⚛️ high-energy theory

Consistent Evaluation of the No-Boundary Proposal

By consistently applying the gravitational path integral to compute both amplitudes and normalization norms, this paper demonstrates that the Hartle-Hawking no-boundary proposal predicts probabilities of nearly or exactly one for closed universes, implying that all relevant cosmological states are effectively parallel to the Hartle-Hawking state.

Original authors: Ahmed I. Abdalla, Stefano Antonini, Raphael Bousso, Luca V. Iliesiu, Adam Levine, Arvin Shahbazi-Moghaddam

Published 2026-02-04

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CC0 1.0

Original authors: Ahmed I. Abdalla, Stefano Antonini, Raphael Bousso, Luca V. Iliesiu, Adam Levine, Arvin Shahbazi-Moghaddam

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

The Big Picture: Trying to Predict the Universe's Future

Imagine you are a meteorologist trying to predict the weather. You have a theory (the "No-Boundary Proposal") that says the universe started from nothing—a smooth, featureless point with no edges. You want to use this theory to calculate the probability of our current universe (with its stars, galaxies, and you) existing.

For decades, scientists have tried to do this calculation. They usually followed a recipe that looked something like this:

Calculate the "amplitude" (a raw score) for the universe to start from nothing and become what it is today.
Square that score to get a probability.
Assume the "rarity" of the starting point doesn't change the math.

This paper says that recipe is broken. The authors, Ahmed Abdalla and colleagues, went back to the drawing board and fixed the math. They found that when you do the calculation correctly, the result is shocking: The theory predicts that any closed universe is almost guaranteed to exist.

It's as if you asked, "What are the odds of rolling a 6?" and the answer was, "Well, it's 100%." But then you asked, "What are the odds of rolling a 1?" and the answer was also, "100%." The theory loses its ability to distinguish between different outcomes.

The Core Problem: The "Normalization" Mistake

To understand why the old math failed, imagine you are judging a talent show.

The Old Way: You look at a contestant (our universe) and give them a score based on how well they performed. You then divide by a "standard difficulty factor" that you assumed was the same for everyone.
The New Way: The authors realized that the "difficulty factor" (called the norm) actually changes depending on who the contestant is.

In quantum physics, to get a real probability, you have to divide the "score" of your specific universe by the "score" of the universe itself (its norm).

The Mistake: Previous scientists assumed this "norm" was a constant number, like a fixed tax rate.
The Reality: The authors calculated that the "norm" is actually a massive, complex number that depends heavily on the specific details of the universe.

When you fix the math and divide the score by the correct (massive) norm, the result changes dramatically. The probability of finding any specific closed universe state becomes nearly 1 (or 100%).

The Two Ways to Look at the Math

The paper explores two different ways of interpreting the "Gravitational Path Integral" (the giant math formula used to sum up all possible universes).

1. The "Conventional" Approach (One Universe)

Imagine you are looking at a single, specific universe.

The Result: The math shows that the "No-Boundary" state (the starting point) is almost perfectly parallel to every possible ending state.
The Analogy: Imagine a giant library where every single book is written in the exact same language, with the exact same plot. If you pick up any book, it looks exactly like the "No-Boundary" book.
The Consequence: The theory cannot tell the difference between a universe with life and a universe that is just empty space. They are all "nearly parallel." The probability of finding our universe is ~1, but so is the probability of finding an empty universe. The theory fails to make useful predictions.

2. The "Statistical" Approach (An Ensemble of Universes)

Imagine the math isn't describing just one universe, but an average over a huge collection (an ensemble) of different possible universes.

The Result: In this view, the Hilbert space (the mathematical room where all states live) is only one-dimensional.
The Analogy: Imagine a room with only one chair. No matter where you stand in the room, you are sitting in that one chair. There is no "other" chair to compare it to.
The Consequence: In this scenario, the probability of any state is exactly 1. It's not just "almost" 1; it is mathematically forced to be 1. The theory predicts that in every single version of reality in this collection, the universe exists.

The Inflation Example: Why the Old Theory Failed

The authors tested this on Cosmic Inflation (the theory that the universe expanded rapidly right after the Big Bang).

The Old Prediction: Using the broken math, the theory predicted that the most likely universe is one that barely inflates at all, or one that is just a boring, empty bubble of space with no stars or galaxies. It essentially predicted an "empty universe."
The New Reality: When they applied the correct math (dividing by the proper norm), they found that the probability of any inflationary universe (including the one we live in) is nearly 100%.
The Twist: This doesn't mean the theory is "better" at predicting us; it means the theory has lost its power to discriminate. It says, "Everything is equally likely," which is the same as saying, "I can't predict anything."

The "Empty" Conclusion

The paper concludes with a somewhat ironic realization:

The states are all the same: The quantum states representing different universes are so similar (nearly parallel) that they are indistinguishable in the math.
The "No-Boundary" state is everywhere: The starting point of the universe is essentially the same as the ending point, no matter what the ending looks like.
The Fix: To make the theory useful again, scientists would have to change the rules. They would need to "project out" the No-Boundary state, effectively saying, "Let's ignore the starting point and only look at the differences between universes." But doing that requires making arbitrary choices that the original theory was supposed to avoid.

Summary in One Sentence

The authors found that when you calculate the odds of the universe correctly, the "No-Boundary Proposal" predicts that every possible closed universe is almost guaranteed to exist, rendering the theory unable to distinguish between a universe like ours and an empty, lifeless void.

