In a previous paper (Chavanis, 2015) we have introduced a new cosmological model that we called the logotropic model. This model involves a fundamental constant Λ which is the counterpart of Einstein’s cosmological constant in the Λ CDM model. The logotropic model is able to account, without free parameter, for the constant surface density of the dark matter halos, for their mass–radius relation, and for the Tully-Fisher relation. In this paper, we explore other consequences of this model. By advocating a form of “strong cosmic coincidence” we predict that the present proportion of dark energy in the Universe is Ωde,0=e∕(1+e)≃0.731 which is close to the observed value. We also remark that the surface density of dark matter halos and the surface density of the Universe are of the same order as the surface density of the electron. This makes a curious connection between cosmological and atomic scales. Using these coincidences, we can relate the Hubble constant, the electron mass and the electron charge to the cosmological constant. We also suggest that the famous numbers 137 (fine-structure constant) and 123 (logotropic constant) may actually represent the same thing. This could unify microphysics and cosmophysics. We study the thermodynamics of the logotropic model and find a connection with the Bekenstein-Hawking entropy of black holes if we assume that the logotropic dark fluid is made of particles of mass mΛ∼ħΛ∕c2=2.08×10−33eV∕c2 (cosmons). In that case, the universality of the surface density of the dark matter halos may be related to a form of holographic principle (the fact that their entropy scales like their area). We use similar arguments to explain why the surface density of the electron and the surface density of the Universe are of the same order and justify the empirical Weinberg relation. Finally, we combine the results of our approach with the quantum Jeans instability theory to predict the order of magnitude of the mass of ultralight axions m∼10−23eV∕c2 in the Bose–Einstein condensate dark matter paradigm.
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