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(ρ/ρ*, where ρ is the rest-mass density, ρ * is a reference density, and A is the logotropic temperature. The energy density e is the sum of two terms: a rest-mass energy term ρ c 2 that mimics dark matter and an internal energy term u(ρ) = −P(ρ) − 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. In the early universe, the rest-mass energy dominates and the dark fluid behaves as pressureless dark matter (P ≃ 0, e ∝ a −3. In the late universe, the internal energy dominates and the dark fluid behaves as dark energy (P ∼ −e, e ∝ ln a. The logotropic model depends on a single parameter B = A /ρ Λ c 2 (dimensionless logotropic temperature), where ρ Λ = 6.72 × 10−24 g m−3 is the cosmological density. For B = 0, we recover the ΛCDM model with a different justification. For B > 0, we can describe deviations from the ΛCDM model. Using cosmological constraints, we find that 0 ≤ B ≤ 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 (Burkert, Astrophys. J. 447, L25 (1995)) up to the halo radius. It decreases as r −1 at large distances, similarly to the profile of dark matter halos close to the core radius (Burkert, arXiv:1501.06604). Interestingly, if we assume that all the dark matter halos have the same logotropic temperature B, we find that their surface density Σ 0 = ρ0 r h is constant. This result is in agreement with the observations (Donato et al., Mon. Not. R. Astron. Soc. 397, 1169 (2009)) where it is found that Σ 0 = 141 M ⊙/pc2 for dark matter halos differing by several orders of magnitude in size. Using this observational result, we obtain B = 3.53 × 10−3. Then, we show that the mass enclosed within a sphere of fixed radius r u = 300 pc has the same value M 300 1.93 × 107 M ⊙ for all the dwarf halos, in agreement with the observations (Strigari et al., Nature 454, 1096 (2008)). Finally, assuming that ρ * = ρ P , where ρ P = 5.16 × 1099 g m−3 is the Planck density, we predict B = 3.53 × 10−3, in perfect agreement with the value obtained from the observations. We approximately have B ≃ 1/ln(ρ P /ρ Λ ∼ 1/[123ln(10)], where 123 is the famous number occurring in the ratio ρ P /ρ Λ ∼ 10123 between the Planck density and the cosmological density. This value of B is sufficiently low to satisfy the cosmological bound 0 ≤ B ≤ 0.09425 and sufficiently large to differ from CDM (B = 0 and avoid density cusps in dark matter halos. It leads to a Jeans length at the beginning of the matter era of the order of Λ J =40.4 pc which is consistent with the minimum size of dark matter halos observed in the universe. Therefore, a logotropic equation of state is a good candidate to account both for galactic and cosmological observations. This may be a hint that dark matter and dark energy are the manifestation of a single dark fluid. If we assume that the dark fluid is made of a self-interacting scalar field, representing for example Bose-Einstein condensates, we find that the logotropic equation of state arises from the Gross-Pitaevskii equation with an inverted quadratic potential, or from the Klein-Gordon equation with a logarithmic potential. We also relate the logotropic equation of state to Tsallis generalized thermodynamics and to the Cardassian model motivated by the existence of extra-dimensions.