# Early Universe Thermodynamics and Evolution in Nonviscous and Viscous Strong and Electroweak Epochs: Possible Analytical Solutions

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Geometry and Field Equations

## 3. Cosmic Evolution in Non-Viscous Approach

#### 3.1. Hadronic Era

- At vanishing k, which is the case at ${\beta}_{1}=-1/3$, we have analytical solutions. Then, Equation (12) can be solved as,$$\begin{array}{ccc}\hfill a\left(t\right)& =& {c}_{2}cosh{\left[\sqrt{{C}_{2}(1-{C}_{1})}(t+{c}_{1})\right]}^{1/(1-{C}_{1})},\hfill \end{array}$$$$\begin{array}{ccc}\hfill H\left(t\right)& =& \frac{\dot{a}\left(t\right)}{a\left(t\right)}=\sqrt{\frac{{C}_{2}}{1+{C}_{1}}}\phantom{\rule{0.277778em}{0ex}}tanh\left[\sqrt{{C}_{2}(1+{C}_{1})}(t+{c}_{1})\right].\hfill \end{array}$$
- At non-vanishing k, there is no direct analytical solution for $a\left(t\right)$. When assuming that $u={\dot{a}}^{2}\left(t\right)$ and substituting this into Equation (12),$$\begin{array}{ccc}\hfill \frac{du\left(a\right(t\left)\right)}{da\left(t\right)}+2{C}_{1}\phantom{\rule{0.166667em}{0ex}}\frac{u\left(a\right(t\left)\right)}{a\left(t\right)}+2{C}_{2}a\left(t\right)+2{C}_{1}\frac{k}{a\left(t\right)}& =& 0,\hfill \end{array}$$$$\begin{array}{ccc}\hfill u\left(a\right(t\left)\right)& =& \dot{a}{\left(t\right)}^{2}\hfill \\ & =& {c}_{1}a{\left(t\right)}^{-2{C}_{1}}-\frac{{C}_{2}}{1+{C}_{1}}a{\left(t\right)}^{2}-\frac{k}{{C}_{1}+1}a{\left(t\right)}^{2}.\hfill \end{array}$$The physical solution is the one assuring that,$$\begin{array}{ccc}\hfill a\left(t\right)& <& {\left[\frac{{c}_{3}}{{C}_{2}}(1+{C}_{1})\right]}^{\frac{1}{2(1-{C}_{1})}}.\hfill \end{array}$$Hence, the Hubble parameter can be deduced as$$\begin{array}{ccc}\hfill H\left(t\right)& =& {\left\{{c}_{1}a{\left(t\right)}^{-2{C}_{1}}-\frac{{C}_{2}+k}{{C}_{1}+1}a{\left(t\right)}^{2}\right\}}^{1/2}\frac{1}{a\left(t\right)}.\hfill \end{array}$$By solving the second-order differential Equation (16), an analytical expression for the scale factor $a\left(t\right)$ can also be deduced,$$\begin{array}{ccc}\hfill a\left(t\right)& =& {\left[\sqrt{\frac{{C}_{2}+k}{{c}_{1}({C}_{1}+1)}}\right]}^{\frac{1}{-(1+{C}_{1})}},\hfill \end{array}$$$$\begin{array}{ccc}\hfill H\left(t\right)& =& {c}_{1}{\left[{\left[\sqrt{\frac{{C}_{2}+k}{{c}_{1}({C}_{1}+1)}}\right]}^{-\frac{1}{1+{C}_{1}}}\right]}^{-(1+2{C}_{1})}+\frac{{C}_{2}+k}{{C}_{1}+1}{\left[\sqrt{\frac{{C}_{2}+k}{{c}_{1}({C}_{1}+1)}}\right]}^{\frac{1}{-(1+{C}_{1})}}.\phantom{\rule{22.76219pt}{0ex}}\hfill \end{array}$$Apparently, all coefficients involved in it can be determined.

