  Modeling the stochastic gravitational wave background from various
astrophysical sources is a key objective in view of upcoming observations with
ground- and space-based gravitational wave observatories such as Advanced LIGO,
VIRGO, eLISA and PTA. We develop a synthetic model framework that follows the
evolution of single and binary compact objects in an astrophysical context. We
describe the formation and merger rates of binaries, the evolution of their
orbital parameters with time and the spectrum of emitted gravitational waves at
different stages of binary evolution. Our approach is modular and allows us to
test and constrain different ingredients of the model, including stellar
evolution, black hole formation scenarios and the properties of binary systems.
We use this framework in the context of a particularly well-motivated
astrophysical setup to calculate the gravitational wave background from several
types of sources, including inspiraling stellar-mass binary black holes that
have not merged during a Hubble time. We find that this signal, albeit weak,
has a characteristic shape that can help constrain the properties of binary
black holes in a way complementary to observations of the background from
merger events. We discuss possible applications of our framework in the context
of other gravitational wave sources, such as supermassive black holes.
