We use the extensive catalog of dark matter haloes from the Millennium simulation to investigate the statistics of the mass accretion histories ( MAHs ) and accretion rates of \sim 500 , 000 haloes from redshift z = 0 to 6 .
We find only about 25 % of the haloes to have MAHs that are well described by a 1-parameter exponential form .
For the rest of the haloes , between 20 % ( Milky-Way mass ) to 50 % ( cluster mass ) experience late-time growth that is steeper than an exponential , whereas the remaining haloes show plateau-ed late-time growth that is shallower than an exponential .
The haloes with slower late-time growth tend to reside in denser environments , suggesting that either tidal stripping or the “ hotter ” dynamics are suppressing the accretion rate of dark matter onto these haloes .
These deviations from exponential growth are well fit by introducing a second parameter : M ( z ) \propto ( 1 + z ) ^ { \beta } e ^ { - \gamma z } .
The full distribution of \beta and \gamma as a function of halo mass is provided .
From the analytic form of M ( z ) , we obtain a simple formula for the mean accretion rate of dark matter , \dot { M } , as a function of redshift and mass .
At z = 0 , this rate is 42 M _ { \odot } { yr } ^ { -1 } for 10 ^ { 12 } M _ { \odot } haloes , which corresponds to a mean baryon accretion rate of \dot { M } _ { b } = 7 M _ { \odot } { yr } ^ { -1 } .
This mean rate increases approximately as ( 1 + z ) ^ { 1.5 } at low z and ( 1 + z ) ^ { 2.5 } at high z , reaching \dot { M } _ { b } = 27 , 69 , and 140 M _ { \odot } { yr } ^ { -1 } at z = 1 , 2 , and 3 .
The specific rate depends on halo mass weakly : \dot { M } / M \propto M ^ { 0.127 } .
Results for the broad distributions about the mean rates are also discussed .