Environmental dependence in the ellipsoidal collapse model

N-body simulations have demonstrated a correlation between the properties of haloes and their environment. In this paper, we assess whether the ellipsoidal collapse model, whose dynamics includes the tidal shear, can produce a similar dependence. First, we explore the statistical correlation that originates from Gaussian initial conditions. We derive analytic expressions for a number of joint statistics of the shear tensor and estimate the sensitivity of the local characteristics of the shear to the global geometry of the large-scale environment. Next, we concentrate on the dynamical aspect of the environmental dependence using a simplified model that takes into account the interaction between a collapsing halo and its environment. We find that the tidal force exerted by the surrounding mass distribution alters the axes collapse and causes haloes embedded in overdense regions to virialize earlier. The environment density is the key parameter in determining the virialization redshift, while the environment asphericity primarily contributes to the increase in the scatter of the critical collapse density. An effective density threshold whose shape depends on the large-scale density provides a good description of this environmental effect. Such an interpretation has the advantage that the excursion set formalism can be applied to quantify the environmental dependence of halo properties. We show that, using this approach, a correlation between formation redshift, large-scale bias and environment density naturally arises. The strength of the effect is comparable, albeit smaller, to that seen in simulations. It is largest for low-mass haloes (M << M-star), and decreases as one goes to higher mass objects (M > M-star). Furthermore, haloes that formed early are substantially more clustered than those that assembled recently. On the other hand, our analytic model predicts a decrease in median formation redshift with increasing environment density, in disagreement with the trend detected in overdense regions. However, our results appear consistent with the behaviour inferred in relatively underdense regions. We argue that the ellipsoidal collapse model may apply in low-density environments where non-linear effects are negligible.