Fuzzy Euclidean wormholes in the inflationary universe

Abstract

In this paper, we investigate complex-valued Euclidean wormholes in the Starobinsky inflation. Due to the properties of the concave inflaton potential, the classicality condition at both ends of the wormhole can be satisfied, as long as the initial condition of the inflaton field is such that it is located sufficiently close to the hilltop. We compare the probabilities of classicalized wormholes with the Hartle–Hawking compact instantons and conclude that the Euclidean wormholes are probabilistically preferred than compact instantons, if the inflation lasts more than 50 e-foldings. Our result assumes that the Euclidean path integral is the correct effective description of quantum gravity. This opens a new window for various future investigations that can be either confirmed or refuted by future experiments.

Introduction

Understanding the origin of our universe is an important task in cosmology. From recent cosmological observations [1], we now confirm the existence of an inflation epoch in the very beginning of the universe [2], which is very well described by a concave inflaton potential exemplified by the Starobinsky model [3].

There remains, however, several problems. For example, the problems of the initial singularity [4] and the proper initial conditions that give rise to the inflation remain unresolved [5], [6]. It is reasonable to expect that a future full-blown quantum theory of gravity might be able to address these issues, based on which the observed cosmic microwave background (CMB) parameters would be shown as typical or natural consequence of such initial conditions.

In short of a final quantum gravity theory, in this paper we invoke the Euclidean path-integral as the wave function of the universe [7] to address the issue. The Euclidean path-integral has many nice properties including its satisfying the Wheeler–DeWitt equation [4], [7]. In addition, if one restricts the path-integral solutions to compact instantons only, that is, if one invokes the no-boundary proposal [7], then the wave function corresponds to the ground state and provides a consistent thermodynamical limit of the stochastic distribution of universes [8]. On the other hand, this ansatz has a weak point: the wave function exponentially prefers a small number of $e$-foldings of inflation that is in conflict with the observations [9].

Because of this reason, there have been some authors have raised doubts about the no-boundary proposal [9]. With that in mind, there have been attempts to attain large $e$-foldings within the Euclidean path-integral formalism. For example, Hartle, Hawking, and Hertog proposed to include the volume-weighting to the wave function as a means to extend the inflation $e$-foldings [5]. Other attempts include fine-tunings of inflation models [6], [10], modified gravity [11], and the introduction of an additional, more massive field [12].

In the present work, we will not specifically choose a certain hypothesis that explains the large $e$-foldings in the Euclidean path-integral approach. Rather, we will demonstrate that, so long as one imposes the restriction that the inflation must be lasted for more than 50 $e$-foldings, the wave function of the universe would prefer non-compact instantons over compact ones. The latter correspond to instantons proposed by Hartle and Hawking, while the former correspond to the so-called Euclidean wormholes [13], [14], [15]. This implies that our universe is more likely to have emerged from a Euclidean wormhole rather than a compact instanton, which in turn would give raise to several interesting implications.

This paper is organized as follows. In Section 2, we summarize our previous investigations about Euclidean wormholes. In Section 3, we consider the classicality condition in the Starobinsky inflation model, and calculate the probability distribution. We then compare the probability of wormholes to that of compact instantons. We conclude that, as long as one restricts duration of the inflation be longer than 50 $e$-foldings, then the Euclidean wave function prefers not compact but non-compact instantons. Finally, in Section 4, we summarize our results and comment on possible future topics.