⚛️ high-energy theory

Probability measures for de Sitter JT gravity

This paper demonstrates that Jackiw-Teitelboim gravity with a positive cosmological constant yields a flat probability distribution for the size of expanding de Sitter universes through holography-consistent double-trumpet amplitudes, contrasting with the suppression of large universes predicted by no-boundary wavefunctions.

Original authors: Wilfried Buchmuller, Alexander Westphal

Published 2026-10-09

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CC BY 4.0

Original authors: Wilfried Buchmuller, Alexander Westphal

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

Imagine trying to understand how a vast, expanding universe could emerge from a single, tiny quantum state. This is one of the deepest puzzles in modern physics, sitting at the intersection of gravity and quantum mechanics. To tackle such a massive problem, scientists often use simplified models, much like a cartographer drawing a flat map of a curved world to understand its basic shape before tackling the full globe. One such model is a theory of gravity that exists in just two dimensions, where space and time are reduced to a simple sheet. In this simplified world, researchers can perform calculations that would be impossible in our full four-dimensional reality. The goal is to see if these models can explain why our universe is so large and why it is expanding, a process known as inflation. A central question in this field is how to assign a probability to the size of the universe: does the math suggest that small universes are common and large ones are rare, or is the opposite true?

In a recent study, physicists Wilfried Buchmüller and Alexander Westphal used this two-dimensional model to investigate how the universe might grow from a quantum beginning. They focused on a specific mathematical object called a density matrix, which acts like a statistical map of all possible universes, rather than a single wave function that describes just one. In their approach, they considered a "connected" contribution to this map, which represents a universe evolving continuously from a specific starting point, rather than a disconnected, isolated possibility. By analyzing the behavior of these connected universes, they discovered a surprising result: the probability of finding a universe of any given size is essentially the same. In other words, the math predicts a flat distribution, meaning that large, expanding universes are just as likely to emerge as small ones. This finding aligns perfectly with the theory of inflation, which requires large universes to be common.

This result stands in sharp contrast to other popular theories that rely on a different mathematical tool called the no-boundary wave function. Those theories suggest that the probability of a universe existing drops off as it gets larger, effectively making huge, inflationary universes extremely unlikely. The authors of this study argue that the no-boundary approach misses a crucial piece of the puzzle: the continuous connection between the initial quantum state and the later expanding universe. By using a method that respects the holographic principle—a concept suggesting that the physics of a volume of space can be described by information on its boundary—they identified a specific type of wave function that behaves correctly at the edges of their model. When they used this holographically consistent wave function to build their density matrix, the suppression of large universes disappeared.

The researchers also examined the mathematical structure of the space where these universes live, known as a Hilbert space, which is the framework used to define probabilities in quantum mechanics. They found that many of the standard wave functions used in the past have mathematical singularities, or points where the values blow up to infinity, making it impossible to define a proper probability space for them. However, the specific wave functions they championed, which are derived from the holographic boundary behavior, avoid these fatal flaws in a way that allows for a consistent definition of probability. While the final probability distribution cannot be normalized to a total of one because the universe can be infinitely large, the relative probabilities remain clear: the chance of finding a universe does not decrease as it grows.

Ultimately, this work suggests that if we look at the emergence of the universe through the lens of a connected density matrix derived from holographic principles, the universe has no preference for being small. It treats all sizes equally, allowing for the vast, expanding cosmos we observe today without requiring a miraculous coincidence. The study does not claim to have solved the mystery of the universe's origin, but it provides a robust mathematical argument that the rules of quantum gravity, when applied correctly to this simplified model, naturally favor the existence of large, inflationary universes. This offers a compelling alternative to older ideas that struggled to explain why our universe is so big, pointing the way toward a more consistent understanding of how the cosmos might have begun.

Technical Summary: Probability Measures for de Sitter JT Gravity

Problem Statement
The emergence of classical expanding universes from a quantum state remains a central unsolved problem in quantum gravity and cosmology. While "wavefunctions of the universe" (Euclidean and Lorentzian) have been proposed to address this, deriving testable predictions consistent with observed cosmic inflation has proven difficult. Specifically, standard probability distributions derived from the Klein-Gordon (KG) current for no-boundary wavefunctions typically predict a suppression of large universes. This suppression is inconsistent with inflationary models, which require a high probability for large universe sizes. Furthermore, defining a consistent Hilbert space for Jackiw-Teitelboim (JT) gravity in de Sitter (dS) space is complicated by singularities in exact solutions of the Wheeler-DeWitt (WDW) equation at finite universe sizes.

