Quantum non-linear evolution of inflationary tensor perturbations
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Springer
Received: April 4, 2019 Accepted: April 23, 2019 Published: May 3, 2019
Jinn-Ouk Gonga and Min-Seok Seob a
Korea Astronomy and Space Science Institute, Daejeon 34055, Korea b Department of Physics Education, Korea National University of Education, Cheongju 28173, Korea
E-mail: [email protected], [email protected] 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. Keywords: Cosmology of Theories beyond the SM, Effective Field Theories ArXiv ePrint: 1903.12295
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)021
JHEP05(2019)021
Quantum non-linear evolution of inflationary tensor perturbations
Contents 1 Introduction
1
2 Non-linear evolution of reduced density matrix
3
3 Interaction Hamiltonian for tensor perturbations
6 11 11 14
5 Evolution of reduced density matrix 5.1 Solution of reduced density matrix 5.2 Decoherence rate
17 17 20
6 Conclusions
22
A Quadratic evolution of tensor perturbations A.1 Solutions of operators A.2 Evolution of the vacuum state
23 23 26
B Full reduced density matrix
28
C Reduction of density matrix elements
33
D Direct computation of reduced density matrix
35
1
Introduction
While inflation [1–3] is postulated to resolve initial condition problems of the early universe, it also gives a natural account of how to generate the initial perturbations on super-horizon scales as observed in the cosmic microwave background (CMB) [4]. That is, during inflation the primordial quantum fluctuations leave the horizon and become classical perturbations as we observe today such as the temperature anisotropies in the CMB and inhomogeneously distributed galaxies [5–9]. As for this aspect, the quantum-to-classical transition is known to arise as the states for the scalar and tensor perturbations take a form of specific coherent superposition of excitations, called the squeezed state [10]. More concretely, through time evolution, some quadratic terms reflecting curved space-time make the state “squeezed” (see appendix A) such that the non-commutative effect between the perturbation field variable
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