anisotropy is usually written as (Bond & Efstathiou (1987), White et al (1994)):
\DeltaT (`; OE)
T
=
1
X
l=0
l
X
m=\Gammal
a l;m Y l;m (`; OE) (1)
where Y l;m (`; OE) are the spherical harmonics:
Y l;m (`; OE) =
s
2l +1
4ú
(l \Gamma m)!
(l +m)! P m
l (cos `)e imOE (2)
We note that in the above equations all the dependence on ` is in the Legendre polynomials, while
all the dependence on OE is in the exponential factor. Provided that the anisotropies are a Gaussian
random field, the a l;m coefficients are randomly distributed variables with zero mean and variance
given by the C l , namely C l = hja l;m j 2 i where C l is the angular power spectrum obtained from a given
cosmological model. The Legendre polynomials can be evaluated by the standard recurrence relations
(see Press et al. (1987)). These relations in some cases are inadequate to evaluate the P m
l , especially
at high values of l, and we have worked with a slightly modified set of polynomials, p m
l , and the
corresponding recurrence relations, related to the Legendre polynomials by:
p m
l =
s
(l \Gamma m)!
(l +m)! P m
l : (3)
In addition, the reality of the temperature anisotropies imposes the condition:
a l;m Y l;m +a l;\Gammam Y l;\Gammam 2 IR ; (4)
taking into account that Y \Lambda
l;m = (\Gamma1) m Y l;\Gammam , it results a \Lambda
l;m = (\Gamma1) m a l;\Gammam .
With this condition, we have that:
a l;m Y l;m +a l;\Gammam Y l;\Gammam = 2! (a l;m Y l;m ) ; (5)
and therefore:
\DeltaT (`; OE)
T
=
l max
X
l=2
r
2l +1
4ú p 0
l (x)!(a l;0 )
+
l max
X
m=1
l max
X
l=Maxf2;mg
2
r
2l +1
4ú p m
l (x) [!(a l;m ) cos(mOE) \Gamma =(a l;m ) sin(mOE)] : (6)
We have chosen to neglect here the cosmological dipole term, l = 1, which is dominated by the Doppler
effect due to the relative motion of the earth respect to the CMB.
In equation 6 only the polynomials p m
l depends on `, while all the dependence on OE is in the square
bracket part. This particular feature makes the choice of the pixelization (i.e. the set of f` i ; OE j g where
to calculate \DeltaT =T ) a crucial parameter for the computational cost of the simulation. Namely, if one
is not particularly interested in having an equal--area pixelization, the best choice is a pixelization in
which the fOE j g is the same for all f` i g; in this way it is possible to evaluate the sin(mOE) and cos(mOE)
once for all and memorise them at the beginning, so diminishing the computational time by a factor
10. In addition it allows one to use FFT techniques to generate a full sky map and this further reduces
the required amount of computation (see for example Muciaccia et al. (1997)). We have chosen to
keep a general routine with no particular assumption on the pixelization but two simple symmetry
properties: 1) if ` k 2 f` i g , then also \Gamma` k 2 f` i g; 2) if OE k 2 fOE j g then also (OE k + ú) 2 fOE j g. The
symmetry property for ` is useful because of the parity in (l \Gamma m) of the polynomials p m
l (x). This
property, together with the trigonometric properties of sines and cosines, allow us to divide by four
the computational time because the temperature anisotropy can be computed in four points of the sky
at the same time. The ``standard'' COBE­CUBE pixelization satisfies these conditions and we have
2

used that for our beam tests. It offers the advantage of good equal­area conditions, hierarchic and
that Galaxy maps and software are presently available for that pixelization scheme (for an improved
scheme see Gorski et al. (1997)).
From a simulated map we can compute the correlation function C(`)map :
C(`)map =
Ü \DeltaT (bn)
T
\DeltaT (bn 0 )
T
AE
; (7)
where cos(`) = b n \Delta b n 0 .
Directly from the a l;m , and the corresponding C l;map , used for generating a given map we can have
the correlation function C(`) al;m defined by:
C(`) al;m = 1
4ú
l max
X
l=2
(2l + 1)C l;map P l (cos`) : (8)
We have generated maps at COBE­CUBE resolution 9, 10 and 11 (i.e. with typical distances between
two pixels of about 18 0 , 9 0 , 5 0 respectively) and with l up to 1200 and we have verified the goodness of
the maps obtained with our code by comparing the correlation functions obtained from the two above
equations, in order avoid the ambiguity due to the cosmic variance. In addition we have checked that
the average of the correlation functions obtained by few tens of maps tends to that obtained from the
equation 7 when the theoretical prescription for the C l is used.
