Quantum Creation of an Open Inflationary Universe

TL;DR

The paper critically compares the Hartle–Hawking and tunneling wave functions in the context of quantum creation of open inflationary universes, arguing that HH favors small $\Omega$ and often fails to describe creation itself, whereas the tunneling approach more naturally yields inflation with $\Omega \approx 1$ in typical chaotic-inflation settings. It connects these viewpoints to the stochastic inflation framework, clarifying how stationary diffusion distributions relate to but do not directly encode creation probabilities. It analyzes Hawking–Turok open-universe instantons and their dependence on wave-function choice, showing that, while HH predicts $\Omega\sim 0.01$ for open universes, the tunneling picture supports $\Omega=1$ in most models, with a few classes permitting $\Omega<1$. The discussion extends to models with a four-form field and to self-reproducing inflation, highlighting that the observable curvature outcome is highly model- and measure-dependent and that open-inflation scenarios remain an active area of investigation. Overall, the work emphasizes the need for careful interpretation of quantum cosmology formalisms when predicting the present-day curvature of the universe.

Abstract

We discuss a dramatic difference between the description of the quantum creation of an open universe using the Hartle-Hawking wave function and the tunneling wave function. Recently Hawking and Turok have found that the Hartle-Hawking wave function leads to a universe with Omega = 0.01, which is much smaller that the observed value of Omega > 0.3. Galaxies in such a universe would be about $10^{10^8}$ light years away from each other, so the universe would be practically structureless. We will argue that the Hartle-Hawking wave function does not describe the probability of the universe creation. If one uses the tunneling wave function for the description of creation of the universe, then in most inflationary models the universe should have Omega = 1, which agrees with the standard expectation that inflation makes the universe flat. The same result can be obtained in the theory of a self-reproducing inflationary universe, independently of the issue of initial conditions. However, there exist two classes of models where Omega may take any value, from Omega > 1 to Omega << 1.

Figures (4)

• Figure 1: A possible interpretation of the Hawking-Moss tunneling from $\phi_0$ to $\phi_1$.
• Figure 2: Tunneling from the minimum at $\phi_0$ occurs not to the points $\phi_2$ or $\phi_3$, which, according to the naive estimates based on the instanton action, would be much more probable, but to the nearby maximum at $\phi_1$.
• Figure 3: Instanton in the theory ${m^2\over 2} \phi^2$ describing creation of the universe with $\phi = 0.1 M_p$. In this case the scalar field rapidly changes in the Euclidean space, and the universe does not inflate at all after the tunneling. If one considers greater values of $\phi$ at $\tau = 0$, the scalar field becomes almost constant, but then it diverges logarithmically when $\tau$ approaches its maximal value.
• Figure 4: Instanton in hybrid inflation model based on supergravity, with $V(\phi) = M^4 (1 - Q\phi^2 + Q^2\phi^4)\, e^{Q\phi^2}$. Everything is expressed in Planck units; $M \sim 10^{-3}$, $Q = 4\pi$, see Ref. Riotto.