# Viscous Matter in FRW Cosmology

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## Abstract

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## 1. Introduction

## 2. Bulk Viscosity in FRW Cosmology

## 3. FRW Dynamics as Motion of a Particle in the Potential Well

## 4. Viscous Dark Energy Models—Dark Energy Models with Dissipation

**Theorem**

**1.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The phase portrait of the cosmological model with Chaplygin gas ($m=-3/2$) where $x=a$, $y=da/dt$. We postulate additionally the presence of non-interacting pressureless baryonic matter. Please note that the phase portrait is structurally stable and is topologically equivalent to the $\Lambda $CDM one.

**Figure 2.**The illustration of a generic mechanism of avoiding the initial singularity in the model with viscous matter and without the cosmological parameter $\Lambda $. This mechanism can be interpreted in terms of the Milne atomic time.

**Table 1.**Cosmological models with a substantial form of dark energy and with modified gravity. The potential $V\left(a\right)$ is obtained from $V\left(a\right)=-{H}^{2}{a}^{2}/2$; $1+z={a}^{-1}$.

Type | Cosmological Models |
---|---|

models with substantial dark energy | $\Lambda $CDM${w}_{\mathbf{X}}=-1$$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{m},0}{(1+z)}^{3}+(1-{\Omega}_{\mathrm{m},0})$ |

constant E.Q.S.${w}_{\mathbf{X}}={w}_{0}<-1$$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{m},0}{(1+z)}^{3}+(1-{\Omega}_{\mathrm{m},0}){(1+z)}^{3(1+{w}_{0})}$ | |

dynamic E.Q.S.${w}_{\mathbf{X}}={w}_{0}+{w}_{1}(1-a)$$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{m},0}{(1+z)}^{3}+(1-{\Omega}_{\mathrm{m},0}){(1+z)}^{3({w}_{0}+{w}_{1}+1)}exp[-\frac{3{w}_{1}z}{1+z}]$ | |

quintessence${\overline{w}}_{\mathbf{X}}\left(a\right)=\int {w}_{\mathbf{X}}\left(a\right)\mathbf{d}(lna)/\int \mathbf{d}(lna)\equiv {w}_{0}{a}^{\alpha}$$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{m},0}{(1+z)}^{3}+(1-{\Omega}_{\mathrm{m},0}){(1+z)}^{3(1+{w}_{0}{(1+z)}^{-\alpha})}$ | |

oscillating E.Q.S.${w}_{\mathbf{X}}\left(z\right)=-1+{(1+z)}^{3}\left[Ccos(ln(1+z\left)\right)\right]$$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\Lambda ,0}exp\left({(1+z)}^{3}{D}_{2}cos(ln(1+z))\right)+{\Omega}_{\mathrm{m},0}{(1+z)}^{3}$ | |

models with modified gravity | interacting DE & DM$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{m},0}{(1+z)}^{3}+{\Omega}_{\mathrm{int},0}{(1+z)}^{n}+1-{\Omega}_{\mathrm{m},0}-{\Omega}_{\mathrm{int},0}$ |

bounce$\Lambda $CDM$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{m},0}{(1+z)}^{3}-{\Omega}_{n,0}{(1+z)}^{n}+1-{\Omega}_{\mathrm{m},0}+{\Omega}_{n,0}$ | |

Cardassian$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{r},0}{(1+z)}^{4}+{\Omega}_{\mathrm{m},0}{(1+z)}^{4}\left[\frac{1}{1+z}+{(1+z)}^{k}\left(\frac{{\Omega}_{\mathrm{C},0}}{{\Omega}_{\mathrm{m},0}}\right)E\left(z\right)\right]$ | |

DGP$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\left[\sqrt{{\Omega}_{\mathrm{m},0}{(1+z)}^{3}+{\Omega}_{\mathrm{rc},0}}+\sqrt{{\Omega}_{\mathrm{rc},0}}\right]}^{2}$, ${\Omega}_{\mathrm{rc},0}=\frac{{(1-{\Omega}_{\mathrm{m},0})}^{2}}{4}$ | |

Sahni-Shtanov brane I$\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={\Omega}_{\mathrm{m},0}{(1+z)}^{3}+{\Omega}_{\sigma ,0}+2{\Omega}_{l,0}-2\sqrt{{\Omega}_{l,0}}P\left(z\right)$ |

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Szydłowski, M.; Krawiec, A.
Viscous Matter in FRW Cosmology. *Symmetry* **2020**, *12*, 1269.
https://doi.org/10.3390/sym12081269

**AMA Style**

Szydłowski M, Krawiec A.
Viscous Matter in FRW Cosmology. *Symmetry*. 2020; 12(8):1269.
https://doi.org/10.3390/sym12081269

**Chicago/Turabian Style**

Szydłowski, Marek, and Adam Krawiec.
2020. "Viscous Matter in FRW Cosmology" *Symmetry* 12, no. 8: 1269.
https://doi.org/10.3390/sym12081269