Anomalous diffusion models for the formation of dark matter haloes

Abstract

We studied the formation of dark matter haloes in terms of the excursion set model in the case of Gaussian density perturbations. We used a more general form for the autocorrelation function of overdensities than that corresponds to a k-sharp filter and applying a path integral approach we constructed first crossing distributions, and consequently mass functions of dark matter haloes. The resulting first crossing distributions depend on a Hurst exponent H. Thus, normal diffusion models (H = 1), subdiffusion models with (H < 1) and superdiffusion models with (H > 1) are predicted. We found values of the parameters so that all models approximate very well the multiplicity functions resulting from N-body simulations. Finally, we calculated distributions of formation redshifts and compared them with the predictions of N-body simulations too. We found that the above models are not able to predict better fits for the distributions of formation redshifts than other well-known analytical models. Thus, the question about the existence of an analytical model that can satisfactorily reproduce both multiplicity functions and formation times distributions derived in N-body simulations remains unanswered.