  A common way to avoid divergent integrals in homogeneous spatially
non-compact gravitational systems is to introduce a fiducial cell by
cutting-off the spatial slice at a finite region $V_o$. This is usually
considered as an auxiliary regulator to be removed after computations by
sending $V_o\to\infty$. In this paper, we analyse the dependence of the
classical and quantum theory of homogeneous, isotropic and spatially flat
cosmology on $V_o$. We show that each fixed $V_o$ regularisation leads to a
different canonically independent theory. At the classical level, the dynamics
of observables is not affected by the regularisation on-shell. For the quantum
theory, however, this leads to a family of regulator dependent quantum
representations and the limit $V_o\to\infty$ becomes then more subtle. First,
we construct a novel isomorphism between different $V_o$-regularisations, which
allows us to identify states in the different $V_o$-labelled Hilbert spaces to
ensure equivalent dynamics for any value of $V_o$. The $V_o\to\infty$ limit
would then correspond to choosing a state for which the volume assigned to the
fiducial cell becomes infinite as appropriate in the late-time regime. As
second main result of our analysis, quantum fluctuations of observables smeared
over subregions $V\subset V_o$, unlike those smeared over the full $V_o$,
explicitly depend on the size of the fiducial cell through the ratio $V/V_o$
interpreted as the (inverse) number of subcells $V$ homogeneously patched
together into $V_o$. Physically relevant fluctuations for a finite region, as
e.g. in the early-time regime, which would be unreasonably suppressed in a
na\"ive $V_o\to\infty$ limit, become appreciable at small volumes. Our results
suggest that the fiducial cell is not playing the role of a mere regularisation
but is physically relevant at the quantum level and complement previous
statements in the literature.
