Characterization of P LANCK-LFI signal components

The simulated cosmological and astrophysical components are generated according to the methods described in Burigana et al. (2000) and the data stream and noise generation as in Burigana et al. (1997b), Seiffert et al. (1997) and Maino et al. (1999). We summarize here below the basic points.

$\bullet$ Modeling the CMB pattern - The CMB monopole and dipole have been generated by using the Lorentz invariance of photon distribution functions, $\eta$, in the phase space (Compton-Getting effect): $\eta _{{\rm obs}} (\nu_{{\rm obs}},\vec n) = \eta_{{\rm CMB}} (\nu _{{\rm CMB}}) \,$, where $\nu _{{\rm obs}}$ is the observation frequency, $\nu _{{\rm CMB}} = \nu_{{\rm obs}} (1+\vec \beta \times \vec n) / \sqrt{1-\beta^2}$ is the corresponding frequency in the CMB rest frame, $\vec n$ is the unit vector of the photon propagation direction and $\vec \beta = \vec v/c$ the observer velocity. A blackbody spectrum at $T_{\rm0}=2.725$ K (Mather et al. 1999) is assumed for $\eta$. For Gaussian models, the CMB anisotropies at $l \ge 2$ can be simulated by following the standard spherical harmonic expansion (see, e.g., Burigana et al. 1998) or by using FFT (Fast Fourier Transform) techniques which take advantage of equatorial pixelization (Muciaccia et al. 1997).

$\bullet$ Modeling the Galaxy emission - The Haslam map at 408 MHz (Haslam et al. 1982) is the only full-sky map currently available albeit large sky areas are sampled at 1420 MHz (Reich 1986) and at 2300 MHz (Jonas et al. 1998). To clean these maps from free-free emission we use a 2.7 GHz compilation of $\sim 7000$ HII sources from C. Witebsky (1978, unpublished) at resolution of $\sim 1^{\circ}$. They are subtracted for modeling the diffuse components and then re-added to the final maps. We use a spectral index $\beta_{{\rm ff}} = 2.1$ from 2.7 to 1 GHz and $\beta_{{\rm ff}} = 0$ below 1 GHz. We then combine the synchrotron maps producing a spectral index map between 408 - 2300 MHz with a resolution of $\lesssim 2^{\circ} \div 3^{\circ}$ ( $<\beta_{{\rm sync}}>\,\sim 2.8$). This spectral index map is used to scale the synchrotron component down to $\sim 10$ GHz. In fact, for typical (local) values of the galactic magnetic field ($\sim 2.5$  $\mu{\rm G}$), the knee in the electron energy spectrum in cosmic rays ($\sim 15$ GeV) corresponds to $\sim~10$ GHz (Platania et al. 1998). From the synchrotron map obtained at 10 GHz and the DMR 31.5 GHz map we derive a high frequency spectral index map for scaling the synchrotron component up to PLANCK frequencies. These maps have a poor resolution and the synchrotron structure needs to be extrapolated to PLANCK angular scales. An estimate of the synchrotron angular power spectrum and of its spectral index, $\gamma$ ( $C_l \propto l^{-\gamma}$), has been provided by Lasenby et al. (1998); we used $\gamma = 3$for the angular structure extrapolation (Burigana et al. 1998). Schlegel et al. (1998) provided a map of dust emission at 100  $\mu{\rm m}$ merging the DIRBE and IRAS results to produce a map with IRAS resolution ($\simeq 7'$) but with DIRBE calibration quality. They also provided a map of dust temperature, $T_{{\rm d}}$, by adopting a modified blackbody emissivity law, $I_\nu \propto B_\nu(T_{{\rm d}}) \nu^{\alpha}$, with $\alpha =2$. This can be used to scale the dust emission map to PLANCK frequencies using the dust temperature map as input for the $B_\nu(T_{{\rm d}})$ function. Unfortunately the dust temperature map has a resolution of $\simeq 1^{\circ}$; again, we use an angular power spectrum $C_l \propto l^{-3}$ to scale the dust skies to the PLANCK proper resolution. Merging maps at different frequencies with different instrumental features and potential systematics may introduce some internal inconsistencies. More data on diffuse galactic emission, particularly at low frequency, would be extremely important.

