Euclidean action for vacuum decay in a de Sitter universe

Abstract

The behavior of the action of the instantons describing vacuum decay in a de Sitter is investigated. For a near-to-limit instanton (a Coleman-de Luccia instanton close to some Hawking-Moss instanton) we find approximate formulas for the Euclidean action by expanding the scalar field and the metric of the instanton in the powers of the scalar field amplitude. The order of the magnitude of the correction to the Hawking-Moss action depends on the order of the instanton (the number of crossings of the barrier by the scalar field): for instantons of odd and even orders the correction is of the fourth and third order in the scalar field amplitude, respectively. If a near-to-limit instanton of the first order exists in a potential with the curvature at the top of the barrier greater than 4 $\times$ (Hubble constant)$^2$, which is the case if the fourth derivative of the potential at the top of the barrier is greater than some negative limit value, the action of the instanton is less than the Hawking-Moss action and, consequently, the instanton determines the outcome of the vacuum decay if no other Coleman-de Luccia instanton is admitted by the potential. A numerical study shows that for the quartic potential the physical mode of the vacuum decay is given by the Coleman-de Luccia instanton of the first order also in the region of parameters in which the potential admits two instantons of the second order.Comment: 16 pages, 3 figures, references adde

(Higgs) vacuum decay during inflation

“…(3.2). In particular, if V (2) (hmax) vanishes, it gets replaced by an average around the top of the barrier [30][31][32]. Figure 3.…”

False vacuum decay in a brane world cosmological model

“…we have the explicit approximative formulas for the instanton and its action, [18] 3 False vacuum decay on a brane -elementary discussion of the instanton equations…”

“…First of all: this difference vanishes as σ → ∞ as it must be because in Einstein's relativity theory the difference between the actions of our CdL instanton and related HM instanton is of the fourth-order in k as it is shown in [18]. For positive k 2 the difference (25) is negative (this coincides with the general-relativistic result [18]), this means that the vacuum decay via our CdL instanton is more probable than the decay via HM instanton.…”

“…Motivated by the earlier works [15] and [16] the authors of the papers [17], [18] and [19] have achieved information about the solution(s) of the instanton equations interesting for us:…”

“…It is obvious that for V M /σ ≪ 1 the theory of vacuum decay on our brane reduces to the theory of vacuum decay in four-dimensional space-time. The first-order CdL instanton plays the most important role in the Einstein's theory of gravity [16], [18], therefore we write down explicitly:…”