# On the Geometry of No-Boundary Instantons in Loop Quantum Cosmology

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## Abstract

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## 1. Introduction

## 2. The Hartle-Hawking Proposal Revisited

#### 2.1. The Wheeler-de Witt Equation and Boundary Conditions

- The Hartle-Hawking proposal [2]—If we choose the exponentially growing mode for $a<1/\sqrt{\overline{V}}$, then the wave function becomes a superposition of in-going and out-going modes for $a>1/\sqrt{\overline{V}}$.
- The tunneling proposal [19]—If we choose the out-going mode for $a>1/\sqrt{\overline{V}}$, then the wave function becomes a superposition of growing and decaying modes for $a<1/\sqrt{\overline{V}}$.

#### 2.2. Euclidean Path Integral

- The Euclidean path integral can be interpreted as the partition function of a thermal system [28]$$\begin{array}{c}\hfill Z=\mathrm{Tr}exp\left(-\beta \widehat{\mathcal{H}}\right)=\int \mathcal{D}\left[g\right]\mathcal{D}\left[\varphi \right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{e}^{-{S}_{\mathrm{E}}[g,\varphi ]},\end{array}$$$$\begin{array}{c}\hfill Z=exp\left(-\beta \mathcal{F}\right)\simeq exp\left(+\frac{3}{8{V}_{0}}\right)={e}^{\mathcal{A}/4},\end{array}$$
- Building on this result, one may consider other semi-classical effects in dS space as well. The classical (Lorentzian) equation of motion for a scalar field on a fixed dS background is given by$$\begin{array}{c}\hfill \ddot{\varphi}=-3H\dot{\varphi}-{V}^{\prime}\phantom{\rule{0.166667em}{0ex}},\end{array}$$$$\begin{array}{c}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\varphi =-\frac{{V}^{\prime}}{3H}+\frac{{H}^{3/2}}{2\pi}\xi \left(t\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$$$$\begin{array}{ccc}\hfill \langle \xi \left(t\right)\rangle & =& 0\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \langle \xi \left(t\right)\xi \left({t}^{\prime}\right)\rangle & =& \delta (t-{t}^{\prime})\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$Then, the probability to have the field value $\varphi $ at t will follow the Fokker-Planck equation [31]$$\begin{array}{c}\hfill \frac{\partial P(\varphi ,t)}{\partial t}=\frac{2\sqrt{2}}{3\sqrt{3\pi}}\frac{\partial}{\partial \varphi}\left[{V}^{3/4}\left(\varphi \right)\frac{\partial}{\partial \varphi}\left({V}^{3/4}\left(\varphi \right)P(\varphi ,t)\right)+\frac{3{V}^{\prime}\left(\varphi \right)}{8{V}^{1/2}\left(\varphi \right)}P(\varphi ,t)\right]\phantom{\rule{0.166667em}{0ex}},\end{array}$$$$\begin{array}{c}\hfill \frac{\partial P(\varphi ,t|\chi )}{\partial t}=\frac{2\sqrt{2}}{3\sqrt{3\pi}}\left[{V}^{3/4}\left(\chi \right)\frac{\partial}{\partial \chi}\left({V}^{3/4}\left(\chi \right)\frac{\partial P(\varphi ,t|\chi )}{\partial \chi}\right)-\frac{3{V}^{\prime}\left(\chi \right)}{8{V}^{1/2}\left(\chi \right)}\frac{\partial P(\varphi ,t|\chi )}{\partial \chi}\right].\end{array}$$In the static limit, a solution that satisfies both of these equations is given by$$\begin{array}{c}\hfill P(\varphi ,t|\chi )\sim {V}^{-3/4}\left(\varphi \right)exp\left[\frac{3}{8V\left(\varphi \right)}-\frac{3}{8V\left(\chi \right)}\right]\phantom{\rule{0.166667em}{0ex}}.\end{array}$$We can interpret this as the tunneling probability of a homogeneous part of a universe that tunnels from the field value $\chi $ to $\varphi $ via stochastic quantum fluctuations when the wavelength is of the order of the Hubble radius.This wave function is consistent with the Euclidean path integral approximated by the Hawking-Moss instantons [32,33]. On further normalizing the initial boundary, one can obtain the no-boundary wave function. Therefore, we can conclude that the Euclidean path integral describes the stationary limit, or thermal equilibrium, of quantum fluctuations of the Hubble-scale wavelength modes consistently, whereas in many situations, such a thermal equilibrium is coincident with the ground state of the system [30].

