Quantum-to-classical Transition of Cosmological Perturbations for Non-vacuum Initial States

TL;DR

This work extends the quantum-to-classical transition of cosmological perturbations to non-vacuum initial states, showing that in the high-squeezing limit the perturbations admit a classical stochastic description with non-Gaussian statistics tied to the initial state. By analyzing Schrödinger, Heisenberg, and Wigner formalisms, it demonstrates a consistent emergence of classicality dominated by the quasi-isotropic growing mode, without requiring environmental decoherence or semiclassical gravity. The Wigner function becomes concentrated along classical trajectories as $|r_k|\to\infty$, while non-Gaussian features persist in the stochastic amplitudes for non-vacuum initial conditions. Observationally, the paper proposes measuring the CMB temperature kurtosis to detect non-Gaussian signatures from initial non-vacuum states, though such effects can be obscured by cosmic variance and mode averaging.

Abstract

Transition from quantum to semiclassical behaviour and loss of quantum coherence for inhomogeneous perturbations generated from a non-vacuum initial state in the early Universe is considered in the Heisenberg and the Schrödinger representations, as well as using the Wigner function. We show explicitly that these three approaches lead to the same prediction in the limit of large squeezing (i.e. when the squeezing parameter $|r_k|\to \infty$): each two-modes quantum state (k, -k) of these perturbations is equivalent to a classical perturbation that has a stochastic amplitude, obeying a non-gaussian statistics which depends on the initial state, and that belongs to the quasi-isotropic mode (i.e. it possesses a fixed phase). The Wigner function is not everywhere positive for any finite $r_k$, hence its interpretation as a classical distribution function in phase space is impossible without some coarse graining procedure. However, this does not affect the transition to semiclassical behaviour since the Wigner function becomes concentrated near a classical trajectory in phase space when $|r_k|\to \infty$ even without coarse graining. Deviations of the statistics of the perturbations in real space from a Gaussian one lie below the cosmic variance level for the N-particles initial states with N=N(|k|) but may be observable for other initial states without statistical isotropy or with correlations between different k modes. As a way to look for this effect, it is proposed to measure the kurtosis of the angular fluctuations of the cosmic microwave background temperature.

Figures (2)

• Figure 1: The kurtosis $Q$ as a function of $\log (N_0)$ for a squeezed thermal state specified by $N_0$ (for given ${\bf k}$).
• Figure 2: The probability distribution as a function of $\log (N_0)$ again for a squeezed thermal state. In order to visualize the shape of the distribution and not its width, we plot ${\cal P}$, for given $N_0$, as a function of $|y|/\sqrt{<y~y^{\dagger}>}$. We see that for $N_0\ll 1$, it is basically Gaussian.