Hawking and Hertog (JHEP 04 (2018) 147) proposed a dual description of scalar-field eternal inflation in the no-boundary state of quantum gravity: a deformed Euclidean conformal field theory living on the threshold (exit) surface, whose inverse partition function measures the amplitude of different exit-surface geometries. In toy models this amplitude is low for surfaces far from the round three-sphere and essentially zero for negative-curvature surfaces, motivating the conjecture that the exit from eternal inflation yields a finite, reasonably smooth universe rather than an infinite fractal-like multiverse. The open problem is to establish or refute this conjecture beyond toy models within a controlled quantum-gravitational realization of the no-boundary state, and to extend the holographic measure to realistic cosmologies containing a post-inflationary decelerating (radiation- or matter-dominated) phase, which requires identifying the appropriate dual field theories and whether inflation corresponds to an infrared fixed point of the dual theory.

Background

In eternal inflation, a deep near-de Sitter regime of slow-roll inflation in which quantum fluctuations of the inflaton's energy density dominate over the classical rolling, the standard semiclassical and stochastic account (Linde, Linde and Mezhlumian 1996) predicts a self-reproducing universe that is highly irregular on the largest scales, with exceedingly large or infinite constant-density surfaces — the familiar 'fractal multiverse' picture. Because quantum diffusion wipes out the separation between classical background and quantum fluctuations on which that account relies, a fundamental treatment must come from quantum cosmology.

The no-boundary wave function of the universe (Hartle and Hawking 1983) is a proposal for the quantum state of the universe in which the wave function on a spacelike three-geometry is given by a path integral over compact, regular histories with no boundary in the past. The dS/CFT correspondence (Strominger 2001) conjectures a dual description of an asymptotically de Sitter cosmology in terms of a three-dimensional Euclidean conformal field theory (CFT, a quantum field theory invariant under local scale transformations) living on the future boundary. Hertog and Hartle (2012) combined these ideas into a 'holographic no-boundary measure': the saddle points of the no-boundary wave function admit a representation in terms of Euclidean anti-de Sitter domain walls, so that Euclidean AdS/CFT relates their weighting to the (inverse) partition function of a deformed dual field theory, where a partition function is the weighted sum over field configurations with specified external sources.

Hawking and Hertog (2018) applied this to scalar-field eternal inflation. They placed a dual, deformed Euclidean CFT on a global constant-density surface at the threshold of scalar-field eternal inflation — the 'exit surface' — with the bulk inflaton acting as a complex source for a low-dimension operator. The inverse of the dual partition function specifies the amplitude of different shapes of this exit surface in the no-boundary state, yielding a holographic measure on the global structure of eternally inflating universes. In a toy model (a mass-deformed interacting O(N) vector model, i.e. a theory of N scalar fields with O(N)-symmetric interactions, on squashed three-spheres, i.e. three-spheres deformed by unequal scale factors, computed in the companion work of Conti, Hertog and Vreys), the amplitude is low for geometries far from the round conformal structure and exponentially small for large squashings with negative scalar curvature. A general argument extends this heuristically: every CFT contains a conformal coupling of the form Rϕ2R\phi^2 between the boundary Ricci scalar RR and the field ϕ\phi, which is expected to make the partition function diverge on boundary geometries whose Yamabe invariant is negative — the Yamabe invariant of a conformal class of metrics being the infimum, over conformal rescalings, of the normalized total scalar curvature, which is negative precisely in conformal classes containing a metric of constant negative scalar curvature. Negative-Yamabe conformal classes are exactly the highly irregular constant-density surfaces of the semiclassical eternal-inflation picture, so their amplitude should be strongly suppressed. On this basis the authors conjectured that the exit from eternal inflation produces classical universes that are finite and reasonably smooth on the largest scales — 'radically different', as they note, from the usual fractal multiverse — with important consequences for anthropic reasoning, since a significantly constrained multiverse would leave anthropic arguments applicable only to continuously varying parameters.

The evidence for this conjecture nevertheless rests on toy models and a general but unproven argument. The paper itself points to open directions and limitations: (a) its cosmologies transition directly from eternal inflation to a cosmological-constant-dominated phase, whereas applying the holographic measure to realistic cosmologies that include a decelerating radiation- or matter-dominated phase after inflation 'requires further development of holographic cosmology' — in particular identifying appropriate dual field theories, and settling whether in such cosmologies inflation corresponds to an infrared fixed point (the long-distance endpoint of renormalization-group flow) of the dual theory, in which case the partition function of that IR theory might specify the amplitude of exit surfaces; (b) the analysis assumes for simplicity a sharp inner boundary at the threshold of eternal inflation, and the authors note that one can imagine models with a fuzzy boundary, in which the detailed exit is encoded in the coupling between the field-theory degrees of freedom on the exit surface and the classical bulk dynamics; (c) the dual Euclidean description amounts to a significant departure from the original no-boundary idea; and (d) the smooth-exit conjecture itself is not proven — its supporting evidence consists of the toy-model computation and a general but heuristic argument, and the authors note that evaluating the relevant partition functions for general large metric deformations is beyond the current state of the art. The no-boundary state on which the whole construction rests has also been contested on mathematical grounds (the Lorentzian Picard-Lefschetz critique of Feldbrugge, Lehners and Turok 2017, Picard-Lefschetz theory being a method for defining oscillatory integrals rigorously by decomposing them into convergent complex contour integrals), with responses from the no-boundary collaboration specifying the wave function as a sum over regular complex saddle points (Diaz Dorronsoro et al. 2018; Halliwell, Hartle and Hertog 2019), so any full solution must make the underlying quantum state and saddle ensemble precise.

Problem Statement

The overarching objective is to determine what the exit from scalar-field eternal inflation produces in the no-boundary state of quantum gravity, and to place that determination under holographic control in realistic cosmologies. Within a controlled realization of the holographic no-boundary framework (a specified bulk gravitational theory with scalar-field eternal inflation, treated in the no-boundary state, together with its dual field theory on the exit surface) — or an equally fundamental quantum-gravitational treatment of the same regime — the problem comprises two targets:

(i) Establish, refute, or sharply delimit the Hawking-Hertog smooth-exit conjecture beyond toy models: determine whether the amplitude measure on exit-surface geometries — the inverse partition function of the dual field theory deformed by the bulk inflaton — is suppressed for surfaces far from the round three-sphere and vanishes for negative-scalar-curvature (negative-Yamabe) conformal classes, and, where suppression is not universal, determine the class of deformations on which it fails; a sharp delimitation must state that class explicitly rather than a trend along a single family of deformations. The measure claim is the direct target; the cosmological conclusion that the exit from eternal inflation then produces a finite, reasonably smooth universe rather than an infinite, fractal-like multiverse is the physical stake the conjecture attaches to it, and establishing the measure claim for the stated classes resolves this target. A refutation would exhibit, within the framework, irregular exit geometries whose amplitude is not suppressed, or a failure of the duality/measure construction in the eternal-inflation regime.

(ii) Extend the holographic measure to realistic cosmologies in which inflation is followed by a decelerating radiation- or matter-dominated phase: identify the appropriate dual field theory and the deformation sourced by the bulk inflaton, and settle whether inflation corresponds to an infrared fixed point of the dual theory whose partition function specifies the amplitude of exit surfaces.

The two targets trace distinct open directions stated by the source paper: target (i) concerns the smooth-exit conjecture itself, whose supporting evidence rests on toy models and a general but unproven argument, while target (ii) concerns the extension of the holographic measure to realistic cosmologies with a post-inflationary decelerating phase, which the paper states 'requires further development of holographic cosmology'.

References

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