Perturbations in quantum cosmology: The continuum limit in Fourier space

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.