Multiplicity Functions of Dark Matter Haloes from Fractional Brownian Motion

We studied the formation of dark matter haloes assuming a Fractional Brownian Motion (FBM) of trajectories in the plane ( S , δ ). S is the variance of the smoothed overdensity δ . The values of δ are calculated using a filter of radius R that defines a mass scale M , and thus S is considered as a function of mass M . Focusing on a specific point of the initial Universe and calculating δ as a function of the radius R of the filter a random walk on the plane ( S , δ ) is established. If the above evolution is a FBM motion then there exist correlations between various mass scales. These correlations depend on a Hurst exponent H . Various mass scales are not correlated for H =1 and the evolution is reduced to the classical Brownian motion. Following Zhang and Hui (Astrophys. J. 641:641, 2006 ), we constructed a Volterra integral equation for the distribution of trajectories after a first crossing of an ellipsoidal barrier. The integral equation is solved numerically and multiplicity functions of dark matter haloes were derived and compared to the results of N-body simulations. Our results show that for H =1.05 the resulting multiplicity functions are in excellent agreement, much better than the agreement predicted by any other model, with the predictions of N-body simulations. This shows that FBM model is a very promising tool for the study of the formation of structures in the Universe.