A method of generating initial conditions for cosmological N-body simulations

We investigate the possibility of generating initial conditions for cosmological N-body simulations by simulating a system whose correlations at thermal equilibrium approximate well those of cosmological density perturbations. The system is an appropriately modified version of the standard “one component plasma” (OCP). We show first how a well-known semianalytic method can be used to determine the potential required to produce the desired correlations, and then verify our results for some cosmological type spectra with simulations of the full molecular dynamics. The advantage of the method, compared to the standard one, is that it gives by construction an accurate representation of both the real and reciprocal space correlation properties of the theoretical model. Furthermore the distributions are also statistically homogeneous and isotropic. We discuss briefly the modifications needed to implement the method to produce configurations appropriate for large N-body simulations in cosmology, and also the generation of initial velocities in this context.

Figure 1

Correlation function of the OCP with interacting Coulomb potential for different temperatures (recall that $\Gamma \sim 1/T$).

Figure 2

Power spectrum of the OCP with interacting Coulomb potential. Note that this is a linear-linear plot.

Figure 3

The PS of a $1/r^2$ OCP for two different values of the coupling parameter ${\Gamma }^{\prime }$. Very good agreement is observed between the predictions from the HNC and MD in the range where they overlap. For the weak coupling case the HZ form for the PS $S(k)\propto k$ is clearly evident. The units are normalized to the ionic radius a. Note that the plot is on a linear-linear scale.

Figure 4

The correlation function in real space for the $1/r^2$ OCP for the same cases as in the previous figure. For the smaller coupling one has anticorrelation at all scales ($g(r)<1$) while for the larger coupling one sees, just as in the standard OCP, the correlation appear with the first neighbor (which becomes more localized as the temperature is lowered).

Figure 5

The theoretical PS $P(k)$ and its discretization $S(k)$, for the parameters choices given in the text. They differ as described by the chosen smoothing function $D(k)$, for $ka\gtrsim 1$, with $S(k)$ going asymptotically to $1/n_0$, where $n_0=3/(4\pi a^3)$ is the particle density for the discretization.

Figure 6

The theoretical input two-point correlation function ${\xi }_c(r)$ and that of the corresponding discretization $h(r)$.

Figure 7

The window function in real space $|W_{k_d,n_0}(r)|$.

Figure 8

The correlation function, discretized correlation function, and interaction potential obtained by the inversion of the HNC for a PS as given in Eq. (58), with $\mathcal{N}=20a^4$, $A\approx 0.69a$ $k_c=2.7/a$ and $k_d=1.55/a$. Note that the spikes arise from the changes in sign of the quantities, of which the modulus is plotted in each case.

Figure 9

The PS measured in a simulation of the molecular dynamics of 8788 particles for the potential shown in the previous figure. Also shown is the input PS i.e. the PS of a system at equilibrium with this potential as calculated in the HNC.

Figure 10

The real space correlation function for the same cases as in the previous figure. The abrupt break to anticorrelation at about $r/a\approx 0.25$ comes from the hard core introduced to ensure stability of the system.

Figure 11

Correlation functions and interaction potential obtained for the cosmological CDM spectrum described in the text.