Path integral approach for merger rates of dark matter haloes

We use path integrals in order to estimate merger rates of dark matter haloes using the Extended Press-Schechter approximation (EPS) for the Spherical Collapse (SC) and the Ellipsoidal Collapse (EC) models. Merger rates have been calculated for masses in the range $10^{10}M_{\odot}\mathrm{h}^{-1}$ to $10^{14}M_{\odot}\mathrm{h}^{-1}$ and for redshifts $z$ in the range 0 to 3. A detailed comparison between these models is presented. Path approach gives a better agreement with the exact solutions for constrained distributions than the approach of \cite{shto02}. Although this improvement seems not to be very large, our results show that the path approach is a step to the right direction. Differences between the two widely used barriers, spherical and ellipsoidal, depend crucially on the mass of the descendant halo. These differences become larger for decreasing mass of the descendant halo. The use of additional terms in the expansion used in the path approach, other improvements as well as detailed comparisons with the predictions of N-body simulations, that could improve our understanding about the important issue of structure formation, are under study.