Dark Matter Candidates: WIMPs

Well, I keep having trouble getting my science posts on a regular schedule, so here’s another one. This is the first of several posts on possible dark matter candidates.

Creation of Dark Matter in the Early Universe

In my last post in this series (which was a while ago), I reviewed some of the basic properties that we believe dark matter should have. It should have mass and should probably be nonrelativistic. It shouldn’t interact much with normal matter or with itself. It should be stable (more or less). We also need a way to create it in the early universe in such a way as to get the right relic density.

Perhaps the most natural way to create dark matter is to use the same processes leading to the relic densities of Standard Model particles. That is, let us assume that dark matter is produced as a thermal relic of the Big Bang. Thus, dark matter starts at thermal equilibrium with everything else, then adiabatic expansion of the universe leads to cooling of the universe and a reduction in density. Eventually the universe is too cool to produce dark matter particles in sufficient numbers to maintain thermal equilibrium. Then, dark matter is constructed in such a way that at some point after the loss of thermal equilibrium, annihilation no longer occurs (and the lifetime is long enough that decay isn’t an issue over the scale of billions of years). This causes the comoving density to level off at the value needed

Thermal production of dark matter is an attractive production mechanism because the basic principle leads to very good predictions when applied to Standard Model particles (in particular, the light nuclei). All we’re doing is adding an extra particle to an already well-motivated and tested model of the early universe.

Setting the Properties of Thermal Relic Dark Matter

So, how do we determine the possible properties of this kind of dark matter?

First, we need the right relic density. This is known from cosmological measurements to be Ωch2 ≈ 0.112. The “c” is for “cold dark matter.” The small h is a unitless version of the Hubble constant, while Ω = ρ/ρc is the fraction of the critical density \rho_c = 3H^2/8\pi G . The critical density is the energy density of a flat universe given the Hubble constant H.

After a series of somewhat complicated calculations, this can be related to the WIMP-WIMP annihilation cross section by the equation

\Omega_c h^2 \simeq \frac{10^{-27}}{\langle \sigma_a v\rangle}

where the denominator is the velocity averaged product of cross section and velocity (the event rate is proportional to both interaction cross section and the particle flux). This is only an approximate expression. For a more detailed view of how to get this expression, I think Dodelson’s Modern Cosmology has a good description. It’s too complicated for me to feel comfortable deriving here (plus, it’s already in some textbooks).

From here, we note that the typical velocity of a WIMP around the time of freeze out is a nontrivial fraction of the speed of light. The annihilation cross section will typically be related to some unitless coupling constant, the mass of the dark matter particles, and a characteristic mass scale that is often going to be the mass of a particle that mediates the interaction. We can reduce these three constants to two: the dark matter mass mχ and an effective coupling constant G that has units of inverse mass squared. From this, we can get an approximate expression for the denominator in the equation above:

\langle \sigma_a v \rangle \simeq G^2 m^2_\chi \hbar^2 c^3 \simeq 10^{-26}~\rm cm^2/s

Again, these are all only approximate, as a more complete theory would be needed to get precise numbers, but these are probably equal to within a few orders of magnitude.

Weakly Interacting Massive Particles

In a rather remarkable coincidence, it turns out that if we plug in typical values from weak interactions – couplings close to the Fermi coupling GF and masses of tens to hundreds of GeV, we get the results we want. This coincidence has led many to believe that it isn’t a coincidence at all. Maybe dark matter is somehow related to the Standard Model weak interaction. Thus, the idea of Weakly Interacting Massive Particles, or WIMPs for short, was born. Thermally produced stable particles with couplings and masses characteristic of the electroweak sector of the Standard Model look remarkably like what we expect for dark matter.

Models With WIMPs

Now that we’ve established that WIMPs look like a good dark matter candidate, we need to figure out where they may come from in different extensions of the Standard Model. It turns out that this is quite easy. WIMPs pop up in all sorts of models, even ones constructed to solve other problems with the Standard Model. In many of the most popular supersymmetric models, the lightest supersymmetric particle (LSP) is a stable neutralino. Neutralinos are the supersymmetric partners of the neutral electroweak bosons. In the minimal supersymmetric standard model (MSSM), there are the usual Z boson and photon but also several neutral Higgs bosons. The interactions of both these and their superpartners are set by the scale of electroweak interactions, just as we want for our WIMP candidate. Models of things like extra dimensions can also give rise to WIMP candidates (assuming something forces one of the new particles to be stable). From an experimental point of view, the specifics of the model don’t really matter. Searches try to be as generic as possible so that a wide variety of models can be tested all at once.

For more information, the Particle Data Group (PDG) is always a good resource, especially for overviews of experimental searches. Particle Dark Matter, edited by Bertone (there is both a book and a preprint on the arXiv) is another great option to learn more about this, as well as nearly everything else you might want to know. There are plenty of other papers and reviews as well, such as Jungman, Kamionkowski, and Griest’s Supersymmetric Dark Matter (published in Physics Reports).