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

TL;DR

The paper tackles whether inflationary perturbations retain quantum birthmarks beyond Gaussianity by analyzing the Wigner function in an EFT of inflation framework. It develops a non-perturbative, separate-universe-like matching for the long-wavelength homogeneous mode and computes the Wigner function for the Goldstone mode χ, including primordial non-Gaussianities. In constant-roll backgrounds, especially ultra-slow-roll, the Wigner function exhibits pronounced interference fringes and regions of negativity that grow roughly as $a^2$, revealing enduring quantum interference effects that squeezing alone cannot erase. These results imply that genuinely quantum signatures of the universe’s origins may be detectable in cosmological observables, motivating further exploration of gradient corrections and open-system effects to quantify observable quantum features.

Abstract

Inflationary perturbations are quantum in origin. Yet, when computing cosmological observables, they are often treated as classical stochastic fields. Do they nevertheless retain quantum birthmarks? A hallmark of genuinely quantum behaviour is quantum interferences, arising from phase coherence between distinct branches of the wavefunction. Such interference is diagnosed by the non-positivity of the Wigner function, and according to Hudson's theorem, the only pure states with positive Wigner functions are Gaussian states. Consequently, any departure from Gaussianity necessarily implies a non-positive Wigner function, precluding a description in terms of a classical distribution. This motivates us to compute the Wigner function of curvature perturbations, accounting for primordial non-Gaussianities, using the EFT of inflation. We find that the Wigner function develops pronounced interference fringes on super-Hubble scales, and in particular, its negativity grows as $a^2$ in ultra-slow-roll backgrounds. These results demonstrate that quantum effects can remain significant at late times, and that squeezing alone does not ensure classicality, contrary to standard lore. This suggests that the prospects for detecting genuinely quantum signatures of the universe's origins in cosmological observables may be less bleak than previously thought.

Figures (7)

• Figure 1: Wigner function in a SR background for a representative benchmark -- RB in Eq. \ref{['eq:RB']} -- evaluated at $\Delta N_\star \equiv N - N_\star = 0$$e$-folds after the matching time. Cross sections correspond to ellipses in phase space centred at the origin. Note that we plot the dimensionless combinations $H \chi_0$ and $\pi_0/H$.
• Figure 2: Wigner function in a USR ($\epsilon_2 = -6$) background for the reference benchmark in Eq. \ref{['eq:RB']} evaluated at $\Delta N_\star = 0$ (left column) and $\Delta N_\star = 0.72$ (right column) $e$-folds after the matching time. The bottom panels show a different angle from which it is clear that the Wigner function assumes non-positive values.
• Figure 3: USR Wigner function evaluated at three sample time steps: $\Delta N_\star = 0$ (top left), $\Delta N_\star = 0.17$ (top right), and $\Delta N_\star = 0.35$ (bottom), where $\Delta N_\star = N - N_\star$ is the number of $e$-folds after the matching time. Note that at later times, the full distribution extends beyond the small inset shown here (see Fig. \ref{['fig:USRWigner']}). Other parameters fixed to RB values -- see Eq. \ref{['eq:RB']}. A video of the time evolution is available https://www.youtube.com/watch?v=sd88gwogfAg.
• Figure 4: Wigner negativity $\mathcal{N}$ of Eq. \ref{['eq:negativitydef']} as a function of the number of $e$-folds since matching, $\Delta N_\star = N - N_\star$, for our reference benchmark of Eq. \ref{['eq:RB']} (red) and a second benchmark with a larger scalar amplitude $(\bar{\mathcal{P}}_\mathcal{R}, \sigma) = (0.28,1)$ (blue). The negativity evolution is well-fit by an exponential growth law $\mathcal{N} = c_1 e^{2 \Delta N_\star} + c_2$. For the reference benchmark, the fitting coefficients are empirically found to be $(c_1, c_2) = (0.15,0.035)$; for the second benchmark, $(c_1, c_2) = (0.17,-0.034)$.
• Figure 5: Left: USR Wigner function for RB values, $(\bar{\mathcal{P}}_{\mathcal{R}}, \sigma) = (0.14,1)$, at $\Delta N_\star = 0$. Right: Same plot for $(\bar{\mathcal{P}}_{\mathcal{R}}, \sigma) = (0.014,1)$, also evaluated at $\Delta N_\star = 0$.