Methodology
The authors investigate these issues within the framework of 2d de Sitter JT gravity, a toy model that is exactly solvable and contains all physical information in its minisuperspace version. The study proceeds through the following steps:

Density Matrix Formulation: Instead of relying solely on wavefunctions, the authors utilize a conditional density matrix defined by a gravitational path integral. This matrix describes a mixed state, incorporating both "disconnected" contributions (related to no-boundary wavefunctions) and "connected" contributions (arising from geometries like double-trumpets and bra-ket wormholes).
Trumpet Wavefunctions: The authors analyze various trumpet wavefunctions, which represent transition amplitudes between a geodesic circle (initial size $\sqrt{h_0}$ $h_{0}$ ) and a boundary circle (final size $\sqrt{h}$ $h$ ). They compare wavefunctions derived from:
- Bulk path integrals with Dirichlet boundary conditions.
- Boundary Schwarzian path integrals (holographic approach).
- Different operator orderings (Henneaux vs. flat ordering).
Holographic Consistency: The authors identify the specific trumpet wavefunction ( $\psi_T$ ) that is consistent with holography, meaning its semiclassical limit matches results derived from Schwarzian boundary fluctuations. This wavefunction satisfies the WDW equation with Henneaux factor ordering.
Probability Measure Derivation: The diagonal elements of the conditional density matrix are interpreted as probability densities for the size of the universe ( $\sqrt{h}$ ) at a fixed dilaton value ( $\phi_b$ ). These are compared against probability densities derived from the KG current and no-boundary wavefunctions.
Hilbert Space Analysis: The paper examines the normalizability of these wavefunctions under the KG inner product to assess the viability of defining a Hilbert space for dS JT gravity, noting singularities at finite universe sizes.

Key Contributions and Results

Flat Probability Distribution from Holography: The primary result is that the conditional density matrix constructed from the holographically consistent trumpet wavefunction ( $\psi_T$ ) leads to a probability distribution that is flat with respect to the size of the universe ( $a = \sqrt{h}$ ). Specifically, $p_+(h|\phi_b) \sim \phi_b / \sqrt{h}$ , which implies $dP_+ \sim \phi_b d\sqrt{h}$ .
Consistency with Inflation: This flat distribution is consistent with models of inflation, which favor large universes. This stands in sharp contrast to predictions from no-boundary wavefunctions and standard KG currents. The disconnected contribution yields a probability density scaling as $p^{\text{dis}}_+(h|\phi_b) \sim (\phi_b / \sqrt{h})^3$ , while the KG current derived from the no-boundary wavefunction scales as $p^{\text{KG}}_+(h|\phi_b) \sim (\phi_b / \sqrt{h})^4$ . Both of these fall off rapidly with increasing $h$ (suppression $\sim h^{-3/2}$ and $h^{-2}$ respectively), thereby disfavoring large universes.
Trajectory Independence: The probability density derived from the connected density matrix is constant along classical trajectories in the semiclassical regime. This property allows it to be interpreted as a valid probability measure on the space of classical trajectories (integration constants). In contrast, probability densities derived from other wavefunctions (such as those with flat operator ordering) vary along trajectories and fail to define a consistent measure.
Hilbert Space Limitations: The authors demonstrate that exact solutions to the WDW equation in dS JT gravity possess singularities at finite universe sizes ( $h = \alpha^2$ or $h = h_c$ ). Consequently, these wavefunctions are not square-integrable under the standard KG inner product, posing a serious challenge for defining a standard Hilbert space for the theory. While an alternative inner product (group averaging) can render certain wavefunctions orthonormal, the singularity issue remains a fundamental obstacle.

Significance
The paper claims that the discrepancy between standard no-boundary predictions and inflationary requirements can be resolved by shifting the focus from pure wavefunctions to a conditional density matrix derived from holography. By summing over double-trumpet amplitudes consistent with Schwarzian boundary fluctuations, the model naturally yields a flat probability distribution for the universe's size. This suggests that the "connected" contributions to the gravitational path integral (associated with wormhole geometries) are essential for recovering the correct cosmological predictions in quantum gravity toy models. The work highlights that the choice of path integral measure and operator ordering is critical, as only the holographically consistent choice yields a probability measure compatible with an inflationary universe.

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