It is generally expected that decoherence processes will erase the quantum properties of the inflationary primordial spectra. However, given the weakness of gravitational interactions, one might end up with a distribution which is only partially decohered. Below a certain critical change, we show that the inflationary distribution retains quantum properties. We identify four of these: a squeezed spread in some direction of phase space, non-vanishing off-diagonal matrix elements, and two properties used in quantum optics called non-P -representability and non-separability. The last two are necessary conditions to violate Bell's inequalities. The critical value above which all these properties are lost is associated to the 'grain' of coherent states. The corresponding value of the entropy is equal to half the maximal (thermal) value. Moreover it coincides with the entropy of the effective distribution obtained by neglecting the decaying modes. By considering backreaction effects, we also provide an upper bound for this entropy at the onset of the radiation dominated era.

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'...

We propose an operational definition of the entropy of cosmological perturbations based on a truncation of the hierarchy of Green functions. The value of the entropy is unambiguous despite gauge invariance and the renormalization procedure. At the first level of truncation, the reduced density matrices are Gaussian and the entropy is the only intrinsic quantity. In this case, the quantum-to-classical transition concerns the entanglement of modes of opposite wave-vectors, and the threshold of classicality is that of separability. The relations to other criteria of classicality are established. We explain why, during inflation, most of these criteria are not intrinsic. We complete our analysis by showing that all reduced density matrices can be written as statistical mixtures of minimal states, the squeezed properties of which are less constrained as the entropy increases. Pointer states therefore appear not to be relevant to the discussion. The entropy is calculated for various models in paper II.

As a simple model for unknown Planck scale physics, we assume that the quantum modes responsible for producing primordial curvature perturbations during inflation are placed in their instantaneous adiabatic vacuum when their proper momentum reaches a fixed high energy scale M . The resulting power spectrum is derived and presented in a form that exhibits the amplitude and frequency of the superimposed oscillations in terms of H/M and the slow roll parameter ǫ.The amplitude of the oscillations is proportional to the third power of H/M . We argue that these small oscillations give the lower bound of the modifications of the power spectrum if the notion of free mode propagation ceases to exist above the critical energy scale M .