We study the clustering properties of galaxy clusters expected to be observed by various forthcoming surveys both in the X-ray and sub-mm regimes by the thermal Sunyaev-Zeldovich effect. Several different background cosmological models are assumed, i
ncluding the concordance $Lambda$CDM and various cosmologies with dynamical evolution of the dark energy. Particular attention is paid to models with a significant contribution of dark energy at early times which affects the process of structure formation. Past light cone and selection effects in cluster catalogs are carefully modeled by realistic scaling relations between cluster mass and observables and by properly taking into account the selection functions of the different instruments. The results show that early dark-energy models are expected to produce significantly lower values of effective bias and both spatial and angular correlation amplitudes with respect to the standard $Lambda$CDM model. Among the cluster catalogues studied in this work, it turns out that those based on emph{eRosita}, emph{Planck}, and South Pole Telescope observations are the most promising for distinguishing between various dark-energy models.
We use semi-analytic modelling of the galaxy-cluster population and its strong lensing efficiency to explore how the expected abundance of large gravitational arcs on the sky depends on $sigma_8$. Our models take all effects into account that have be
en shown to affect strong cluster lensing substantially, in particular cluster asymmetry, substructure, merging, and variations in the central density concentrations. We show that the optical depth for long and thin arcs increases by approximately one order of magnitude when $sigma_8$ increases from 0.7 to 0.9, owing to a constructive combination of several effects. Models with high $sigma_8$ are also several orders of magnitude more efficient in producing arcs at intermediate and high redshifts. Finally, we use realistic source number counts to quantitatively predict the total number of arcs brighter than several magnitude limits in the R and I bands. We confirm that, while $sigma_8sim0.9$ may come close to the known abundance of arcs, even $sigma_8sim0.8$ falls short by almost an order of magnitude in reproducing known counts. We conclude that, should $sigma_8sim0.8$ be confirmed, we would fail to understand the strong-lensing efficiency of the galaxy cluster population, and in particular the abundance of arcs in high-redshift clusters. We argue that early-dark energy or non-Gaussian density fluctuations may indicate one way out of this problem.
