Self-Similar Solutions of a Gravitating Dark Fluid
Abstract
:1. Introduction
2. The Model
- (i)
- means the EoS for ordinary non-relativistic ‘matter’ (e.g., cold dust);
- (ii)
- means ultra-relativistic ‘radiation’ (including neutrinos) and, in the very early universe, other particles that later become non-relativistic;
- (iii)
- is the simplest case and describes the expanding universe, hypothetical phantom energy would cause Big Rip;
- (iv)
- means quintessence as hypothetical fluid;
- (v)
- is responsible for the flatness of the Big Bang;
- (vi)
- A scalar field can be viewed as a sort of perfect fluid with EoS ofwhich makes scalar fields useful models for any phenomena in cosmology.
3. Analytic Solution with the Sedov–Ansatz
- The linear algebraic equation system which contains the exponents is overdetermined, and leads to a contradiction. In this case, the system has inherently no physically self-similar power-law decaying or exploding solutions. Such systems are very rare; however, damped wave equations like the telegraph equations are like this.
- All exponents have given numerical values. In that case, the analysis of the solutions is simple. The derived coupled nonlinear ODE system can be analyzed, sometimes it can be decoupled and in best cases all variables can be written down in analytic formulas.
- The given linear algebraic equation system is under-determined, leaving usually one or two self-similar exponents completely free, this means extra free parameters in the obtained ODE system. Such systems have a very rich mathematical structure. The derived free exponent can have either negative or positive sign. Negative values in most cases result in power-law divergent or exploding solutions on the other side; positive exponents usually mean power-law decaying solutions, which are very desirable for dissipative systems.
4. Results
- case is seen in the first line panels with setting the kinetic-to-potential energy ratio to −50. Both physical quantities changed radically. The feature that the center of the fluid gains more and more kinetic and potential energy density is much more suppressed; on the other side, at small times, the kinetic energy density of the total fluid enhances. The larger the radius, the larger the energy density gain. Note that the values of the maximum linearly enhance with time. One may say that this phenomenon is a kind of “delayed” acceleration and deceleration of the fluid due to the enhanced potential energy density. This effect is not visible for larger initial velocities, where the initial kinetic energy densities dominate the potential energy. In this scenario, the strength of the two energy density terms lies almost in the same range. The figure of the total energy density function is drastically different to the former case, since it becomes negative everywhere. The effect of the kinetic energy density for small time values at any radius is clearly seen as a "bump" which lifts the potential energy density. The same statements are valid also for the panel of the third line panels, where the dynamics for a small positive initial velocity case, is presented.
- case describes the dynamics of a fluid with zero initial velocity (middle panels). Here, the effect of the potential energy density can be studied in a non-perturbed way. The “bump” of the kinetic energy color density for small times at an all fluid radius—a short acceleration and deceleration—of the fluid is originated by the gravitational potential of the fluid itself.
5. Discussion
6. Summary and Outlook
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
Density | |
u | Radial velocity |
p | Pressure field |
w | Strength of the EoS |
Adiabatic speed of sound | |
f | Shape function of the radial velocity |
g | Shape function of the density |
The argument of the shape functions | |
Kinetic energy | |
Potential energy |
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Barna, I.F.; Pocsai, M.A.; Barnaföldi, G.G. Self-Similar Solutions of a Gravitating Dark Fluid. Mathematics 2022, 10, 3220. https://doi.org/10.3390/math10183220
Barna IF, Pocsai MA, Barnaföldi GG. Self-Similar Solutions of a Gravitating Dark Fluid. Mathematics. 2022; 10(18):3220. https://doi.org/10.3390/math10183220
Chicago/Turabian StyleBarna, Imre Ferenc, Mihály András Pocsai, and Gergely Gábor Barnaföldi. 2022. "Self-Similar Solutions of a Gravitating Dark Fluid" Mathematics 10, no. 18: 3220. https://doi.org/10.3390/math10183220