Wineglass/No-Boundary Instantons in Quantum Cosmology

Updated 5 July 2026

Wineglass/no-boundary instantons are semiclassical saddle points defined by regular, compact Euclidean geometries with a single boundary that evolve into a classical, Lorentzian universe.
They utilize complex time contours that transition from Euclidean through complex segments to Lorentzian regimes, ensuring analytical continuity and classicality via WKB conditions.
Studies show that factors like mass hierarchies, charge limits, and loop quantum effects critically influence instanton actions, impacting inflationary, ekpyrotic, and cyclic cosmological scenarios.

Wineglass/no-boundary instantons are semiclassical saddle points of the Euclidean path integral in quantum cosmology. In the Hartle–Hawking no-boundary proposal, the relevant saddle is a regular compact geometry with only one boundary, specified by final three-geometry and matter data and smooth closure in the interior. In the broader family now discussed in the literature, “wineglass” geometries are Euclidean wormhole-like saddles whose scale factor develops a stem and a rim, with analytic continuation at a local maximum producing an expanding Lorentzian universe. Recent work has emphasized that these are not disjoint constructions: in explicit charged wormhole models, the zero-charge limit pinches off the throat and yields a disconnected no-boundary instanton, placing wineglass wormholes and no-boundary instantons in a common family of Euclidean solutions (Hwang et al., 2014, Lavrelashvili et al., 11 Mar 2026, Lavrelashvili et al., 11 May 2026).

1. Semiclassical definition and contour structure

The no-boundary wave function is defined by a Euclidean path integral over regular compact geometries with only one boundary. In minisuperspace this is typically written as

$\Psi(h_{ij},\chi)=\int_{\mathcal C}\delta g\,\delta\phi\, e^{-S_E(g_{\mu\nu},\phi)},$

or, after symmetry reduction,

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$

with $b$ and $\chi$ denoting the final scale factor and scalar data (Battarra et al., 2014, Yeom, 2021).

For the $O(4)$ -symmetric ansatz

$ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$

or its lapse-generalized version, regularity at the South Pole requires smooth closure of the geometry. In the single-field setting this is imposed as

$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$

while in the two-field inflationary model the corresponding conditions are

$a(0)=0,\qquad \dot a(0)=1,\qquad \dot\phi_i(0)=0.$

The final boundary data are fixed at some $\tau_f$ by $a(\tau_f)=b$ and $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 0, or $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 1, $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 2 in the multi-field case (Battarra et al., 2014, Hwang et al., 2014).

A defining feature of the modern treatment is that the relevant saddles are generally not purely real. The contour in complex time $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 3 typically begins at the South Pole, traverses a Euclidean or complex segment, and then turns into a Lorentzian branch. In the two-field inflationary analysis, the contour runs along a Euclidean segment to a turning point $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 4 and then vertically into the Lorentzian regime. In ekpyrotic cosmology, an $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 5-shaped contour is often used, with an initial Euclidean-like segment and a final vertical segment on which the solution becomes classical. The action is contour-independent provided singularities and branch cuts are avoided (Hwang et al., 2014, Battarra et al., 2014).

This contour structure is central to the topic. Wineglass and no-boundary instantons are therefore best understood not as purely Euclidean geometries in the old sense, but as complex saddles whose Euclidean, complex, and Lorentzian domains are connected within one analytic solution.

2. Complexification, classicality, and deformed instanton geometry

A recurrent conclusion of the no-boundary literature is that dynamical cosmological instantons are necessarily complexified. The review of fuzzy instantons states this explicitly: if instantons are dynamical, then instantons are necessarily complexified, and the complexification is precisely what allows a no-boundary quantum state to evolve into a classical universe (Yeom, 2021).

Classicality is formulated in WKB language. In the no-boundary setting, one requires that the phase of the wave function vary much more rapidly than its amplitude, equivalently

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 6

so that the late-time history obeys an approximate Hamilton–Jacobi relation and becomes effectively Lorentzian. When this condition holds, the semiclassical probability is

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 7

with $\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 8 controlling classical evolution (Yeom, 2021, Hwang et al., 2014).

