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 = A ln(rho/rho(P)), where rho is the rest-mass density, rho(P) = 5.16 x 10(99) gm(-3) is the Planck density, and Lambda is the logotropic temperature. The energy density epsilon is the sum of a rest-mass energy term rho c(2) proportional to a(-3) mimicking dark matter and an internal energy term u(rho) = -P(rho) - A = 3A ln a + C mimicking dark energy (a is the scale factor). The logotropic temperature is approximately given by A similar or equal to rho(Lambda) c(2)/ln (rho(P)/rho(Lambda)) similar or equal to rho(Lambda) c2/[123 ln(10)], where rho(Lambda) = 6.72 x 10(-24) gm(-3) is the cosmological density and 123 is the famous number appearing in the ratio rho(P)/rho(Lambda) similar to 10(123) between the Planck density and the cosmological density. More precisely, we obtain A = 2.13 x 10(-9) gm(-1) s(-2) that we interpret as a fundamental constant. At the cosmological scale, our model fulfills the same observational constraints as the Lambda CDM 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 lambda(J) = 40.4 pc, 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 Sigma(0) = rho(0)r(h) = 141 M-circle dot/pc(2) and that the mass of dwarf galaxies enclosed within a sphere of fixed radius r(u) = 300 pc has the same value M-300 = 1.93 x 10(7) M-circle dot, in remarkable agreement with the observations [Donato et al. [10], Strigari et al. [13]]. It also implies the Tully-Fisher relation M-b/v(h)(4) = 44 M-circle dot km(-4) s(4). We stress that our model has no free parameter. (C) 2016 The Author. Published by Elsevier B.V.