Clustering dark energy imprints on cosmological observables of the
  gravitational field
  We study cosmological observables on the past light cone of a fixed observer
in the context of clustering dark energy. We focus on observables that probe
the gravitational field directly, namely the integrated Sachs-Wolfe and
non-linear Rees-Sciama effect (ISW-RS), weak gravitational lensing,
gravitational redshift and Shapiro time delay. With our purpose-built $N$-body
code "$k$-evolution" that tracks the coupled evolution of dark matter particles
and the dark energy field, we are able to study the regime of low speed of
sound $c_s$ where dark energy perturbations can become quite large. Using ray
tracing we produce two-dimensional sky maps for each effect and we compute
their angular power spectra. It turns out that the ISW-RS signal is the most
promising probe to constrain clustering dark energy properties coded in
$w-c_s^2$, as the $\textit{linear}$ clustering of dark energy would change the
angular power spectrum by $\sim 30\%$ at low $\ell$ when comparing two
different speeds of sound for dark energy. Weak gravitational lensing, Shapiro
time-delay and gravitational redshift are less sensitive probes of clustering
dark energy, showing variations of a few percent only. The effect of dark
energy $\textit{non-linearities}$ in all the power spectra is negligible at low
$\ell$, but reaches about $2\%$ and $3\%$, respectively, in the convergence and
ISW-RS angular power spectra at multipoles of a few hundred when observed at
redshift $\sim 0.85$. Future cosmological surveys achieving percent precision
measurements will allow to probe the clustering of dark energy to a high degree
of confidence. Clear evidence for clustering dark energy at any scale would
rule out a cosmological constant as the leading contender for dark energy, and
may help to distinguish between different dark energy models.
