We study the perturbative stability of Euclidean axion-dilaton wormholes that asymptotically approach flat space, both with a massless and a massive dilaton, and focus on homogeneous perturbations. We find massless wormholes to always be perturbatively stable. The phenomenologically more relevant case of a massive dilaton presents us with a wide variety of wormhole solutions, depending on the dilaton coupling and mass, and on the axion charge. We find that the solutions with the smallest dilaton potential are perturbatively stable and dominant, even in cases where the wormhole solutions are not continuously connected to the massless case by decreasing the mass. For branches of solutions emanating from a bifurcation point, one side of the branch always contains a negative mode in its spectrum, rendering such solutions unstable. The existence of classes of perturbatively stable wormhole solutions with massive dilaton sharpens the puzzles associated with Euclidean wormholes.

Belgium

Caroline Jonas

Georgia

George Lavrelashvili

Germany

Jean-Luc Lehners

Stability of axion-dilaton wormholes

Many cosmological models assume or imply that the total size of the universe is very large, perhaps even infinite. Here we argue instead that the universe might be comparatively small, in fact not much larger than the currently observed size. A concrete implementation of this idea is provided by the no-boundary proposal, in combination with a plateau-shaped inflationary potential. In this model, opposing effects of the weighting of the wave function and of the criterion of allowability of the geometries conspire to favour small universes. We point out that a small size of the universe also fits well with swampland conjectures, and we comment on the relation with the dark dimension scenario.

Canada

Jerome Quintin

A small Universe

As was discovered some time ago by Giddings and Strominger (GS), an axion can support a wormhole geometry in the presence of a massless dilaton, as long as the dilaton coupling remains below a critical value. We find that when the dilaton becomes massive, the set of solutions is vastly increased: not only do solutions exist above the critical value of the coupling, but new branches of solutions with several minima in the geometry also appear. All of these generalized GS-like solutions possess the property that, when analytically continued, they lead to a contracting baby universe. We show that in addition there exist families of solutions which, upon analytic continuation, lead to expanding baby universes. A curious property of axion-dilaton wormhole families is that their Euclidean action often decreases when the solutions acquire additional oscillations in the fields. When we replace the dilaton by an ordinary scalar field with a double well potential, we find analogous wormhole families leading to expanding baby universes. This time the Euclidean action has the expected behavior of increasing with the number of oscillations in the fields, although it also contains a puzzling aspect in that some solutions possess a negative action.

Zoo of axionic wormholes

String compactifications typically require fluxes, for example in order to stabilize moduli. Such fluxes, when they thread internal dimensions, are topological in nature and take on quantized values. This poses the puzzle as to how they could arise in the early universe, as they cannot be turned on incrementally. Working with string inspired models in six and eight dimensions, we show that there exist no-boundary solutions in which internal fluxes are present from the creation of the universe onwards. The no-boundary proposal can thus explain the origin of fluxes in a Kaluza-Klein context. In fact, it acts as a selection principle since no-boundary solutions are only found to exist when the fluxes have the right magnitude to lead to an effective potential that is positive and flat enough for accelerated expansion. Within the range of selected fluxes, the no-boundary wave function assigns higher probability to smaller values of flux. Our models illustrate how cosmology can act as a filter on a landscape of possible higher-dimensional solutions.

United Kingdom

Rahim Leung

K. S. Stelle

How to create universes with internal flux

Recently there have been discussions about which complex metrics should be allowable in quantum gravity. These discussions assumed that the matter fields were real valued. We make the observation that for compactified solutions it makes sense to demand convergence of the theory’s path integral in the higher-dimensional parent theory. Upon compactification this allows for more general matter configurations in the lower-dimensional theory, in particular it allows for complex scalar fields, with a bound on their imaginary parts. Similar considerations apply to metric rescalings in the presence of higher curvature corrections. We illustrate this effect with the example of the no-boundary proposal, in which scalar fields are typically required to take complex values. We find that complex no-boundary solutions exist, and satisfy the derived bound, if the potential is sufficiently flat. For instance, for a compactification from <math><mi>D</mi></math> dimensions, the bound on the imaginary part <math><mi>Im</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo></math> of the internal volume modulus reads <math><mo>|</mo><msub><mi>V</mi><mrow><mo>,</mo><mi>ϕ</mi></mrow></msub><mo>|</mo><mo>/</mo><mi>V</mi><mo>&lt;</mo><msqrt><mfrac><mrow><mi>D</mi><mo>−</mo><mn>4</mn></mrow><mrow><mi>D</mi><mo>−</mo><mn>2</mn></mrow></mfrac></msqrt><mo>/</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></math>. This leads to a mild tension with swampland conjectures.

