# Study of a Viscous ΛWDM Model: Near-Equilibrium Condition, Entropy Production, and Cosmological Constraints

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## Abstract

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## 1. Introduction

## 2. Exact Analytical Solution in Eckart’s Theory with CC

## 3. Near-Equilibrium Condition and Entropy Production

#### 3.1. Near-Equilibrium Condition

#### 3.2. Entropy Production

## 4. Study of the Exact Solution

#### 4.1. Near-Equilibrium Condition of the Exact Solution

#### 4.2. Entropy Production of the Exact Solution

## 5. Cosmological Constraints

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DM | Dark matter |

DE | Dark energy |

CC | Cosmological constant |

EoS | Equation of state |

## References

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**Figure 1.**Numerical behavior of $E\left(\tau \right)$ obtained from Equation (11) at late times, for ${\mathrm{\Omega}}_{{\mathsf{\Lambda}}_{0}}=0.69$, ${\mathrm{\Omega}}_{{\xi}_{0}}=0.001$ and $\gamma =1.002$. For comparison, we also plotted the $\mathsf{\Lambda}$CDM model obtained from Equation (8).

**Figure 2.**Behavior of l, given by Equation (16), for $0\le {\mathrm{\Omega}}_{{\xi}_{0}}\le 0.02$ and $\sqrt{{\mathrm{\Omega}}_{{\mathsf{\Lambda}}_{0}}}\le E\le 4$. We also consider the fixed values of ${\mathrm{\Omega}}_{{\mathsf{\Lambda}}_{0}}=0.69$ and $\gamma =1.002$. The green line represents the near-equilibrium condition when ${\mathrm{\Omega}}_{{\xi}_{0}}=0.001$ and the red zone represents the values for which we are far from the near-equilibrium condition ($l>1$).

**Figure 3.**Numerical behavior of T, given by Equation (39), for $0.5\le a\le 3.5$. We also consider the fixed values of ${T}_{0}=1$, ${\mathrm{\Omega}}_{{\xi}_{0}}=0.001$, and ${\mathrm{\Omega}}_{{\mathsf{\Lambda}}_{0}}=0.69$.

**Figure 4.**Numerical behavior of $dS/d\tau $, given by Equation (43), for $-0.4\le \tau \le 0.4$. We also consider the fixed values of ${\mathrm{\Omega}}_{{\mathsf{\Lambda}}_{0}}=0.69$ and $\gamma =1.002$. The red dashed line represents the time ${\tau}_{s}$, given by Equation (44), in which the Big-Rip singularity occurs.

**Figure 5.**Joint and marginalized regions of the free parameters h and ${\mathrm{\Omega}}_{m0}$ for the $\mathsf{\Lambda}$CDM model, obtained in the MCMC analysis described in Section 5. The admissible regions presented in the joint regions correspond to $1\sigma (68.3\%)$, $2\sigma (95.5\%)$ and $3\sigma (99.7\%)$ of confidence level (CL), respectively. The best-fit values for each model-free parameter are shown in Table 2.

**Figure 6.**Joint and marginalized regions of the free parameters h, ${\mathrm{\Omega}}_{m0}$, ${\mathrm{\Omega}}_{{\xi}_{0}}$ and $\gamma $ for the viscous $\mathsf{\Lambda}$WDM model, obtained in the MCMC analysis described in Section 5. The admissible regions presented in the joint regions correspond to $1\sigma (68.3\%)$, $2\sigma (95.5\%)$ and $3\sigma (99.7\%)$ of confidence level (CL), respectively. The best-fit values for each model-free parameter are shown in Table 2.

**Table 1.**Final values of the total number of steps, mean acceptance fraction (MAF) and autocorrelation time ${\tau}_{corr}$ for each model-freeparameters, obtained when the convergence test described in Section 5 is fulfilled for a MCMC analysis with 30 chains or “walkers”. The values of the MAF are obtained for a value of the stretch move of $a=7$ for the $\mathsf{\Lambda}$CDM model, and a value of $a=3$ for the viscous $\mathsf{\Lambda}$WDM model.

