458 Numerical simulations in cosmology
Figure 15.15.The mass function for distinct halos (top) and for sub-halos bottom). Raw
counts are marked by symbols with error bars. The curves are Schechter-function fits. The
Press–Schechter (dotted) and Sheth–Tormen (dashes) predictions for distinct halos are also
shown. On scales below 10^14 h−^1 M the mass function is close to a power law with slope
α≈− 1 .8. There is no visible difference in the slope for sub-halos and that for distinct
halos. (After Sigadet al2000.)
For each halo one can measure the maximum circular velocityVmax.Inmany
cases (especially for sub-halos)Vmaxis a better measure of the size of the halo.
It is also related more closely with the observed properties of galaxies hosted
by halos. Figure 15.16 presents the velocity distribution functions of different
types of halo. In addition to distinct halos and sub-halos, we also show isolated
halos and halos in groups and clusters. Here isolated halos are defined as halos
with a mass less than 10^13 h−^1 M , which are not inside a larger halo and which
do not have sub-halos more massive than 10^11 h−^1 M. The velocity function is
approximated by a power law dn=∗Vmaxβ dVmaxwith slopeβ≈− 3 .8for
distinct halos. The slope depends on the environment:β≈− 3 .1 for halos in
groups andβ≈−4 for isolated halos. Klypinet al(1999b) and Ghignaet al
(1999) found that the slopeβ≈− 3 .8–4 of the velocity function extends to much
smaller halos with velocities down to 10 km s−^1.