Allowable complex scalars from Kaluza-Klein compactifications and metric rescalings

Complex metrics are a double-edged sword: they allow one to replace singular spacetimes, such as those containing a big bang, with regular metrics, yet they can also describe unphysical solutions in which quantum transitions may be more probable than ordinary classical evolution. In the cosmological context, we investigate a criterion proposed by Witten (based on works of Kontsevich &amp; Segal and of Louko &amp; Sorkin) to decide whether a complex metric is allowable or not. Because of the freedom to deform complex metrics using Cauchy’s theorem, deciding whether a metric is allowable in general requires solving a complicated optimisation problem. We describe a method that allows one to quickly determine the allowability of minisuperspace metrics. This enables us to study the off-shell structure of minisuperspace path integrals, which we investigate for various boundary conditions. Classical transitions always reside on the boundary of the domain of allowable metrics, and care must be taken in defining appropriate integration contours for the corresponding gravitational path integral. Perhaps more surprisingly, we find that proposed quantum (‘tunnelling’) transitions from a contracting to an expanding universe violate the allowability criterion and may thus be unphysical. No-boundary solutions, by contrast, are found to be allowable, and moreover we demonstrate that with an initial momentum condition an integration contour over allowable metrics may be explicitly described in arbitrary spacetime dimensions.

Uses of complex metrics in cosmology

Recent works have suggested that the no-boundary proposal should be defined as a sum over regular, not necessarily compact, metrics. We show that such a prescription can be implemented in the presence of a scalar field. For concreteness, we consider the model of Garay et al., in which the potential is a sum of exponentials, and which lends itself to an analytical treatment. Compared to the earlier implementation, we find that saddle points with unstable fluctuations can be eliminated by imposition of an appropriate regularity condition. This leads to the appearance of additional saddle points, corresponding to unclosed geometries. We argue that such saddles will occur generically, though we also find in our example that they are subdominant to the closed, Hartle-Hawking, saddle points. When the potential is positive, classical spacetime is only predicted for inflationary histories. When the potential is negative, we recover the anti–de Sitter gravitational path integral, with a stable scalar field included. One puzzle that we find is that in general the path integral must be restricted to sum only over specific, discrete and late time dependent initial values of the scalar field. Only when the scalar is required to take real values is this puzzle eliminated, a situation that moreover leads to advantageous phenomenological characteristics.

Vincent Meyer

Revisiting the no-boundary proposal with a scalar field

Kontsevich and Segal (K-S) have proposed a criterion to determine which complex metrics should be allowed, based on the requirement that quantum field theories may consistently be defined on these metrics, and Witten has recently suggested that their proposal should also apply to gravity. We explore this criterion in the context of gravitational path integrals, in simple minisuperspace models, specifically considering de Sitter (dS), no-boundary and anti–de Sitter (AdS) examples. These simple examples allow us to gain some understanding of the off-shell structure of gravitational path integrals. In all cases, we find that the saddle points of the integral lie right at the edge of the allowable domain of metrics, even when the saddle points are complex or Euclidean. Moreover the Lefschetz thimbles, in particular the steepest descent contours for the lapse integral, are cut off as they intrude into the domain of nonallowable metrics. In the AdS case, the implied restriction on the integration contour is found to have a simple physical interpretation. In the dS case, the lapse integral is forced to become asymptotically Euclidean. We also point out that the K-S criterion provides a reason, in the context of the no-boundary proposal, for why scalar fields would start their evolution at local extrema of their potential.

Allowable complex metrics in minisuperspace quantum cosmology

We elaborate on the correspondence between the canonical partition function in asymptotically AdS universes and the no-boundary proposal for positive vacuum energy. For the case of a pure cosmological constant, the analytic continuation of the AdS partition function is seen to define the no-boundary wave function (in dS) uniquely in the simplest minisuperspace model. A consideration of the AdS gravitational path integral implies that on the dS side, saddle points with Hawking-Moss/Coleman-De Luccia-type tunneling geometries are irrelevant. This implies that simple topology changing geometries do not contribute to the nucleation of the universe. The analytic AdS/dS equivalence holds up once tensor fluctuations are added. It also works, at the level of the saddle point approximation, when a scalar field with a mass term is included, though in the latter case, it is the mass that must be analytically continued. Our results illustrate the emergence of time from space by means of a Stokes phenomenon, in the case of positive vacuum energy. Furthermore, we arrive at a new characterization of the no-boundary condition, namely that there should be no momentum flux at the nucleation of the universe.

Wave function of simple universes analytically continued from negative to positive potentials

We study a self-interacting scalar field theory coupled to gravity and are interested in spherically symmetric solutions with a regular origin surrounded by a horizon. For a scalar potential containing a barrier, and using the most general spherically symmetric ansatz, we show that in addition to the known static, oscillating solutions discussed earlier in the literature there exist new classes of solutions which appear in the strong field case. For these solutions the spatial sphere shrinks either beyond the horizon, implying a collapsing universe outside of the cosmological horizon, or it shrinks already inside of the horizon, implying the existence of a black hole surrounding the scalar lump in all directions. Crucial for the existence of all such solutions is the presence of a scalar field potential with a barrier that satisfies the swampland conjectures.