Technical Summary: Consistent Evaluation of the No-Boundary Proposal

Problem Statement
The Hartle-Hawking no-boundary proposal posits that the quantum state of the universe, $|\Psi\rangle$ , is the state with no boundaries ( $|\emptyset\rangle$ ). To extract physical predictions, one must compute the probability of observing specific cosmological data $J$ (such as a specific metric and field configuration on a closed universe $\Sigma$ ) using the Born rule. This requires evaluating the normalized inner product:
$P(\emptyset \to J) = \frac{|\langle J | \emptyset \rangle|^2}{\langle J | J \rangle \langle \emptyset | \emptyset \rangle}$
Historically, calculations of this probability relied on two implicit assumptions: (1) that the set of states defined by boundary conditions $J$ are mutually orthogonal, allowing the unnormalized amplitude $\Psi(J) \propto G(J)$ to be interpreted directly as a wavefunction; and (2) that the norm $\langle J | J \rangle$ is independent of $J$ . These assumptions led to predictions, such as an exponentially suppressed probability for a universe with sufficient inflation or an empty de Sitter universe, which conflict with observational data. The central problem addressed in this paper is the consistent evaluation of the gravitational path integral (GPI) to determine the norms and inner products without these unjustified assumptions.

Methodology
The authors employ the gravitational path integral (GPI) defined over complex manifolds $M$ with boundary conditions $J$ on $\Sigma = \partial M$ . They analyze the GPI under two distinct interpretative frameworks:

The Conventional Approach: The GPI $G(J_1, J_2)$ is interpreted as a single unnormalized transition amplitude (inner product) between states defined by boundary conditions. The probability is computed by explicitly evaluating the numerator $|G(J)|^2$ and the denominator $G(J^*, J)G(\emptyset)$ as separate path integrals. The evaluation utilizes a saddle-point approximation in the semiclassical limit ( $G_N \to 0$ ), distinguishing between disconnected geometries (products of separate spacetimes) and connected geometries (spacetimes bridging the boundaries).
The Statistical Approach: The GPI is interpreted as computing an ensemble average over a collection of theories. In this view, $G(J_1, \dots, J_n)$ represents the average of products of inner products over an ensemble of one-dimensional Hilbert spaces. This approach has gained prominence in recent years to resolve factorization puzzles in holography.

The authors apply these methods to a model of single-field slow-roll inflation, utilizing both Dirichlet boundary conditions (fixing the scale factor $a$ and inflaton $\phi$ ) and conformal boundary conditions (fixing the trace of the extrinsic curvature $K$ and $\phi$ ). They explicitly calculate the contributions of disconnected "hemisphere" saddles and connected "pancake" or "cylinder" saddles to the inner products.

Key Contributions and Results

Re-evaluation of Norms and Probabilities: The paper demonstrates that the norm $\langle J | J \rangle$ is not independent of $J$ . In the semiclassical limit, the dominant contribution to the norm $G(J^*, J)$ comes from disconnected geometries (two separate spacetimes), which scale as $|G(J)|^2$ . The connected geometries, which would distinguish different states, are suppressed by factors of $\exp(-L^2/G_N)$ , where $L$ is the characteristic curvature length.
Conventional Approach Result: When the GPI is evaluated consistently, the probability for any closed universe state $|J\rangle$ in the Hartle-Hawking state is found to be:
$P(\emptyset \to J) \approx 1$
Specifically, the states $|J\rangle$ are nearly parallel to the Hartle-Hawking state $|\emptyset\rangle$ up to non-perturbative corrections. The "wavefunction" interpretation $\Psi(J) \propto G(J)$ fails because the normalization factor cancels the amplitude's dependence on $J$ to leading order.
Statistical Approach Result: Under the statistical interpretation, where the GPI computes averages over an ensemble of theories, the Hilbert space of closed universes in each realization is one-dimensional. Consequently, all normalized inner products are pure phases, and the probability for any state $J$ is exactly 1:
$P(\emptyset \to J) = 1$
Inflationary Example: Applying these results to slow-roll inflation, the authors show that the historical prediction of an empty universe (or a universe with negligible inflation) arises from the inconsistent assumption that $\langle J | J \rangle$ is constant. When calculated consistently, the probability of observing a universe with specific inflationary parameters is nearly unity, rendering the proposal unable to discriminate between different cosmological histories.
Orthogonality and State Construction: The paper highlights that classically distinct data $J$ do not define orthogonal quantum states in the GPI framework. To recover a non-trivial Hilbert space where states are distinguishable, one must either project out the no-boundary state from all physical states or modify the GPI to include only connected geometries ( $G_{conn}$ ).

Significance
The paper claims that the Hartle-Hawking no-boundary proposal, when evaluated consistently using the gravitational path integral, fails to provide useful probabilistic predictions for cosmology. The result that $P(\emptyset \to J) \approx 1$ (or exactly 1) implies that the proposal does not discriminate between naively different closed universes.

The authors argue that this outcome is a direct consequence of the structure of the gravitational path integral, where the disconnected contributions dominate the norms. This finding upends the extant literature that relies on the assumption of $J$ -independent norms. While the statistical approach yields a trivial result (all probabilities are 1) consistent with the one-dimensional Hilbert space of closed universes, the conventional approach also yields a trivial result (probabilities near 1) because the Hilbert space is effectively trivialized by the near-parallelism of all states to the no-boundary state.

The paper concludes that for the no-boundary proposal to yield distinct predictions, the mapping of boundary conditions to quantum states must be fundamentally modified, such as by projecting out the no-boundary state or restricting the path integral to connected geometries. Without such modifications, the proposal cannot account for the observed properties of our universe, such as the duration of inflation, as it assigns nearly equal probability to all possible closed universe configurations.