  In our previous work, we derive the CR-IGIMF: a new scenario for a variable
stellar initial mass function (IMF), which combines numerical results on the
role played by cosmic rays in setting the thermal state of star forming gas,
with the analytical approach of the integrated galaxy-wide IMF. In this work,
we study the implications of this scenario for the properties of local
Early-Type galaxies (ETG), as inferred from dynamical, photometric and
spectroscopic studies. We implement a library of CR-IGIMF shapes in the
framework of the Galaxy Evolution and Assembly (GAEA) model. Our realization
includes a derivation of synthetic spectral energy distribution for each model
galaxy, allowing a direct derivation of the mass fraction in the mean IMF of
low-mass stars (i.e. the dwarf-to-giant ratio - $\rm f_{dg}$) and a comparison
with IMF sensitive spectral features. The predictions of the GAEA model
implementing the CR-IGIMF confirm our previous findings: it correctly
reproduces both the observed excess of z$\sim$0 dynamical mass (mass-to-light
ratios) with respect to spectroscopic (photometric) estimates assuming a
universal, MW-like, IMF, and the observed increase of [$\alpha$/Fe] ratios with
stellar mass in spheroidal galaxies. Moreover, this realization reproduces the
increasing trends of $\rm f_{dg}$, and IMF-sensitive line-strengths with
velocity dispersion, although the predicted relations are significantly
shallower than the observed ones. Our results show that the CR-IGIMF is a
promising scenario that reproduces at the same time dynamical, photometric and
spectroscopic indications of a varying IMF in local ETGs. The shallow relations
found for spectral indices suggest that either a stronger variability as a
function of galaxy properties or additional dependences (e.g. as a function of
star forming gas metallicity) might be required to match the strength of the
observed trends.
