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.

*Modeling the CMB pattern* - The CMB monopole and
dipole have been generated by using the Lorentz invariance of
photon distribution functions, ,
in the phase space
(Compton-Getting effect):
,
where
is the observation
frequency,
is the corresponding frequency in the CMB rest
frame,
is the unit vector of the photon propagation
direction and
the observer velocity. A
blackbody spectrum at
K (Mather et al. 1999) is
assumed for .
For Gaussian models, the CMB anisotropies at
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).

*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 HII sources
from C. Witebsky (1978, unpublished)
at resolution of
.
They are subtracted for modeling the
diffuse components and then re-added to the final maps. We use a
spectral index
from 2.7 to 1 GHz and
below 1 GHz. We then combine the synchrotron maps
producing a spectral index map between 408 - 2300 MHz with a
resolution of
(
). This spectral index map is used to
scale the synchrotron component down to
GHz. In fact, for
typical (local) values of the galactic magnetic field
(
), the knee in the electron energy spectrum in cosmic rays
( GeV) corresponds to 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,
(
), has
been provided by Lasenby et al. (1998); we used
for the angular structure extrapolation
(Burigana et al. 1998). Schlegel et al. (1998)
provided a map of dust emission at 100
merging the DIRBE and
IRAS results to produce a map with IRAS resolution ()
but with DIRBE calibration quality. They also provided a map of
dust temperature,
,
by adopting a modified blackbody
emissivity law,
,
with
.
This can be used to scale the dust emission map to
PLANCK frequencies using the dust temperature map as input for the
function. Unfortunately the dust temperature map has
a resolution of
;
again, we use an angular power
spectrum
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.

*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
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
for compact sources up to
GHz and a
break to
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
GHz, whereas radio selected sources should dominate at lower
frequencies (Toffolatti et al. 1998).

*Instrumental noise* - The white noise depends on
instrumental performances (bandwidth
,
system temperature
), on
the observed sky signal,
,
dominated by CBM monopole, and on the
considered integration time, ,
according to:

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

where and

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
.
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
,
we transform
them and obtain a real noise stream which has to be normalized to the white
noise level
(Maino et al. 1999).

*Modeling the observed signal* -
We produce full sky maps,
,
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*-
data sample in the form
,
where *N*_{i} is the instrumental noise generated
as described above.

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*_{i}and the non reduced time variation of
have to be much
smaller than the PLANCK nominal sensitivity. Thus, we generate the
"observed'' map assuming a constant value,
,
of
for all the data samples. We note that possible constant
small off-sets in
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.

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