  We establish a well-posedness theory for the f(R) theory of modified gravity,
which is a generalization of Einstein's theory of gravitation. The scalar
curvature R of the spacetime, which arises in the integrand of the
Einstein-Hilbert functional, is replaced by a function f=f(R). The field
equations involve up to fourth-order derivatives of the spacetime metric, and
the challenge is to understand the structure of these high-order terms. We
propose a formulation of the initial value problem in modified gravity when the
initial data are prescribed on a hypersurface. In addition to the induced
metric and second fundamental form of the initial slice and the initial matter
content, an initial data set must also provide the spacetime scalar curvature
and its time-derivative. We introduce an augmented conformal formulation, as we
call it, in which the spacetime curvature is an independent variable. In the
so-called wave gauge, the field equations of modified gravity reduce a coupled
system of nonlinear wave-Klein-Gordon equations with defocusing potential,
whose unknowns are the conformally-transformed metric and the scalar curvature,
as well as the matter fields. We are able to establish the existence of maximal
globally hyperbolic developments when the matter is represented by a scalar
field. We analyze the so-called Jordan coupling and work with the Einstein
metric, which is conformally equivalent to the physical metric --the conformal
factor depending upon the scalar curvature. Our analysis leads us to a rigorous
validation of the theory of modified gravity. We derive quantitative estimates,
which are uniform in terms of the nonlinearity f(R), and we prove that
asymptotically flat spacetimes of modified gravity are `close' to Einstein
spacetimes, when the function f(R) in the action functional of modified gravity
is `close' to the Einstein-Hilbert integrand R.
