Is the Universe logotropic?

Abstract: We consider the possibility that the universe is made of a single dark fluid described by a logotropic equation of state $P=A\ln(\rho/\rho_)$, where $\rho$ is the rest-mass density, $\rho_$ is a reference density, and $A$ is the logotropic temperature. The energy density $\epsilon$ is the sum of two terms: a rest-mass energy term $\rho c^2$ that mimics dark matter and an internal energy term $u(\rho)=-P(\rho)-A$ that mimics dark energy. This decomposition leads to a natural, and physical, unification of dark matter and dark energy, and elucidates their mysterious nature. The logotropic model depends on a single parameter $B=A/\rho_{\Lambda}c^2$ where $\rho_{\Lambda}$ is the cosmological density. For $B=0$, we recover the $\Lambda$CDM model. Using cosmological constraints, we find that $0\le B\le 0.09425$. We consider the possibility that dark matter halos are described by the same logotropic equation of state. When $B>0$, pressure gradients prevent gravitational collapse and provide halo density cores instead of cuspy density profiles, in agreement with the observations. The universal rotation curve of logotropic dark matter halos is consistent with the observational Burkert profile up to the halo radius. Interestingly, if we assume that all the dark matter halos have the same logotropic temperature $B$, we find that their surface density $\Sigma=\rho_0 r_h$ is constant. This result is in agreement with the observations where it is found that $\Sigma_0=141\, M_{\odot}/{\rm pc}^2$ for dark matter halos differing by several orders of magnitude in size. Using this observational result, we obtain $B=3.53\times 10^{-3}$. Assuming that $\rho_*=\rho_P$, where $\rho_P$ is the Planck density, we predict $B=3.53\times 10^{-3}$, in perfect agreement with the value obtained from the observations.