#
On Spacetime Duality and Bounce Cosmology of a Dual Universe^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Extended Field Equations for a Conformally Curved Universe

## 3. Bounce from a Closed Early Universe

## 4. Evolution of Universe Worldlines

^{−1}∙Mpc

^{−1}and a phase transition of expansion at the Universe’s age of $~10$ Gyr were applied to tune the integration constants of the derived model, where the predicted energy density parameter is $~1.16$.

## 5. Spiral Galaxy Formation and Rotation under External Fields

## 6. Early Universe Boundary Contribution

## 7. Conclusions and Future Works

## Supplementary Materials

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The hypersphere of a positively curved early Universe plasma expansion upon the CMB emissions. ${r}_{p}$ is the reference radius of the intrinsic curvature and ${a}_{p}$ is the reference scale factor of the early Universe at the corresponding reference time ${t}_{p}$. ${\widehat{n}}_{u}$ and ${\overrightarrow{t}}_{v}$ are the normal and tangential vectors on the manifold boundary respectively regarding the extrinsic curvature.

**Figure 2.**(

**a**) Evolution of the wavefunction of matter/antimatter of one side of the Universe, and a radiation only wavefunction, in addition to the straight line of light cone (diagram is not to scale). (

**b**) The evolution of Hubble parameter, $H$, and its rate. The orange curves show a deceleration during the first $~10$ Gyr followed by an accelerated expansion rate.

**Figure 3.**(

**a**) A 2D schematic of the predicted cosmic topology of both sides at the first phase away from the early plasma, while the second phase corresponds to the reversal of the expansion direction. The future third phase corresponds to a spatial contraction, leading to a Big Crunch. (

**b**) The apparent topology during the first and second phases caused by gravitational lensing effects.

**Figure 4.**Schematic of the 3D spatial and 1D temporal dimensions of both sides, according to the wavefunction of space-time worldlines. (

**a**) In the first phase, both sides expand away from the early plasma. (

**b**) In the second stage, both sides expand in reverse directions and free-fall towards each other under gravitational acceleration. In the third phase, both sides contract leading to the big Crunch. Blue circles represent 3D slices of the Universe that are not necessarily simple path connected slices.

**Figure 5.**The hypersphere of a compact core of a galaxy (the red-orange 4D hypersphere representing the local relativistic space-time of the cloud forming a galaxy) along its travel and spin through the conformal space-time (the blue-purple 4D independent background) of a pre-existing curvature that evolves over cosmic time.

**Figure 6.**(

**a**) External fields exerted on a galaxy as it travels through conformally curved space-time. Green curves represent the divergence in the conformal space-time curvature over cosmic time. Blue curves represent the simulated space-time continuum flux. (

**b**) Simulation of a spiral galaxy rotation under external fields, where the blue represents the slowest tangential speeds, and red represents the fastest speeds.

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**MDPI and ACS Style**

Al-Fadhli, M.B.
On Spacetime Duality and Bounce Cosmology of a Dual Universe. *Phys. Sci. Forum* **2021**, *2*, 61.
https://doi.org/10.3390/ECU2021-09291

**AMA Style**

Al-Fadhli MB.
On Spacetime Duality and Bounce Cosmology of a Dual Universe. *Physical Sciences Forum*. 2021; 2(1):61.
https://doi.org/10.3390/ECU2021-09291

**Chicago/Turabian Style**

Al-Fadhli, Mohammed B.
2021. "On Spacetime Duality and Bounce Cosmology of a Dual Universe" *Physical Sciences Forum* 2, no. 1: 61.
https://doi.org/10.3390/ECU2021-09291