Oscillating bounce solutions and vacuum tunneling in de Sitter spacetime

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 $k_{\mathrm{m}\mathrm{a}\mathrm{x}}$ 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.