⚛️ general relativity

When inflationary perturbations refuse to classicalise: the role of non-Gaussianity in Wigner negativity

By computing the Wigner function of inflationary curvature perturbations using the EFT of inflation, this paper demonstrates that primordial non-Gaussianities generate persistent quantum interference fringes and growing Wigner negativity on super-Hubble scales, proving that squeezing alone is insufficient to ensure classicality and suggesting that genuinely quantum signatures of the universe's origins may be detectable.

Original authors: Aurora Ireland, Vincent Vennin

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Aurora Ireland, Vincent Vennin

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

The Big Question: Did the Universe Start as a Quantum Ghost or a Classical Rock?

Imagine the very beginning of the universe. According to the standard story, the universe started as a tiny, quantum "fuzz" of energy. As it expanded rapidly (a period called inflation), these tiny quantum fluctuations were stretched out to become the seeds of galaxies, stars, and us.

For decades, physicists have treated these seeds as if they were classical objects (like little rocks or waves on a pond) once they got big enough. The logic was: "They got so big and stretched out that they must have 'decided' to become real, solid things." This is called classicalisation.

This paper asks a bold question: What if they never actually became classical? What if they are still quantum ghosts, even today?

The Detective Tool: The Wigner Function

To solve this mystery, the authors use a special mathematical tool called the Wigner function. Think of this as a "quantum X-ray" or a "probability map" of a system.

The Classical Map: If a system is truly classical, this map looks like a smooth, positive hill. It's like a weather map showing where rain is likely to fall. The numbers are always positive (you can't have -50% chance of rain).
The Quantum Map: If a system is truly quantum, this map can have negative values. In the world of probability, "negative probability" sounds impossible, but in quantum mechanics, it represents interference. It's like two waves crashing into each other: where they meet, they cancel out (negative) or boost up (positive).

The Golden Rule (Hudson's Theorem): The paper relies on a famous rule: If a quantum state is perfectly smooth and Gaussian (bell-shaped), its map is always positive. But if the state gets "weird" (non-Gaussian), the map must have negative spots.

The Experiment: Stretching the Universe

The authors simulated the universe during inflation, but they didn't just look at the "slow and steady" expansion (Slow Roll). They looked at a more chaotic, fast-expanding phase called Ultra-Slow Roll (USR).

They used a method called the Effective Field Theory (EFT) to track the quantum state of these fluctuations as they grew.

The Analogy: The Trampoline and the Mirror

Imagine the quantum fluctuations are a ball bouncing on a trampoline.

Standard View: As the trampoline stretches (inflation), the ball gets squished into a long, thin line. Physicists thought this "squeezing" made the ball act like a classical particle.
This Paper's View: The authors realized that because the universe's expansion isn't perfectly smooth (it has "kinks" and non-linearities), the ball doesn't just stretch; it starts to bounce off invisible walls.

In their math, these "walls" are boundaries where the quantum wave must vanish. When a wave hits a wall, it reflects. The original wave and the reflected wave crash into each other, creating a standing wave pattern.

The Discovery: Interference Fringes and "Negative" Spots

When they looked at the "Wigner X-ray" of these fluctuations, they found something shocking:

Interference Fringes: Instead of a smooth hill, the map looked like a zebra crossing or a barcode. There were alternating stripes of positive and negative values.
The "Boomerang" Shape: The shape of the probability cloud wasn't an ellipse (like a stretched balloon); it bent into a boomerang shape.
Growing Negativity: As time went on (as the universe expanded more), these negative spots didn't disappear. In fact, they grew stronger.

What does this mean?
Those negative spots are the "quantum fingerprints." They prove that the fluctuations are still in a superposition (existing in multiple states at once) and are interfering with themselves. They have not turned into classical rocks.

Why This Matters: The "Squeezing" Myth

For a long time, physicists believed that squeezing (stretching the quantum state until it looks thin) was enough to make things classical.

The Paper's Verdict: Squeezing is like stretching a rubber band. It makes the band thin, but it doesn't stop the rubber from being rubber. You can still see the quantum "fuzz" if you look closely enough.
The Takeaway: The universe didn't "wake up" and become classical just because it got big. It might still be holding onto its quantum nature, especially in the chaotic, fast-expanding phases of the early universe.

