Equations and dimensionless variables

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$\dot{\mathbf}$x we write the equations for particle momenta p :

 

This choice of ``velocity'' simplifies the equations of motion by removing a few terms with $\dot{a}$/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:

 

$${d\mathbf{p}\over da} {\nabla\phi\over\dot a} {d\mathbf{x}\over da} {\mathbf{p}\over \dot a a^2}$$ (2)

 

$$\nabla^{2}_{} \phi \pi \Omega_{\rm m}^{} \rho_{\rm cr}^{} \delta \pi \Omega_{0}^{} \rho_{\rm cr,0}^{} {\delta\over a} \delta \equiv {\rho(\mathbf{x})-\rho_b\over\rho_b}$$ (3)

 

$$\dot{a} \sqrt{a} \sqrt{\Omega_0 +\Omega_{\rm curv,0}a +\Omega_{\Lambda,0}a^3} \Omega_{0}^{} \Omega_{\rm curv,0}^{} \Omega_{\Lambda,0}^{}$$ (4)

where $\Omega_{0}^{}$ = $\Omega_{m}^{}$(z = 0) , $\Omega_{\rm curv,0}^{}$ , and $\Omega_{\Lambda,0}^{}$ 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 x₀ and by measuring the time in units of 1/H₀ :

 

 

 

$$\rho {\tilde\rho\over a^3} {3H_0^2\over 8\pi G} \Omega_{0}^{}$$ (7)

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

 

$${d\tilde\mathbf{p}\over da} \tilde{\nabla} \tilde{\phi} {d\tilde\mathbf{x}\over da} {\tilde\mathbf{p}\over a^2}$$ (8)

 

$$\tilde{\nabla}^{2}_{} \tilde{\phi} {3\over 2} {\Omega_0\over a} \tilde{\rho}$$ (9)

where

 

$$\equiv \dot{a} \left({ \Omega_0 +\Omega_{\rm curv,0}a+\Omega_{\Lambda,0}a^3} \over a \right)^$$ (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 $\Omega_{0}^{}$ of the critical density, then the transformations from dimensionless variables given by the code to dimensional variables are given by

 

$$\tilde{x} {L\over N_{\rm grid}} \left({L_{\rm Mpc}\over N_{\rm grid}/128}\right),$$ (11)

 

$${\tilde p\over a} {\rm km\over s} \cdot {\tilde p\over a} \cdot {L_{\rm Mpc}h\over N_{\rm grid}/128}$$ (12)

 

$$\cdot \Omega_{0}^{} \rho_{\rm cr,0}^{} \left({L\over N_{\rm row}}\right)^ \cdot \Omega_{0}^{} \left({L_{\rm Mpc}\over N_{\rm row}/128}\right)^$$ (13)