The fractional Brownian motion and the halo mass function

The 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 mass function (PS) is a special case when $\alpha=1/2$. Although the new mass function is computed assuming spherical collapse, comparison with the Sheth-Tormen fitting function (ST) shows that the new mass function of $\alpha\approx 0.435$ agrees with ST remarkably well in high mass regime, while predicts more small mass halos than the ST but less than the PS. The index $\alpha$ is the Hurst exponent, which exact value in context of structure formation is modulated by properties of the smoothing window function and the shape of power spectrum. It is conjectured that halo merging rate and merging history in the fractional set theory might be imprinted with the interplay between halos at small scales and their large scale environment. And the mass function in high mass regime can be a good tool to detect the non-Gaussianity of the initial density fluctuation.