The Dark Matter is Out There

In the last 10 years evidence piled up that the largest part of the Universe is not made by ordinary (baryonic) matter. Atoms and neutrinos account to about 5%, while the remaining fraction is partly due to another form of mass that does not interact electromagnetically, called Dark Matter (25%). There are indications that it is made by a yet-to-be-discovered non baryonic species that does not interact electromagnetically, generally known as weakly-interactive massive particles (WIMPs).
The remaining part is explained by the vacuum energy, called Dark Energy, that adds up to about 70%.

In this post we will explore that main reasons behind the Dark Matter hypothesis. We will simulate a typical galaxy cluster with the help of Processing and the Traer Physics library. You can download the PDE from my google drive space (remember to install the library first!). For another application, take a look at my previous post about the discovery of Neptune.

The first evidence of the existence of some sort of matter that does not emit light is due to Zwicky (1933). While studying the gravitational stability of the Coma galaxy cluster, the astronomer measured an inconsistency between the average velocity of the galaxies and the hypothesis that the mean field potential was only due to gravity alone – we still have good reasons to believe the latter to be true. The measurement hinges on the virial theorem, that states that the time average of the kinetic energy of a system is twice the time average potential:

2 <T> = <V>

We know that T = 1/2 M v^2 and in the following we suppose that V = -G m_i m_j / r_ij^2 . To simplify a bit the calculation, we have to make some additional hypothesis. First, we do not make a gross error if we take that the cluster mass is uniformly distributed in a sphere of radius R around the center of mass.
Eventually, the cluster mass can be estimated by measuring the average velocity and the cluster radius:

GM/R ~ v^2

Zwicky discovered that the measured mass in the Coma cluster was about 100 times than expected – the visible mass can be estimated by the total emitted light.
In the 70s, simulations performed by Peebles & Ostriker, and observations made by Ford & Rubin pointed out that the velocity distribution of stars as a function of the distance from the center of spiral galaxies are not consistent with the “keplerian” hypothesis. In fact, assuming that most of the mass is confined in the bulge at the center of the galaxy, the radial average velocity should drop as r^-{1/2}. Observations showed that in fact it remains practically constant. How is that possible? The simplest solution is that there is more mass distributed uniformly across the galaxy, but this additional mass cannot be observed by light alone.
Between 2005-2010, a large scale N-body simulation of the evolution of the Universe called Millennium Run (Springel et al.) suggested that without Dark Matter galaxies do not even form!

A laptop-scale simulation

With the aid of Processing, we will try to reproduce the impact of Dark Matter on a cluster of 50 galaxies. As first step, we generate a cluster with no dark matter. The mass of each galaxy is set to 1 in unit of Milky Way’s mass, with a random position that follows a gaussian distribution with standard deviation equal to the cluster radius R. The initial velocity of each galaxy is distributed as a gaussian with average:

<v>^2 = G m / R

While running the simulation, we can measure the potential exponent parameter n

n = -2\frac{<T>}{V}

and the cluster virial mass as defined before.

As you can see, after few iterations the exponent approaches -1, as expected with a newtonian potential, and the average mass is in the order of 1, the “injected” galaxy mass. As soon as these number become stable, the system is said to be virialized, i.e. in equilibrium. Actually, you can see a slight drift towards lower values of n and M: this means that the system is slowly “evaporating”.

Enter Dark Matter

We now set up the cluster so that more objects are present. To make things look more similar to reality, those dark particles (WIMPs) are now drawn, but you can still measure their presence! In fact, as the simulation runs, the virial mass is about 10 times the average galaxy mass. What’s happening? The WIMPs are interacting among themselves and with the visible galaxies: the potential well is deeper, and the exponent is definitely less than -1. If the cluster is in equilibrium, and if gravity is the only force in action, there must be more invisible mass you’re not taking into account to balance the virial theorem.
Can you tell how much Dark Matter is there, relative to visible mass?


Now take a look at the last video with Dark Matter objects drawn as blue dots. Impressive, isn’t it?

Conclusions

Dark Matter is one of the greatest mysteries in modern science. I think that is extremely funny to see it in action on your laptop!

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