Linear and nonlinear clusterings of Horndeski-inspired dark energy models with fast transition

We analyze time-dependent dark energy equations of state through linear and nonlinear structure formation and their quintessence potentials, characterized by fast, recent transitions, inspired by parameter space studies of Horndeski models. The influence of dark energy on structures comes from modifications to the background expansion rate and from perturbations as well. In order to compute the structures growth, we employ a generalization of the \emph{spherical collapse} formalism that includes perturbations of fluids with pressure. We numerically solve the equations of motion for the perturbations and the field. Our analysis suggests that a true Heaviside step transition is a good approximation for most of the considered models, since most of the quantities weakly depend on the transition speed. We find that transitions occurring at redshifts $z_{\rm t}\gtrsim 2$ cannot be distinguished from the $\Lambda$CDM model if dark energy is freezing, i.e., the corresponding equation of state tends to $-1$. For fast, recent transitions, the redshift at which the properties of dark energy have the most significant effect is $z=0.6\pm 0.2$. We also find that in the freezing regime, the $\sigma_8$ values can be lowered by about $8\%$, suggesting that those models could relieve the $\sigma_8$-tension. Additionally, freezing models generally predict faster late-time merging rates but a lower number of massive galaxies at $z=0$. Finally, the matter power spectrum for smooth dark energy shows a low-wavenumber peak which is absent in the clustering case.