The cosmological necklace problem

Abstract: We investigate 3d de Sitter axion wormhole which contribute to the no-boundary density matrix. We identify "cosmological necklace" solutions: an infinite series of Euclidean saddles corresponding with repeated bounces. This results in an unbound gravitational entropy, and a divergent path integral. To remedy this, we study the gravitational path integral using a (mostly) Lorentzian lapse contour. Within a minisuperspace steepest-descent analysis, we find that a single necklace dominates, leading to a finite entropy. A crucial element is to take into account an $(a\to -a)$ redundancy in the FLRW path integral, where $a$ is the scale factor. Surprisingly, the dominant solution is not purely Euclidean. Its entropy turns out to be independent of the axion flux, and equals the empty de Sitter entropy. We also study higher-dimensional necklaces, sourced by either an axion flux or by Yang-Mills instantons, and argue for qualitatively similar results: the single necklace solution dominates along a Lorentzian lapse contour.

• The paper presents a solution to the necklace instability problem in the nuclear continuum
• By proposing a Lorentzian contour of the lapse $N=\varepsilon+i R$
• The result allows for the exclusion of higher necklaces and retention

Problem formulation

The paper identifies a failure of the Euclidean gravitational path integral in de Sitter minisuperspace and proposes a Lorentzian lapse contour as a resolution. The setting is a closed FLRW universe with positive cosmological constant and matter whose Euclidean stress tensor supports a cosmological wormhole. The principal example is three-dimensional de Sitter gravity coupled to an axion flux, supplemented by higher-dimensional axion and Yang–Mills constructions.

The central issue is not merely the conventional conformal-factor instability. The authors exhibit an infinite sequence of real Euclidean classical solutions with the same boundary conditions. A basic Euclidean wormhole can be periodically repeated $k$ times, producing a “necklace” with lapse $N=kN_0$ and action

$I_k=k I_0,$

where $I_0<0$. Consequently, each successive necklace has a larger semiclassical weight $e^{-I_k}$, and the sum over $k$ has no dominant saddle and diverges. This is a divergence generated by a discrete family of on-shell geometries rather than by an unbounded off-shell direction.

The authors therefore reject the prescription of summing indiscriminately over all real Euclidean solutions. Their proposed definition integrates the lapse over the regulated Lorentzian contour

$N=\varepsilon+i\mathbb{R},$

with $\varepsilon>0$. The contour determines which saddles are represented in the Picard–Lefschetz decomposition. The principal claim is that this contour excludes the higher necklaces while retaining a single $k=1$ saddle, yielding a finite entropy compatible with the Gibbons–Hawking result.

The necklace instability

For the three-dimensional axion model, the minisuperspace metric and action are

$ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$

and

$N=kN_0$0

The Hamiltonian constraint gives

$N=kN_0$1

For $N=kN_0$2, the Euclidean solution oscillates between a minimum and maximum scale factor. Imposing equal maximal boundary sizes produces the discrete family

$N=kN_0$3

with $N=kN_0$4. The corresponding Euclidean configurations are repeated copies of the elementary wormhole. Their on-shell actions are

$N=kN_0$5

Because the action becomes increasingly negative with $N=kN_0$6, the naïve positive-real lapse contour produces a divergent semiclassical series. The pathology is particularly severe because the repeated solutions are covers or necklaces of the same local geometry. No local condition on the equations of motion can retain the elementary wormhole while excluding all of its repetitions.

This observation motivates the paper’s broader methodological conclusion: the admissible set of gravitational saddles cannot be specified solely by requiring real Euclidean classical solutions. The off-shell integration cycle is part of the definition of the theory.

Exact three-dimensional analysis

The strongest part of the paper is the exact treatment of the three-dimensional model. The scale factor is regarded as the coordinate of a Wheeler–DeWitt quantum mechanics with potential

$N=kN_0$7

At fixed lapse, the propagator is

$N=kN_0$8

The authors solve the corresponding Schrödinger equation exactly and obtain a modified-Bessel representation involving $N=kN_0$9. The choice of $I_k=k I_0,$0, rather than the alternative $I_k=k I_0,$1 solution, is fixed by the short-time condition

$I_k=k I_0,$2

The alternative would produce the wrong short-time distribution, proportional to $I_k=k I_0,$3.

A crucial structural point is that the physical metric depends on $I_k=k I_0,$4, so $I_k=k I_0,$5 and $I_k=k I_0,$6 describe the same spatial geometry. The gravitational amplitude must therefore include both sectors,

$I_k=k I_0,$7

The first contains the even necklaces, while the second contains the odd necklaces. This decomposition is not optional bookkeeping: it is required by the $I_k=k I_0,$8 redundancy of the minisuperspace variable.

