Mostly taken from Andrey’s PM notes

In proper coordinates, gravitational potential and equations of motion for particles:

time and spatial derivatives are also with respect to proper coordinates.

Now we change to comoving variables and make them dimensionless:

where , is the peculiar velocity and is the peculiar potential defined as (Peebles 1980)

where

To make the variables dimensionless we define:

Now using the scale factor as the time variable, the Poisson equation and the equations of motion are:

here is the overdensity in comoving coordinates and is given by:

In dimensionless variables these equations become:

where and

To do the time evolution of the DM particles, these last equations are used in the three main steps of the PM code:

  • Solve the Poisson equation using the density field estimated with the current particle positions.

  • Advance the momentum of the particles using the new potential.

  • Update particle positions using the new momenta ( Leap-Frog scheme ).