  Parameterised Post-Newtonian Cosmology (PPNC) is a theory-agnostic framework
for testing gravity in cosmology, which connects gravitational physics on small
and large scales in the Universe. It is a direct extension of the Parameterised
Post-Newtonian (PPN) approach to testing gravity in isolated astrophysical
systems, and therefore allows constraints on gravity from vastly different
physical regimes to be compared and combined. We investigate the application of
this framework to a class of example scalar-tensor theories of gravity in order
to verify theoretical predictions, and to investigate for the first time the
scale-dependence of the gravitational couplings that appear within its
perturbation equations. In doing so, we evaluate the performance of some simple
interpolating functions in the transition region between small and large
cosmological scales, as well as the uncertainties that using such functions
would introduce into the calculation of observables. We find that all
theoretical predictions of the PPNC framework are verified to high accuracy in
the relevant regimes, and that simple interpolating functions perform well (but
not perfectly) between these regimes. This study is an important step towards
being able to use the PPNC framework to analyse cosmological datasets, and to
thereby test if/how the gravitational interaction has changed as the Universe
has evolved.