In the semiclassical limit, the corresponding effective actions are

$I_k=k I_0,$9

For maximal boundary size, the relevant expressions are

$I_0<0$0

and

$I_0<0$1

The two sectors have complementary analytic structures. The $I_0<0$2 action has saddles at even multiples of the elementary period and poles at odd multiples. The $I_0<0$3 action has saddles at odd multiples and a pole at $I_0<0$4. That pole is decisive: in the short-time limit, propagation from $I_0<0$5 to $I_0<0$6 requires finite displacement in vanishing time and therefore divergent kinetic energy. Semiclassically,

$I_0<0$7

so the effective action contains a $I_0<0$8 singularity.

Lorentzian contour and saddle selection

The contour $I_0<0$9 is analyzed by Picard–Lefschetz theory. A saddle contributes only if its steepest-ascent cycle intersects the defining contour. This global criterion eliminates the higher necklaces even though they remain legitimate classical solutions.

In the standard $e^{-I_k}$0 sector, the contour picks up the $e^{-I_k}$1 configuration and saddles with negative winding, whose actions are positive and therefore exponentially suppressed. In the tunneling $e^{-I_k}$2 sector, the pole at $e^{-I_k}$3 changes the contour topology. The regulated contour passes to the right of the pole and intersects the ascent cycle of the positive saddle at $e^{-I_k}$4. The dominant contribution is consequently the $e^{-I_k}$5 saddle.

Figure 1: The complex lapse plane for the standard $e^{-I_k}$6 sector, showing poles, saddles, steepest-descent contours, and the regulated Lorentzian contour.

The pole also explains why a saddle with negative Euclidean action can contribute despite the predominantly Lorentzian contour. Away from the pole, the Lorentzian action is imaginary and the usual deformation argument would appear to forbid saddles with $e^{-I_k}$7. The $e^{-I_k}$8 singularity invalidates that argument by allowing the contour to pass onto the relevant thimble.

Figure 2: The complex lapse plane for the tunneling $e^{-I_k}$9 sector, where the pole at $k$0 allows the $k$1 saddle to contribute.

For the higher odd saddles, the authors provide an analytic confinement argument in the exactly solvable model. Defining

$k$2

the ascent line from the $k$3-th saddle has height greater than $k$4. Meanwhile, vertical lines above the singularities at even multiples of the elementary period have fixed height $k$5. The ascent lines of higher odd saddles are therefore confined to regions that cannot intersect the defining contour. The result is a finite saddle expansion dominated by the single necklace.

Figure 3: Height-function barriers confining ascent lines of higher necklaces away from the defining Lorentzian contour.

The same mechanism is visible in the numerical higher-dimensional analysis: any ascent line reaching the origin must cross a locus with height below the threshold required for a higher necklace. Thus, higher-necklace contributions are excluded by the global contour geometry rather than by a local instability criterion.

Figure 4: Numerical steepest-descent structures in higher-dimensional axion models, showing the absence of higher-necklace intersections with the defining contour.

Dominant geometry and entropy

At the dominant $k$6 saddle, the tunneling configuration begins at $k$7, passes through a Lorentzian segment near the would-be singularity at $k$8, and ends at $k$9. For nonzero axion flux, the transition is implemented by deforming the $N=\varepsilon+i\mathbb{R},$0 contour around the complexified big-bang/big-crunch singularity. Since $N=\varepsilon+i\mathbb{R},$1 behaves locally as a square root near the singularity, one winding changes its sign:

$N=\varepsilon+i\mathbb{R},$2

The contour winding contributes an additional term to the action. This contribution precisely cancels the explicit $N=\varepsilon+i\mathbb{R},$3-dependence of the ordinary Euclidean action. The resulting dominant action is

$N=\varepsilon+i\mathbb{R},$4

and hence

$N=\varepsilon+i\mathbb{R},$5

This equals the empty three-dimensional de Sitter entropy in the units used by the paper and is independent of the axion flux $N=\varepsilon+i\mathbb{R},$6. The flux changes the local wormhole geometry, but not the final entropy selected by the contour.

The paper interprets this independence through a contour deformation toward late Lorentzian time. At late times, the axion contribution redshifts away and the solution approaches empty de Sitter space. The action can then be evaluated on a contour whose nontrivial matter-dependent portion cancels, leaving the de Sitter area term. The result is consistent with cosmic no-hair behavior, although the authors explicitly regard the exact $N=\varepsilon+i\mathbb{R},$7-independence as puzzling and leave its broader interpretation open.

The claim is stronger than a recovery of the empty de Sitter limit: the proposed observer entropy is exactly flux-independent throughout the three-dimensional axion family, not merely asymptotically independent as $N=\varepsilon+i\mathbb{R},$8.

