Halo abundance and properties

The linear density fluctuation field is assumed to be Gaussian. The Press-Schechter (1974) assumption states that the probability that a random mass element is part of a dark halo of mass exceeding M at a given redshift z is just twice the probablity that a surrounding sphere of mass M in the initial conditions has a linearly extrapolated overdensity greater than 1.686 at that redshift. Although its physical basis is still unclear, the assumption has been found to describe the simulation results remarkably well. This assumption enables the calculation of the halo abundance (co-moving number density) of a given mass at a given redshift directly from the Gaussian distribution. The result is:

n(M,z)dM = -(2/)1/2(/M)(/)(dln/dlnM)e-(/2)2(dM/M),

where is the current mean density, =1.686(1+z) is the overdensity just collapsing at z linearly extrapolated to present time, and the variance within the given scale is determined by the CDM fluctuation power spectrum, especially by its normalization where the bias factor, which describes how galaxy distribution deviates from that of the dark matter, is also involved. For a nice description of the form and the normalization of the CDM spectrum, see Peebles (1993), section 25, "The Cold Dark Matter Model".

Another statistical approach to obtain the halo abundance is the Peaks Formalism by Bardeen et al. (1986).

An isothermal model for the dark halo density profile is found to describe the simulation results well. This means that the size, the mass, the variance , the circular velocity, and the virial temperature are identical variables for a given halo and CDM spectrum. More recent modelings follow the Navarro, Frenk, White (NFW) profile obtained from high-resolution simulations.

References page
Introduction page

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Fan Fang       Last update: 26-July-99