2/3^n

The quantities 2/3, 2/9, 2/27 show up in the world of Koide numerology as angles (in radian units) appearing in mass formulas.

Last year, Marni Sheppeard had a paper in which she tried to derive Louise Riofrio's cosmological dark-sector numerology from the "2/9".

This year, the Planck satellite has given us some new estimates for the dark-sector density fractions. As already noted here, one quantity is close to 1 - 1/π, which could be the starting point for a new cosmic numerology.

However, it amused me to notice that, with much less precision, the density fractions for dark energy, dark matter, and baryonic matter, are not too far from that sequence, 2/3, 2/9, 2/27.

In conventional cosmology, all these density fractions evolve throughout the history of the universe, such that it shouldn't make much sense to focus on their values at a specific moment in cosmic history (like now) as a clue to anything deep.

However, it's also true that conventional cosmology has a notorious multiple coincidence problem in explaining why those density fractions are even of the same order of magnitude. (Some conventional papers trying to explain these coincidences: 1 2. A less conventional paper: 3.)

While it's very unlikely that the cosmic densities are actually such a direct echo of whatever it is that produces the Koide patterns... it might still be worth trying to make a model in which that is the case, because it would take us in new directions and make us think of new things.

So I'm recording here an idea about where to start in such an effort. It's simply the nuMSM of Asaka and Shaposhnikov, coupled to the cosmon field of Wetterich, which for him serves as both inflaton and dynamical dark energy. And in the quest for a deeper theory, one might start with Q6 symmetry.

What is the logic of this proposal? The angle 2/9, like Koide's original formula, applies to leptons. But here we want to see it show up as the cosmic density fraction for dark matter; and in the nuMSM, the dark matter is leptonic, a keV-mass sterile neutrino.

Meanwhile, Wetterich's cosmology is wacky enough that the universe might even be shrinking, rather than expanding; and yet (he says; I haven't verified) it can be transformed through a change of frame into a far more standard cosmology. It's the best opportunity I can see, to start with something that's at least semi-orthodox, but in which it might actually make sense to model the time evolution of the density fractions as "2/3^n + ϵ_n(t)", the second term being a time-dependent correction with - one hopes - a deeper explanation. 

Finally, the finite group Q6 shows up in an attempt to explain the nuMSM's odd neutrino mass spectrum. The hope would be that this could be joined up with an explanation of Koide relations, perhaps via the group S3.