Quantum-to-classical transition of cosmological perturbations for non-vacuum initial states

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 | → ∞): 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 | → ∞ 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 0 the kurtosis of the angular fluctuations of the cosmic microwave background temperature.

Single field inflation and non-Gaussianity

“…In our case, another difference consists in the fact that the oscillations in the spectrum studied in Ref. [7] are not present due to the monotony condition on the function h(k). In the following, in order to perform concrete calculations, we will choose an analytical form forh(k) which mimics the behavior of the spectrum considered in Ref.…”

“…In Ref. [7] the allowed range of parameters is 0.8 < p < 1.7 with an especially good agreement for the inverted step p < 1. In our case, another difference consists in the fact that the oscillations in the spectrum studied in Ref.…”

“…Then, if the function h(k) is chosen such that it contains a preferred scale, see the Introduction, and such that it is approximatively constant on both sides, the model becomes very similar to the one presented in Ref. [7] for, in the notation of that article, p > 1. In Ref.…”

“…In the following, in order to perform concrete calculations, we will choose an analytical form forh(k) which mimics the behavior of the spectrum considered in Ref. [7] with p > 1. Interestingly enough, we will see that the final result does not strongly depend on the values of the free parameters that describe the functionh(k).…”

Holographic discreteness of inflationary perturbations

“…The field quanta in the original fluctuations convert to classical perturbations as they pass through the de Sitter-like event horizon; they are then parametrically amplified by an exponential factor during the many subsequent e-foldings of inflation, creating an enormous number of coherent quanta in phase with the original quantum seed perturbation; these in turn create large scale perturbations in the classical gravitational gauge-invariant potential φ m [16], leading to observable anisotropy and large scale structure. All stages of this process are under good calculational control, even the conversion of quantum to classical regimes [17][18][19][20][21][22]. The phase and amplitude of the modes observed today are a direct result of the quantum field activity during inflation; indeed, the hot and and cold spots of microwave anisotropy on the largest scales correspond to faithfully amplified images of the field configurations as they were during inflation.…”