Back in March 2013, the Planck collaboration announced a new estimate for the fraction of today's universe that consists of dark energy; about 68%. As chronicled here, someone soon noticed that this number is approximately 1-1/π. A few months later, it was noticed that this number is also approximately ln(2). If we combine these two observations, we discover what I shall call the "dark identity"
ln(2) + 1/π ≈ 1
The terms on the left-hand side correspond to "dark energy" and "everything else", respectively.
This little gem takes crackpot physics numerology to a new level, because even just as mathematics, it's anomalous; but it also encodes the universe.
One mission of this blog is to look at such relationships and to not immediately dismiss them. But so far, it has been a matter of asking whether particular numerical relationships between quantities from physics could be something other than coincidence.
Here, we first need to ask this of the strictly numerical aspect of the anomaly. Could this near-relationship between π, the number 1, and the natural logarithm of 2, conceivably be something other than a coincidence?
For this to be so, I think it would have to derive from some simple and significant mathematical object. It needs to be simple because numerically, the coincidence is not supernaturally close.
Suppose we learned, for example, that this formula approximates an exact identity concerning the discriminants of an obscure type of number field. This might be unimpressive if there are thousands of equally obscure number fields. But it would be impressive if it was a very special number field, which e.g. played a role in some important construction.
Obviously one should look for mathematical topics where both π and the natural logarithm of 2 play a role.
So much for pure math. What about the physics? Right away a problem presents itself: 68% is the density fraction of dark energy now; but this number changes as the universe expands. It is not impossible that this number means something fundamental; but it will require some unusual assumptions about physics for this to be so, and one must take care, in constructing such unusual models, that one does not violate the more conventional assumptions that the Planck team used to derive the number in the first place.
Similarly, one has to suppose that the hypothesized number field (or whatever) which is mathematically responsible for the dark identity, also plays a role in the mathematics of the physics of the dark sector. This is another constraint on model building; potentially a very stringent constraint, if no mathematical explanation of the identity is forthcoming.
Having spelled out some of the reasons why this may be a barren line of inquiry, now let me say why it's still interesting to think about. Mostly it's interesting because it offers an unexpected perspective on the coincidence problem of dark energy - why the amount of dark energy is about the same as the amount of everything else, rather than being different by orders of magnitude. If you're completely in the dark about how to solve a problem, then even a crazy idea with delirious logic can be a good starting point; in trying to make it work, you may discover some other idea that is similar, but better.
Also, this is a reminder of how one wants physics to work - a harmony in which deep mathematical facts are responsible for deep physical facts.
Finally, let me note that I may not have ever noticed this "dark identity", were it not for L. Edgar Otto's musings on approximations of π, which are apparently part of a larger poetic metaphysics which also includes "dark fluid dynamics".