# Dark matter and SO(10) grand unification

$SO(10)$ grand unified theories nicely accommodate a massive $\sim 10^{14-15}\; GeV$ sterile neutrino. Would this be a viable dark matter candidate? I haven't found any specific material regarding this possibility and I was wondering if there is some simple reason to exclude it. GUT's based on $SU(5)$ have mostly been refuted, so the next candidate according the Occam's razor would be $SO(10)$. I know the energy range to test it is not accessible (and won't be for a long time) but I find it odd that $SO(10)$ GUT's don't feature much in people explanations regarding beyond the Standard Model physics.

On a related note, would the absence of any proton decay in the Hyper-Kamiokande refute $SO(10)$? If proton decay is detected would this mean GUT's are the right way to go or do other classes of theories better incorporate proton decay?

• So I haven't looked into this at all, but assuming that an SO(10) unification is not already ruled out, and assuming that it has a massive sterile neutrino that is stable and could be produced in the appropriate amounts in the early universe to be dark matter today, I can tell you why it wouldn't be commonly discussed. Such a particle falls into the category of 'dark matter that only interacts gravitationally (or at very very high scales).' There's no way a direct detection experiment could measure it. It could very well be true, but isn't discussed much because it can't be tested directly. – Andrew Nov 5 '13 at 17:00
• in SO(10) models, the right-handed neutrino would have interactions other than gravity with heavy gauge bosons, no? although it is singlet under the subgroup SU(5), SM etc. – innisfree Nov 5 '13 at 17:14
• I don't know why you haven't found any material on this, it is discussed commonly enough. It is in Wikipedia under "sterile neutrino" and has been covered by all the popular science news mags. Just google on SO(10) sterile neutrino dark matter. – Philip Gibbs - inactive Nov 5 '13 at 18:21
• You're right, it is mentioned but mostly in passing. What I haven't found is a theoretical paper dedicated to the issue. If we were to assume that all dark matter/energy is due to a massive sterile neutrino would this be give any restriction to a GUT? and similar questions. – Sebby Nov 6 '13 at 1:53
• How massive a neutrino? One thing that comes to mind is we call the standard cosmological model $\Lambda$CDM for a reason - the dark matter has to be cool enough in the right epochs to match observed large-scale structure. – user10851 Nov 7 '13 at 20:30

I think that there are at least two things that might not work for these models. The first would be if they don't give rise to the right abundance, e.g. not enough is produced before the particles decouple from the plasma. The other issue is (even providing the right amount) the cold dark matter particle is cold (non-relativistic) enough not so spoil the formation of cosmic structure (relativistic matter tends to erase the cosmological inhomogeneities that end up producing galaxies.

I understand that sterile neutrinos only interact through gravity with other particles (although they mix with other neutrinos). In that case they wouldn't be produced by any known physics and therefore the above points would be difficult to assess, making the model uninteresting.

In the following I explain how the decoupling temperature would be computed for a particle which has a standard interaction with the others. I hope it is useful (although the computation is inconclusive, as I don't know the relevant parameters).

Whether or not a species is in equilibrium or not with the rest in the early universe depends on the ratio between the expansion rate $$H\propto \frac{\rho^{1/2}}{M_p}\propto \frac{T^2}{M_p}\,,$$ (because the energy density is dominated by relativistic particles) and the interaction rate of the sterile neutrinos $$\Gamma=n\sigma|v|\,,$$ given by the particle number density (which scales as $T^{-3}$ and the cross section. The interaction rate as inversely proportional to the interaction time: if $\Gamma\lesssim H$, then the universe doubles its size in the time it takes (on average) for a sterile neutrino to interact with the rest. In that time the number density would reduce considerably, making the interaction even less likely.

Let's look at the cross section for the interaction of sterile neutrinos with other particles. It will be determined by whatever interaction keeps the sterile neutrinos in thermal equilibrium. If that interaction is mediated by a particle with $M_X \ll T$, we can neglect its mass and the cross section will be roughly $\sigma \sim \alpha^2/T^2$ with $\alpha$ regulating the interaction strenght. On the other hand, if $M_X \gg T$ we have an analogue of Fermi's weak interaction $\sigma \sim \alpha^2/M_X^4$.

For the neutrinos to be cold dark matter, we need that this happens when $T_D\ll M_\nu$ (i.e. they are non-relativistic). If we assume that the mass of the sterile neutrinos and the other GUT particles is of the same scale $M_\nu\sim M_X$, the decoupling will occur approximately at a temperature $T_D$ such that $$\frac{\Gamma}{H} \sim \alpha^2\frac{M_p}{M_X^4}T_D^3 \sim 1 \,.$$ Again assuming that $M_X\sim M_\nu$, the above requirements translate into $M_p\alpha^2 \gg M_\nu$. Therefore $\alpha\gg (M_\nu/M_p)^{1/2}\sim 0.006$ would be a requirement for the particles to be cold since their decoupling. This limit could be lowered by a large factor, as the relevant time at which the particles should have become non-relativistic is equality (with $T_{eq}\sim eV$). You'd have to plug the values of a concrete model to see if it works out.

I hope that was not too messy. There might as well be all kind of subtleties with the $O(10)$ GUTs, the mass of the mediators and/or the properties of sterile neutrinos.