Probabilities in the landscape: The decay of nearly flat space

TL;DR

The paper investigates how probabilities are defined in eternal inflation landscapes by examining Coleman–De Luccia tunneling with gravity. It demonstrates that the decay rate from false vacua varies continuously as the false-vacuum energy $V_{\rm F}$ crosses zero, ruling out anomalous stabilization of near-flat space and arguing that negative and zero cosmological constant sectors cannot be neglected. Through analytic control of singular CDL solutions, a thorough classification of the CDL solution space, and comprehensive numerical evidence, the authors show that regular instantons appear and disappear in a structured way, with the number of field passes changing by one at transitions. The results challenge proposals that extremely long-lived low-energy de Sitter vacua dominate dynamics and support a more nuanced picture of the landscape where all vacua, including those with $\Lambda\le 0$, contribute to cosmological probabilities. Overall, the work clarifies the global structure of CDL tunneling and its implications for predicting observed vacua in the string landscape.

Abstract

We discuss aspects of the problem of assigning probabilities in eternal inflation. In particular, we investigate a recent suggestion that the lowest energy de Sitter vacuum in the landscape is effectively stable. The associated proposal for probabilities would relegate lower energy vacua to unlikely excursions of a high entropy system. We note that it would also imply that the string theory landscape is experimentally ruled out. However, we extensively analyze the structure of the space of Coleman-De Luccia solutions, and we present analytic arguments, as well as numerical evidence, that the decay rate varies continuously as the false vacuum energy goes through zero. Hence, low-energy de Sitter vacua do not become anomalously stable; negative and zero cosmological constant regions cannot be neglected.

Figures (13)

• Figure 1: The Euclidean potentials we are interested range from that shown in (a) to that in (b).
• Figure 2: A diagram showing solutions with starting point $\phi(0)$ at the regular pole, in a potential shifted vertically by $V_{\rm F}$. The numbers indicate the number of passes for each solution before the field escapes to infinity. The lines divide regions with different numbers of passes and represent regular solutions. In this example, tunneling is allowed for $V_{\rm F} = 0$. The instanton of interest is the thick line. For positive $V_{\rm F}$, the instanton is compact, and the part of the line at negative $\phi(0)$ represents the same solution, thinking of the opposite pole as the starting point. For $V_{\rm F} \leq 0$, the instanton becomes noncompact so there is only one origin of polar coordinates.
• Figure 3: Here, tunneling is forbidden for $V_{\rm F} \leq 0$. The single-pass instanton is the thick line; for a given $V_{\rm F}$, there are two values of $\phi(0)$ which represent the field values at the two poles. For negative $V_{\rm F}$ the instanton remains compact, so it no longer mediates decay.
• Figure 4: A graph of the potential $v(x)$ for $b=1$, and $z = \{1,0,-1\}$.
• Figure 5: A graph of the potential $v(x)$ for $b=0.3$, and $z = \{1,0,-1\}$.