Evolution of cosmological simulations over the last 50 years


I recently scanned the literature for the purpose of following and plotting the number of particles used in \(N\)-body simulations over the last five decades.

Direct summation simulations

Cosmological simulations started with as few as 100 particles just before 1970. Until particle-mesh methods became popular in the 1980s, "direct summation", i.e. explicit computation of all gravitational interactions between pairs of particles, was the preferred method [1-7]. Direct summation is now mostly used in the context of simulating star clusters or the Galactic centre, with detailed interactions between stellar systems and the central supermassive black hole [30,36,45,56]. The number of particles used in some of these simulations is presented in figure 1.

Number of particles in direct summation simulations as a function of timeFigure 1: Evolution of the number of particles used in direct \(N\)-body simulations as a function of year of publication. The green line shows the trend as compared to Moore's law for hardware performance (black line).

Direct \(N\)-body methods scale as \(\mathcal{O}(N^2)\). For this reason, despite the enhancement provided by the use of accelerators such as GPUs [see e.g. 47], it is understandable that the number of particles does not follow Moore's law. A linear regression of \(\log N\) shows that \(N\) increases by around 18% per year, as compared to 59% per year for computer chip performance, which roughly doubles every 18 months according to Moore's law.

Sofware improvements

In order to circumvent the problem of the long-range nature of gravitational interactions, much of the work on numerical cosmology since 1980 has focused on algorithms (such as mesh, tree, and multipole methods) that reduce the need for communications across the full computational volume. The number of particles in some \(N\)-body simulations run using such techniques is plotted in figure 2, with the symbol and colour indicating the technique employed: particle-particle-particle-mesh (P3M) and adaptive P3M (AP3M); parallel or vectorized P3M; Tree; TreePM; and particle-mesh with adaptive mesh refinement (PM AMR).

Number of particles in direct summation simulations as a function of timeFigure 2: Evolution of the number of particles used in \(N\)-body simulations as a function of year of publication. The symbols and colours indicate the technique used for gravity. Hydrodynamic simulations are represented in a black square.

Because of software improvements the number of particles in cosmological simulations has increased much more rapidly than with the direct summation method. Since 1990, a super-exponential trend can further be observed for gravity-only simulations (it is illustrated by a quadratic regression of \(\log N\) in figure 2). It cannot be explained only by the increase in computer speed.

Downloading, reproducing, and citing these plots

The latest versions of the above plots are available to download from a Github repository made public along with this blog post:

  • Main plot, with all the data points: [pdf] [png]
  • Direct summation data points only: [pdf] [png]
  • Direct summation data points, linear regression, and Moore's law: [pdf] [png]
  • All data points and Moore's law: [pdf] [png]
  • All data points, Moore's law, and quadratic regression of gravity-only data points: [pdf] [png]

These plots can be reproduced using this Jupyter notebook. The references for all of the data points are listed below, and the latest bibliography is also provided in an ASCII file which can be easily loaded with python:

import numpy as np
A=np.loadtxt("references.txt",dtype={'names': ('id','authors', 'year', 'N', 'physics',
'method', 'ref'), 'formats': ('i4','S1000', 'i4', 'i8', 'S8', 'S8', 'S1000')},
delimiter=" ; ")

You can use all of these products freely, but giving credit, for instance by citing this blog post ( and/or the address of the companion Github repository (, would be appreciated. Everything is provided under the GNU General Public License version 3.

To submit corrections or updates, please do not hesitate to contact me using my usual email address, or to fork the Github repository and create pull requests.


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