2.2 Galactic Emission
In the spectral range of interest here, the Galaxy emission is mainly due to three different physical
mechanisms: synchrotron emission from cosmic rays electrons accelerated into galactic magnetic fields,
free--free or thermal bremsstrahlung emission and dust emission.
Both synchrotron and free--free spectral shape can be described, in terms of antenna temperature,
by simple power laws, T (š) / š \Gammaff , with spectral indices: ff syn = 2:8 \Gamma 3:1 and ff ff = 2:1. While free--
free emission is a well known mechanisms and ff ff has relatively small uncertainties, the synchrotron
emission is still rather unknown and it seems that a steeping of the spectral index will occur at
higher frequencies, as expected from the theory. It is also expected a spatial variation of ff syn due
to its dependence upon electrons energy density and galactic magnetic field which change in the
Galaxy (Lawson et al. (1987), Banday & Wolfendale (1990), Banday & Wolfendale (1991), Kogut et
al. (1996), Platania et al. (1997)).
Dust emission spectral shape can be described by a simple modified blackbody law I š / š ff B š (T )
where ff is the emissivity and B š (T ) is the brightness of a blackbody of temperature T . Recent
works, based upon COBE­DMR and DIRBE data (Kogut et al. (1996)), give values of ff ¸ 1:8 \Gamma 2 and
T ¸ 18 K, but also a model with two dust temperatures is possible.
Figure 1 shows a schematic representation of these spectral shapes.
In order to build up a realistic model of galactic emission we have to know both spatial and spectral
behaviour of the three emission mechanisms. The best situation would be when we have measurements
of galactic emission in those spectral regions where only one of these emission mechanisms is dominant.
From Figure 1 it is clear that this is possible only for synchrotron emission (at very low frequencies)
and for dust emission (at very high frequencies). For free--free emission is more complicated to derive
detailed informations, because does not exist a spectral region where it is the dominant component
of the signal. At the moment, our model does not yet include free--free emission in our Galaxy. We
use a synchrotron spectral index constant in the sky, but our code allows to change its value within
an appropriate range. For the dust model we took a two dust temperature model, but we are able to
select different models.
The model we constructed is based upon two full­sky maps: the map of Haslam et al. (1982) at
408 MHz and DIRBE map at 240 ¯m. Both maps have nearly the same angular resolution: 0.85 deg
and 0.6 deg respectively. It is also clear that both maps lack the proper angular resolution to simulate
directly the PLANCK observations (30 0 \Gamma 10 0 ).
Studies on the spatial distribution and power spectrum of galactic emission (Gautier et al. (1992),
Kogut et al. (1996)) show that dust and at least one component of the free--free follow a power law
3

Figure 1: Schematic representation of the three different components of Galaxy emission compared to
the CMB emission and its typical fluctuations in terms of antenna temperature (in mK).
4

with index \Gamma3: P l / l \Gamma3 . For synchrotron emission the situation is still under discussion: the index
will probably range from \Gamma2 to \Gamma3.
We then extended in power our maps to match the proper PLANCK resolution. In order to do
that we build square region of about 20 \Theta 20 deg 2 with different indices of the power law spectrum.
We then ``cover'' the sky with many of these fundamental patches with the condition that the signal
in the extended map on the smaller angular scales provided by the original ``true'' map (Haslam et al.
(1982) or DIRBE) must be the signal in the ``true'' map. In this way we are able to construct maps
of galactic emission with the desired angular resolution for each of the PLANCK frequencies where
the signals are scaled in frequency according to their spectral shape. In particular we built two maps
of galactic emission at 30 and 100 GHz, that we have used for the present beam tests.
3 Convolution of the simulated map with the beam response
A first application of our simulation code is the study of the effect of beam distortions on the measured
sky temperatures. The sky is simulated adding the CMB and galactic components as described in
section 3.
Different feedhorns must be located on different parts of the focal plane. The magnitude and the
kind of beam distortion depend on several parameters: the beam FWHM, the observational frequency,
the telescope optical scheme and the beam location with respect to the optical axis. Optical simulations
(Nielsen & Pontoppidan (1996)) show that the main expected distortion in the off­axis beams has an
elliptical shape, with more complex asymmetries in the sidelobe structure. Here we have assumed
that the beam is located along the optical axis, but that it can have an elliptical shape, i.e. the curves
of equal response are ellipses.