Higher-dimensional axion and Yang–Mills extensions

The higher-dimensional analysis is performed numerically in a kinetic gauge in which the scale factor has a canonical kinetic term and the $N=\varepsilon+i\mathbb{R},$9 symmetry remains manifest. The authors study four- and five-dimensional axion potentials of the form

$\varepsilon>0$0

and

$\varepsilon>0$1

The $\varepsilon>0$2 behavior near the origin produces a universal potential wall in this gauge. The authors assume that the same topological sectors identified exactly in three dimensions continue to organize the higher-dimensional propagators: standard trajectories contribute to $\varepsilon>0$3, while tunneling trajectories contribute to $\varepsilon>0$4.

The numerical Picard–Lefschetz analysis finds that the contour $\varepsilon>0$5 intersects the first tunneling saddle and does not intersect the ascent lines of higher necklaces. This supports the conclusion that the single necklace dominates in four and five dimensions.

However, the paper qualifies this result carefully. The exact propagator is known only in the three-dimensional model. In higher dimensions, the relevant saddle sectors are inferred from the three-dimensional analysis and then studied numerically. The authors do not establish that the proposed tunneling trajectories are the correct quantum-mechanical propagator contributions in arbitrary dimension.

This limitation is consequential. The pole contribution to the entropy is linear in $\varepsilon>0$6 in every dimension, whereas the ordinary necklace action has a dimension-dependent, generally nonlinear dependence on $\varepsilon>0$7. The exact cancellation that produces a flux-independent entropy in three dimensions therefore need not occur in higher dimensions. At $\varepsilon>0$8, the authors note that the putative tunneling contribution can exceed the empty de Sitter entropy for $\varepsilon>0$9, apparently violating an entropy bound. They explicitly identify this as evidence that the assumed higher-dimensional tunneling sector may not be the correct quantum amplitude.

The paper also constructs magnetic necklaces supported by an $k=1$0 Yang–Mills configuration on $k=1$1. The Yang–Mills stress tensor produces a negative Euclidean energy density scaling as $k=1$2, generating a bounce analogous to the axion case. The classical solutions are periodic and can again be repeated arbitrarily, so the Euclidean necklace divergence persists. Numerical analysis in four dimensions indicates that the same regulated Lorentzian contour selects the Gibbons–Hawking saddle.

Figure 5: Real-action trajectories for higher-dimensional axion necklaces, illustrating the repeated-bounce structure in kinetic gauge.

Figure 6: Constant-height curves showing that ascent lines reaching the origin cannot originate from higher-necklace saddles.

Alternative contour and unresolved consistency conditions

The contour prescription is not unique. The alternative contour

$k=1$3

or the half-contour $k=1$4, selects the opposite orientation and does not capture the positive $k=1$5 Gibbons–Hawking saddle. Its leading contribution is instead the trivial $k=1$6 configuration, yielding no entropy of order $k=1$7.

Thus, the result depends on the side from which the Lorentzian contour avoids the $k=1$8 pole. The authors associate $k=1$9 with stable matter fluctuations but acknowledge that the positive regulator leaves the conformal-mode problem unresolved. A strictly Lorentzian contour avoids wrong-sign conformal fluctuations, whereas the regulated contour retains them near the dominant saddle. Conversely, the Vilenkin-type contour resolves the conformal issue at the cost of unstable matter fluctuations.

The dominant tunneling saddle also violates the standard Kontsevich–Segal–Witten allowability criterion because its complex scale factor passes through regions with $ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$0. The paper suggests that a weaker spectral criterion might still permit the saddle, but this is not established. Nor is it shown that the minisuperspace intersection numbers remain unchanged after including inhomogeneous gravitational and matter fluctuations.

The central open questions are therefore specific:

• Does the $ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$1-independent entropy persist beyond three-dimensional minisuperspace?
• Does a full fluctuation analysis preserve the saddle intersection numbers?
• Can the $ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$2 redundancy and the associated tunneling sector be formulated directly in an observer-centered static-patch Hilbert space?
• Is there a precise allowability or stability criterion under which the complex contour winding around $ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$3 is admissible?

Conclusion

The paper argues that cosmological wormholes expose a discrete version of the Euclidean gravitational path-integral problem. Repeated real Euclidean bounces generate necklaces with increasingly negative action, making the unrestricted Euclidean sum divergent. In the exactly solvable three-dimensional axion model, the authors resolve this problem by treating $ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$4 as a gauge redundancy, decomposing the amplitude into standard and tunneling sectors, and integrating the lapse over $ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$5.

Picard–Lefschetz theory then selects a single $ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$6 tunneling necklace while excluding higher repetitions. Its action is

$ds^2=N^2d\tau^2+a(\tau)^2d\Omega_2^2,$7

so the resulting entropy equals the empty de Sitter entropy and is independent of the axion flux. Higher-dimensional axion and Yang–Mills calculations provide numerical support for the same saddle-selection mechanism, but their entropy interpretation remains conditional because the relevant propagator sectors are not derived exactly. The paper’s principal conclusion is therefore well established within three-dimensional minisuperspace and suggestive, but not yet conclusive, beyond it.