⚛️ general relativity

Dimensional (In)dependence of Lorentzian Quantum Cosmology

This paper demonstrates that within the de Sitter minisuperspace approximation of Einstein gravity, the Lorentzian path integral combined with Picard-Lefschetz theory and Borel-Padé resummation consistently yields a well-defined transition amplitude favoring the tunneling proposal for universe creation across various spacetime dimensions, including the large- $D$ limit.

Original authors: Masazumi Honda, Hiroki Matsui, Kota Numajiri, Kazumasa Okabayashi, Takahiro Terada

Published 2026-09-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Masazumi Honda, Hiroki Matsui, Kota Numajiri, Kazumasa Okabayashi, Takahiro Terada

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

To understand the birth of the universe, physicists must first accept a strange truth: space and time are not a fixed stage where events play out, but a dynamic actor that changes shape and size. In the realm of quantum cosmology, scientists try to calculate the probability of the universe appearing from nothing, moving from a state of non-existence to the vast, expanding cosmos we see today. This calculation relies on a mathematical tool called a path integral, which sums up every possible history the universe could have taken to reach its current state. For decades, researchers struggled with this tool because the standard way of calculating these sums led to results that were mathematically unstable or ambiguous, leaving them unsure whether the universe emerged through a smooth, continuous process or a sudden quantum jump.

A team of researchers has now tackled this problem by applying a sophisticated mathematical technique to the question of how the universe began, testing their method across different numbers of spatial dimensions. They found that when the universe is modeled as emerging from nothing, the most likely path is not a smooth, gentle birth, but a sudden quantum tunneling event. This result holds true whether the universe has three, four, or five dimensions, and even when the researchers pushed their calculations to a theoretical limit with a very large number of dimensions. Their work suggests that the mechanism for the universe's creation is a universal feature of gravity, independent of the specific number of dimensions in which it exists, and that the universe likely began by tunneling through a barrier of impossibility rather than growing smoothly from a tiny seed.

The core of this investigation lies in how physicists handle the mathematics of gravity. When they try to calculate the transition from nothing to something, they encounter a problem where the equations do not settle on a single, clear answer. Instead, the math produces multiple possible paths, some of which correspond to a universe that expands smoothly from a tiny, regular point, and others that correspond to a universe that appears suddenly through a quantum leap. In the past, researchers often relied on a method that involved turning time into a mathematical imaginary number to make the equations easier to solve. However, this approach introduced its own confusion, as different choices in how to handle these imaginary numbers led to completely different stories about the universe's origin.

To resolve this, the authors used a modern mathematical framework known as Picard-Lefschetz theory. This method allows scientists to deform the path of their calculation into the complex number plane, effectively finding the most stable and direct routes through the mathematical landscape. By doing this, they could identify which specific histories of the universe actually contribute to the final answer. They applied this technique to models of the universe in three, four, and five dimensions, as well as a theoretical scenario with a very large number of dimensions. In each case, they set up the conditions to represent the universe starting from a state of zero size and growing to a finite size, a scenario known as creation from nothing.

The results were strikingly consistent across all the dimensions they tested. In every case, the mathematical analysis pointed to a single dominant path: the tunneling proposal. This scenario describes the universe appearing through a quantum tunneling process, where it jumps from a state of non-existence to an expanding state, bypassing the classical rules that would normally forbid such a transition. The researchers found that the alternative idea, known as the no-boundary proposal, which suggests the universe began as a smooth, rounded-off shape without a sharp edge, was not the favored outcome under these specific conditions. Instead, the mathematics consistently favored the tunneling path, characterized by a specific type of mathematical behavior that indicates a sudden emergence rather than a gradual one.

The study also addressed a subtle complication that arises in these calculations. In certain configurations, the mathematical paths can become ambiguous, meaning it is unclear which specific route the universe should take. This is similar to a fork in the road where the signs are missing. The researchers showed that this ambiguity is not a flaw in the theory but a feature that can be resolved by combining their path analysis with another advanced technique called resurgence theory. This method allows them to look at the small, leftover errors in their calculations and use them to clarify the main result. By applying this combined approach, they demonstrated that the apparent confusion in the math cancels itself out, leaving a single, unambiguous prediction: the universe emerged via tunneling.

This finding is significant because it suggests that the way our universe began is not a fluke of our specific four-dimensional reality. Even when the researchers tested the theory in a world with many more dimensions, the tunneling mechanism remained the preferred path. They did find that if they changed the way they scaled the physical constants in their large-dimension model, a smooth, classical transition could become possible, but under the standard conditions used to describe the universe's birth, the tunneling path was the robust winner. This implies that the instability often associated with tunneling scenarios in previous studies is a fundamental characteristic of gravity itself, rather than an artifact of having too few or too many dimensions.

The authors conclude that the quantum creation of the universe from nothing is a universal phenomenon governed by the laws of gravity, regardless of the number of spatial dimensions involved. Their work provides a clearer, more rigorous picture of how the universe might have started, moving beyond the ambiguities of older methods. While the tunneling scenario they identified comes with its own theoretical challenges regarding stability, the consistency of their results across different dimensions suggests that this is the most reliable description of the universe's origin currently available within the framework of Einstein's gravity. The study does not claim to have solved every mystery of the cosmos, but it has firmly established that if the universe began from nothing, it likely did so by tunneling into existence.

Technical Summary: Dimensional (In)dependence of Lorentzian Quantum Cosmology

Problem Statement
The construction of a consistent quantum theory of gravity remains a fundamental challenge, particularly regarding the quantum creation of the universe from nothing. In the Lorentzian formulation, the transition amplitude is defined by a gravitational path integral over metrics. However, this integral is notoriously ill-defined due to the unbounded nature of the Euclidean action (stemming from the wrong-sign kinetic term of the conformal factor) and the ambiguity in choosing the integration contour in the complexified lapse plane. Historically, these ambiguities have led to competing cosmological proposals, most notably the Hartle-Hawking no-boundary proposal and Vilenkin's tunneling proposal, which predict opposite exponential behaviors for the wave function of the universe.

