Nucleating an Inflationary Universe: Euclidean Wormholes and their No-Boundary Limit

Abstract

No-boundary instantons and Euclidean "wineglass" wormholes have both been proposed as providing suitable initial conditions for the current expanding phase of our universe, and in particular for providing conditions that are favorable to an inflationary phase. These finite action solutions have generally been regarded as unrelated, and enacting different scenarios - in one case the creation of spacetime from nothing, and in the other up-tunneling from a Euclidean Anti-de Sitter vacuum. By studying explicit solutions of both axionic and magnetic wineglass wormholes, we find that in the zero-charge limit the throat of the wormholes pinches off, leaving a no-boundary instanton that disconnects from the asymptotic Anti-de Sitter region. Thus wormholes and no-boundary instantons are part of a common family of Euclidean solutions. Along the way, we resolve the long-known puzzle that the action of wineglass wormholes can become negative. Moreover, small-charge wormholes lead to a longer inflationary phase than large-charge solutions, while no-boundary instantons dominate the probability distribution overall.

Figures (6)

• Figure 1: Left: A wineglass wormhole interpolating between EAdS and a local maximum of the scale factor. Upon analytic continuation this leads to an expanding universe (indicated by the dashed lines). Right: As the axionic or magnetic charge tends to zero, the throat of the wormhole pinches off, leaving a disconnected no-boundary instanton.
• Figure 2: The scalar potential $V(\phi)$ contains two negative AdS extrema and one maximum at positive values. The range of our wormhole solutions is indicated in orange -- these interpolate from either of the AdS extrema to a location just over the potential barrier provided by the positive maximum.
• Figure 3: Explicit examples of magnetically charged wineglass wormholes, with $Q_m=4$ (dot-dashed line interpolating to AdS maximum from $\phi_0=-8.64505963673769,$ dashed line interpolating to AdS minimum from $\phi_0=-8.64424753320004$) and $Q_m=0.1$ (solid line, $\phi_0=-8.21540632864282,$ interpolating to the AdS minimum). The evolution of the scalar field is shown on the left (the locations of the potential extrema are indicated by the light gray lines), and that of the scale factor on the right.
• Figure 4: Optimized values of the scalar field at $\tau=0,$ as a function of the axionic or magnetic charge. The solid lines are for solutions reaching the AdS minimum, and the dashed for those reaching the AdS maximum.
• Figure 5: Sizes of the "mouth" $a_0$ and the "throat" $a_{throat}$ of a family of magnetic wormhole solutions interpolating to the AdS minimum, as a function of the charge.