Incidence of eternal inflation at small field scales and in multifield random-potential landscapes

Greenwood and Aguirre (2022) measured how often inflation is eternal — by the stochastic, false-vacuum tunneling, or topological mode — across ensembles of single-field inflationary potentials drawn as Gaussian random fields under explicit measures, finding that genericity depends strongly on the measure, the initialization, and the field scale. Their incidence statistics leave linked regimes untouched: observationally viable small-field potentials (field scales below roughly an order of magnitude under the Planck mass, where viable models are too rare for native sampling), multifield (N-dimensional random) landscapes, and robustness of any verdict under physically motivated measures. This problem asks for the incidence statistics of the three eternal modes extended to those regimes — via importance-sampled or targeted sampling of rare viable models and rare post-tunneling histories, an approach suggested in the source, and via multifield ensembles — with the measure-conditioned character of every genericity claim made explicit, including whether a natural measure on initial conditions emerges from the statistics of high-dimensional landscapes.

Background

Inflation — a postulated epoch of quasi-exponential expansion of the very early universe, driven by a slowly rolling scalar field (the inflaton) with potential $V(\varphi)$ — is the standard mechanism for explaining the observed homogeneity and flatness of the cosmos and the origin of primordial structure. In many models this expansion is eternal: quantum fluctuations of the inflaton, or long-lived metastable vacua, keep some physical volume inflating forever, producing an unbounded ensemble of causally disconnected pocket universes (a Level II multiverse). Whether eternality is a generic consequence of inflation or requires fine-tuning bears directly on how much credence to place in multiverse-based and anthropic reasoning.

Eternal inflation can arise by three modes. (i) Stochastic eternal inflation: per Hubble time $H^{-1}$ (where $H$ is the Hubble parameter), Hubble-scale quantum fluctuations of amplitude $\delta\varphi_q = H/2\pi$ compete with the classical slow-roll drift $|\dot{\varphi}|/H$; if fluctuations up the potential win often enough, some Hubble volumes inflate indefinitely — a practical sufficient criterion being $V(\varphi)^{3/2} \gtrsim 6.6\,|V'(\varphi)|\,M_P^3$, with $M_P$ the reduced Planck mass. (ii) False-vacuum (tunneling) eternal inflation: a metastable vacuum with positive energy is locally de Sitter space, decaying by bubble nucleation through the barrier via a Coleman–de Luccia (CDL) instanton — a solution of the Euclidean field equations describing the nucleating bubble — or by a Hawking–Moss fluctuation over the barrier; if the per-four-volume decay rate satisfies $\lambda \lesssim 9H_F^4/4\pi$ (with $H_F$ the false-vacuum Hubble rate), inflating volume never disappears. (iii) Topological eternal inflation: if the field settles into different minima in neighboring regions, a domain wall forms near the potential maximum, and when the barrier is sufficiently flat ($\eta_V < (4-\epsilon)/3$, where $\epsilon = -\dot{H}/H^2$ is the first slow-roll parameter and $\eta_V = M_P^2\,V''(\varphi)/V(\varphi)$ the second potential slow-roll parameter) the wall inflates forever.

Greenwood and Aguirre asked: by what modes and to what extent is inflation generically eternal with respect to explicit measures over single-field models? Following Tegmark's Monte Carlo tradition, they drew the potential as a one-dimensional Gaussian random field (GRF) — a random smooth function with jointly Gaussian values, here a truncated Fourier series — with a vertical mass scale $m_v$ setting the amplitude and a horizontal mass scale $m_h$ setting the field range over which the potential varies: $V(\varphi) = (m_v M_P)^4\, f(\varphi/(m_h M_P))$. They instantiated three initial-condition measures: A (start at local maxima, weighted by adjacent basin widths), B (uniform in field value), and C (displaced $H_{\max}/2\pi$ from maxima, with $H_{\max}$ the Hubble rate at the maximum), and two mass-weighting schemes (democratic, and epektacratic — weighted by expansion). Simulating inflationary histories, including instanton computations for vacuum transitions, they found: topological eternal inflation is generic only when initialized atop the barrier; stochastic eternal inflation is generic at large field scales in Measure A and in pockets of Measure C parameter space, but generically absent among models conditioned on minimal viable inflation; and CDL or Hawking–Moss transitions in viable histories generically result in eternal inflation.

These incidence statements are conditional on the adopted measures and on the sampled regime. Although the nominal sampling range extends down to $m_h \sim 10^{-2}$, observationally viable small-field realizations are so rare under the native distribution that no meaningful incidence statistics exist at field scales below roughly an order of magnitude under $M_P$. The ensemble is also strictly single-field, while realistic theory landscapes involve many fields. Related multifield random-landscape studies — Masoumi, Vilenkin and Yamada on initial conditions for successful slow roll, and Low, Easther and Hotchkiss on the statistics of inflationary saddles — address the success of slow-roll inflation, not the incidence of the eternal modes, leaving the typicality of eternal inflation in the small-field and multifield regimes unmeasured.

Problem Statement

Extend the measure-conditioned incidence statistics of eternal inflation to the regimes that native sampling cannot reach, as one coherent objective: quantify, with controlled uncertainties, the fraction of observationally viable inflationary histories that are eternal by each of the three modes (stochastic, false-vacuum tunneling, topological), under explicit measures over potentials and initial conditions, in the following linked regimes.

(i) Small field scales: potentials varying on field ranges below roughly one order of magnitude under the reduced Planck mass (horizontal scale $m_h \lesssim 0.1$), where viable small-field realizations are rare under the native measure distribution; the source suggests an importance-sampling approach to probe the measure over viable small-field models rather than sampling the measure's native distribution, and correct reweighting of any such targeted sampling is part of the controlled-uncertainty requirement stated above.

(ii) Multifield landscapes: potentials given by $N$-dimensional Gaussian random fields rather than a single field, including how the incidence of each eternal mode changes with $N$ and with the reduction of multifield dynamics to effectively single-field behavior along approach trajectories; in particular, whether a natural measure on initial conditions emerges from the statistical properties of the high-dimensional landscape — identifying, along those approach trajectories and in their field-space coordinates, where the potential becomes effectively one-dimensional. Here a measure on initial conditions counts as emerging from the landscape statistics when it is generated by the statistics of the approach trajectories in field space under no externally imposed weighting, and the operative test is whether the incidence fractions computed under it differ from those under measures A, B, and C; a definite answer either way — that no such measure emerges, as well as that one does — discharges this sub-question.

(iii) Measure robustness: whether any genericity or fine-tuning verdict obtained in (i) and (ii) survives under physically motivated priors over potentials and initial conditions, or under a wider class of measures — or, failing that, a quantified characterization of the measure dependence of the verdict.

The objective includes quantifying the contribution of rare post-tunneling histories — Coleman–de Luccia or Hawking–Moss transitions followed by enough e-folds (each an expansion of the universe by a factor of $e$; of order 70 in total) to be observationally viable — to the relative incidence of observable non-eternal versus eternal histories, since these tails dominate observational viability in the small-field regime. Every incidence claim must state explicitly the measure under which it holds.

Requirements (i)–(iii) and this post-tunneling quantification are jointly constitutive of the objective and are each mandatory; a result that addresses only a proper subset of them is a partial contribution and does not by itself resolve the problem.

References

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