  We use the most up-to-date cosmological evolution models of star-forming (SF)
galaxies and radio sources to compute the extragalactic number counts and the
cosmic background from 408MHz to 12THz. The model of SF galaxies reproduces the
constraints obtained by Spitzer, Herschel, and ground-based submm/mm
experiments: number counts, redshift distribution of galaxies, cosmic
background intensity and anisotropies. The template SEDs of this model are
extrapolated to the radio adding synchrotron, free-free, and spinning dust
emissions. To fix the synchrotron intensity, we use the IR/radio flux ratio,
q70, and a spectral index beta=-3. For a constant q70, our model added to the
AGN contribution provides a good fit to the number counts from 12THz to 408MHz
and to the CIB. Spinning dust accounts for up to 20% of the cosmic microwave
background produced by SF galaxies, but for less than 10% of the total
background when AGN are included. The SF galaxies account for 77.5% of the
number counts at 1.4GHz for a flux of 1e-4Jy. However, the model does not
explain the CRB measured with the ARCADE2 experiment. Considering the case when
q70 decreases strongly with redshift, this still does not explain the ARCADE2
results. It also yields to an overestimate of the low-flux number counts in the
radio. Thus, we rule out a steep variation of q70 with redshift at least for
z<3.5. Adding a population of faint SF galaxies at high redshift (Lir<1e11Lsun
and 4<z<6), which would reproduce the ARCADE2 results, leads to predictions of
the CIB much higher than what is observed, ruling out this as the explanation
for the ARCADE2 results. Considering our findings and previous studies, we
conclude that if the radio emission measured by ARCADE2 is astrophysical in
origin, it has to originate in the Galaxy or in a new kind of radio sources
(with no mid- to far-IR counterparts) or emission mechanism still to be
discovered.