Let be s the spin axis vector, which is on the ecliptic plane for the standard scanning strategy, and
p the vector of the direction of the optical axis of the telescope, at an angle ff(= 70 deg) from the spin
axis. We choose two coordinates x and y on the plane tangent to the celestial sphere in the optical
axis direction, with vector u and v respectively; we choose the x axis according to the condition that
the vector u points always toward the satellite spin axis; indeed, for standard PLANCK observational
strategy, this condition is preserved as the telescope scans different sky regions. With this frame of
reference choice, we have that v = p “ s=jp “ sj and u = v “ p=jv “pj (here “ indicates the vector
product). With the simple work assumption that the beam centre is along the optical axis, we have
that the beam (elliptical) response in a given point (x;y) is given by:
K(x;y) = exp
\Theta
\Gamma0:5[(x=oe x ) 2 + (y=oe y ) 2 ]
\Lambda (9)
The ratio r = max(oe x ; oe y )=min(oe x ; oe y ) between major and minor axis of the ellipses of constant
response quantifies the amount of beam distortion (we have chosen the major axis along the x axis,
but we have verified for a suitable number of cases that our conclusions are unchanged if the major
axis is chosen along the y axis). We have convolved the simulated map with this beam kernel up
to the level (x=oe x ) 2 + (y=oe y ) 2 = 9, i.e. up to the 3oe level. The integration has been performed by
using a 2­dimensional Gaussian quadrature with a grid of 46 \Theta 46 points. We have performed the
convolution under the assumption that the telescope points always at the same direction during a
given integration time; this artificially simplifies the analysis, but it is useful to make the study
independent of the scanning strategy and related only to the optical properties of the instrument.
The sky map, obtained by using the COBE­CUBE pixelization, has been interpolated in a standard
way to have the temperature values at the grid points. For maps at resolution 11 (9) we have about
50 (3) pixels within the FWHM (' 30 0 ) at 30 GHz and 6 (less than 1) at 100 GHz (FWHM ' 12 0 );
then the true accuracy of the integration derives not only from the adopted integration technique but
from the map resolution too. For this reason the use of high resolution maps and a careful comparison
between beam test results obtained from maps at different resolutions are recommended.
In order to quantify how the beam distortion affects the anisotropy measurements, we use a
simple estimator: the rms of the difference between the temperature observed by an elliptical beam
and a symmetric one. We use here thermodynamical temperature which does not depend on the
observational frequency; the present results can be translated in terms of antenna temperature with
the relation rmsA = rms th x 2 exp(x)= [exp(x) \Gamma 1] 2 where x = hš=kT 0 (rmsA ' rms th at 30 GHz,
whereas rmsA ' 0:78rms th at 100 GHz).
5

Figure 2: Difference between the (antenna) temperature observed by asymmetric and symmetric beams
for a typical scan circle as function of the scan integration number (top panel) or the corresponding
galactic latitude (bottom panel).
Figure 3: Antenna temperature observed by asymmetric (triangles) and symmetric (crosses) beams
and differences (circles) between the two measurements for a typical scan circle as function of the scan
integration number (top panel) or the corresponding galactic latitude (bottom panel).
4 Results of beam tests
We have considered circle scans that crosses (or not) the region near the galactic centre to check
the effects of Galaxy temperature gradients in different situations. For standard PLANCK scanning
strategy, a scan circle consists of about 680 integrations at 30 GHz and of about 1700 integrations
at 100 GHz. We have carried out simulations that either include or neglect the contribution of the
Galaxy for the both frequencies (we have adopted here the scaling law P l / l \Gamma3 to extrapolate its
fluctuations at small angular scales). It results that the Galaxy does not affect the difference between
the temperature measured by symmetric and elliptical beams. We have performed our tests using
maps at different COBE­CUBE resolution (from 9 to 11) to check the dependence of the result on the
resolution of the map.
6

Table 1: rms th value of (T ell \Gamma T circ ) for different cases.
š (GHz) FWHM (arcmin) r rms th (¯K)
30 30 1.1 0.83
30 30 1.3 2.3
30 30 1.8 5.1
100 12 1.1 0.51
100 12 1.3 1.4
100 12 1.8 3.2
Figures 2 and 3 show our results for a test at 30 GHz, the frequency considered here where the
emission of the Galaxy and its fluctuations are more important (Toffolatti et al. (1995), Danese et
al. (1996)); the distribution of the temperature differences does not depend on the galactic latitude
(see Figure 2) and, even where galactic emission coupled to CMB quadrupole large scale waves pro­
duces significant increases of the observed antenna temperature, the temperature differences remain
practically equal to those obtained in other sky regions (see Figure 3).