Data | Total Steps | MAF | ${\mathit{\tau}}_{\mathit{corr}}$ | |||
---|---|---|---|---|---|---|

$\mathit{h}$ | ${\mathrm{\Omega}}_{\mathit{m}\mathbf{0}}$ | ${\mathrm{\Omega}}_{{\mathbf{\xi}}_{\mathbf{0}}}$ | $\mathbf{\gamma}$ | |||

$\mathsf{\Lambda}$CDM Model | ||||||

SNe Ia | 1050 | $0.370$ | $16.5$ | $17.5$ | ⋯ | ⋯ |

OHD | 1000 | $0.367$ | $14.9$ | $17.1$ | ⋯ | ⋯ |

SNe Ia+OHD | 800 | $0.364$ | $15.8$ | $15.4$ | ⋯ | ⋯ |

Viscous $\mathsf{\Lambda}$WDM Model | ||||||

SNe Ia | 2700 | $0.385$ | $45.8$ | $44.3$ | $49.6$ | $51.9$ |

OHD | 2450 | $0.377$ | $39.6$ | $45.5$ | $48.2$ | $48.9$ |

SNe Ia+OHD | 2700 | $0.379$ | $43.0$ | $45.9$ | $50.5$ | $53.3$ |

**Table 2.**Best-fit values and goodness-of-fit criteria for the viscous $\mathsf{\Lambda}$WDM model with free parameters h, ${\mathrm{\Omega}}_{m0}$, ${\mathrm{\Omega}}_{{\xi}_{0}}$ and $\gamma $; in addition, for the $\mathsf{\Lambda}$CDM model with free parameters h and ${\mathrm{\Omega}}_{m0}$, obtained in the MCMC analysis described in Section 5 for the SNe Ia data, OHD, and in their joint analysis. The uncertainties correspond to $1\sigma (68.3\%)$, $2\sigma (95.5\%)$ and $3\sigma (99.7\%)$ of confidence level (CL), respectively. The best-fit values for the $\mathsf{\Lambda}$CDM model are used for the sake of comparison with the viscous $\mathsf{\Lambda}$WDM model.

Data | Best-Fit Values | Goodness of Fit | ||||
---|---|---|---|---|---|---|

h | ${\mathbf{\Omega}}_{\mathit{m}0}$ | ${\mathbf{\Omega}}_{{\mathbf{\xi}}_{0}}(\times {10}^{-2})$ | $\mathit{\gamma}$ | ${\mathit{\chi}}_{\mathit{min}}^{2}$ | BIC | |

$\mathsf{\Lambda}$CDM Model | ||||||

SNe Ia | ${0.740}_{-0.013\phantom{\rule{0.277778em}{0ex}}-0.028\phantom{\rule{0.277778em}{0ex}}-0.040}^{+0.014\phantom{\rule{0.277778em}{0ex}}+0.028\phantom{\rule{0.277778em}{0ex}}+0.043}$ | ${0.299}_{-0.022\phantom{\rule{0.277778em}{0ex}}-0.042\phantom{\rule{0.277778em}{0ex}}-0.058}^{+0.022\phantom{\rule{0.277778em}{0ex}}+0.045\phantom{\rule{0.277778em}{0ex}}+0.064}$ | ⋯ | ⋯ | $1026.9$ | $1040.8$ |

OHD | ${0.720}_{-0.009\phantom{\rule{0.277778em}{0ex}}-0.017\phantom{\rule{0.277778em}{0ex}}-0.025}^{+0.009\phantom{\rule{0.277778em}{0ex}}+0.018\phantom{\rule{0.277778em}{0ex}}+0.026}$ | ${0.241}_{-0.014\phantom{\rule{0.277778em}{0ex}}-0.027\phantom{\rule{0.277778em}{0ex}}-0.037}^{+0.014\phantom{\rule{0.277778em}{0ex}}+0.027\phantom{\rule{0.277778em}{0ex}}+0.038}$ | ⋯ | ⋯ | $28.6$ | $36.5$ |

SNe Ia+OHD | ${0.710}_{-0.008\phantom{\rule{0.277778em}{0ex}}-0.016\phantom{\rule{0.277778em}{0ex}}-0.022}^{+0.008\phantom{\rule{0.277778em}{0ex}}+0.016\phantom{\rule{0.277778em}{0ex}}+0.023}$ | ${0.259}_{-0.012\phantom{\rule{0.277778em}{0ex}}-0.022\phantom{\rule{0.277778em}{0ex}}-0.036}^{+0.012\phantom{\rule{0.277778em}{0ex}}+0.024\phantom{\rule{0.277778em}{0ex}}+0.034}$ | ⋯ | ⋯ | $1058.3$ | $1072.3$ |

