  We explore the possibility of selecting a natural vacuum state for scalar and
tensor gauge-invariant cosmological perturbations in the context of hybrid
quantum cosmology, by identifying those variables for the description of the
perturbations that display a dynamical behavior adapted to the evolution of the
entire cosmology. We make use of a canonical formulation of the whole of the
cosmological system in which the perturbative gauge-invariant degrees of
freedom are identified as canonical variables. Introducing background-dependent
linear canonical transformations on these perturbations and completing them for
the entire system, we are able to characterize a generic collection of
annihilation and creationlike variables that obey the dynamics dictated by a
respective collection of Hamiltonians. We then impose that such Hamiltonians
possess no self-interaction terms so that, in a Fock representation with normal
ordering, they act diagonally on the basis of $n$-particle states. This leads
to a semilinear first-order partial differential equation with respect to the
background for the coefficients that define the annihilation and creationlike
variables for all Fourier modes, as well as to a very precise ultraviolet
characterization of them. Such first-order equation contains, in the imaginary
part of its complex solutions, the complicated second-order field equation that
typically arises for the time-dependent frequency of the perturbations in the
context of quantum field theory in curved spacetimes. We check that the
asymptotic knowledge acquired allows one to select the standard vacua in
Minkowski and de Sitter spacetimes. Finally, we discuss the relation of our
vacuum and the standard adiabatic vacua, and check that our asymptotic
characterization of variables with a diagonal Hamiltonian displays the
properties that would be desirable for an adiabatic state of infinite order.

General Relativity and Quantum Cosmology