Typical results obtained by using maps at COBE­CUBE resolution 11 are tabulated in Table 1 in
terms of rms th for different beam FWHM's and distortion parameters r. The results do not change
significantly by using maps at different resolutions.
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. These two parameters are the most
important for what concerns here, whereas the observational frequency is directly not relevant (a part
for the obvious relationship between the considered frequency and the beam properties), due to the
very small effect of Galaxy fluctuations.
5 Discussion: implications of beam tests
In the previous section we have shown that the typical difference between the temperature by sym­
metric and distorted beams is about few ¯K, depending on the eccentricity of the beam response
shape and on the beam FWHM. Although the rms value of this effect is not very high, for some
``pixels'' the effect may be significantly higher than the average. In addition it can be not significantly
reduced by repeating the observation with different spin axis directions, because a given sky region
is typically observed (for scanning strategy similar to that proposed for the PLANCK mission) with
similar orientation of the plane x;y; therefore this kind of distortion produces systematic and not
statistic errors in the temperature measurements of any given pixel. Deconvolution techniques are
generally developed for symmetric beams and it is presently not clear how deconvolving an observed
map in the case of asymmetric beams, particularly when the orientation in the sky of plane x;y may
change with time. On the other hand, by averaging maps deconvolved with standard methods and
obtained from different channels at the same frequency, this effect may be reduced, provided that
detectors at the same frequency observe the same sky region with different beam orientations. Then,
we can hope that the final product will be less sensitive to beam distortions.
The present analysis shows that, for the same beam distortion parameter r, the effect is more
important for the low frequency beams than for high frequency ones, due to the different beam
widths. This fact indicates that a good solution may be to arrange low frequency feedhorns near
to the optical axis and high frequency ones in the outer regions of the focal plane, because beam
distortions typically increase with its distance from the optical axis. On the contrary, when this
distance is fixed, the distortion parameter r is typically larger for high frequency detectors than for
low frequency ones (Nielsen & Pontoppidan (1996)) as recently studied by Villa (1997) for the case of
the coma distortion; this fact suggests a focal plane arrangement that goes in a direction opposite to
that delined above. Given the present knowledge of beam distortion as function of the distance from
optical axis and of the frequency and the numerical estimates quoted in Table 1, this second choice
seems to be more advantageous. On the other hand, the goodness of the focal plane arrangement must
be checked by evaluating the average global temperature effect, rms, and by minimising the resulting
7

potential errors introduced by all the distortion effects.
Finally, an other crucial point is the comparison between the value of rms due to beam distortion
and the sensitivity of different receivers. For PLANCK LSI, low frequency channels are more sensitive
(Bersanelli et al. (1996)); on the other hand, high frequency channels are more efficient for the primary
cosmological goal.
We intend to complete the present analysis in the next future, to examine other CMB anisotropy
maps, different extrapolation laws for the Galaxy contamination and the effect of discrete sources and
to explore a more complete set of distortion parameters, FWHM's and map resolutions.
Acknowledgements
It is a pleasure to thank P. Platania, G. Smoot and F. Villa for useful discussions and J. Aymon for
his exhaustive suggestions on the use of the COBE software.
References
Banday A.J. & Wolfendale A.W., 1990, MNRAS, 245, 182.
Banday A.J. & Wolfendale A.W., 1991, MNRAS, 248, 705.
Bersanelli et al., 1996, ESA, COBRAS/SAMBA Report on the Phase A Study, D/SCI(96)3.
Bond J.R. & Efstathiou G., 1987, MNRAS, 226, 655.
Danese et al., 1996, Astro. Lett. & Comm., 33, 257.
Gautier T.N. et al., 1992, AJ, 103, 1313.
Gorski K., 1997, this Conference.
Haslam C.G.T. et al., 1982, AASS, 47, 1
Kogut A. et al., 1996, ApJ, 460, 1.
Lawson et al., 1987, MNRAS, 225, 307.
Muciaccia P.F. et al., 1997, preprint astro­ph/9703084.
Nielsen P.H.& Pontoppidan K., 1996, ``Design & analysis of the COBRAS/SAMBA telescope'', TICRA
Rep. S­699­02.
Platania P. et al., 1997, ApJ, to be submitted.
Press W.H. et al., 1987, ``Numerical Recipes'', Cambridge University Press (Cambridge, UK).
Toffolatti et al., 1995, Astro. Lett. & Comm., 32, 125.
White M. et al., 1994, ARAA, 32, 319.
Villa F., 1997, communication at the ``PLANCK Focal Plane Meeting'', ITESRE/CNR -- Bologna,
18/6/1997.
8