Universality of the Volume Bound in Slow-Roll Eternal Inflation

TL;DR

The paper proves a universal bound on the total volume produced after slow-roll inflation, showing that the probability of generating a finite volume larger than $e^{S_{dS}/2}$ vanishes up to nonperturbative quantum gravity effects, with the bound saturated at the eternal-inflation phase transition ($\Omega=1$). It demonstrates the bound’s universality across spacetime dimensionality, robustness to higher-derivative corrections (via Wald entropy and modified inflaton kinetics), and validity in multifield inflation by formulating a functional Laplace transform framework for the volume distribution. The authors test the bound in two explicit multifield scenarios (waterfall and tilted waterfall), finding that the presence of extra fields generally strengthens the bound and that saturation occurs only in the effectively single-field limit. These results support a holographic/thermodynamic perspective on inflationary volume growth and provide a formalism to study vacuum-population questions in the landscape.

Abstract

It has recently been shown that in single field slow-roll inflation the total volume cannot grow by a factor larger than e^(S_dS/2) without becoming infinite. The bound is saturated exactly at the phase transition to eternal inflation where the probability to produce infinite volume becomes non zero. We show that the bound holds sharply also in any space-time dimensions, when arbitrary higher-dimensional operators are included and in the multi-field inflationary case. The relation with the entropy of de Sitter and the universality of the bound strengthen the case for a deeper holographic interpretation. As a spin-off we provide the formalism to compute the probability distribution of the volume after inflation for generic multi-field models, which might help to address questions about the population of vacua of the landscape during slow-roll inflation.

Figures (4)

• Figure 1: The stretched horizon (in blue) in de Sitter space, with the unit vectors $u^\mu\propto \xi^\mu$ and $n^\mu$ and the region $\Upsilon$ (in red) spanned by the area section $\Sigma(t)$, as defined in the text.
• Figure 2: Depending on the starting point, inflaton trajectories in field space (in magenta) drift towards different points of the reheating surface (in red), which in general correspond to different vacua. The field space is often confined by boundary regions (in blue) where, for instance, the energy density of the scalar potential become Planckian.
• Figure 3: Average volume distribution $\langle V(y_r)\rangle$ as a function of $y_r$ for the two inflationary examples: the waterfall (left) and the tilted waterfall (right). The red-dashed lines refer to the classical evolution. Near the classical limit, for large $\Omega$, the distribution (in blue) is peaked around the classical exit point. Near the phase transition to eternal inflation, at $\Omega_x \gtrsim 1$, the distribution (in magenta) broadens (left) and drifts towards smaller values of $y_r$ (right).
• Figure 4: Each node corresponds to the insertion of a Riemann tensor, each link to the contraction of a pair of indices. Closed orientable loops can be identified such that all nodes will end up having an equal number of incoming and outgoing lines. Hence any scalar contraction of Riemann tensors can be written just in terms of Riemann tensors with two upper and two lower indices contracted without using the metric.