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Introductory caveat: the following is my first attempt, in this forum, to summarise in a few hundred words something as general and technical as the material in the Millennium Simulation paper. It will inevitably contain inaccuracies, over-simplifications, and so on. However, I hope it will not be too misleading or distort the relevant physics and models too much. I would very much like sol invictus, ben m, and zosima, who have (I think) indicated that they are professional physicists, and MattusMaximus, who has (I think) indicated that he is a science teacher, to comment on it, pointing especially to parts that they think are wrong, seriously misleading, or badly incomplete.
There are three domains at play in the part of the paper we are discussing, physics, mathematics, and computing science (numerical simulation). They are, of course, closely inter-woven, so it is not always easy to talk about each separately … but I'll try. In this post, I will not address the simulation aspects; if anyone's interested, I'd be happy to have a go at doing so later (or maybe someone else would like to?)
The physics which is explicitly used in the paper is Newton's three laws of motion, Newtonian gravity, and General Relativity (GR). Since Newton's day, a great deal of work has been done to express the three laws, and Newtonian gravity, in more general mathematical forms; three conservation laws thus encompass the three laws of motion – conservation of energy, conservation of momentum, and conservation of angular momentum.
There is a long tradition of how these parts of physics (and others, of course) are expressed in symbolic form, such as the use of the symbol
t for time, and the use of
bold to mean a vector quantity; this paper follows convention, making it relatively easy to connect what's in the paper to earlier work on systems of particles interacting via gravity (due to their mass), and to earlier work on universes where GR rules. This earlier work can be found in the papers explicitly cited, and in standard physics textbooks.
So, what this simulation does is start with a lot of 'point masses' (particles) - ~10 billion of them – distributed throughout a cube in a particular way (the
xi at
t = 0) and with particular velocities (the
xi with a dot over the
x at
t = 0). The
i-th particle has a mass of
mi.
The particles are cold, dark matter (CDM): 'dark' because the only way they interact with each other is via gravity due to their mass; 'cold' meaning that none ever has a speed large enough to require (special) relativity (SR) to describe their motion (another way to say this is that the simulation does not include the physics of SR).
The CDM particles interact with each other, and the chunk of the universe 'evolves'. The only interactions are each particle's gravitational effect on every other particle.
How does GR enter into the simulation? That’s "the background cosmology". If you assume an isotropic, homogeneous universe in which GR rules, you have the Friedmann-Lemaître-Robertson-Walker (FLRW) metric as an exact solution to the Einstein field equations (of GR). That’s also why
xi are called the 'comoving coordinate vectors', and that's how the scale factor (
a(t)) enters the picture.
How the model represents – accurately - the effect of the gravity of each of the particles on all the others, and how all those particles move – accurately - as a result, as this chunk of Friedmann-Lemaître model universe evolves (
t increases), is the numerical simulation part; needless to say, there are quite a lot of really, really neat things involved in this!
