A new logotropic model based on a complex scalar field with a logarithmic potential

We introduce a new logotropic model based on a complex scalar field with a logarithmic potential that unifies dark matter and dark energy. The scalar field satisfies a nonlinear wave equation generalizing the Klein-Gordon equation in the relativistic regime and the Schr\"odinger equation in the nonrelativistic regime. This model has an intrinsically quantum nature and returns the $\Lambda$CDM model in the classical limit $\hbar\rightarrow 0$. It involves a new fundamental constant of physics $A/c^2=2.10\times 10^{-26}\, {\rm g}\, {\rm m}^{-3}$ responsible for the late accelerating expansion of the Universe and superseding the Einstein cosmological constant $\Lambda$. The logotropic model is almost indistinguishable from the $\Lambda$CDM model at large (cosmological) scales but solves the CDM crisis at small (galactic) scales. It also solves the problems of the fuzzy dark matter model. Indeed, it leads to cored dark matter halos with a universal surface density $\Sigma_0^{\rm th}=5.85\,\left ({A}/{4\pi G}\right )^{1/2}=133\, M_{\odot}/{\rm pc}^2$. This universal surface density is predicted from the logotropic model without adjustable parameter and turns out to be close to the observed value $\Sigma_0^{\rm obs}=141_{-52}^{+83}\, M_{\odot}/{\rm pc}^2$. We also argue that the quantities $\Omega_{\rm dm,0}$ and $\Omega_{\rm de,0}$, which are usually interpreted as the present proportion of dark matter and dark energy in the $\Lambda$CDM model, are equal to $\Omega_{\rm dm,0}^{\rm th}=\frac{1}{1+e}(1-\Omega_{\rm b,0})=0.2559$ and $\Omega_{\rm de,0}^{\rm th}=\frac{e}{1+e}(1-\Omega_{\rm b,0})=0.6955$ in very good agreement with the measured values $\Omega_{\rm dm,0}^{\rm obs}=0.2589$ and $\Omega_{\rm de,0}^{\rm obs}=0.6911$ (their ratio $2.669$ is close to the pure number $e=2.71828...$).