Spherical collapse in DHOST theories and EFT of dark energy

We study the nonlinear evolution of matter overdensities using the spherical collapse model in degenerate higher-order scalar-tensor (DHOST) theories beyond Horndeski, employing the effective field theory (EFT) of dark energy approach. We investigate the impact of the EFT parameters characterising DHOST theories on the formation of large-scale structure. We identify the parameter space in which the collapse of the spherical overdensity is prevented by the scalar field turning imaginary at some moment, which allows us to place constraints on the model parameters. We show how the collapse time and the critical density contrast depend on the EFT parameters. To assess the observational implications, we compute the halo mass function using the Press-Schechter formalism. We find that the number density of halos is suppressed compared to the $\Lambda$CDM model due to ``beyond Horndeski'' effects, upon imposing the stability of linear perturbations.