The Formulation of Scaling Expansion in an Euler-Poisson Dark-fluid Model

Abstract

We present a dark fluid model described as a non-viscous, non-relativistic, rotating, and self-gravitating fluid. We assumed that the system has spherical symmetry and the matter can be described with the polytropic equation of state. The induced coupled non-linear partial differential equation system was solved by using a self-similar time-dependent ansatz introduced by L. Sedov and G. I. Taylor. These kinds of solutions were successfully used to describe blast waves induced by an explosion since the Guderley-Landau-Stanyukovich problem. We showed that these kinds of solutions can provide new solutions that are consistent with the Newtonian cosmological framework. We have found that such solutions can be applied to describe normal-to-dark energy on the cosmological scale.

Figures (7)

• Figure S1: Numerical solutions of the shape functions, the integration was started at $\zeta_{0} = 0.001$, and the initial conditions of $f(\zeta_{0}) = 0.05$, $g(\zeta_{0}) = 0.053$, $h(\zeta_{0}) = 0$, and $h'(\zeta_{0}) = 1$ were used. For better visibility, the function $g(\zeta)$ was scaled up with a factor of 100. The values are given in geometrized unit.
• Figure S2: Different radial (left) and time (right) projections of the velocity flow (1st row), density (2nd row), and gravitational potential (3rd row) for the non-rotating case, respectively. A detailed explanation is given in the main text. The domain range is given in geometrized unit.
• Figure S3: Numerical solutions of the velocity flow $u(r,t)$, density flow $\rho(r,t)$, and gravitational potential $\Phi(r,t)$ as a function of the spatial and time coordinates in case of a non-rotating system. We also present the distribution of the total and kinetic energy density. For the numerical integration, we used $\zeta_{0} = 0.001$, and the initial conditions were $f(\zeta_{0}) = 0.05$, $g(\zeta_{0}) = 0.053$, $h(\zeta_{0}) = 0$, and $h'(\zeta_{0}) = 1$.
• Figure S4: The time and radial projections of the velocity flow (1st row), density (2nd row), and gravitational potential (3rd row) respectively for the rotating system $(\omega = 0.1535)$. For the numerical integration we used $\zeta_{0} = 0.001$, and the initial conditions were $f(\zeta_{0}) = 0.05$, $g(\zeta_{0}) = 0.053$, $h(\zeta_{0}) = 0$, and $h'(\zeta_{0}) = 1$.
• Figure S5: The maximal angular velocity $\omega$ dependence of the space and time evolution. Different lines correspond to different angular velocity values, $\omega$. The curves were evaluated at a particular time (left) and radial (right) coordinates were given on the vertical axis. A detailed explanation is given in the main text.