The Future: Hunting for Quantum Ghosts

The authors suggest that this is great news for cosmology. If the universe is still quantum, we might be able to find evidence of it in the Cosmic Microwave Background (CMB)—the afterglow of the Big Bang.

The Search: We need to look for specific patterns (like the "zebra stripes" in the math) in the distribution of galaxies or black holes.
The Hope: Instead of thinking, "Oh no, the quantum effects are gone forever," we can say, "The quantum effects are still there, hiding in the negative spots of the Wigner function, waiting to be found!"

Summary in One Sentence

This paper shows that the seeds of our universe didn't lose their quantum magic when they grew up; they are still interfering with themselves like waves, and we might be able to see those "quantum ghosts" in the stars today.

1. Problem Statement

The standard paradigm of cosmology posits that cosmic structures originate from quantum vacuum fluctuations during inflation. While these perturbations are fundamentally quantum, they are routinely treated as classical stochastic fields in the calculation of cosmological observables (e.g., CMB anisotropies). This "classicalization" is typically justified by two arguments:

Squeezing: As modes exit the Hubble horizon, the quantum state becomes highly squeezed, suppressing the commutator between the field and its momentum, making them effectively commute.
Decoherence: Interaction with environmental degrees of freedom (other fields or sub-horizon modes) suppresses quantum interferences.

However, the authors argue that squeezing alone does not guarantee classicality. According to Hudson's theorem, a pure quantum state has a non-negative Wigner function (a necessary condition for a classical probability distribution) if and only if the state is Gaussian. Since General Relativity is non-linear, primordial perturbations inevitably develop non-Gaussianities at higher orders in perturbation theory. The paper investigates whether these non-Gaussianities, arising from the non-linear dynamics of inflation, prevent the Wigner function from becoming positive, thereby preserving "quantum birthmarks" (interferences) even at late times.

2. Methodology

The authors employ a non-perturbative approach to compute the Wigner function of curvature perturbations, specifically focusing on backgrounds where non-Gaussian effects are significant.

Effective Field Theory (EFT) of Inflation: They utilize the EFT of inflation, which describes the Goldstone boson ( $\chi$ ) associated with the spontaneous breaking of time translation invariance. This framework allows for a systematic inclusion of non-linear interactions to all orders.
Separate Universe Approach & Matching: To handle the transition from sub-Hubble (where full quantum field theory is needed) to super-Hubble scales (where the homogeneous mode dominates), they adopt a matching prescription inspired by the $\delta N$ $δ N$ formalism.
- Modes are treated perturbatively until they cross a coarse-graining scale $\lambda_\sigma$ .
- Beyond this scale, the system is described by a homogeneous Schrödinger equation for the coarse-grained field $\chi_0$ .
- Initial conditions for the wavefunction are set by matching the two-point correlation functions of the linear theory to the homogeneous mode at the matching time $t_\star$ .
Constant-Roll Inflation: The authors specialize to Constant-Roll (CR) inflation, where the second Hubble-flow parameter $\epsilon_2 = \dot{\epsilon}_1 / (H\epsilon_1)$ $ϵ_{2} = \overset{ϵ}{˙}_{1} / (H ϵ_{1})$ is constant. This includes:
- Slow Roll (SR): $\epsilon_2 = 0$ .
- Ultra-Slow Roll (USR): $\epsilon_2 = -6$ .
  The USR case is particularly interesting as it features a growing mode for the curvature perturbation that scales as $a^3$ , leading to large fluctuations.
Wavefunction Construction:
- They derive the action for the Goldstone boson and perform a non-linear canonical transformation to a variable $Y$ where the potential is quadratic.
- The Schrödinger equation for $Y$ admits Gaussian solutions. However, the physical field $\chi_0$ is related to $Y$ via a non-linear transformation.
- Crucially, the domain of $Y$ is truncated (due to the non-linear mapping of $\chi_0 \in (-\infty, \infty)$ ), requiring a Dirichlet boundary condition at a finite boundary $Y_b$ .
- The wavefunction is constructed using the method of images: a superposition of the original Gaussian and a reflected Gaussian to satisfy the boundary condition.
Wigner Function Calculation: The Wigner function is computed by transforming the wavefunction back to the original variables $(\chi_0, \pi_0)$ . Due to the non-linear mapping, the resulting Wigner function is non-Gaussian and must be evaluated numerically (or via series expansions involving Bessel functions).

