Blanco-Pillado, Deng and Vilenkin argued that eternal inflation survives in 'swampy' string landscapes - where de Sitter vacua are rare and isolated by AdS/Minkowski minima, or absent altogether - through flyover transitions and the nucleation of inflating domain walls, with new inflating regions hidden inside black holes or AdS bubbles. The argument, however, is parametric: transition rates are known only to order of magnitude in the exponent, the energy bookkeeping of the flyover fluctuation is unexplained, the domain-wall multiverse has no volume measure, and the anthropic prediction of the cosmological constant rests on the untested condition that the number of de Sitter vacua satisfy NdS≫10284N_{dS}\gg 10^{284}. The open problem is to make this picture quantitative and predictive: controlled transition rates including prefactors and gravitational backreaction, clarification of the energy-compensation mechanism and of upward transitions from Minkowski vacua, a measure prescription yielding the prior Pprior(Λ)P_{\rm prior}(\Lambda) for the wall-dominated multiverse, and the de Sitter vacuum landscape statistics these require.

Background

In the string-landscape picture, quantum gravity admits an enormous number of metastable vacua, and inflation can populate all of them, producing a 'multiverse' in which low-energy parameters such as the cosmological constant Λ\Lambda vary between regions. Eternal inflation - inflation that never ends globally because inflating volume regenerates faster than it decays - makes this picture anthropically testable: Weinberg's successful prediction of a small nonzero Λ\Lambda requires weighting vacua by a probability measure. The 'measure problem' is that the infinite number of pocket universes must be regulated before any prediction can be made; one comparatively well-behaved regulator is the scale-factor cutoff measure, which counts events on hypersurfaces of constant cosmological scale factor.

The swampland program conjectures that consistent quantum gravity strongly restricts which effective theories exist. In particular, the (refined) de Sitter conjecture forbids, or severely limits, metastable de Sitter (dS) vacua - positive vacuum-energy states driving exponential expansion - and its refined form also excludes the sufficiently flat maxima and saddle points needed for hilltop eternal inflation. If these conjectures hold, or if dS vacua merely require strong fine-tuning and so are vastly outnumbered by anti-de Sitter (AdS, negative vacuum energy) and Minkowski (zero vacuum energy) minima, the landscape is 'swampy': inflation is possible only during slow roll on the slopes of the potential, and possibly only in rare dS pockets.

In such a landscape the standard machinery of vacuum transitions breaks down. Inter-vacuum transitions are usually described by Coleman-DeLuccia (CdL) instantons - Euclidean solutions mediating quantum decay of a metastable vacuum - or by Hawking-Moss jumps across shallow barriers. But if dS vacua are isolated by AdS or Minkowski minima, interpolating CdL instantons need not exist. Brown and Dahlen emphasized that transitions still occur through rare quantum fluctuations of the scalar-field velocity: a large fluctuation of ϕ˙\dot\phi in a finite region lets the field 'fly over' intervening barriers and minima and settle in a distant vacuum. Blanco-Pillado, Deng and Vilenkin quantified these 'flyover' transitions by combining the quantum probability of the initial velocity fluctuation with classical numerical evolution: for downward transitions the rate has the same parametric exponential form as tunneling, but upward transitions between dS vacua are parametrically enhanced, in tension with the standard thermal picture of de Sitter space. In flat spacetime, upward transitions are forbidden outright by energy conservation, and the energy bookkeeping of the initial fluctuation itself - its positive kinetic energy must be compensated by a negative vacuum-energy fluctuation - was left as an open mechanism.

Against this backdrop, Blanco-Pillado, Deng and Vilenkin ('Eternal Inflation in Swampy Landscapes', JCAP 2020) argued that eternal inflation survives even in swampy landscapes. Where dS vacua are rare, flyover transitions connect them, with the newly created inflating regions hidden inside black holes or AdS bubbles. Where no dS vacua exist, quantum creation from nothing and nucleation of inflating domain walls during slow roll seed infinitely many inflating open universes, again hidden behind horizons. However, the analysis is parametric: transition rates are estimated only to order of magnitude in the exponent (prefactors are omitted, and for potential differences ΔV\Delta V much larger than the parent vacuum energy no estimate at all is attempted because the initial fluctuation is strongly gravitating); the volume fractions are computed with rate equations that apply only 'roughly' when new inflating regions form inside black holes; and the wall-nucleation multiverse receives no measure treatment whatsoever. The anthropic prediction for Λ\Lambda then rests on the prior Pprior(Λ)P_{\rm prior}(\Lambda) under the scale-factor cutoff and on the condition ΔΛ fSM NdS≫Sˉ\Delta\Lambda\, f_{SM}\, N_{dS}\gg \bar S (with NdSN_{dS} the number of dS vacua, fSMf_{SM} the fraction with roughly Standard-Model microphysics, and Sˉ\bar S a maximal de Sitter entropy factor), yielding the conservative bound NdS≫10284N_{dS}\gg 10^{284} - a bound whose validity depends on the unknown statistical distribution and localization of dS vacua within the landscape.

Problem Statement

Turn the parametric picture of eternal inflation in swampy landscapes into a quantitative and predictive theory. The common objective is to decide whether - and what - anthropic predictions, beginning with the distribution of the observed cosmological constant Λ\Lambda, survive in landscapes compatible with the swampland conjectures. For string-inspired or controlled multi-field potentials in which dS vacua are rare and separated by AdS/Minkowski minima, or absent altogether, determine:

(a) Transition dynamics. Compute the inter-vacuum transition rates beyond leading-exponential accuracy: flyover exponents and tunneling/wall-nucleation instanton actions including prefactors and gravitational backreaction, in particular for large potential differences ΔV≫Vp\Delta V\gg V_p (with VpV_p the vacuum-energy density of the parent vacuum), where the initial fluctuation is strongly gravitating and the field can no longer be treated as free in the parent vacuum. Clarify the energy-compensation mechanism of the flyover initial condition - how the positive kinetic energy of the velocity fluctuation is balanced by a negative vacuum-energy fluctuation, especially near Minkowski limits - and determine under what circumstances upward transitions out of Minkowski vacua are forbidden.

(b) Measure and predictions. Construct a well-defined global volume/measure prescription for the multiverse dominated by nucleated inflating domain walls, in which new inflating regions form inside black holes or AdS bubbles and the standard rate-equation bookkeeping of volume fractions fails, and derive from it the prior Pprior(Λ)P_{\rm prior}(\Lambda) - the probability distribution of the effective cosmological constant across the vacua of the landscape - together with its phenomenological consequences.

(c) Landscape statistics. Determine how the statistical distribution and localization of dS vacua within a realistic landscape - including possible correlations between cosmological-constant fine-tuning and Standard-Model parameter fine-tuning - enter (a) and (b), thereby refining or refuting the conservative condition NdS≫10284N_{dS}\gg 10^{284} (with NdSN_{dS} the number of metastable de Sitter vacua in the landscape) for the successful anthropic prediction of Λ\Lambda.

These components are jointly load-bearing: (a) supplies the transition rates and (c) the landscape input that any measure in (b) needs in order to yield a prediction; an answer that makes only one component quantitative while leaving the others parametric does not close the problem.

References

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