Toolkit for Scalar Fields in Universes with finite-dimensional Hilbert
  Space
  The holographic principle suggests that the Hilbert space of quantum gravity
is locally finite-dimensional. Motivated by this point-of-view, and its
application to the observable Universe, we introduce a set of numerical and
conceptual tools to describe scalar fields with finite-dimensional Hilbert
spaces, and to study their behaviour in expanding cosmological backgrounds.
These tools include accurate approximations to compute the vacuum energy of a
field mode $\mathbf{k}$ as a function of the dimension $d_{\mathbf{k}}$ of the
mode Hilbert space, as well as a parametric model for how that dimension varies
with $|\mathbf{k}|$. We show that the maximum entropy of our construction
momentarily scales like the boundary area of the observable Universe for some
values of the parameters of that model. And we find that the maximum entropy
generally follows a sub-volume scaling as long as $d_{\mathbf{k}}$ decreases
with $|\mathbf{k}|$. We also demonstrate that the vacuum energy density of the
finite-dimensional field is dynamical and decays between two constant epochs in
our fiducial construction. These results rely on a number of non-trivial
modelling choices, but our general framework may serve as a starting point for
future investigations of the impact of finite-dimensionality of Hilbert space
on cosmological physics.