While recent advances utilizing Picard-Lefschetz theory have successfully resolved these ambiguities in four-dimensional (4D) Einstein gravity and two-dimensional (2D) Jackiw-Teitelboim (JT) gravity, the results differ: 4D Einstein gravity favors the tunneling proposal under Dirichlet boundary conditions, whereas 2D JT gravity can realize the no-boundary proposal. It remains unclear whether the tunneling scenario is a robust feature of Einstein gravity across all spacetime dimensions or if the favored scenario depends nontrivially on dimensionality. Furthermore, in specific regimes (Euclidean-to-Euclidean transitions), the Stokes phenomenon introduces ambiguities in the selection of contributing saddle points, requiring a resurgent analysis to ensure the continuity of the physical amplitude.

Methodology
The authors investigate the dimensional dependence of Lorentzian quantum cosmology within the de Sitter minisuperspace approximation of Einstein gravity. The study covers spacetime dimensions $D = 3, 4, 5$ , and the large- $D$ limit.

Setup: The authors employ a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) ansatz. They utilize two distinct canonical parametrizations (Option I and Option II) for the scale factor and lapse function to simplify the action for specific dimensions. The path integral is reduced to an integral over the complexified lapse variable $\bar{N}$ (or $N$ ) with Dirichlet boundary conditions on the scale factor ( $q_0$ at $t=0$ and $q_1$ at $t=T$ ).
Picard-Lefschetz Theory: To define the path integral rigorously, the integration contour is deformed onto Lefschetz thimbles (steepest-descent paths) associated with saddle points in the complex lapse plane. This method identifies which saddle points (classical, tunneling, or no-boundary) contribute to the transition amplitude based on the boundary conditions ( $q_0, q_1$ ) relative to the critical scale $q_{crit}$ .
Resurgence and Borel-Padé Resummation: In regimes where the Stokes phenomenon occurs (specifically Euclidean-to-Euclidean transitions where saddle points become degenerate), the choice of contributing saddles becomes ambiguous. The authors apply resurgence theory to resolve this. They perform a numerical analysis using Borel-Padé resummation on the asymptotic expansion of the amplitude around the unambiguous saddle. This technique demonstrates that the ambiguity in the thimble decomposition (Stokes ambiguity) is exactly canceled by the ambiguity arising from the Borel resummation of the perturbative series, rendering the total amplitude continuous and well-defined.
Large- $D$ Limit: The authors analyze the $D \to \infty$ limit, both with and without rescaling the cosmological constant and lapse function, to understand how the saddle-point structure simplifies or changes in higher dimensions.

Key Contributions and Results

Dimensional Independence of the Tunneling Proposal: The primary finding is that for the creation of the universe from nothing ( $q_0 = 0 < q_1$ ) in Einstein gravity, the tunneling proposal is universally realized for $D \geq 3$ . This holds for $D=3, 4, 5$ , and the large- $D$ limit (under specific parameter scalings). The favored saddle point consistently has $\text{Im}[N] > 0$ (anti-Wick rotation), which is required to ensure the convergence of the path integral due to the negative kinetic term in the Einstein-Hilbert action. This contrasts with 2D JT gravity, where the no-boundary proposal can be realized.
Resolution of Stokes Ambiguities: The paper explicitly demonstrates that in $D=3$ and $D=4$ (and by extension $D=5$ ), the Stokes phenomenon in Euclidean-to-Euclidean transitions leads to an ambiguity in the number of contributing tunneling saddles. Through numerical Borel-Padé resummation, the authors show that the discontinuity in the thimble contribution is precisely compensated by the Borel ambiguity in the resummation of the perturbative series. This confirms that the full transition amplitude is unambiguous and continuous across Stokes lines.
Large- $D$ Analysis:
- Without parameter rescaling, the large- $D$ limit yields only tunneling saddles; classical transitions are prohibited because the effective potential becomes negative.
- With appropriate rescaling of the cosmological constant (fixing the de Sitter radius) and the lapse, the effective potential can become positive, allowing for classical transitions if the spatial curvature and cosmological constant satisfy specific conditions ( $1 > k\ell^2$ ).
- The analysis reveals that as $D$ increases, the number of independent semiclassical saddles decreases. Saddles found in lower dimensions either merge or are pushed to infinity in the complex plane in the strict $D \to \infty$ limit, suggesting a connection between the complex saddle structures of lower dimensions and the simplified structure of the large- $D$ limit.

Significance
The paper establishes that the preference for the tunneling scenario in the quantum creation of the universe is a robust feature of Einstein gravity that persists across spacetime dimensions $D \geq 3$ . This suggests that the perturbative instability associated with the tunneling proposal (due to the mismatch between the anti-Wick rotation required for gravity and the Wick rotation required for matter fields) is an intrinsic limitation of Einstein gravity-based cosmology, regardless of dimensionality.

Consequently, the authors argue that simply varying the spacetime dimensionality is insufficient to resolve the instability issues of the tunneling scenario or to realize a stable no-boundary universe. Instead, realizing a stable creation-from-nothing scenario likely requires introducing additional gravitational degrees of freedom (such as a dilaton, as seen in 2D JT gravity) or nontrivial couplings. The work also validates the utility of resurgence theory combined with Picard-Lefschetz theory as a powerful tool for defining Lorentzian path integrals in quantum cosmology beyond the special cases previously studied.

More like this