Viscous $\mathsf{\Lambda}$WDM Model | ||||||

SNe Ia | ${0.741}_{-0.014\phantom{\rule{0.277778em}{0ex}}-0.028\phantom{\rule{0.277778em}{0ex}}-0.044}^{+0.014\phantom{\rule{0.277778em}{0ex}}+0.029\phantom{\rule{0.277778em}{0ex}}+0.044}$ | ${0.293}_{-0.022\phantom{\rule{0.277778em}{0ex}}-0.043\phantom{\rule{0.277778em}{0ex}}-0.064}^{+0.023\phantom{\rule{0.277778em}{0ex}}+0.048\phantom{\rule{0.277778em}{0ex}}+0.066}$ | ${0.980}_{-0.716\phantom{\rule{0.277778em}{0ex}}-0.943\phantom{\rule{0.277778em}{0ex}}-0.979}^{+1.232\phantom{\rule{0.277778em}{0ex}}+2.954\phantom{\rule{0.277778em}{0ex}}+4.318}$ | ${1.023}_{-0.011\phantom{\rule{0.277778em}{0ex}}-0.019\phantom{\rule{0.277778em}{0ex}}-0.021}^{+0.015\phantom{\rule{0.277778em}{0ex}}+0.031\phantom{\rule{0.277778em}{0ex}}+0.045}$ | $1026.9$ | $1054.7$ |

OHD | ${0.721}_{-0.010\phantom{\rule{0.277778em}{0ex}}-0.020\phantom{\rule{0.277778em}{0ex}}-0.029}^{+0.009\phantom{\rule{0.277778em}{0ex}}+0.018\phantom{\rule{0.277778em}{0ex}}+0.026}$ | ${0.237}_{-0.017\phantom{\rule{0.277778em}{0ex}}-0.033\phantom{\rule{0.277778em}{0ex}}-0.045}^{+0.017\phantom{\rule{0.277778em}{0ex}}+0.037\phantom{\rule{0.277778em}{0ex}}+0.053}$ | ${1.026}_{-0.738\phantom{\rule{0.277778em}{0ex}}-0.988\phantom{\rule{0.277778em}{0ex}}-1.019}^{+1.250\phantom{\rule{0.277778em}{0ex}}+2.641\phantom{\rule{0.277778em}{0ex}}+3.763}$ | ${1.023}_{-0.012\phantom{\rule{0.277778em}{0ex}}-0.019\phantom{\rule{0.277778em}{0ex}}-0.022}^{+0.014\phantom{\rule{0.277778em}{0ex}}+0.030\phantom{\rule{0.277778em}{0ex}}+0.048}$ | $28.5$ | $44.2$ |

SNe Ia+OHD | ${0.709}_{-0.009\phantom{\rule{0.277778em}{0ex}}-0.017\phantom{\rule{0.277778em}{0ex}}-0.027}^{+0.009\phantom{\rule{0.277778em}{0ex}}+0.017\phantom{\rule{0.277778em}{0ex}}+0.026}$ | ${0.261}_{-0.015\phantom{\rule{0.277778em}{0ex}}-0.030\phantom{\rule{0.277778em}{0ex}}-0.041}^{+0.015\phantom{\rule{0.277778em}{0ex}}+0.030\phantom{\rule{0.277778em}{0ex}}+0.047}$ | ${1.633}_{-1.059\phantom{\rule{0.277778em}{0ex}}-1.544\phantom{\rule{0.277778em}{0ex}}-1.627}^{+1.381\phantom{\rule{0.277778em}{0ex}}+2.871\phantom{\rule{0.277778em}{0ex}}+4.186}$ | ${1.026}_{-0.013\phantom{\rule{0.277778em}{0ex}}-0.021\phantom{\rule{0.277778em}{0ex}}-0.024}^{+0.015\phantom{\rule{0.277778em}{0ex}}+0.031\phantom{\rule{0.277778em}{0ex}}+0.045}$ | $1056.9$ | $1084.9$ |

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**MDPI and ACS Style**

Cruz, N.; González, E.; Jovel, J. Study of a Viscous ΛWDM Model: Near-Equilibrium Condition, Entropy Production, and Cosmological Constraints. *Symmetry* **2022**, *14*, 1866.
https://doi.org/10.3390/sym14091866

**AMA Style**

Cruz N, González E, Jovel J. Study of a Viscous ΛWDM Model: Near-Equilibrium Condition, Entropy Production, and Cosmological Constraints. *Symmetry*. 2022; 14(9):1866.
https://doi.org/10.3390/sym14091866

**Chicago/Turabian Style**

Cruz, Norman, Esteban González, and Jose Jovel. 2022. "Study of a Viscous ΛWDM Model: Near-Equilibrium Condition, Entropy Production, and Cosmological Constraints" *Symmetry* 14, no. 9: 1866.
https://doi.org/10.3390/sym14091866