  The higher order generalisation of the clockwork mechanism to gravitational
interactions provides a means to generate an exponentially suppressed coupling
to matter from a fundamental theory of multiple interacting gravitons, without
introducing large hierarchies in the underlying potential and without the need
for a dilaton, suggesting a possible application to the hierarchy problem. We
work in the framework of ghost free multi-gravity with "nearest-neighbour"
interactions, and present a formalism by which one is able to construct
potentials such that the theory will always exhibit this clockwork effect. We
also consider cosmological solutions to the general theory, where all metrics
are of FRW form, with site-dependent scale factors/lapses. We demonstrate the
existence of multiple deSitter vacua where all metrics share the same Hubble
parameter, and we solve the modified Einstein equations numerically for an
example clockwork model constructed using our formalism, finding that the
evolution of the metric that matter couples to is essentially equivalent to
that of general relativity at the modified Planck scale. It is important to
stress that while we focus on the application to clockwork theories, our work
is entirely general and facilitates finding cosmological solutions to any ghost
free multi-gravity theory with "nearest-neighbour" interactions. Moreover, we
clarify previous work on the continuum limit of the theory, which is
generically a scalar-tensor braneworld, using the Randall-Sundrum model as a
special case and showing how the discrete-clockwork cosmological results map to
the continuum results in the appropriate limit.