3. Key Contributions

Non-Perturbative Framework: The paper provides a rigorous non-perturbative method to compute the Wigner function for inflationary perturbations, overcoming the limitations of standard perturbation theory which fails to capture sign changes (negativity) in the Wigner function.
Explicit Demonstration of Wigner Negativity: They explicitly show that non-Gaussianities induced by non-linear dynamics lead to regions of Wigner negativity, proving that the state remains genuinely quantum.
Refutation of Squeezing as a Classicality Criterion: The work demonstrates that even in highly squeezed states (like USR), the system does not become classical if non-Gaussianities are present. Squeezing is shown to be coordinate-dependent and insufficient to diagnose classicality.
Distinction between SR and USR: The authors highlight a qualitative difference between Slow Roll and Ultra-Slow Roll backgrounds. While SR backgrounds remain effectively Gaussian (and thus positive Wigner) in the homogeneous limit, USR backgrounds develop pronounced non-Gaussian features and Wigner negativity.

4. Key Results

Interference Fringes: In the USR background, the Wigner function develops pronounced interference fringes on super-Hubble scales. These fringes arise from two sources:
1. Interference between the original and reflected Gaussian wavepackets (due to the boundary condition).
2. The non-linear canonical transformation mapping the Gaussian $Y$ -space to the physical $\chi_0$ -space.
Growth of Negativity: The Wigner negativity volume ( $N$ $N$ ), a measure of the quantumness of the state, grows monotonically with time.
- In the USR regime, the negativity grows approximately as $N \propto a^2$ (or $e^{2\Delta N_\star}$ ) at early times.
- This growth indicates that quantum effects do not decay but rather become more significant as the universe expands in these specific backgrounds.
Phase Space Structure: The Wigner function in USR evolves from a localized "boomerang" shape into a structure with a cascade of rapidly oscillating fringes at large negative momenta. The envelope of these fringes mirrors the central structure, but the oscillations become finer and more frequent.
Dependence on Parameters:
- Amplitude ( $\bar{P}_R$ ): Larger curvature perturbation amplitudes lead to stronger non-perturbative effects and generally larger negativity (though the scaling is complex due to the interplay of perturbative and non-perturbative regions).
- Coarse-graining ( $\sigma$ ): The results are robust to moderate variations in the matching scale, confirming the physical relevance of the non-linear effects.

5. Significance and Implications

Quantum Signatures in Cosmology: The results suggest that the prospects for detecting genuinely quantum signatures of the universe's origin are less bleak than previously thought. If the early universe underwent phases like Ultra-Slow Roll (which are theoretically motivated for enhancing small-scale power), the resulting non-Gaussianities could preserve quantum interferences.
Challenging the Classical Paradigm: The paper challenges the standard lore that inflationary perturbations naturally classicalize. It shows that squeezing is not a sufficient condition for classicality; non-Gaussianity is the critical factor that maintains quantum coherence.
Observational Prospects: The presence of Wigner negativity close to the origin in phase space suggests that observables sensitive to large-density objects (e.g., Primordial Black Holes or Ultra-Compact Dark Matter Halos) might be able to detect these quantum signatures.
Future Directions: The authors outline extensions to include gradient corrections (to handle transitions between SR and USR) and open-system effects (decoherence), suggesting that the developed framework is a robust tool for exploring the quantum-to-classical transition in the early universe.

In conclusion, this work establishes that primordial fluctuations in non-slow-roll inflationary backgrounds retain their quantum nature through non-Gaussian induced Wigner negativity, offering a potential pathway to observationally test the quantum origins of cosmic structure.