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.