Spherical collapse of dark matter haloes in tidal gravitational fields

We study the spherical collapse model in the presence of external gravitational tidal shear fields for different dark energy scenarios and investigate the impact on the mass function and cluster number counts. While previous studies of the influence of shear and rotation on $\delta_\mathrm{c}$ have been performed with heuristically motivated models, we try to avoid this model dependence and sample the external tidal shear values directly from the statistics of the underlying linearly evolved density field based on first order Lagrangian perturbation theory. Within this self-consistent approach, in the sense that we restrict our treatment to scales where linear theory is still applicable, only fluctuations larger than the scale of the considered objects are included into the sampling process which naturally introduces a mass dependence of $\delta_\mathrm{c}$. We find that shear effects are predominant for smaller objects and at lower redshifts, i. e. the effect on $\delta_\mathrm{c}$ is at or below the percent level for the $\Lambda$CDM model. For dark energy models we also find small but noticeable differences, similar to $\Lambda$CDM. The virial overdensity $\Delta_\mathrm{V}$ is nearly unaffected by the external shear. The now mass dependent $\delta_c$ is used to evaluate the mass function for different dark energy scenarios and afterwards to predict cluster number counts, which indicate that ignoring the shear contribution can lead to biases of the order of $1\sigma$ in the estimation of cosmological parameters like $\Omega_\mathrm{m}$, $\sigma_8$ or $w$.