Inflationary spectra and violations of Bell inequalities

Abstract

In spite of the macroscopic character of the primordial fluctuations, the standard inflationary distribution (that obtained using linear mode equations) exhibits inherently quantum properties, that is, properties which cannot be mimicked by any stochastic distribution. This is demonstrated by a Gedanken experiment for which certain Bell inequalities are violated. These violations are in principle measurable because, unlike for Hawking radiation from black holes, in inflationary cosmology we can have access to both members of correlated pairs of modes delivered in the same state. We then compute the effect of decoherence and show that the violations persist provided the decoherence level (and thus the entropy) lies below a certain non-vanishing threshold. Moreover, there exists a higher threshold above which no violation of any Bell inequality can occur. In this regime, the distributions are "separable" and can be interpreted as stochastic ensembles of fluctuations. Unfortunately, the precision which is required to have access to the quantum properties is so high that, in practice, an observational verification seems excluded.The inflationary paradigm [1] successfully accounts for the properties of primordial spectra revealed by the combined analysis of CMBR temperature anisotropy and Large Scale Structure spectra [2]. In particular, it predicts that the distribution of primordial fluctuations is homogeneous, isotropic and Gaussian, and that the power spectrum is nearly scale invariant (simply because the Hubble radius was slowly varying during inflation).Surprisingly, inflation implies that density fluctuations arise from the amplification of vacuum fluctuations [3]; because of backreaction effects, the vacuum is indeed the only possible initial state [4]. In addition of being amplified, the modes of opposite wave-vectors k and −k end up highly correlated. More precisely, using linear mode equations, the vacuum evolves into a product of twomode squeezed states [5,6,7,8]. The highly squeezed character of the distribution implies the vanishing of the variance in one direction in phase space. This direction is that of the decaying mode [7]. The observational consequence of this squeezing are the acoustic peaks in the temperature anisotropy spectrum [9,10].In spite of the macroscopic character of the mode amplitudes, we shall show that the inflationary distribution is still entangled in a quantum mechanical sense. To prove this, we shall provide observables able to distinguish quantum correlations from stochastic correlations. At this point, it is important to notice that, unlike for Hawking radiation from black holes, we have in principle access to the purity of the state since, both members of two-mode sectors in the same state can be simultaneously observed on the last scattering surface [11].Another important element should now be discussed: the linear mode equation is only approximate. Indeed, even in the simplest inflationary models there exists gravitational interactions which couple sectors with different k'...

Spectrum and entanglement of phonons in quantum fluids of light

“…In addition to the negative peak at x = x due to the repulsive interparticle interactions, we see a propagating correlation pattern which is governed by the group velocity v gr = ∂ k ω k where ω k is given in Eq. (25). As in the case of equilibrium condensates [8,21], the fast oscillations at large separations are due to the superluminal form of the dispersion relation Eq.…”

“…For instance, one may consider observing the emitted pairs associated with the (analogous) cosmological pair creation or dynamical Casimir (DCE) effects [6][7][8], as well as those produced by the (analogous) Hawking effect [9][10][11][12][13][14][15]. Most remarkably, accurate measurements of the observables are expected to distinguish quantum (spontaneous) from classical (stimulated) correlations by looking at the non-separability of the outgoing state [16][17][18][19][20][21][22], or violations of Cauchy-Schwarz [23,24] or even Bell inequalities [25].…”

Violation of Cauchy-Schwarz inequalities by spontaneous Hawking radiation in resonant boson structures

“…The purpose of this paper is to show that, despite the remaining difficulties, the violation of CS inequalities in HR is comparatively much easier to observe near the peaks characteristic of resonant radiation. The quest for CS violation here proposed complements those approaches relying on the direct detection of entanglement, as applied to inflationary cosmology [20], BHs [21], general relativistic quantum fields [22], or BH analogs [23][24][25].…”

Decoherence from isocurvature perturbations in inflation

“…Various arguments and calculations suggesting that a form of such environmental decoherence can indeed occur for inflationary perturbations have been put forward in [8,9,10,11,12,13,14,15,16]. In [9,10,11,12,13,14,16], the environment is taken to consist of the short wavelength modes which are coupled to the long wavelength modes via non-linear couplings. In this paper, we point out that the decoherence of cosmological perturbations can occur in a precisely defined dynamical setting if the role of the environment is played by unobservable isocurvature perturbations generated during multi-field inflation.…”