# Cosmological Solutions for the Geometrical Scalar-Tensor with the Potential Determined by the Noether Symmetry Approach

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## Abstract

**:**

## 1. Introduction

## 2. The Model and the Weyl Transformations

## 3. Pointlike Lagrangian and FRW and Klein-Gordon Equations

## 4. Noether Symmetry

## 5. Solutions of the Field Equations

#### 5.1. Case $\lambda <0$:

#### 5.2. Case $\lambda >0$:

## 6. Cosmological Solutions in Each Frame

#### 6.1. Jordan Frame

#### 6.2. Einstein Frame

## 7. A Particular Case: $\mathbf{\omega}=\mathbf{1}/\mathbf{2}$

#### Jordan Frame vs. Einstein Frame

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FRW | Friedmann-Robertson-Walker |

ST | Scalar-Tensor |

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**Figure 1.**Scale factors ${a}_{\lambda <0}^{(-)}$ and ${\overline{a}}_{\lambda <0}^{(+)}$ as functions of time $\sqrt{\Lambda t}$.

**Figure 2.**Scale factor velocities ${\dot{a}}_{\lambda <0}^{(-)}/\sqrt{\Lambda}$ and ${\dot{\overline{a}}}_{\lambda <0}^{(+)}/\sqrt{\Lambda}$ as functions of time $\sqrt{\Lambda t}$.

**Figure 3.**Deceleration parameters ${q}_{\lambda <0}^{(+)}$ and ${\overline{q}}_{\lambda <0}^{(-)}$ as functions of time $\sqrt{\Lambda t}$.

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**MDPI and ACS Style**

Barreto, A.B.; Kremer, G.M.
Cosmological Solutions for the Geometrical Scalar-Tensor with the Potential Determined by the Noether Symmetry Approach. *Symmetry* **2020**, *12*, 1110.
https://doi.org/10.3390/sym12071110

**AMA Style**

Barreto AB, Kremer GM.
Cosmological Solutions for the Geometrical Scalar-Tensor with the Potential Determined by the Noether Symmetry Approach. *Symmetry*. 2020; 12(7):1110.
https://doi.org/10.3390/sym12071110

**Chicago/Turabian Style**

Barreto, Adriano B., and Gilberto M. Kremer.
2020. "Cosmological Solutions for the Geometrical Scalar-Tensor with the Potential Determined by the Noether Symmetry Approach" *Symmetry* 12, no. 7: 1110.
https://doi.org/10.3390/sym12071110