The geometry of the saddle can be substantially more intricate than the original half-sphere intuition suggests. In the Bianchi IX analysis of anisotropic inflationary universes, no-boundary instantons continue to exist for arbitrarily large anisotropies, but their complex contours must be deformed to avoid singularities in the complex time plane. For small anisotropy, the saddle resembles the familiar no-boundary cap. For larger anisotropy, the contour may need to run first vertically and then horizontally, and the shape becomes significantly different from Hawking’s original instanton. The resulting histories still inflate and the anisotropies decay away, but the wave function classicalizes more slowly in the anisotropy directions than in the isotropic scale factor and scalar directions (Bramberger et al., 2017).

This suggests that “wineglass” should not be read too narrowly as one fixed shape. In the literature it denotes a broader class of complex Euclidean-to-Lorentzian geometries whose common feature is regularity plus later classicalization, while the detailed contour geometry can vary sharply with matter content, anisotropy, and quantum-gravity corrections.

3. Inflationary no-boundary instantons and multi-field effects

In inflationary minisuperspace, the basic no-boundary problem is to identify those initial scalar values and contours for which a complex saddle classicalizes and yields sufficient inflation. For a quadratic inflaton potential, the fuzzy-instanton review summarizes the standard difficulty: the no-boundary measure favors small potential energy,

$\Psi(b,\chi)\sim \sum e^{-S_E(b,\chi)},$ 9

and in a simple quadratic model with $b$ 0 the preferred number of $b$ 1-folds is of order $b$ 2, far below the $b$ 3 usually required observationally (Yeom, 2021).

The two-field analysis with unequal masses modifies this conclusion in a precise way. The model uses

$b$ 4

with an $b$ 5-symmetric minisuperspace metric

$b$ 6

After rescaling, the problem depends only on the mass ratio $b$ 7, and the no-boundary wave function is approximated by a steepest-descent sum over complex saddle points (Hwang et al., 2014).

The central result is that a relatively massive direction is harder to classicalize. In the Euclidean problem the potential acts with the opposite sign, so a larger mass implies a sharper instability in that direction. As a consequence, if $b$ 8 is heavy and $b$ 9 is the light slow-roll direction, then classicality of the heavy mode requires the instanton to start from a larger vacuum energy along the slow direction. Quantitatively, with $\chi$ 0 treated as a slow-roll background, the effective mass parameter governing classicality of $\chi$ 1 is

$\chi$ 2

and the classicality requirement is $\chi$ 3. This gives

$\chi$ 4

Since chaotic inflation gives

$\chi$ 5

one obtains

$\chi$ 6

The numerical fit reported in the paper is

$\chi$ 7

consistent with the abstract statement that the most probable $\chi$ 8-foldings are approximately

$\chi$ 9

in the $O(4)$ 0 limit (Hwang et al., 2014).

The physical implication is sharply counterintuitive but explicit in the paper: a sufficient mass hierarchy pushes the most probable initial condition to larger values of the light field, and the no-boundary wave function can thereby explain large $O(4)$ 1-foldings, including more than $O(4)$ 2 $O(4)$ 3-foldings, without extra ad hoc weighting factors.

4. Ekpyrotic and cyclic no-boundary instantons

The 2014 ekpyrotic papers overturn the earlier expectation that inflation is required to obtain classical no-boundary histories. For the steep negative potential

$O(4)$ 4

with

$O(4)$ 5

they identify a new class of complex no-boundary instantons that begin at a regular South Pole, pass through a complex Euclidean regime, and then approach a real Lorentzian contracting universe during the ekpyrotic phase. The crucial condition is

$O(4)$ 6

which is the ekpyrotic attractor condition (Battarra et al., 2014, Battarra et al., 2014).

