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Predictions from the logotropic model: the universal surface density of dark matter halos and the present proportion of dark matter and dark energy

The logotropic model [P.H. Chavanis, Eur. Phys. J. Plus {\bf 130}, 130 (2015)] may be an interesting alternative to the $Λ$CDM model. It is able to account for the present accelerating expansion of the universe while solving at the same time the core-cusp problem of the CDM model. In the logotropic model, there is a single dark fluid. Its rest-mass plays the role of dark matter and its internal energy plays the role of dark energy. We highlight two remarkable predictions of the logotropic model. It yields cored dark matter halos with a universal surface density equal to $Σ_0^{\rm th}=0.01955 c\sqrtΛ/G=133\, M_{\odot}/{\rm pc}^2$ without free parameter in very good agreement with the observational value $Σ_0^{\rm obs}=141_{-52}^{+83}\, M_{\odot}/{\rm pc}^2$. It also predicts the present ratio of dark energy and dark matter to be the pure number $Ω_{\rm de,0}^{\rm th}/Ω_{\rm dm,0}^{\rm th}=e=2.71828...$ in very good agreement with the observations giving $Ω_{\rm de,0}^{\rm obs}/Ω_{\rm dm,0}^{\rm obs}=2.669\pm 0.08$. Using the measured present proportion of baryonic matter $Ω_{\rm b,0}^{\rm obs}=0.0486\pm 0.0010$, we find that the values of the present proportion of dark matter and dark energy are $Ω_{\rm dm,0}^{\rm th}=\frac{1}{1+e}(1-Ω_{\rm b,0})=0.2559$ and $Ω_{\rm de,0}^{\rm th}=\frac{e}{1+e}(1-Ω_{\rm b,0})=0.6955$ in very good agreement with the observational values $Ω_{\rm dm,0}^{\rm obs}=0.2589\pm 0.0057$ and $Ω_{\rm de,0}^{\rm obs}=0.6911\pm 0.0062$ within the error bars. These theoretical predictions are obtained by advocating a mysterious strong cosmic coincidence (dubbed "dark magic") implying that our epoch plays a particular role in the history of the universe.