  We analyze the passage to a continuum limit of the mode spectrum of
primordial perturbations around flat cosmological spacetimes in Quantum
Cosmology, showing that this limit can be reached even if one starts by
considering a finite fiducial cell as spatial slice. Whereas the resulting
system can be described in an invariant way under changes of the fiducial
volume using appropriate variables, both for the background cosmology and the
perturbations, obtaining in this way a discrete mode spectrum owing to the
compactness of the fiducial cell, we show that the desired continuum limit for
the perturbations can still be established by means of scaling transformations
of the physical volume when this volume grows unboundedly. These
transformations lead to a model with a continuum of modes and independent of
any scale of reference for the physical volume. For the sake of comparison, we
also consider an alternative road to the continuum in Fourier space that has
been employed in geometrodynamics and is based on the use of scaling
transformations of the fiducial volume, together with variables that are
independent of them.
