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Vlasov-Poisson in 1D Cosmology: the Particle-Mesh (PM) N-body code Vlafroid

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Authors of the code: S. Colombi & A. Taruya
Publication introducing the code (to be cited when using Vlafroid):

Download current version of Vlafroid: froid1D.1.7.tar.gz (replaces froid1D.1.5.tar.gz)

Table of Contents

The Vlasov-Poisson equations in the cosmological case

To write Vlasov-Poisson equations in the cosmological context, it is convenient to introduce the superconformal dimensionless time τ defined by (Doroshkevich et al. 1972; Martel & Shapiro 1998)

the comoving coordinate x and the normalized peculiar velocity v given by

In these equations, r and u are respectively the usual physical coordinate and physical velocity, a is the expansion factor of the Universe, H the Hubble parameter and H0 the Hubble constant (the value of the Hubble parameter at present time). With this choice of units, Vlasov-Poisson equations remain nearly the same as in physical coordinates:

where Ω0 is the matter density parameter of the Universe and the projected density ρ is normalised such that its average is unity.

The particle-mesh algorithm in 1D

In Vlafroid, we resolve dynamically Vlasov-Poisson equations in the one dimensional case using an ensemble of macro-particles to represent the phase-space density: large-scale structure dynamics is described by the gravitational interaction of massive parallel infinite planes moving left and right along a fixed axis, while Hubble expansion is taking place as usual in all the directions following standard Friedman-Lemaître equations. Each plane is represented by a particle moving along x axis in a periodic box of size L and following the standard Lagrangian equations of motion:

These equations are resolved numerically using a standard predictor-corrector with slowly varying time step (more details can be found in section 5.1 of Taruya & Colombi 2017). To compute the acceleration γ of each particle, we compute the projected density ρ on a grid using Cloud-In-Cell (CIC) interpolation (Hockney & Eastwood 1981) and solve Poisson equation by Fast Fourier Transform. The acceleration is computed on the grid from Fourier derivative of the potential, then interpolated back to the particles using linear interpolation. The calculation could also be performed directly in configuration space, as the acceleration can be directly expressed as

where θ is the Heaviside step function.

Initial conditions

While Vlafroid could be used to simulate warm systems, where the phase-space distribution function is initially a cloud of particles in phase-space, its initial conditions generator only consider the cold case, i.e. the case where initial velocity dispersion is zero. In this framework, the phase-space distribution function is represented in the continuous limit as a non self-intersecting curve moving in phase-space. Here, the curve is sampled by an ensemble of particles initially disposed on x axis and regularly spaced. The positions and velocities of particles are then perturbed using Zel'dovich approximation (i.e. Lagrangian perturbation theory at linear order, Zel'dovich 1970), which is exact in the one dimensional case prior to shell-crossing. In this case, initial positions of particles, designed by the Lagrangian coordinate q, are perturbed as follows,

while initial velocities are given by


where δL=ρL-1 is the initial density contrast linearly extrapolated to present time and D+ is the linear growing mode normalised to unity at present time. Choosing initial conditions consists in specifying δL. Too kind of initials conditions are considered in the public version of the code:

  • the single sine wave (figure 1), with

where the linear amplitude A is chosen by the user;

Figure 1: Evolution of the phase-space distribution function of an initial sine wave. The simulation (red curves) is compared to predictions from post-collapse perturbation theory (blue curves) and Zel'dovich solution (green dots). On each panel, the lower insert shows as well the projected density. From Taruya & Colombi (2017).

  • random initial conditions specified by the power-spectrum P(k) of δL (figure 2). This power-spectrum can be a power-law, or a Cold Dark Matter (CDM) like one (see, e.g., McQuinn & White 2016):

where P3D is the standard 3-dimensional CDM power-spectrum linearly normalised at present time using the so-called quantity σ8, which represents the r.m.s. of the density fluctuations in a sphere of radius 8 Mpc/h, with h=H0/100. To set P3D, we included in Vlafroid routines written by Eisenstein & Hu (1998).

Figure 2: Phase-space distribution function in a CDM like 1D Universe. The simulation (in black) is compared to predictions from post-collapse perturbation theory and Zel'dovich solution as indicated on the panel.

(Post-collapse) Perturbation theory predictions

In addition to numerically solving the actual equations of the dynamics, the code also computes the predictions given by the Zel'dovich approximation. Optionally, it can also compute predictions by various variants of post-collapse perturbation theory, with and without adaptive smoothing of initial conditions, as described in Taruya & Colombi (2017). This will be detailed later in a Theory and analysis item.


Outputs of the code consist of positions and velocities of particles, power-spectrum of the projected density, 2-point correlation function, and this for the simulation, the Zel'dovich solution and other perturbative models if required, as well as various quantities estimated on the computational mesh, such as the gravitational potential, the acceleration, the projected density and velocity moments of the phase-space distribution function.


  1. Doroshkevich A. G., Ryaben'kii V. S., Shandarin S. F., 1973, Astrophysics 9, 144
  2. Eisenstein D. J., Hu W., 1998, ApJ 496, 605
  3. Hockney R. W., Eastwood J. W., 1981, Computer Simulation Using Particles, New York: McGraw-Hill
  4. Martel H., Shapiro P. R., 1998, MNRAS 297, 467
  5. McQuinn M., White M., 2016, J. Cosmology Astropart. Phys. 1601, 043
  6. Zel'dovich Y. B., 1970, A&A 5, 84