# Cosmic Time and the Initial State of the Universe

## Abstract

**:**

## 1. The Cosmic Clock and Intrinsic Time Geometrodynamics

#### 1.1. Symplectic Potential of Geometrodynamics, and Generalized DeWitt Supermetric

#### 1.2. Generalized Hamiltonian Constraint of Geometrodynamics; and Global Intrinsic Time Interval

#### 1.3. Cosmic Time and the Reduced Physical Hamiltonian

## 2. Quantum Gravity as Schrödinger–Heisenberg Quantum Mechanics

## 3. Origin of the Universe

#### 3.1. Bekenstein–Hawking Black Hole Entropy and Penrose’s Weyl Curvature Hypothesis

#### 3.2. Hartle–Hawking No-Boundary Proposal

## 4. Chern–Simons Initial State

#### 4.1. Semiclassical Robertson–Walker Beginning

#### 4.2. A Hot Beginning

## 5. Quantum Origin as Chern–Simons Hartle–Hawking State

#### Relative Chern–Simons Functional

## 6. Quantum Metric Fluctuations and Two-Point Correlation Functions of the Chern–Simons Hartle–Hawking State

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CMC | constant mean curvature |

d.o.f. | degree(s) of freedom |

GR | General Relativity |

ITG | Intrinsic Time Geometrodynamics or Intrinsic Time Gravity |

TT | transverse traceless |

## Appendix A. Lichnerowicz–York Equation with Addition of Cotton–York Term and Penrose’s Weyl Curvature Hypothesis

## Appendix B. Gravitational Chern–Simons Functional, Cotton–York Tensor, and Pontryagin Invariant

## Notes

1 | Appendix A contains related discussions; and a comparison of York’s extrinsic time formulation and ITG can be found in Ref. [12]. |

2 | The inverse of ${\overline{q}}_{ij}$ is denoted by ${\overline{q}}^{ij}$; also $\int \phantom{\rule{0.166667em}{0ex}}\tilde{\pi}\delta ln{q}^{\frac{1}{3}}=\int \phantom{\rule{0.166667em}{0ex}}\frac{1}{3}\frac{\tilde{\pi}}{q}\delta q$. |

3 | The explicit commutation relations are [8] $\left[{\overline{q}}_{ij}\left(x\right),{\overline{\pi}}_{l}^{k}\left(y\right)\right]=i\hslash (\frac{1}{2}\left({\delta}_{m}^{i}{\overline{q}}_{jn}+{\delta}_{n}^{i}{\overline{q}}_{jm}\right)-\frac{1}{3}{\delta}_{j}^{i}{\overline{q}}_{mn}){\delta}^{3}(x,y)$; $\left[{\overline{\pi}}_{j}^{i}\left(x\right),{\overline{\pi}}_{l}^{k}\left(y\right)\right]=\frac{i\hslash}{2}\left({\delta}_{j}^{k}{\overline{\pi}}_{l}^{i}-{\delta}_{l}^{i}{\overline{\pi}}_{j}^{k}\right){\delta}^{3}(x,y)$. |

4 | |

5 | Time-dependence of a Hamiltonian is allowed in quantum mechanics; it does not spoil unitarity which is ensured by the self-adjointness of the Hamiltonian. |

6 | Even if all black holes were to eventually evaporate away after eons, it is possible a very small cosmological constant, ergo very large cosmological horizon, and entropy, remains. In a de Sitter manifold, the area of the cosmological horizon is inversely proportional to the value of the cosmological constant. |

7 | Notes on gravitational Chern–Simons functional, Cotton–York tensor, and Euclidean Pontryagin invariant which may be helpful to understanding the arguments are delegated to Appendix B. |

8 | It is known that every compact simply connected conformally flat manifold is conformally diffeomorphic to the n-sphere [43]. Grigori Perelman proved the Poincaré Conjecture, which says that every simply connected, closed three-manifold is homeomorphic to the three-sphere. Topological and differentiable isomorphism is the same in dimension three and below, so the generalized smooth Poincaré Conjecture that a manifold which is a homotopy three-sphere is, in fact, diffeomorphic to the standard three-sphere holds. |

9 | Explicitly, ${\left({Q}_{CS}\right)}_{j}^{i}{\Psi}_{CS}=\left({e}^{-g{W}_{CS}}{\overline{\pi}}_{j}^{i}{e}^{g{W}_{CS}}\right){\Psi}_{CS}={e}^{-g{W}_{CS}}{\overline{\pi}}_{j}^{i}{N}^{\prime}=0$, provided ${N}^{\prime}$ is a topological invariant which satisfies $\frac{\delta {N}^{\prime}}{\delta {\overline{q}}_{ij}}=0$. In the metric representation, the self-adjoint momentric operator can be expressed as [8] ${\overline{\pi}}_{i}^{j}=\frac{\hslash}{i}(\frac{1}{2}\left({\delta}_{m}^{i}{\overline{q}}_{jn}+{\delta}_{n}^{i}{\overline{q}}_{jm}\right)-\frac{1}{3}{\delta}_{j}^{i}{\overline{q}}_{mn})\frac{\delta}{\delta {\overline{q}}_{mn}}$. |

10 | This is analogous to the situation in the simple harmonic oscillator in which the action of the complex annihilation operator, a, on the vacuum state yields $\langle 0\left|a\right|0\rangle =0$, consequently $\langle 0\left|x\right|0\rangle =\langle 0\left|p\right|0\rangle =0$. |

11 | |

12 | $|{\Psi}_{CS}\rangle $ has a norm which is essentially just the partition function of the relative Chern–Simons action, $\langle {\Psi}_{CS}|{\Psi}_{CS}\rangle \propto Z=\int \mathfrak{D}\overline{q}\phantom{\rule{0.166667em}{0ex}}{e}^{-2g({W}_{CS}[\Gamma ]-{W}_{CS}\left[{\Gamma}_{o}\right])}$. |

13 | In usual practice, g is absorbed in the quadratic “propagator” term by redefinition, and $\frac{1}{\sqrt{g}}$ is considered to be “the coupling constant” in the interaction terms. So, in usual QFT language, a large value for g implies weak coupling, and vice versa. |

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**Figure 1.**Origin of the universe from Euclidean tunneling. (Semiclassical picture of the quantum Chern–Simons Hartle–Hawking state).

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Soo, C.
Cosmic Time and the Initial State of the Universe. *Universe* **2023**, *9*, 489.
https://doi.org/10.3390/universe9120489

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Soo C.
Cosmic Time and the Initial State of the Universe. *Universe*. 2023; 9(12):489.
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**Chicago/Turabian Style**

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2023. "Cosmic Time and the Initial State of the Universe" *Universe* 9, no. 12: 489.
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