Fractional Brownian motion and the halo mass function

Fractional Brownian motion with index alpha is introduced to construct the fractional excursion set model. A new mass function with single parameter alpha is derived within the formalism, of which the Press-Schechter (PS) mass function is a special case when alpha=1/2. Although the new mass function is computed assuming spherical collapse, comparison with the Sheth-Tormen (ST) fitting function shows that the new mass function of alpha approximate to 0.435 agrees with the ST function remarkably well in the high-mass regime, while it predicts more low-mass haloes than the ST function but less than the PS function. The index alpha is the Hurst exponent, the exact value of which in the context of structure formation is modulated by properties of the smoothing window function and the shape of the power spectrum. It is conjectured that the halo merging rate and merging history in the fractional set theory might be imprinted with the interplay between haloes at small scales and their large-scale environment. Also, the mass function in the high-mass regime can be a good tool to detect the non-Gaussianity of the initial density fluctuation.