  Temperature anisotropies in the Cosmic Microwave Background (CMB) are
affected by the late Integrated Sachs-Wolfe (lISW) effect caused by any
time-variation of the gravitational potential on linear scales. Dark energy is
not the only source of lISW, since massive neutrinos induce a small decay of
the potential on small scales during both matter and dark energy domination. In
this work, we study the prospect of using the cross-correlation between CMB and
galaxy density maps as a tool for constraining the neutrino mass. On the one
hand massive neutrinos reduce the cross-correlation spectrum because
free-streaming slows down structure formation; on the other hand, they enhance
it through their change in the effective linear growth. We show that in the
observable range of scales and redshifts, the first effect dominates, but the
second one is not negligible. We carry out an error forecast analysis by
fitting some mock data inspired by the Planck satellite, Dark Energy Survey
(DES) and Large Synoptic Survey Telescope (LSST). The inclusion of the
cross-correlation data from Planck and LSST increases the sensitivity to the
neutrino mass m_nu by 38% (and to the dark energy equation of state w by 83%)
with respect to Planck alone. The correlation between Planck and DES brings a
far less significant improvement. This method is not potentially as good for
detecting m_nu as the measurement of galaxy, cluster or cosmic shear power
spectra, but since it is independent and affected by different systematics, it
remains potentially interesting if the total neutrino mass is of the order of
0.2 eV; if instead it is close to the lower bound from atmospheric
oscillations, m_nu ~ 0.05 eV, we do not expect the ISW-galaxy correlation to be
ever sensitive to m_nu.
