Cosmological Constant Numerology

While this particular hypothesis is probably wrong, I have a soft spot in my heart for "numerology" attempting to get to the bottom of why fundamental physical constants have the values that they do, in part because these explorations deepen one's understanding of these constants even if they don't bear out. 

In the early-mid 20th century Dirac and Zel'dovich were among the first scientists to suggest an intimate connection between cosmology and atomic physics. Though a revolutionary proposal for its time, Dirac's Large Number Hypothesis (1937) adopted a standard assumption of the day, namely, the non-existence of the cosmological constant term (Λ=0). As a result, its implementation necessitated extreme violence to the theory of general relativity -- something few physicists were prepared to sacrifice in favour of `numerology' -- requiring a time-dependent gravitational coupling of the form G(t)∼1/t. Zel'dovich's insight (1968) was to realise that a small but nonzero cosmological term (Λ>0) allowed the present day radius of the Universe to be identified with the de Sitter radius, rU≃ldS≃1/Λ‾‾√, which removed the need for time-dependence in the fundamental couplings. Thus, he obtained the formula Λ≃m6G2/ℏ4, where m is a mass scale characterising the relative strengths of the gravitational and electromagnetic interactions, which he identified with the proton mass mp. In this paper, we review a number of recent arguments which, instead, suggest the identification m=me/αe, where me is the electron mass and αe=e2/ℏc≃1/137 is the usual fine structure constant. We note that these are of a physical nature and, therefore, represent an attempt to lift previous arguments a la Dirac from the realm of numerology into the realm of empirical science. If valid, such arguments suggest an intimate connection, not only between the macroscopic and microscopic worlds, but, perhaps even more surprisingly, between the very essence of "dark" and "light" physics.

Matthew J. Lake, "Is there a connection between "dark" and "light" physics?" (July 21, 2017).

The body of the paper notes that four independent researchers have come up with this formula since 1968 by different, although not inconsistent methods.

The cosmological constant appears in Einstein's field equation in the form:

and the value of Λ, the cosmological constant of General Relativity, is known so imprecisely that pretty much any formulation the produces a result with the right order of magnitude can be consistent with observation. 

The least precisely known physically observed constant that is important to know in order to estimate the value of the cosmological constant is Hubble's constant, for which, according to the Particle Data Group, the leading direct measurement has a 5% margin of error, the leading indirect measurement has a 1.5% margin of error (the two leading measurements are in tension with each other, although not beyond the two sigma threshold usually set as the standard for inconsistent measurements):

The most recent derivation based on this approach utilizes the maser-based distance to NGC4258 to re-calibrate its Cepheid distance scale to obtain H0 = 72.0 ± 3.0 kms−1 Mpc−1. The major sources of uncertainty in this result are due to the heavy element abundance of the Cepheids and the distance to the fiducial nearby galaxy, the Large Magellanic Cloud, relative to which all Cepheid distances are measured. 

The indirect determination of H0 by the Planck Collaboration found a lower value, H0 = 67.8 ± 0.9 kms−1 Mpc−1. As discussed in that paper, there is strong degeneracy of H0 with other parameters, e.g., Ωm and the neutrino mass. The tension between the H0 from Planck and the traditional cosmic distance-ladder methods is under investigation.

The best fit value for the cosmological constant based upon currently available data according to the paper is:

Λ = 1.114 ×10⁻⁵⁶ cm⁻² (which is stated to spurious accuracy by one or two orders of magnitude). 

Wikipedia quotes a value of it has the value (converting units from meters to cm) of:

Λ = 1.19×10⁻⁵⁶ cm⁻² 

The proposed formula yields a cosmological constant value of:

Λ = 1.366×10⁻⁵⁶ cm⁻² (which is stated in a manner consistent with the precision of the least accurately measured constant that goes into it "G"). 

The value produced by the proposed formula, of course, is 14.7% higher than the experimentally measured value and is clearly inconsistent with it given the margins of error with the measured value. 

There has been a controversial suggestion that one of the important measurements that goes into determining the cosmological constant directly is wrong because it conflates two different kinds of Type 1a Supernova, which are hard to distinguish from each other at optical wavelengths, but are quite distinct at other wave lengths. But, if that is the case, the cosmological constant would actually be smaller than the best fit measured value shown above, not somewhat larger like this theoretically determined value. That paper is J.T. Nielsen, A. Guffanti an S. Sarkar, "Marginal evidence for cosmic acceleration from Type Ia supernovae" 6 Scientific Reports 35596 (October 21, 2016) (open access).