  Einstein's theory of gravity has been extensively tested on solar system
scales, and for isolated astrophysical systems, using the perturbative
framework known as the parameterized post-Newtonian (PPN) formalism. This
framework is designed for use in the weak-field and slow-motion limit of
gravity, and can be used to constrain a large class of metric theories of
gravity with data collected from the aforementioned systems. Given the
potential of future surveys to probe cosmological scales to high precision, it
is a topic of much contemporary interest to construct a similar framework to
link Einstein's theory of gravity and its alternatives to observations on
cosmological scales. Our approach to this problem is to adapt and extend the
existing PPN formalism for use in cosmology. We derive a set of equations that
use the same parameters to consistently model both weak fields and cosmology.
This allows us to parameterize a large class of modified theories of gravity
and dark energy models on cosmological scales, using just four functions of
time. These four functions can be directly linked to the background expansion
of the universe, first-order cosmological perturbations, and the weak-field
limit of the theory. They also reduce to the standard PPN parameters on solar
system scales. We illustrate how dark energy models and scalar-tensor and
vector-tensor theories of gravity fit into this framework, which we refer to as
"parameterized post-Newtonian cosmology" (PPNC).
