We construct merger trees from the largest database of dark matter haloes to date provided by the Millennium simulation to quantify the merger rates of haloes over a broad range of descendant halo mass ( 10 ^ { 12 } \la M _ { 0 } \la 10 ^ { 15 } M _ { \odot } ) , progenitor mass ratio ( 10 ^ { -3 } \la \xi \leq 1 ) , and redshift ( 0 \leq z \la 6 ) .
We find the mean merger rate per halo , B / n , to have very simple dependence on M _ { 0 } , \xi , and z , and propose a universal fitting form for B / n that is accurate to 10-20 % .
Overall , B / n depends very weakly on the halo mass ( \propto M _ { 0 } ^ { 0.08 } ) and scales as a power law in the progenitor mass ratio ( \propto \xi ^ { -2 } ) for minor mergers ( \xi \la 0.1 ) with a mild upturn for major mergers .
As a function of time , we find the merger rate per Gyr to evolve roughly as ( 1 + z ) ^ { n _ { m } } with n _ { m } = 2 - 2.3 , while the rate per unit redshift is nearly independent of z .
Several tests are performed to assess how our merger rates are affected by , e.g .
the time interval between Millennium outputs , binary vs multiple progenitor mergers , and mass conservation and diffuse accretion during mergers .
In particular , we find halo fragmentations to be a general issue in merger tree construction from N -body simulations and compare two methods for handling these events .
We compare our results with predictions of two analytical models for halo mergers based on the Extended Press-Schechter ( EPS ) model and the coagulation theory .
We find that the EPS model overpredicts the major merger rates and underpredicts the minor merger rates by up to a factor of a few .