Final Report Summary - EDECS (Exploring Dark Energy through Cosmic Structures: Observational Consequences of Dark Energy Clustering) The universe is currently undergoing a phase of accelerated expansion driven by an exotic form of matter, dubbed “dark energy”. This makes up to 70% of the cosmic energy budget and, unlike standard matter, it is characterized by a negative pressure that causes the cosmic expansion to accelerate. A simple Cosmological Constant in Einstein’s equation of General Relativity can account for this phenomenon. However, the unnatural discrepancy between theoretical expectations and the measured value of the constant may point to a complete different origin. To date the nature of dark energy remains unknown. A key question to address is whether this phenomenon is dynamical or time-independent. In other words, whether the equation of state given by the ratio of the pressure to the dark energy density is a constant set to the Cosmological Constant value or a function of cosmic time. In the latter case dark energy may be the manifestation of a new form of matter not described by the Standard Model of particle physics, or a sign of departure from General Relativity on cosmic scales. Observations of the distribution of matter in the universe may provide key information, since a dynamical dark energy phenomenon can alter the late time clustering of matter and leave a characteristic imprint on the formation and evolution of cosmic structures. The goal of the EDECS project has been to investigate such imprints through the realization and analysis of large volume high-resolution N-body simulations of non-standard dynamical dark energy models. In particular, we have investigated the relation between the underlying properties of dark energy and the clustering of matter well into the non-linear regime of gravitational collapse. The upcoming generation of galaxy surveys aims to constrain dark energy scenarios through accurate mapping of the distribution of structures across a wide range of scales and redshifts. These observations will be sensitive to the non-linearity of the clustering of matter, thus it will necessary to accurately model these effects to perform unbiased data analysis. The studies we have carried out during the EDECS project have shown that indeed non-linear effects carry cosmological information on the dynamical nature of dark energy. As an example, non-linearities alter the cosmic distance information encoded in the Baryon Acoustic Oscillations signal in the matter power spectrum. The amplitude and redshift evolution of such distortions depends on the underlying properties of dark energy. On the one hand, the detection of such effects can provide additional information on dark energy. On the other hand, such non-linearities induce mode correlations that need to be taken. To this purpose we have provided unprecedented accurate evaluations of the power spectrum covariance through the realization of the largest suite of cosmological N-body simulations. However, dark energy leaves a signature not only on the cosmic matter distribution, but also on the late time formation and evolution of dark matter halos, which today host galaxies and cluster of galaxies. We have shown that such information can be accessed through measurements of the internal mass distribution of galaxy clusters as well as the statistical sampling of the extreme pairwise velocities of colliding clusters. The late time abundance of dark matter halos, their internal structures and the hierarchical halo formation processes through mergers of smaller mass halos play a fundamental role in the formation of galaxies. As dark energy affects these processes through the gravitational interaction with matter, it is natural to ask whether a trace of dark energy properties are also imprinted in the late time galaxy formation. This is a new direction of investigation emerged from the finalization of the EDECS project and which we hope to address further in future studies.