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Quantum Gravitational Non-Singular Tunneling Wavefunction Proposal^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. k = 1 Loop Quantum Cosmology: Effective Dynamics

## 3. Effective Minisuperspace Potential

#### 3.1. $\tilde{B}\left(v\right)=1$

#### 3.2. $\tilde{B}\left(v\right)\ne 1$

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Effective minisuperspace potential for Wheeler–DeWitt quantum cosmology. We set $G=1$ and $\mathrm{\Lambda}=0.03$.

**Figure 2.**Effective minisuperspace potential, including holonomy and inverse scale factor corrections (just $\tilde{A}\left(v\right)$ term), and schematic behavior of Hartle-Hawking wavefunction for $\mathrm{\Lambda}=0.03$ and ${p}_{\varphi}=600$ (

**top**), and $\mathrm{\Lambda}=11$ and ${p}_{\varphi}=35$ (

**bottom**). Points B and R denote the bounce and recollapse turnaround points, respectively.

**Figure 3.**Effective minisuperspace potential, including holonomy and inverse scale factor corrections (just $\tilde{A}\left(v\right)$ term), and the schematic behavior of the Hartle–Hawking wavefunction for $\mathrm{\Lambda}=0.2$ and ${p}_{\varphi}=110$ (

**top**), and $\mathrm{\Lambda}=11$ and ${p}_{\varphi}=20$ (

**bottom**). Points B and R denote the bounce and recollapse turnaround points, respectively.

**Figure 4.**Effective minisuperspace potential, including both holonomy and inverse scale factor corrections (both $\tilde{A}\left(v\right)$ and $\tilde{B}\left(v\right)$ terms), and the schematic behavior of the Hartle–Hawking wavefunction (

**top**) and the Vilenkin wavefunction (

**bottom**) for $\mathrm{\Lambda}=0.03$ and ${p}_{\varphi}=2$. Point B denotes the classical bounce turnaround point.

**Figure 5.**Effective minisuperspace potential, including both holonomy and inverse scale factor corrections (both $\tilde{A}\left(v\right)$ and $\tilde{B}\left(v\right)$ terms), and the schematic behavior of the Hartle–Hawking wavefunction (

**top**) and the Vilenkin wave–function (

**bottom**) for $\mathrm{\Lambda}=13$ and ${p}_{\varphi}=3$. Points B and R denote the classical bounce and recollapse turnaround points, respectively.

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**MDPI and ACS Style**

Motaharfar, M.; Singh, P.
Quantum Gravitational Non-Singular Tunneling Wavefunction Proposal. *Phys. Sci. Forum* **2023**, *7*, 44.
https://doi.org/10.3390/ECU2023-14101

**AMA Style**

Motaharfar M, Singh P.
Quantum Gravitational Non-Singular Tunneling Wavefunction Proposal. *Physical Sciences Forum*. 2023; 7(1):44.
https://doi.org/10.3390/ECU2023-14101

**Chicago/Turabian Style**

Motaharfar, Meysam, and Parampreet Singh.
2023. "Quantum Gravitational Non-Singular Tunneling Wavefunction Proposal" *Physical Sciences Forum* 7, no. 1: 44.
https://doi.org/10.3390/ECU2023-14101