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We extend the analysis of Pace et al., JCAP, 2019, 060, by considering thevirialization process in the extended spherical collapse model for clusteringdark-energy models, i.e., accounting for dark-energy fluctuations. Differentlyfrom the standard approach, here virialization is naturally achieved byproperly modelling deviations from sphericity due to shear and rotation inducedby tidal interactions. We investigate the time evolution of the virialoverdensity $\Delta_\mathrm{vir}$ in seven clustering dynamical dark energymodels and compare the results to the $\Lambda$CDM model and to thecorresponding smooth dark-energy models. Taking into account all theappropriate corrections, we deduce the abundance of convergence peaks for RubinObservatory-LSST and Euclid-like weak-lensing surveys, of Sunyaev-Zel'dovichpeaks for a Simon Observatory-like CMB survey, and of X-ray peaks for aneROSITA-like survey. Despite the tiny differences in $\Delta_\mathrm{vir}$between clustering and smooth dark-energy models, owing to the large volumescovered by these surveys, five out of seven clustering dark-energy models canbe statistically distinguished from $\Lambda$CDM. The contribution ofdark-energy fluctuation cannot be neglected, especially for theChevallier-Polarski-Limber and Albrecht-Skordis models, provided theinstrumental configurations provide high signal-to-noise ratio. These resultsare almost independent of the tidal virialization model.

Tidal virialization of dark matter haloes with clustering dark energy