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

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## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Scaling Solution and Sedov–Taylor Ansatz

## 4. Results

#### 4.1. Non-Rotating System

#### 4.2. Rotating System

## 5. Connection to Newtonian Friedmann Equation

#### 5.1. Connection to the Expansion Rate

#### 5.2. Connection to the Critical Density

## 6. Discussion

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Relevant physical quantities in SI and geometrized units. Geometrized units can be converted into SI units using the factors provided.

Variable | SI Unit | Geometrized Unit | Factor |
---|---|---|---|

mass | kg | m | ${c}^{2}{G}^{-1}$ |

length | m | m | 1 |

time | s | m | ${c}^{-1}$ |

density | kg ${\mathrm{m}}^{-1}$ | ${\mathrm{m}}^{-2}$ | ${c}^{2}G$ |

velocity | m ${\mathrm{s}}^{-1}$ | 1 | c |

acceleration | m ${\mathrm{s}}^{-2}$ | m${}^{-1}$ | ${c}^{2}$ |

force | kg m ${\mathrm{s}}^{-2}$ | 1 | ${c}^{4}{G}^{-1}$ |

energy | kg m^{2} s^{−2} | m${}^{-1}$ | ${c}^{4}{G}^{-1}$ |

energy density | kg m^{−1} s^{−2} | m${}^{-2}$ | ${c}^{4}{G}^{-1}$ |

Variable | Astronomical Unit | SI Unit |
---|---|---|

length | ly | $9.46073047258\xb7{10}^{15}$ m |

length | Gly | $9.46073047258\xb7{10}^{24}$ m |

length | kPc | $3.08567758128\xb7{10}^{19}$ m |

time | Gy | $3.1556926\xb7{10}^{16}$ s |

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**Figure 1.**Numerical solutions of the shape functions; integration was started at ${\zeta}_{0}=0.001$ and initial conditions of $f\left({\zeta}_{0}\right)=0.5$, $g\left({\zeta}_{0}\right)=0.01$, $h\left({\zeta}_{0}\right)=0$, and ${h}^{\prime}\left({\zeta}_{0}\right)=1$ were used. For better visibility, the $g\left(\zeta \right)$ function was upscaled by a factor of 200. The values are provided in geometrized units.

**Figure 2.**Different radial (left) and time (right) projections of the velocity flow (first row), density (second row), and gravitational potential (third row), respectively, for the non-rotating case. A detailed explanation is provided in the main text. The domain range is provided in geometrized units.

**Figure 3.**Numerical solutions of the velocity flow $u(r,t)$, density flow $\rho (r,t)$, and gravitational potential $\mathsf{\Phi}(r,t)$ as a function of the spatial and time coordinates in the case of a non-rotating system, additionally showing the distribution of the total and kinetic energy densities. We used ${\zeta}_{0}=0.001$ for numerical integration and initial conditions $f\left({\zeta}_{0}\right)=0.5$, $g\left({\zeta}_{0}\right)=0.01$, $h\left({\zeta}_{0}\right)=0$, and ${h}^{\prime}\left({\zeta}_{0}\right)=1$.

**Figure 4.**Time and radial projections of the velocity flow (first row), density (second row), and gravitational potential (third row) for the rotating system $(\omega =0.2535)$. We used ${\zeta}_{0}=0.001$ for numerical integration and initial conditions $f\left({\zeta}_{0}\right)=0.5$, $g\left({\zeta}_{0}\right)=0.01$, $h\left({\zeta}_{0}\right)=0$, and ${h}^{\prime}\left({\zeta}_{0}\right)=1$.

**Figure 5.**Dependence of the space and time evolution on the maximal angular velocity $\omega $; the different lines correspond to different values of the angular velocity $\omega $. The curves were evaluated at a particular time (left), with radial coordinates provided on the vertical axis (right). A detailed explanation is provided in the main text.

**Figure 6.**Numerical solutions of the velocity flow $u(r,t)$, density flow $\rho (r,t)$, and gravitational potential $\mathsf{\Phi}(r,t)$ as a function of the spatial and time coordinates for the rotating case. In addition, we present the distribution of the total and kinetic energy densities. We used ${\zeta}_{0}=0.001$ for numerical integration, $\omega =0.2535$, and initial conditions $f\left({\zeta}_{0}\right)=0.5$, $g\left({\zeta}_{0}\right)=0.01$, $h\left({\zeta}_{0}\right)=0$, and ${h}^{\prime}\left({\zeta}_{0}\right)=1$.

**Figure 7.**Analytical (non-rotating) and numerical (rotating) solutions of the expansion rate of the universe; integration started at ${\zeta}_{0}=0.001$, and the initial conditions were $f\left({\zeta}_{0}\right)=0.5$, $g\left({\zeta}_{0}\right)=0.008$, $h\left({\zeta}_{0}\right)=0$, and ${h}^{\prime}\left({\zeta}_{0}\right)=1$. These results match well with the data from [28].

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**MDPI and ACS Style**

Szigeti, B.E.; Barna, I.F.; Barnaföldi, G.G.
The Formulation of Scaling Expansion in an Euler-Poisson Dark-Fluid Model. *Universe* **2023**, *9*, 431.
https://doi.org/10.3390/universe9100431

**AMA Style**

Szigeti BE, Barna IF, Barnaföldi GG.
The Formulation of Scaling Expansion in an Euler-Poisson Dark-Fluid Model. *Universe*. 2023; 9(10):431.
https://doi.org/10.3390/universe9100431

**Chicago/Turabian Style**

Szigeti, Balázs Endre, Imre Ferenc Barna, and Gergely Gábor Barnaföldi.
2023. "The Formulation of Scaling Expansion in an Euler-Poisson Dark-Fluid Model" *Universe* 9, no. 10: 431.
https://doi.org/10.3390/universe9100431