⚛️ high-energy theory

Runaways in the no-boundary wavefunction

This paper investigates the no-boundary wavefunction in Einstein gravity with a positive cosmological constant, identifying families of weakly curved bulk saddles that exhibit runaway behavior threatening state normalizability, while exploring their validity through the Kontsevich-Segal-Witten criterion and proposing a contour integral model to exclude specific runaway contributions.

Original authors: Jordan Cotler, Prateksh Dhivakar, William Harvey, Victor Ivo, Kristan Jensen

Published 2026-10-19

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CC BY 4.0

Original authors: Jordan Cotler, Prateksh Dhivakar, William Harvey, Victor Ivo, Kristan Jensen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the grand effort to understand the universe, physicists often ask a simple but profound question: how did it begin? One of the most influential ideas in modern cosmology suggests that the universe did not have a sharp starting point, like a singularity, but rather emerged smoothly from a state of pure geometry. This concept, known as the "no-boundary proposal," imagines the early universe as a shape that closes off on itself, much like the smooth curve of a sphere, rather than ending abruptly at a point. To make sense of this, scientists use a mathematical tool called a path integral. Think of this as a way of adding up every possible history the universe could have taken to reach its current state, weighing each history by how likely it is to happen. The goal is to find a single, stable description of the universe that makes sense physically and mathematically. However, for this description to work, the total probability of all these histories must add up to a finite number. If the sum grows without limit, the theory breaks down, and the idea of a well-defined beginning collapses.

A team of researchers has recently taken a deep dive into this problem, specifically looking at the mathematics of Einstein's gravity in a universe with a positive cosmological constant, which is the kind of universe we appear to live in. They discovered that within the standard rules of their theory, there are entire families of possible universe histories that behave dangerously. These histories, which they call "runaways," involve universes that become extremely stretched or anisotropic, meaning they look very different in one direction than in another. As these shapes stretch further, the mathematical weight assigned to them grows exponentially. In plain terms, the theory predicts that these highly distorted universes are not just possible, but overwhelmingly probable. If these runaway histories are allowed to count toward the final answer, the total probability becomes infinite. This would mean the no-boundary proposal, as currently formulated, cannot produce a stable, normalizable state for our universe.

The researchers found that these runaway solutions are not rare or exotic flukes; they appear in many different dimensions and shapes, from simple products of spheres to more complex geometries. They are "weakly curved," meaning they do not involve the extreme, Planck-scale distortions that might be dismissed as a failure of the theory at very small scales. Instead, these runaways occur in a regime where the theory of gravity is supposed to work perfectly. The team calculated the specific weights of these geometries and found that in many cases, the growth is so rapid that it threatens to overwhelm the entire calculation. For instance, in a universe shaped like a product of two spheres, if one sphere becomes much larger than the other, the probability of that configuration explodes. This happens in dimensions ranging from six up to eleven, suggesting the problem is a fundamental feature of the theory rather than a quirk of a specific setup.

To address this crisis, the team explored whether these dangerous histories should actually be included in the final sum. In quantum mechanics, not every mathematical solution to an equation necessarily contributes to the physical reality; some are excluded by the specific path or "contour" used to perform the calculation. The researchers tested a popular rule called the Kontsevich-Segal-Witten criterion, which acts as a filter to decide which complex shapes are allowed in the calculation. They found that this rule successfully excludes many of the runaway families, particularly those involving toroidal shapes or specific types of black holes. However, the rule is not a perfect shield. It excludes some solutions that the researchers' own models suggest should be included, and it fails to exclude others that are clearly problematic. In some cases, the rule removes solutions that are physically reasonable, while in others, it leaves the dangerous runaways untouched.

The team also constructed a simplified model to test how the calculation behaves when the integration path is chosen differently. In this model, they treated the size of a closing circle in the early universe as a variable to be integrated over. By carefully choosing the path of integration, they were able to exclude the runaway solutions that would otherwise make the probability infinite. This suggests that the problem might not be with the existence of these shapes, but with how we choose to count them. The researchers found that a specific, natural path for the calculation—one that includes a known, stable solution called the Nariai endpoint—automatically drops the dangerous runaways. This offers a glimmer of hope: perhaps the universe is stable, but only if we choose the correct mathematical path to describe its birth.

Despite these findings, the question remains open. The researchers have not yet proven that the correct integration path for the full theory of gravity is the one that excludes the runaways. They have shown that such a path exists in simplified models and that it aligns with some, but not all, of the existing rules for what is allowed. The study highlights a deep tension in our understanding of the early universe: the mathematics that describes the birth of the cosmos seems to naturally produce infinite probabilities unless we impose very specific, and sometimes arbitrary-seeming, restrictions on how we calculate them. The work does not settle the debate but rather clarifies the stakes. It shows that the stability of the no-boundary proposal depends entirely on the details of the integration cycle, a mathematical structure that is still not fully understood. Until we know exactly how to sum up the histories of the universe, the possibility remains that the theory predicts a beginning that is mathematically ill-defined, leaving the origin of our cosmos an unsolved puzzle.

Technical Summary: Runaways in the No-Boundary Wavefunction

Problem Statement
This work investigates the normalizability of the Hartle-Hawking no-boundary wavefunction in $d$ -dimensional Einstein gravity with a positive cosmological constant. While the no-boundary proposal aims to define a natural quantum state of the universe via a gravitational path integral (GPI), it remains unclear whether this prescription yields a normalizable state when the metric is dynamical. Previous studies have identified that for certain topologies (e.g., toroidal slices), the sum over saddle points can lead to divergences. The central problem addressed here is the existence of "runaway" saddle points—smooth, weakly curved bulk geometries whose weights $|e^{iS}|^2$ grow exponentially with geometric parameters of the late-time boundary. If these saddles contribute to the GPI, they threaten the normalizability of the no-boundary state within the regime of validity of gravitational effective field theory (EFT).

