Pointer states for primordial fluctuations in inflationary cosmology
C. Kiefer, I. Lohmar, D. Polarski, A. A. Starobinsky
TL;DR
The paper addresses how inflationary primordial fluctuations acquire classical properties via decoherence, showing that the pointer basis is well approximated by narrow Gaussians in the field amplitude. Using the Lindblad open-system formalism, it derives how the Wigner function becomes positive and computes entropies per mode, revealing partial decoherence with entropy bounded below the maximal value by about half the squeezing entropy. For super-Hubble modes, the positivity time is near the Hubble time and largely insensitive to the precise environmental coupling, while for sub-Hubble modes decoherence depends on real dissipation and bath occupancy. These results clarify the quantum-to-classical transition for primordial perturbations and support the field-amplitude basis as the robust classical reference, with implications for the observed acoustic oscillations and the cosmic microwave background.
Abstract
Primordial fluctuations in inflationary cosmology acquire classical properties through decoherence when their wavelengths become larger than the Hubble scale. Although decoherence is effective, it is not complete, so a significant part of primordial correlations remains up to the present moment. We address the issue of the pointer states which provide a classical basis for the fluctuations with respect to the influence by an environment (other fields). Applying methods from the quantum theory of open systems (the Lindblad equation), we show that this basis is given by narrow Gaussians that approximate eigenstates of field amplitudes. We calculate both the von Neumann and linear entropy of the fluctuations. Their ratio to the maximal entropy per field mode defines a degree of partial decoherence in the entropy sense. We also determine the time of partial decoherence making the Wigner function positive everywhere which, for super-Hubble modes during inflation, is virtually independent of coupling to the environment and is only slightly larger than the Hubble time. On the other hand, assuming a representative environment (a photon bath), the decoherence time for sub-Hubble modes is finite only if some real dissipation exists.