Near the crunch, the asymptotic form is controlled by corrections that decay only when $O(4)$ 7. That decay suppresses the complex parts of $O(4)$ 8 and $O(4)$ 9, making the history increasingly real and therefore classical. The role played here is directly analogous to the inflationary attractor in inflationary no-boundary instantons: the attractor classicalizes the Lorentzian branch (Battarra et al., 2014).

The probability measure again takes the semiclassical form

$ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 0

For these ekpyrotic instantons the probability scales as

$ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 1

with $ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 2. Smaller-magnitude potentials are therefore favored, which in the ekpyrotic setting means a longer, shallower ekpyrotic phase. Both papers emphasize that, in a landscape allowing both inflationary and ekpyrotic/cyclic regions, ekpyrotic histories are far more probable than inflationary ones under no-boundary weighting (Battarra et al., 2014, Battarra et al., 2014).

In cyclic cosmology the potential

$ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 3

admits two coexisting branches on the dark-energy plateau: de Sitter-like instantons associated with the dark-energy phase, and ekpyrotic instantons that classicalize only later, near the crunch. The stated conclusion is that the ekpyrotic branch usually has the larger probability (Battarra et al., 2014).

A key caveat is also explicit. The ekpyrotic creation papers do not include a bounce, and because the scalar sector does not violate the null energy condition, every classical ekpyrotic history in the model ends in a crunch. Any connection to the observed expanding universe therefore requires an additional bounce assumption (Battarra et al., 2014).

5. Wineglass wormholes and the no-boundary limit

The modern “wineglass” terminology is most directly associated with Euclidean wormholes that can seed an expanding inflationary universe after analytic continuation. The distinguishing geometric feature is a local maximum of the Euclidean scale factor at the continuation surface. In the 2026 analyses, only such wineglass wormholes are suitable for producing expansion after continuation: the scale factor must first shrink to a local minimum, then re-expand, and finally reach a local maximum at the analytic-continuation surface. Wormholes ending at a local minimum would instead continue to a crunching Lorentzian spacetime (Lavrelashvili et al., 11 Mar 2026, Lavrelashvili et al., 11 May 2026).

These constructions use the $ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 4-symmetric metric

$ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 5

with a homogeneous scalar $ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 6 and either axionic charge $ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 7 or magnetic charge $ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 8. In one formulation the Euclidean action is

$ds^2=d\tau^2+a^2(\tau)\,d\Omega_3^2,$ 9

while the later analysis presents the same matter content with explicit boundary terms and counterterms (Lavrelashvili et al., 11 Mar 2026, Lavrelashvili et al., 11 May 2026).

The central result is the zero-charge limit. As the charge is lowered, the throat or stem shrinks and tends to zero size, while the outer “mouth” or rim remains approximately set by the Hubble radius at the top of the barrier. In this limit, the wormhole disconnects from the asymptotic flat or AdS region and becomes a compact no-boundary instanton. The 2026 papers state this in complementary language: the throat pinches off, the geometry splits into a background spacetime plus a disconnected no-boundary instanton, and the wineglass branch approaches a no-boundary saddle at the top of the barrier with

$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 0

Equivalently, the wineglass/no-boundary distinction becomes a distinction between finite-charge and zero-charge limits of one Euclidean family (Lavrelashvili et al., 11 Mar 2026, Lavrelashvili et al., 11 May 2026).

The action analysis is also important. The renormalized on-shell action is defined using holographic counterterms,

$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 1

followed by subtraction of the background AdS action. This resolves the longstanding puzzle that wineglass wormholes can have negative action: as the charge decreases, the action becomes more negative but remains bounded below by the corresponding no-boundary instanton. Within the wormhole family, smaller charge generally means higher weighting and longer inflation, but the global result reported in the 2026 study is that no-boundary instantons dominate the probability distribution overall (Lavrelashvili et al., 11 Mar 2026).

The broader 2026 analysis extends this picture to asymptotically flat and asymptotically AdS settings, to axionic and magnetic support, and to more exotic multi-extremum solutions. Two-stem and even six-stem wormholes are exhibited, together with multi-barrier-crossing solutions in periodic scalar potentials. These enlarge the zoo of wineglass geometries without changing the main conclusion: the dominant small-charge branch flows continuously to the no-boundary limit (Lavrelashvili et al., 11 May 2026).

