Oscillating Fubini instantons in curved space

A Fubini instanton is a bounce solution which describes the decay of a vacuum state located at the top of the tachyonic potential {\it via} the tunneling without a barrier. We investigate various types of Fubini instantons of a self-gravitating scalar field under a tachyonic quartic potential. With gravity taken into account, we show there exist various types of unexpected solutions including oscillating bounce solutions. We present numerically oscillating Fubini bounce solutions in anti-de Sitter and de Sitter spaces. We construct the parametric phase diagrams of the solutions, which is the extension of our previous work. Of particular significance is that there always exist solutions in all parameter spaces in anti-de Sitter space. The regions are divided depending on the number of oscillations. On the other hand, de Sitter space allows solutions with codimension-one in parameter spaces. We numerically evaluate semiclassical exponents which give the finite tunneling probabilities.