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 galactic emission.

2.1 CMB fluctuation maps

The CMB anisotropy is usually written as (Bond & Efstathiou 1987; White et al. 1994):  

$${{\Delta T(\vartheta,\phi)}\over T} = \sum_{l=0}^{\infty}\, \sum_{m=-l}^{l} a_{lm} \ Y_{lm}(\vartheta,\phi) \,$$ (1)

where $Y_{l,m}(\vartheta,\phi)$ are the usual spherical harmonics (Press et al. 1987). They can be expressed as a product of the Legendre polynomials, P_l^m, where is all the dependence on colatitude $\vartheta$, and an exponential factor that contains the dependence upon longitude $\phi$.Provided that the anisotropies are a gaussian random field, the a_lm coefficients are randomly distributed variables with zero mean and variance C_l=<| a_lm |² >. The coefficients C_l define the angular power spectrum obtained from a given cosmological model. The Legendre polynomials can be evaluated by the standard recurrence relations (e.g. Press et al. 1987). These relations in some cases are inadequate to evaluate the P_l^m, expecially at high values of l, and we have worked with a slightly modified set of polynomials, p_l^m, and the corresponding recurrence relations, related to the Legendre polynomials by:

$$p_l^m(\mu) = \sqrt{ {(l-m)!} \over {(l+m)!}} P_l^m(\mu) \, ,$$ (2)

where $\mu={\rm cos}(\vartheta)$.With this notation, from the reality of the temperature anisotropies we have that the temperature fluctuations can be expressed as:

$$\begin{eqnarray} \frac{\Delta T(\vartheta,\phi)} {T} =&& \sum_{l=2}^{ \l _{\r... ...\nonumber \\ &\times & \cos(m\phi)- \Im m (a_{lm}) \sin(m \phi) ].\end{eqnarray}$$ (3)

We have choosen to neglect here the cosmological dipole term, l = 1, which is dominated by the Doppler effect due to the relative motion of the earth relative to the CMB.

In Eq. (3) only the polynomials p_l^m depend on $\vartheta$, while all the dependence on $\phi$ is in the square bracket part. This particular feature makes the choice of the pixelisation (i.e. the set of $\{ \vartheta_i, \phi_j \}$ where to calculate $\Delta T /T$) a crucial parameter for the computational cost of the simulation. The "standard'' COBE-cube pixelisation satisfies two simple symmetry properties: 1) if $\vartheta_k \in \{ \vartheta_i \}$ , then also $- \vartheta_k \in \{\vartheta_i\}$; 2) if $\phi_k \in \{ \phi_j \}$ then also $(\phi_k + \pi) \in \{ \phi_j \}$.It allows to divide by four the computational time because the temperature anisotropy can be computed in four points of the sky at the same time. It offers the advantage of good equal-area conditions, hierarchic and also Galaxy maps and software are presently available for that pixelisation scheme (see Gorski (1997) for an improved scheme which also includes the recepies of Muciaccia et al. (1997) that strongly reduce the computational load).

From a simulated map we can compute the usual correlation function $C(\theta)_{{\rm map}}$ (Peebles 1972). Directly from the a_lm, and the corresponding $C_{l,{\rm map}}$ used for generating a given map we can have the correlation function $C(\theta)_{a_{lm}}$. We have generated maps at angular resolutions (i.e. typical pixel dimensions) of about 19', 10', 5', corresponding respectively to COBE-cube resolutions R equal to 9, 10 and 11 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 methods, in order to 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 derived from the theoretical prescription for the C_l.

2.2 Galactic emission

In the spectral range of interest here, the galactic emission is due to three different physical mechanisms: synchrotron emission from cosmic ray electrons accelerated by galactic magnetic fields, free-free or thermal bremsstrahlung emission and dust emission.

To first approximation, both synchrotron and free-free spectral shape can be described, in terms of antenna temperature, by simple power laws, $T(\nu) \propto \nu^{-\alpha}$,with spectral indeces $\alpha_{\rm syn} = 2.8-3.1$ and $\alpha_{\rm ff} = 2.1$ respectively. While free-free emission is a well known mechanism and $\alpha_{\rm ff}$ has relatively small uncertaintes, the synchrotron emission is still rather unknown and, as derived from the theory, a steepening of the spectral index is expected at higher frequencies (Lawson et al. 1987; Banday & Wolfendale 1990, 1991). It is also expected a spatial variation of $\alpha_{\rm syn}$ due to its dependence upon electrons energy density and galactic magnetic field (Lawson et al. 1987; Banday & Wolfendale 1990, 1991; Kogut et al. 1996; Platania et al. 1998).