Loop quantum cosmology and loop quantum gravity introduce a distinct modification of the no-boundary picture: the Euclidean region need not be imposed by Wick rotation but can arise dynamically through signature change. In this framework the effective line element is

$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 2

with $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 3 determined by quantum geometry. When $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 4, the geometry becomes Euclidean. The 2018 and 2020 loop papers argue that this non-singular signature change naturally cures the perturbative instability found in Lorentzian path-integral treatments of the no-boundary proposal, because the dangerous branch cuts are shifted away and the perturbation action remains real and finite (Bojowald et al., 2018, Bojowald et al., 2020).

A crucial technical point is that the relevant no-boundary saddles in the Lorentzian formulation are off-shell instantons: they satisfy second-order equations but not the first-order Friedmann constraint. The loop analysis shows that $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 5 can still be derived directly for such off-shell instantons, and that signature change can occur even at sub-Planckian densities. This is the basis for the claim that loops rescue the no-boundary proposal (Bojowald et al., 2018, Bojowald et al., 2020).

The geometry of the Euclidean cap is also altered. For a closed $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 6 FLRW universe with a positive cosmological constant, the effective LQC instanton develops an infinite Euclidean tail rather than a finite hemispherical South Pole. In the small- $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 7 regime,

$a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 8

so $a(0)=0,\qquad a'(0)=1,\qquad \phi'(0)=0,$ 9 only as $a(0)=0,\qquad \dot a(0)=1,\qquad \dot\phi_i(0)=0.$ 0. The paper stresses that this does not render the saddle singular; rather, it produces an infinitely stretched throat or tail that still contributes finite action and naturally favors asymptotic closing-off of the geometry (Brahma et al., 2018).

Selection criteria for complex saddles remain unsettled. A 2026 note applies a milder version of the Kontsevich–Segal–Witten allowability criterion to no-boundary instantons and wine-glass geometries. In that analysis, the no-boundary instanton is KSW allowed, while wine-glass geometries are KSW disallowed. The note bases this on a contour criterion for FRW complex metrics and on the observation that the asymptotically Euclidean AdS segment of the wine-glass contour violates the bound immediately. Because the paper explicitly describes its treatment as a simple study using a milder version of the criterion, this result is best read as a sharp proposal rather than a settled consensus (Ailiga et al., 24 Mar 2026).

Finally, several adjacent instanton constructions are related in method but not identical in interpretation. The exact Lee–Weinberg/Linde gravitational instantons for tunneling without barriers are regular $a(0)=0,\qquad \dot a(0)=1,\qquad \dot\phi_i(0)=0.$ 1-symmetric Euclidean bounces describing Minkowski $a(0)=0,\qquad \dot a(0)=1,\qquad \dot\phi_i(0)=0.$ 2 AdS decay, not strict no-boundary creation-from-nothing saddles, though the paper notes a resemblance in the smooth turning region and regular Euclidean geometry (Kanno et al., 2012). Likewise, the 2020 analysis of quantum-regularized $a(0)=0,\qquad \dot a(0)=1,\qquad \dot\phi_i(0)=0.$ 3-invariant instantons with a quantum core does not use the terms “wineglass instanton” or “no-boundary instanton”; its relevance is conceptual rather than terminological (Mukhanov et al., 2020).

Taken together, these developments place wineglass/no-boundary instantons at the intersection of semiclassical gravity, contour deformation, holographic renormalization, cosmological attractors, and quantum-gravity corrections. The common problem is always the same: to identify Euclidean or complex saddles that are regular, have finite action, yield a meaningful weighting in the path integral, and classicalize into a Lorentzian cosmology. The main divergence in the literature concerns which saddles should count as admissible—compact no-boundary instantons alone, charged wineglass wormholes in the same family, or only those complex geometries that satisfy additional criteria such as KSW allowability.