  The late integrated Sachs-Wolfe (ISW) effect correlates the Cosmic Microwave
Background (CMB) temperature anisotropies with foreground cosmic large-scale
structures. As the correlation depends crucially on the growth history in the
era of dark energy, it is a key observational probe for constraining the
cosmological model. Here we present a detailed study based on full-sky and deep
light cones generated from very large volume numerical N-body simulations,
which allow us to avoid the use of standard replica techniques, while capturing
the entirety of the late ISW effect on the large scales. We post-process the
light cones using an accurate ray-tracing method and construct full-sky maps of
the ISW temperature anisotropy for three different dark energy models. We
quantify in detail the extent to which the ISW effect can be used to
discriminate between different dark energy scenarios when cross-correlated with
the matter distribution or the CMB lensing potential. We also investigate the
onset of non-linearities, the so-called Rees-Sciama effect which provides a
complementary probe of the dark sector. We find the signal of the
lensing-lensing and ISW-lensing correlation of the three dark energy models to
be consistent with measurements from the Planck satellite. Future surveys of
the large-scale structures may provide cross-correlation measurements that are
sufficiently precise to distinguish the signal of these models. Our methodology
is very general and can be applied to any dark energy or modified gravity
scenario as long as the metric seen by photons can still be characterized by a
Weyl potential.
