# Bounce and Stability in the Early Cosmology with Anomaly-Induced Corrections

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## Abstract

**:**

## 1. Introduction

## 2. Anomaly-Induced Action and the Early Universe

## 3. De Sitter-Like Solutions and Their Stability

## 4. Phase Diagrams

## 5. Bounce and the Stability Analysis

## 6. Conclusions and Discussions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Phase diagram for Equation (44) with the MSSM particle contents. In the left plot we choose $\mathsf{\Lambda}=0.001{M}_{P}^{2}$ and in the right plot, $\mathsf{\Lambda}=0.5{M}_{P}^{2}$.

**Figure 2.**Phase diagram for Equation (44) with the MSM particle contents, corresponding to the unstable exponential solution. In the left plot there is $\mathsf{\Lambda}=0.001{M}_{P}^{2}$ and in the right plot, $\mathsf{\Lambda}=0.5{M}_{P}^{2}$.

**Figure 3.**Solution without radiation for the MSSM particles contents and hence $c>0$. The initial conditions are $\sigma \left(0\right)=0$, $\dot{\sigma}\left(0\right)=-{10}^{-4}\phantom{\rule{0.166667em}{0ex}}{H}_{2}$, $\ddot{\sigma}\left(0\right)=0$ and $\stackrel{\u20db}{\sigma}\left(0\right)=0$. On the left plot we show the interval $-100\le t\le 100$ in the Planck units and on the right plot the interval is ten times smaller.

**Figure 4.**Numerical solution for the conformal factor $\sigma \left(t\right)$ with the small variation in the initial conditions at $t=0$. The left plot shows the behavior of the conformal factor under variations of $\dot{\sigma}\left(0\right)$ and $\stackrel{\u20db}{\sigma}\left(0\right)$. In this case, the initial conditions become $\sigma \left(0\right)=0$, $\dot{\sigma}\left(0\right)=-0.1{H}_{2}$, $\ddot{\sigma}\left(0\right)=0$ and $\stackrel{\u20db}{\sigma}\left(0\right)=0.05$. In the right plot, the variation was performed as $\ddot{\sigma}\left(0\right)$, so that the new initial conditions are $\sigma \left(0\right)=0$, $\dot{\sigma}\left(0\right)=-{10}^{-4}{H}_{2}$, $\ddot{\sigma}\left(0\right)=0.6$ and $\stackrel{\u20db}{\sigma}\left(0\right)=0$. In both cases the bounce solution is maintained.

**Figure 5.**Numerical solution for the conformal factor $\sigma \left(t\right)$ with the non-zero cosmological constant density. In the left graph, we consider $\mathsf{\Lambda}=0.001{H}_{2}^{2}$ and the initial conditions $\sigma \left(0\right)=0$, $\dot{\sigma}\left(0\right)=-0.1{H}_{2}$, $\ddot{\sigma}\left(0\right)=0$ and $\stackrel{\u20db}{\sigma}\left(0\right)=0$. In the right plot, the value is larger, $\mathsf{\Lambda}=0.1{H}_{2}^{2}$ and the initial conditions are $\sigma \left(0\right)=0$, $\dot{\sigma}\left(0\right)=-0.01{H}_{2}$, $\ddot{\sigma}\left(0\right)=0.1$ and $\stackrel{\u20db}{\sigma}\left(0\right)=0$.

**Figure 6.**Numerical solution for the conformal factor $\sigma \left(t\right)$ in the presence of the anomalous radiation term. Here, we use the value $\beta {\overline{F}}^{2}=-0.1$ and the following initial conditions: $\sigma \left(0\right)=1$, $\dot{\sigma}\left(0\right)=-0.01{H}_{0}$, $\ddot{\sigma}\left(0\right)=0.1$, and $\stackrel{\u20db}{\sigma}\left(0\right)=0$. In the left plot, we show the range $-50\le t\le 50$ in Planck units and in the right plot the range is ten times smaller. One can observed that, even considering the quantum contribution to radiation, the bounce solution is still present.

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Silva, W.C.e.; Shapiro, I.L.
Bounce and Stability in the Early Cosmology with Anomaly-Induced Corrections. *Symmetry* **2021**, *13*, 50.
https://doi.org/10.3390/sym13010050

**AMA Style**

Silva WCe, Shapiro IL.
Bounce and Stability in the Early Cosmology with Anomaly-Induced Corrections. *Symmetry*. 2021; 13(1):50.
https://doi.org/10.3390/sym13010050

**Chicago/Turabian Style**

Silva, Wagno Cesar e, and Ilya L. Shapiro.
2021. "Bounce and Stability in the Early Cosmology with Anomaly-Induced Corrections" *Symmetry* 13, no. 1: 50.
https://doi.org/10.3390/sym13010050