Dust emission spectral shape can be described by a simple modified blackbody law $I_{\nu} \propto \nu^{\alpha}B_{\nu}(T)$ where $\alpha$ is the emissivity and $B_{\nu}(T)$ is the brightness of a blackbody of temperature T. Recent works, based upon COBE-DMR and DIRBE data (Kogut et al. 1996), give values of $\alpha \sim 1.8-2$ and $T \sim 18~{\rm K}$;a recent analysis of FIRAS maps (Burigana & Popa 1998) supports a model with two dust temperatures (Wright et al. 1991; Brandt et al. 1994).

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. Useful information can be obtained from measurements in those spectral regions where only one of these emission mechanisms is dominant.

This is possible only for synchrotron emission (at very low frequencies) and for dust emission (at very high frequencies), while free-free emission does not dominate in any frequency range. Our model does not yet include free-free emission in our Galaxy but this does not significantly affect the results of our beam tests. For our simulation of the synchrotron emission we took a spectral index (between 2.8 and 3.1) that is constant on the whole sky, i.e. we did not allow any spatial variation in $\alpha_{\rm syn}$.Also we are able to select different dust models; here we used the two dust temperature model of Brandt et al. (1994).

The simulated maps are based upon two full-sky maps: the map of Haslam et al. (1982) at 408 MHz and DIRBE map at 240 $\mu \rm{m}$. Both maps have nearly the same angular resolution ($\theta \sim 0.85 ^{\circ}\;$ and 0.6$^{\circ}\;$ respectively) which is clearly not sufficient to simulate directly the Planck observations ($\theta \,\lower2truept\hbox{${< \atop\hbox{\raise4truept\hbox{$\sim$}}}$}\,30'$).

Studies on the spatial distribution and angular 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 $C_l \propto l^{-\beta}$, with $\beta =3$. For synchrotron emission the situation is still unclear, the index probably ranging from 2 to 3, although a recent study (Lasenby 1997) indicates a value closer to 2 (our analysis of Haslam map tends to confirm this spectral shape).

In order to match the proper Planck resolution we can extend in power the present galactic maps. A complete, self-consistent approach will require their inversion in order to obtain the coefficients a_lm in the range of l,m covered by the maps resolution. Then, one may extrapolate the coefficients a_lm at large values of l (and |m|), possibly according to some physical, frequency dependent model for Galaxy fluctuations at small angular scales. This analysis is out of the aim of the present work.

In order to generate high resolution galactic maps we adopted here a simple euristic approach which is only a first guess but which is neverthless a reasonable choice. Firstly, we increase the original angular resolution (of about 19') of a given map (Haslam and DIRBE) in an artificial way, by dividing each pixel of this map in smaller pixels (of about 5') with the same temperature of the larger pixel that contains them. We want now the temperature field oscillates within this scale. We then calculate from the original map the RMS fluctuation on a certain angular scale (in our case we took $2.6^{\rm \circ}$). Then we built a suitable number of squared regions of about $20^{\circ}\;\times 20^{\circ}\;$with an "extended'' angular power spectrum $C_l \propto l^{-\beta}$ (we have considered the cases of $\beta=2$ or 3) with a resolution of about 5' (corresponding to the COBE-cube pixelisation at R=11) and considering the multipoles from l corresponding to a scale $2.6^{\circ}\;$ up to l=2000. We randomly "covered'' the whole RMS sky with these patches by locally rescaling them requiring that the RMS in the different regions of $2.6^{\circ}\;$ size in the extended map has to be the same found from the original map; this determines the normalization of the "extended'' angular power spectrum. Finally we add the "extended'' RMS sky to the above artificial "extended'' sky, that were uniform on scales of 19'. In this way we add fluctuations on smaller angular scales starting from what the fluctuations really are on larger angular scales. We have checked that this extended map, degraded at a COBE-cube resolution 9, presents pixels temperatures that differ from those of the original map for only few percent, substantially confirming the stability of the method.

Finally, the signal in these maps is scaled in frequency according to the spectral shapes described above in order to match the Planck frequencies. In particular we built two maps of the galactic emission at 30 and 100 GHz (with both synchrotron and dust emission) that we used for the present beam tests.