  We investigate the ambiguities in the Fock quantization of the scalar
perturbations of a Friedmann-Lema\^{i}tre-Robertson-Walker model with a massive
scalar field as matter content. We consider the case of compact spatial
sections (thus avoiding infrared divergences), with the topology of a
three-sphere. After expanding the perturbations in series of eigenfunctions of
the Laplace-Beltrami operator, the Hamiltonian of the system is written up to
quadratic order in them. We fix the gauge of the local degrees of freedom in
two different ways, reaching in both cases the same qualitative results. A
canonical transformation, which includes the scaling of the matter field
perturbations by the scale factor of the geometry, is performed in order to
arrive at a convenient formulation of the system. We then study the
quantization of these perturbations in the classical background determined by
the homogeneous variables. Based on previous work, we introduce a Fock
representation for the perturbations in which: (a) the complex structure is
invariant under the isometries of the spatial sections and (b) the field
dynamics is implemented as a unitary operator. These two properties select not
only a unique unitary equivalence class of representations, but also a
preferred field description, picking up a canonical pair of field variables
among all those that can be obtained by means of a time-dependent scaling of
the matter field (completed into a linear canonical transformation). Finally,
we present an equivalent quantization constructed in terms of gauge-invariant
quantities. We prove that this quantization can be attained by a mode-by-mode
time-dependent linear canonical transformation which admits a unitary
implementation, so that it is also uniquely determined.
