New predictions from the logotropic model

In a previous paper we have introduced a new cosmological model that we called the logotropic 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 $\Omega_{\rm de,0}=e/(1+e)\simeq 0.731$ 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 to the Bekenstein-Hawking entropy of black holes if we assume that the logotropic fluid is made of particles of mass $m_{\Lambda}\sim \hbar\sqrt{\Lambda}/c^2=2.08\times 10^{-33}\, {\rm eV/c^2}$ (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.