The equations of motion of particles in expanding
coordinates, which are used in our PM code, were presented by Kates
et al.(1991). Different numerical effects (including resolution) were
discussed in Klypin et al.(1996). We use comoving
coordinates ofr particles
**x** = **x** (*t*) , which are related to
proper coordinates by
**r** = *a*(*t*)**x** , where
*a*(*t*) = (1 + *z*)^{- 1} is
the expansion parameter. Instead of using peculiar velocity
**v** _{pec} = *a**x* we write the equations for particle momenta
**p** :

This choice of ``velocity''
simplifies the equations of motion by removing a few terms with /*a* . It
is also convenient to change the time variable from time *t* to the
expansion parameter *a* . The equations governing the motion of particles
are:

= - , = | (2) |

= 4G(t)a^{ 2}(t) = 4G, ,
| (3) |

= H_{0}, + + = 1,
| (4) |

where
= (*z* = 0) ,
, and
are the density of the matter, effective
density of the curvature, and the cosmological constant in units of the
critical density at *z* = 0 . The curvature contribution is positive for
negative curvature.

Dimensionless variables (shown with tildas below) are defined by
introducing the length of a cell of the grid *x*_{0} and by measuring
the time in units of 1/*H*_{0} :

= | (7) |

Equations (2-5) can be rewritten in terms of dimensionless variables:

= - F(a), = F(a)
| (8) |

= ( - 1), | (9) |

F(a) H_{0}/ = -1/2 .
| (10) |

Equations (8 - 9) are solved numerically by the PM code.

If *L* is the length of the computational box at *z* = 0 ,
*N*_{grid} is the
number of grid cells in one direction, and *N*_{row} is the number
of particles in one direction, which contribute a fraction of
the critical density, then the transformations from dimensionless variables
given by the code to dimensional variables are given by

x = x_{0}, x_{0} = = 7.8kpc
| (11) |

v_{pec} = (x_{0}H_{0}) = 0.781 ,
| (12) |

Mass = N_{particles} m_{1}, m_{1} = 3 = 1.32 10^{5}(h^{ 2})3
| (13) |