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

Abstract

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 Schrödinger equation in the nonrelativistic regime. This model has an intrinsically quantum nature and returns the ΛCDM model in the classical limit → 0. It involves a new fundamental constant of physics A/c 2 = 2.10 × 10 −26 g m −3 responsible for the late accelerating expansion of the Universe and superseding the Einstein cosmological constant Λ. The logotropic model is almost indistinguishable from the ΛCDM 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 Σ th 0 = 5.85 (A/4πG) 1/2 = 133 M /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 Σ obs 0 = 141 +83 −52 M /pc 2 . We also argue that the quantities Ω dm,0 and Ω de,0 , which are usually interpreted as the present proportion of dark matter and dark energy in the ΛCDM model, are equal to Ω th dm,0 = 1 1+e (1 − Ω b,0 ) = 0.2559 and Ω th de,0 = e 1+e (1 − Ω b,0 ) = 0.6955 in very good agreement with the measured values Ω obs dm,0 = 0.2589 and Ω obs de,0 = 0.6911 (their ratio 2.669 is close to the pure number e = 2.71828...). We point out, however, important difficulties with the logotropic model, similar to those encountered by the generalized Chaplygin gas model. These problems are related to the difficulty of forming large-scale structures due to an increasing speed of sound as the Universe expands. We discuss potential solutions to these problems, stressing in particular the importance to perform a nonlinear study of structure formation.

Generalized Logotropic Models and Their Cosmological Constraints

“…Interestingly, the logotropic equation of state gives a mathematical meaning to the self-gravitating polytrope of index n = −1 which is otherwise ill-defined [45]. Therefore, the logotropic equation of state nicely completes the family of polytropic equations of state by filling the gap at n = −1 [46]. Furthermore, it is the only equation of state that leads to DM halos with a universal surface density, in agreement with the observations [30].…”

“…In fact, the speed of sound in logotropic models has the unpleasant property of increasing with the scale factor, leading, like for the Chaplygin gas model, to oscillations in the mass power-spectrum that are not detected in observations at the cosmological level [51]. However, there are several possibilities to solve this problem by considering additional effects such as non-linear and non-adiabatic perturbations, or higher order derivatives in K-essence Lagrangians associated with braneworld models, among others (see the discussion in Section XVI.G of [46]). These modifications could solve the problems in the theory of perturbations for structure formation (e.g., by reducing the speed of sound) without affecting the evolution of the cosmological background.…”

“…However, the two-fluid model does not determine the value of the constant A, contrary to the one-fluid model. Indeed, the Planck scale ρ P does not occur in Equation (A24), unlike Equation (46). This is a huge advantage of the one-fluid model with respect to the two-fluid model [30].…”