Quantum non-linear evolution of inflationary tensor perturbations

Abstract

We study the quantum mechanical evolution of the tensor perturbations during inflation with non-linear tensor interactions. We first obtain the Lindblad terms generated by non-linear interactions by tracing out unobservable sub-horizon modes. Then we calculate explicitly the reduced density matrix for the super-horizon modes, and show that the probability of maintaining the unitarity of the squeezed state decreases in time. The decreased probability is transferred to other elements of the reduced density matrix including off-diagonal ones, so the evolution of the reduced density matrix describes the quantum-to-classical transition of the tensor perturbations. This is different from the classicality accomplished by the squeezed state, the suppression of the non-commutative effect, which is originated from the quadratic, linear interaction, and also maintains the unitarity. The quantum-to-classical transition occurs within 5 -10 e-folds, faster than the curvature perturbation. state, they still maintain the trace of the density matrix unity. Then the pure state evolves into the mixed state, giving a non-trivial density matrix elements representing classical probability. This indeed invites us to use the Lindblad operators to describe the information paradox in Hawking radiation [17] and also similar phenomena in the inflationary cosmology. Therefore, we have two different aspects for the quantum-to-classical transition. First, the linear evolution leading to the squeezed state is evidently unitary process. Moreover, it corresponds to the quadratic sector in the action, in which the system-environment interaction does not appear. On the contrary, the Lindblad operators lead to non-unitary time evolution in both decoherence and generation of the mixed state. It comes from the system-environment interaction, which appears through non-linear interactions beyond quadratic order. Tiny non-Gaussianity constrained by recent observations on the CMB [30] implies that the non-linear effect is sub-dominant. Then we expect that the WKB-like "unitary" classicality achieved by the squeezed state, often referred to as the semi-classicality, is slowly converted into the "nonunitary" classicality through the Lindblad operators as a perturbation. The Lindblad operators provide the rate of such conversion, reflecting the specific form of the non-linear interactions.Previous studies on the non-unitary classicality considered the curvature perturbation as an example [25,26,27,28]. The curvature perturbation is interpreted as a Goldstone boson resulting from the spontaneous breaking of dS isometry by quasi dS background, which is parametrized by the slow-roll parameters [31] (see also [32,33]). This implies that small slowroll parameters make the conversion rate of unitary to non-unitary classicality very small, as analyzed in e.g. [27]. For the tensor perturbations, on the other hand, situation is different. Unlike the curvature perturbation, they are well defined even in perfect dS space-time, so the leading effect is irrelevant to the...

Entanglement entropy of cosmological perturbations

“…are not slow-roll suppressed which typically lead them to decohere faster than their scalar counterpart [46]. Anticipating along similar lines, we expect that the entanglement entropy of tensor modes would be somewhat enhanced, and this will be studied in future work.…”

“…As we have shown, the calculation of the entanglement entropy of cosmological perturbations simplifies when done in momentum space. It is easy to appreciate this properly if one compares our result with that for determining the full nonunitary evolution of the density matrix of the system modes as has been done, for instance, in [39] (see [46] for the case of tensor modes). The time evolution of the reduced density matrix involves non-Hamiltonian terms, and might even contain non-Markovian terms, which depend on the so-called Lindblad operator.…”

“…we can write down the interaction Hamiltonian by converting the ζ field to our canonical field χ, and then expanding in terms of the creation and annihilation operators in momentum space. We find the following expression [46]:…”

Entropy production in inflation from spectator loops

“…However, the momentum correlators (37-38) grow as powers of the scale factor, such that they are not necessarily suppressed at the end of inflation. We calculate the Gaussian, von Neumann entropy of the curvature perturbation (47), and show that during inflation and on super-Hubble scales it grows as ∼ 6 ln(a). This rapid growth of the entropy indicates a rapid classicalization of the curvature perturbation on super-Hubble scales during inflation, and it is a consequence of the rapid growth (∝ a 6 , see Eq.…”

“…This means that, before one makes any definite conclusion concerning the strength of decoherence during inflation, one also ought to investigate the effect of the quantum gravitational loops. In fact, there have been several attempts to do precisely that [26,27,30,31,47]. In addition, a lot of work has been invested into a much easier set of problems, namely into studying how the inflaton coupling with the other quantum fields (scalar, fermionic or vector) induces decoherence in the inflaton sector [16,18,31,34].…”