$\bullet$ Modeling the extragalactic source fluctuations - The simulated maps of point sources have been created by an all-sky Poisson distribution of the known populations of extragalactic sources in the $10^{-5}< S(\nu)< 10$ Jy flux range exploiting the number counts of Toffolatti et al. (1998) and neglecting the effect of clustering of sources. The number counts have been calculated by adopting the Danese et al. (1987) evolution model of radio selected sources and an average spectral index $\alpha=0$ for compact sources up to $\simeq 200$ GHz and a break to $\alpha=0.7$ at higher frequencies (see Impey & Neugebauer 1988; De Zotti & Toffolatti 1998), and by the model C of Franceschini et al. (1994) updated as in Burigana et al. (1997a), to account for the isotropic sub-mm component estimated by Puget et al. (1996) and Fixsen et al. (1996). At bright fluxes, far-IR selected sources should dominate the number counts at High Frequency Instrument (HFI) channels for $\nu \gtrsim 300$ GHz, whereas radio selected sources should dominate at lower frequencies (Toffolatti et al. 1998).

$\bullet$ Instrumental noise - The white noise depends on instrumental performances (bandwidth $\Delta \nu$, system temperature $T_{{\rm sys}}$), on the observed sky signal, $T_{{\rm sky}}$, dominated by CBM monopole, and on the considered integration time, $\tau$, according to:

$$% \Delta \mbox{$T_{{\rm sky}}$ } = \frac{\sqrt{2}(\mbox{$T_{{... ...+ \mbox{$T_{{\rm sky}}$ })}{\sqrt{\Delta \nu \,\, \tau}} \cdot$$ (1)

Under certain idealistic assumptions, Burigana et al. (1997b) and Seiffert et al. (1997) provide analytical estimates for the knee frequency, $f_{{\rm k}}$, of LFI radiometers; it is predicted to critically depend also on the load temperature, $T_{{\rm load}}$, according to:

$$% \mbox{$f_{{\rm k}}$ } = \frac{A^2{\Delta \nu}}{8}(1-r)^2\le... ...\mbox{$T_{{\rm sys}}$ }+\mbox{$T_{{\rm sky}}$ }}\right)^2 \, ,$$ (2)

where $r=(\mbox{$T_{{\rm sky}}$ }+\mbox{$T_{{\rm sys}}$ })/(\mbox{$T_{{\rm load}}$ }+\mbox{$T_{{\rm sys}}$ })$ and A is a constant, depending on the state of art of radiometer technology, which has to be minimized for reducing via hardware the knee frequency (current estimates are $A\sim 1.8\,\, 10^{-5}$ for 30 and 44 GHz radiometers and $A\sim 2.5 \,\, 10^{-5}$ for 70 and 100 GHz).

Recent experimental results from M. Seiffert (1999, private communication), show knee frequency values of this order of magnitude, confirming that the present state of art of the radiometer technology is close to reach the ideal case.

A pure white noise stream can be easily generated by employed well tested random generator codes and normalizing their output to the white noise level $\Delta \mbox{$T_{{\rm sky}}$ }$. A noise stream which takes into account both white noise and 1/f noise can be generated by using FFT methods. After generating a realization of the real and imaginary part of the Fourier coefficients with spectrum defined $\mbox{$S_{{\rm noise}}$ }(f) \propto (1 + \mbox{$f_{{\rm k}}$ }/f)$, we transform them and obtain a real noise stream which has to be normalized to the white noise level $\Delta \mbox{$T_{{\rm sky}}$ }$(Maino et al. 1999).

$\bullet$ Modeling the observed signal - We produce full sky maps, $T_{{\rm sky}}$, by adding the antenna temperatures from CMB, Galaxy emission and extragalactic source fluctuations. PLANCK will perform differential measurements and not absolute temperature observations; we then represent the final observation in a given i-${\rm th}$ data sample in the form $T_i = R_i (T_{{\rm sky},i} + N_i - \mbox{$T_{{\rm ref},i}$ }) \,$, where N_i is the instrumental noise generated as described above.

$T_{{\rm ref},i}$ is a reference temperature subtracted in the differential data and R_i is a constant which accounts for the calibration. Of course, the uncertainty on R_iand the non reduced time variation of $T_{{\rm ref},i}$ have to be much smaller than the PLANCK nominal sensitivity. Thus, we generate the "observed'' map assuming a constant value, $T_{{\rm ref},i}$, of $T_{{\rm ref},i}$ for all the data samples. We note that possible constant small off-sets in $T_{{\rm ref},i}$ could be in principle accepted, not compromising an accurate knowledge of the anisotropy pattern. We arbitrarily generate the "observed'' map with R_i=R=1 for all the data samples.