#### 3.2. QCD and EW Era

#### 3.3. Asymptotic Limit

## 4. Cosmic Evolution in Viscous Approaches

#### 4.1. Viscous Equations of State

- The first one is the hadron-QGP domain (Hadron-QGP), which spans over $\rho \left(t\right)\u2a85100\phantom{\rule{3.33333pt}{0ex}}$GeV/fm${}^{3}$. At the beginning, there is a rapid increase in $\zeta \left(t\right)$, i.e., $\zeta \eqsim 1\phantom{\rule{3.33333pt}{0ex}}$GeV${}^{3}$, at $\rho \left(t\right)\simeq 1\phantom{\rule{3.33333pt}{0ex}}$GeV/fm${}^{3}$, which is then followed by a slight increase in $\zeta \left(t\right)$. For example, at $\rho \left(t\right)\simeq 100\phantom{\rule{3.33333pt}{0ex}}$GeV/fm${}^{3}$, $zeta\left(t\right)$ reaches ∼$130\phantom{\rule{3.33333pt}{0ex}}$GeV${}^{3}$. It is apparent that the hadron–parton phase transition seems to take place at $\rho \left(t\right)\u2a850.5\phantom{\rule{3.33333pt}{0ex}}$GeV/fm${}^{3}$[34,35]. At this value, $\zeta \left(t\right)\u2a850.5\phantom{\rule{3.33333pt}{0ex}}$GeV${}^{3}$.
- The second domain, the QGP epoch, seems to be formed, at $0.5\u2a85\rho \left(t\right)\u2a85100\phantom{\rule{3.33333pt}{0ex}}$GeV/fm${}^{3}$, i.e., a much wider $\rho \left(t\right)$ than that of the hadron domain. Thus, we could conclude that over this wide range of $\rho \left(t\right)$, the bulk viscosity is obviously not only finite but rather largely supporting the RHIC discovery of strongly correlated viscous QGP [3,4,6]. At higher $\rho \left(t\right)$, we observe a tendency of a linear increase in $\zeta \left(t\right)$ with further increasing $\rho \left(t\right)$. Thus, the second domain is the one where $100\u2a86\rho \left(t\right)\u2a855\times {10}^{7}\phantom{\rule{3.33333pt}{0ex}}$GeV/fm${}^{3}$ and $80\u2a86\zeta \left(t\right)\u2a85{10}^{6}\phantom{\rule{3.33333pt}{0ex}}$GeV${}^{3}$. In light of this observation, we conclude that the phase transition from QCD to EW domain is very smooth.
- The third domain is also characterized by an almost linear increase in $\zeta \left(t\right)$ with increasing $\rho \left(t\right)$. For ${10}^{8}\u2a85\rho \left(t\right)\u2a85{10}^{15}\phantom{\rule{3.33333pt}{0ex}}$GeV/fm${}^{3}$, there is a nearly steady increase in $\zeta \left(t\right)$ from ${10}^{8}$ to ${10}^{14}\phantom{\rule{3.33333pt}{0ex}}$GeV${}^{3}$.

#### 4.2. Eckart Relativistic Viscous Fluid

#### 4.2.1. Hadron-QGP Era

#### 4.2.2. QCD-EW Era

#### 4.2.3. EW (Asymptotic) Era

#### 4.3. Israel–Stewart Relativistic Viscous Fluid

## 5. Results

#### 5.1. Non-Viscous Fluid

#### 5.2. Eckart-Type Viscous Fluid

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Relativistic Viscous Fluid in the Expanding Early Universe

#### Appendix A.1. Israel—Stewart Second-Order Theory

#### Appendix A.1.1. Hadron Epoch

#### Appendix A.1.2. QGP Epoch

#### Appendix A.1.3. QCD-EW Epoch

#### Appendix A.1.4. EW (Asymptotic) Epoch

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**Figure 1.**The pressure is depicted as a function of the energy density. Both quantities are given in GeV/fm${}^{3}$ units. The dashed lines present the various parameterizations (see text).

**Figure 2.**Top panel depicts the energy–density dependence of the bulk viscosity. Bottom panel illustrates the temperature as a function of energy density. The parameterizations are depicted as curves.

**Figure 3.**The energy–density dependence of the relaxation time. The parameterizations, Equations (39)–(41), are depicted as curves.

**Figure 4.**The dependence of the Hubble parameter on the scale factor in non-viscous cosmic background is depicted for finite (top) and vanishing cosmological constant (bottom panel). The equations of state for hadron, QCD-EW, and asymptotic limit are presented as dashed, dotted, and long dashed curves, respectively.

**Figure 5.**The same as in Figure 4, but here for viscous cosmic geometry (Eckart theory).

Section | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | |
---|---|---|---|---|---|

$\mathtt{Hadron}$ | 3.1 | $\frac{3}{2}(1+{\beta}_{1})-1$ | $4\pi {\alpha}_{1}-\frac{1}{2}(1-{\beta}_{1})\mathsf{\Lambda}$ | ||

$\mathtt{QCD}/\mathtt{EW}$ | 3.2 | $\frac{3}{2}(1+{\beta}_{2})-1$ | $4\pi {\alpha}_{2}-\frac{1}{2}(1-{\beta}_{2})\mathsf{\Lambda}$ | $4\pi {\left(\frac{3}{8\pi k}\right)}^{{\delta}_{2}}{\gamma}_{2}$ | $\frac{\mathsf{\Lambda}}{6}$ |

$\mathtt{Asymp}.$ | 3.3 | $\frac{3}{2}(1+{\gamma}_{3})-1$ | $-\frac{1}{2}(1+{\gamma}_{3})\mathsf{\Lambda}$ |

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Tawfik, A.N.; Greiner, C.
Early Universe Thermodynamics and Evolution in Nonviscous and Viscous Strong and Electroweak Epochs: Possible Analytical Solutions. *Entropy* **2021**, *23*, 295.
https://doi.org/10.3390/e23030295

**AMA Style**

Tawfik AN, Greiner C.
Early Universe Thermodynamics and Evolution in Nonviscous and Viscous Strong and Electroweak Epochs: Possible Analytical Solutions. *Entropy*. 2021; 23(3):295.
https://doi.org/10.3390/e23030295

**Chicago/Turabian Style**

Tawfik, Abdel Nasser, and Carsten Greiner.
2021. "Early Universe Thermodynamics and Evolution in Nonviscous and Viscous Strong and Electroweak Epochs: Possible Analytical Solutions" *Entropy* 23, no. 3: 295.
https://doi.org/10.3390/e23030295