The Logotropic Dark Fluid as a unification of dark matter and dark energy

We propose a heuristic unification of dark matter and dark energy in terms of a single “dark fluid” with a logotropic equation of state P=Aln⁡(ρ/ρP)P=Aln⁡(ρ/ρP), where ρ   is the rest-mass density, ρP=5.16×1099gm−3 is the Planck density, and A is the logotropic temperature. The energy density ϵ   is the sum of a rest-mass energy term ρc2∝a−3ρc2∝a−3 mimicking dark matter and an internal energy term u(ρ)=−P(ρ)−A=3Aln⁡a+Cu(ρ)=−P(ρ)−A=3Aln⁡a+C mimicking dark energy (a   is the scale factor). The logotropic temperature is approximately given by A≃ρΛc2/ln⁡(ρP/ρΛ)≃ρΛc2/[123ln⁡(10)]A≃ρΛc2/ln⁡(ρP/ρΛ)≃ρΛc2/[123ln⁡(10)], where ρΛ=6.72×10−24gm−3 is the cosmological density and 123 is the famous number appearing in the ratio ρP/ρΛ∼10123ρP/ρΛ∼10123 between the Planck density and the cosmological density. More precisely, we obtain A=2.13×10−9gm−1s−2 that we interpret as a fundamental constant. At the cosmological scale, our model fulfills the same observational constraints as the ΛCDM model (they will differ in about 25 Gyrs when the logotropic universe becomes phantom). However, the logotropic dark fluid has a nonzero speed of sound and a nonzero Jeans length which, at the beginning of the matter era, is about λJ=40.4pc, in agreement with the minimum size of the dark matter halos observed in the universe. The existence of a nonzero Jeans length may solve the missing satellite problem. At the galactic scale, the logotropic pressure balances the gravitational attraction, providing halo cores instead of cusps. This may solve the cusp problem. The logotropic equation of state generates a universal rotation curve that agrees with the empirical Burkert profile of dark matter halos up to the halo radius. In addition, it implies that all the dark matter halos have the same surface density Σ0=ρ0rh=141M⊙/pc2 and that the mass of dwarf galaxies enclosed within a sphere of fixed radius ru=300pc has the same value M300=1.93×107M⊙, in remarkable agreement with the observations [Donato et al. [10], Strigari et al. [13]]. It also implies the Tully–Fisher relation Mb/vh4=44M⊙km−4s4. We stress that our model has no free parameter.