# Exact Solution and Exotic Fluid in Cosmology

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## Abstract

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## 1. Introduction

## 2. ΛCDM Model

## 3. de Sitter Nonlinear Sigma Model with Potential

## 4. Stability

- point A$$\begin{array}{c}\hfill {\mu}_{1}=\frac{3}{2}({\omega}_{f}-1),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{2}=\frac{1+3{\omega}_{f}}{2},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{3}=\frac{3({\omega}_{f}+1)}{2}\end{array}$$
- point B$$\begin{array}{c}{\mu}_{1,2}=-\frac{3(1-{\omega}_{f})}{4}\pm \frac{\sqrt{{\overline{\gamma}}^{2}(-1+{\omega}_{f})(24+24{{\omega}_{f}}^{2}+7{\overline{\gamma}}^{2}+{\omega}_{f}(48+9{\overline{\gamma}}^{2}))}}{4{\overline{\gamma}}^{2}}\hfill \end{array}$$$$\begin{array}{c}{\mu}_{3}=\frac{\overline{\gamma}+3{\omega}_{f}\overline{\gamma}+6(1+{\omega}_{f})\overline{\xi}}{2\overline{\gamma}}\hfill \end{array}$$
- point C$$\begin{array}{c}{\mu}_{1,2}=-\frac{3(1-{\omega}_{f})}{4}\pm \frac{\sqrt{{\overline{\xi}}^{2}(-1+{\omega}_{f})(4{(1+3{\omega}_{f})}^{2}+3(5+27{\omega}_{f}){\overline{\xi}}^{2})}}{4{\overline{\xi}}^{2}}\hfill \end{array}$$$$\begin{array}{c}{\mu}_{3}=\frac{\overline{\gamma}+3{\omega}_{f}\overline{\gamma}+6(1+{\omega}_{f})\overline{\xi}}{4\overline{\xi}}\hfill \end{array}$$
- point D$$\begin{array}{c}\hfill {\mu}_{1}=-\frac{6+{\overline{\gamma}}^{2}}{2},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{2}=-3-3{\omega}_{f}-{\overline{\gamma}}^{2},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{3}=-\frac{2+{\overline{\gamma}}^{2}+2\overline{\gamma}\overline{\xi}}{2}\end{array}$$
- point E$$\begin{array}{c}\hfill {\mu}_{1}=-1-3{\omega}_{f}-6{\overline{\xi}}^{2},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{2}=1-3{\overline{\xi}}^{2}-\frac{3\overline{\gamma}\overline{\xi}}{2},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{3}=-2-3{\overline{\xi}}^{2}\end{array}$$
- point F$$\begin{array}{c}{\mu}_{1,2}=-\frac{\overline{\gamma}+3\overline{\xi}}{\overline{\gamma}+2\overline{\xi}}\pm \frac{\sqrt{-(8-6\overline{\gamma}\overline{\xi}{(\overline{\gamma}+2\overline{\xi})}^{2})-3{\overline{\gamma}}^{2}+10\overline{\gamma}\overline{\xi}+33{\overline{\xi}}^{2}}}{\overline{\gamma}+2\overline{\xi}}\hfill \end{array}$$$$\begin{array}{c}{\mu}_{3}=-\frac{\overline{\gamma}+3{\omega}_{f}\overline{\gamma}+6(1+{\omega}_{f})\overline{\xi}}{\overline{\gamma}+2\overline{\xi}}\hfill \end{array}$$

## 5. Exact Cosmological Solution

**Figure 1.**The plot of ${\Omega}_{\Lambda ,0}\left(blue\right)\simeq 0.726\pm 0.015,\phantom{\rule{3.33333pt}{0ex}}{H}_{0}\left(green\right)\simeq 2.28\pm 0.04\times {10}^{-18}{s}^{-1}$ on the (B, t)-plane for ${\omega}_{f}=0.13$(left), ${\omega}_{f}=0.18$(right), ${\omega}_{f}=0.22$(lower). The red band is the current age with uncertainty ${t}_{0}\simeq 4.33\pm 0.04\times {10}^{17}s$.

## 6. Conclusion

## Acknowledgments

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Kouwn, S.; Moon, T.; Oh, P.
Exact Solution and Exotic Fluid in Cosmology. *Entropy* **2012**, *14*, 1771-1783.
https://doi.org/10.3390/e14091771

**AMA Style**

Kouwn S, Moon T, Oh P.
Exact Solution and Exotic Fluid in Cosmology. *Entropy*. 2012; 14(9):1771-1783.
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**Chicago/Turabian Style**

Kouwn, Seyen, Taeyoon Moon, and Phillial Oh.
2012. "Exact Solution and Exotic Fluid in Cosmology" *Entropy* 14, no. 9: 1771-1783.
https://doi.org/10.3390/e14091771