Oscillating bounce solutions and vacuum tunneling in de Sitter spacetime

TL;DR

This work identifies and analyzes a new class of oscillating bounce solutions in Euclidean gravity with a scalar field in a de Sitter background. By combining fixed-background approximations, linearized analyses, nonlinear extensions, and targeted numerics, the authors show that the number of barrier-crossings k is finite for typical potentials and is controlled by the curvature of the potential at the barrier top via the parameter $\beta = |V''(\phi_{\rm top})|/H^2$, with $k_{\max}$ determined by discrete thresholds $\beta > N(N+3)$. For unusually flat barriers, an averaged curvature criterion $\gamma$ governs the appearance of higher-k solutions, indicating that no universal lower bound on $|V''(\phi_{\rm top})|/H^2$ suffices. The results illuminate the spectrum of vacuum-decay channels in de Sitter space, showing a richer structure than Coleman-De Luccia or Hawking-Moss, and highlight the interplay between quantum tunneling and thermal effects in gravitational settings.

Abstract

We study a class of oscillating bounce solutions to the Euclidean field equations for gravity coupled to a scalar field theory with two, possibly degenerate, vacua. In these solutions the scalar field crosses the top of the potential barrier $k>1$ times. Using analytic and numerical methods, we examine how the maximum allowed value of $k$ depends on the parameters of the theory. For a wide class of potentials $k_{\rm max}$ is determined by the value of the second derivative of the scalar field potential at the top of the barrier. However, in other cases, such as potentials with relatively flat barriers, the determining parameter appears instead to be the value of this second derivative averaged over the width of the barrier. As a byproduct, we gain additional insight into the conditions under which a Coleman-De Luccia bounce exists. We discuss the physical interpretation of these solutions and their implications for vacuum tunneling transitions in de Sitter spacetime.

Figures (11)

• Figure 1: The potential for a typical theory with a false vacuum.
• Figure 2: A Coleman-De Luccia bounce solution for the case where the flat space bounce radius is much less than $H_{\rm f}^{-1}$. The picture should be visualized as a four-sphere viewed head-on, with $\xi=0$ being the point at the center of the cross-hatched region where the field is on the true vacuum side of the potential barrier; $\xi=\xi_{\rm max}$ is the antipodal point on the opposite side of the sphere. The dashed line passing through $\xi=0$ (and also through $\xi=\xi_{\rm max}$) denotes a three-sphere corresponding to the spatial hypersurface on which the bubble materializes. The three-sphere denoted by the lower dashed line is roughly analogous to the false vacuum initial-state hypersurface in the flat space problem.
• Figure 3: A schematic plot of the starting points of bounce solutions to a $\lambda > 0$ theory, such as that discussed in Sec. V, with $\beta$ slightly greater than 28. The numbers next to the points represent the number of times the solution crosses the top of the potential barrier.
• Figure 4: Bounce solutions for the asymmetric ($g = {1 \over 2 \sqrt{2}}$) potential discussed in the text. For these solutions $\beta = 70.03$.
• Figure 5: Bounce solutions for the symmetric ($g=0$) potential with two degenerate vacua that is discussed in the text. Again, $\beta = 70.03$.