#### 2.3. Semiclassical Approximation: Instanton Solutions

## 3. Geometry of the Hartle-Hawking Instantons

## 4. No-Boundary Instantons in LQC

#### Numerical Results

## 5. Robustness of the No-Boundary Condition

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | ‘Kinematic’ here refers to the fact that the Gauss and the (spatial) diffeomorphism constraints have been solved whereas ‘physical’ would imply the solution of the Hamiltonian constraint as well. For a minisuperspace model, these distinctions are not very important since the only leftover symmetry of the system is time-reparameterization invariance. |

2. | Note that we choose this normalization at this point for historical reasons and to keep the resulting equations simple. However, we shall change this normalization later on to facilitate comparison with LQC. |

3. | The reason why one traditionally has to consider a $k=1$ universe is because it is the only case which provides a potential barrier for the tunneling of the universe from nothing. In other words, from the Friedmann equation, only for the $k=1$ case can the right hand side go to zero for certain choices of matter (a flat potential or a pure cosmological constant). However, this can be generalized for LQC since one gets a “bounce” in all types of tolopologies for a FLRW universe. We intend to establish this generalization in future work. |

4. | Note the new normalization chosen here to facilitate comparison with LQC later on. |

**Figure 1.**

**Left**: Euclidean path integral that connects from the initial state to the final state.

**Right**: If two states are disconnected at the Euclidean manifold, one can consider a wave function for the only final state.

**Figure 2.**

**Left**: A typical time contour over the complex time, where $X=\pi /2{H}_{0}$.

**Right**: Euclidean and Lorentzian manifold along the given time contour.

**Figure 4.**The black curve is an example of ${\dot{a}}^{2}$ for $\mathrm{\Lambda}=1$, $G=1$, and ${l}_{Pl}=0.1$, where the red dashed curve is the limit of the Einstein gravity with the same ${a}_{\mathrm{max}}$ that satisfies ${\dot{a}}_{\mathrm{max}}=0$. Right is the behavior near the $a=0$ limit.

**Figure 5.**

**Left**: $a\left(\tau \right)$ (red dashed curve is the limit of the Einstein gravity).

**Right**: $loga$ for small a limit. It has an infinitely long throat.

**Figure 6.**

**Left**: ${S}_{\mathrm{E}}/\left|{S}_{\mathrm{E}}^{\mathrm{GR}}\right|$, where for $G=1$, and ${l}_{Pl}=0.1$ by varying $\mathrm{\Lambda}$ (equivalently, varying ${\mathrm{\Lambda}}_{\mathrm{eff}}\equiv 1/{a}_{\mathrm{max}}^{2}$), where ${S}_{\mathrm{E}}^{\mathrm{GR}}$ is the Euclidean action for the corresponding Einstein limit.

**Right**: ${S}_{\mathrm{E}}$ for the same parameters.

**Figure 7.**

**Left**: By varying ${l}_{Pl}$, one can calculate $d\propto {l}_{Pl}^{2}$ numerically.

**Right**: By varying $\gamma $ (with ${l}_{Pl}=0.1$), one can see a linear dependence $d\propto 1/\gamma $. We can numerically conclude that $d\simeq 8.7\times {l}_{Pl}^{2}/\gamma $.

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Brahma, S.; Yeom, D.-h.
On the Geometry of No-Boundary Instantons in Loop Quantum Cosmology. *Universe* **2019**, *5*, 22.
https://doi.org/10.3390/universe5010022

**AMA Style**

Brahma S, Yeom D-h.
On the Geometry of No-Boundary Instantons in Loop Quantum Cosmology. *Universe*. 2019; 5(1):22.
https://doi.org/10.3390/universe5010022

**Chicago/Turabian Style**

Brahma, Suddhasattwa, and Dong-han Yeom.
2019. "On the Geometry of No-Boundary Instantons in Loop Quantum Cosmology" *Universe* 5, no. 1: 22.
https://doi.org/10.3390/universe5010022