  We develop a model of spatially flat, homogeneous and isotropic cosmology in
Lorentzian Regge calculus, employing 4-dimensional Lorentzian frusta as
building blocks. By examining the causal structure of the discrete spacetimes
obtained by gluing such 4-frusta in spatial and temporal direction, we find
causality violations if the sub-cells connecting spatial slices are spacelike.
A Wick rotation to the Euclidean theory can be defined globally by a
complexification of the variables and an analytic continuation of the action.
Introducing a discrete free massless scalar field, we study its equations of
motion and show that it evolves monotonically. Furthermore, in a continuum
limit, we obtain the equations of a homogeneous scalar field on a spatially
flat Friedmann background. Vacuum solutions to the causally regular Regge
equations are static and flat and show a restoration of time reparametrisation
invariance. In the presence of a scalar field, the height of a frustum is a
dynamical variable that has a solution if causality violations are absent and
if an inequality relating geometric and matter boundary data is satisfied. Edge
lengths of cubes evolve monotonically, yielding a contracting or an expanding
branch of the universe. In a small deficit angle expansion, the system can be
deparametrised via the scalar field and a continuum limit of the discrete
theory can be defined which we show to yield the relational Friedmann equation.
These properties are obstructed if higher orders of the deficit angle are taken
into account. Our results suggest that the inclusion of timelike sub-cells is
necessary for a causally regular classical evolution in this symmetry
restricted setting. Ultimately, this works serves as a basis for forthcoming
investigations on the cosmological path integral within the framework of
effective spin foams.
