  Density contrasts in the universe are governed by scalar cosmological
perturbations which, when expressed in terms of gauge-invariant variables,
contain a classical component from scalar metric perturbations and a quantum
component from inflaton field fluctuations. It has long been known that the
effect of cosmological expansion on a quantum field amounts to squeezing. Thus
the entropy of cosmological perturbations can be studied by treating them in
the framework of squeezed quantum systems. Entropy of a free quantum field is a
seemingly simple yet subtle issue. In this paper, as different from previous
treatments, we tackle this issue with a fully developed nonequilibrium quantum
field theory formalism for such systems. We compute the covariance matrix
elements of the parametric quantum field and solve for the evolution of the
density matrix elements and the Wigner functions, and, from them, derive the
von Neumann entropy. We then show explicitly why the entropy for the squeezed
yet closed system is zero, but is proportional to the particle number produced
upon coarse-graining out the correlation between the particle pairs. We also
construct the bridge between our quantum field-theoretic results and those
using probability distribution of classical stochastic fields by earlier
authors. From this we can see the clear advantages of the quantum
field-theoretical approach over the stochastic classical field treatment since
the latter misses out in some important quantum properties, such as
entanglement and coherence, of the quantum field.
