Abstract
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, , 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 à 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.
Highlights
Setting Λ 1/ld2S, where ldS is the de Sitter horizon, this is equivalent to h λp = mpc1/3
The current best fit to the available cosmological data favours a cosmological concordance, or ΛCDM model, in which dark energy takes the form of a positive cosmological constant, which accounts for approximately 69% of the total energy density of the Universe, whereas cold dark matter (CDM) accounts for around 26% and ordinary visible matter for around 5% [8, 9]
A further attempt to explain the connection between cosmological and atomic scales in the LNH, in terms of the stability of fundamental particles in the presence of dark energy, was presented in a series of recent papers [19, 20]. (See [23, 24] for extensions to modified gravity theories.) This approach considered the status of minimum length uncertainty relations (MLURs), motivated by quantum gravity phenomenology, in the presence of a vacuum energy density ρΛ given by Eq (6)
Summary
The relations (9)-(10) were first obtained in 1993 by Nottale [12] who argued that, like other fundamental ‘constants’, the cosmological constant is an explicitly scale-dependent quantity, obeying a renormalization group equation. Where m(r) /(cr) is the effective mass of the particles at scale r, Nottale obtained the scale-dependent formula for the vacuum energy density as ρvac(r) ρPl lPl r (12). Nottale obtained the low-energy asymptotic value of the cosmological constant, which was found to be scale-independent. He argued that the transition between scale-dependence and scale-independence should be identified with the Thomson scattering length (the classical electron radius), given via σT πre, where σT is the scattering cross-section. Re represents the effective electron radius, which is an energy-dependent quantity r(E), evaluated at its own mass scale.
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