Methodology
The authors analyze the no-boundary wavefunction on late-time slices with various topologies, primarily focusing on product spheres $S^p \times S^q$ and toroidal geometries. The methodology involves:

Saddle Point Search: Solving the Einstein equations for complex metrics that smoothly cap off in the early time and asymptote to de Sitter space at late times. The analysis utilizes a minisuperspace approximation for $S^p \times S^q$ and numerical shooting methods for specific cases like $S^2 \times S^2$ .
Weight Calculation: Evaluating the renormalized on-shell action $S$ to determine the weight of each saddle. For "all-minus" geometries (metrics that are minus a real Euclidean asymptotically hyperbolic Einstein metric), the authors relate the weight to the renormalized volume $V_{\text{ren}}$ in even dimensions and the integrated $Q$ -curvature (or Weyl anomaly) in odd dimensions.
Admissibility Criteria: The paper examines whether these saddles contribute to the wavefunction by testing them against two criteria:
- Kontsevich-Segal-Witten (KSW) Criterion: A condition restricting complex metrics to ensure convergent path integrals for quantum fields.
- Contour Integration Models: For $S^1 \times S^q$ and toroidal slices, the authors construct a toy model where the wavefunction is an integral over an off-shell parameter (the cap radius). By choosing specific convergent contours (analogous to the Nariai contour), they determine which saddles have non-zero coefficients in the saddle decomposition.

Key Contributions and Results

Identification of Runaway Families: The authors identify several families of smooth, weakly curved saddles with exponentially growing weights:
- Product Spheres ( $S^p \times S^q$ ): In dimensions $6 \le d \le 11$ , there exist solutions where one sphere caps off and the other becomes large (large anisotropy). For $q \equiv 0, 3 \pmod 4$ , the weight grows as $\exp(c \phi^q / G)$ , where $\phi$ is the anisotropy ratio. This includes $S^2 \times S^3$ in $d=6$ and $S^2 \times S^4$ in $d=7$ .
- $S^1 \times S^q$ and Tori: For $q \equiv 0 \pmod 4$ , the large black hole and black brane solutions (minus Euclidean AdS black holes) exhibit runaway weights at large anisotropy (small circle).
- Hyperbolic Fillings: In $d=4$ , there are infinitely many smooth hyperbolic fillings for certain closed hyperbolic boundaries, with weights growing with the Euler characteristic of the filling.
- Lens Spaces: In $d=4$ , Taub-Bolt saddles for squashed lens spaces $L(p,1)$ have weights that grow without bound at large squashing.
KSW Analysis:
- Most runaway families (product spheres, toroidal black branes, hyperbolic fillings) correspond to "all-minus" geometries that violate the KSW criterion at the cap.
- For lens spaces, the runaway saddles also violate KSW. However, the authors construct explicit KSW-allowable contours for the bounded weight range of these saddles, showing that KSW excludes the runaway regime but allows a subset of the family.
Contour Model for $S^1 \times S^q$ :
- The authors construct a toy model for the GPI as an integral over the cap radius. They demonstrate that a natural convergent contour (the "Nariai contour") excludes the runaway large black holes and black branes for $q \equiv 0 \pmod 4$ and odd $q$ .
- This contour retains the upper Nariai saddle and the small black holes (which fail KSW) but excludes the dangerous runaways. This suggests that the integration cycle of the GPI, rather than the KSW criterion alone, may be responsible for removing the divergences.
Loop Effects and Normalizability:
- The authors argue that ordinary loop corrections cannot cancel the tree-level exponential growth of the runaways, as the growth is of order $1/G$ while loop corrections are subleading.
- For the $S^1 \times S^q$ case, they analyze loop effects near the Nariai endpoint and at large $S^2$ (in $d=4$ ). They find that while thermal graviton gas can enhance the distribution, the contribution from the runaway tail remains exponentially suppressed relative to the Nariai regime within the controlled semiclassical expansion.

Significance and Claims
The paper claims that the existence of these runaway saddles poses a genuine threat to the no-boundary proposal within gravitational EFT. If these saddles contribute, the no-boundary state is not normalizable. The authors do not claim to have definitively resolved whether these saddles contribute in the full theory but offer the following conclusions:

The Problem is Real: Runaway saddles are common in dimensions $d \ge 6$ and are not artifacts of large curvature; they exist within the EFT regime.
KSW is Insufficient/Restrictive: While KSW excludes many runaway families, it also excludes saddles (like the upper Nariai saddle and small black holes) that a natural contour model retains. Thus, KSW alone may be too restrictive to define a rich enough state space.
Integration Cycle is Key: The resolution likely lies in the specific integration cycle of the gravitational path integral. The toy model for $S^1 \times S^q$ demonstrates that a well-chosen contour can exclude runaways while preserving other physically sensible saddles, even those that fail KSW.
Open Questions: The authors emphasize that for $p, q > 1$ , they lack a comparable off-shell family and contour construction to definitively rule out the product-sphere runaways. The question of whether the gravitational integration cycle naturally excludes these dangerous saddles remains open.

In summary, the paper maps the landscape of no-boundary saddles, identifies specific directions where the wavefunction diverges, and argues that the consistency of the no-boundary proposal depends critically on the precise definition of the gravitational integration cycle, potentially requiring a mechanism to exclude runaways that is distinct from the KSW criterion.

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