Much of the structure of cosmological correlators is controlled by their
singularities, which in turn are fixed in terms of flat-space scattering
amplitudes. An important challenge is to interpolate between the singular
limits to determine the full correlators at arbitrary kinematics. This is
particularly relevant because the singularities of correlators are not directly
observable, but can only be accessed by analytic continuation. In this paper,
we study rational correlators, including those of gauge fields, gravitons, and
the inflaton, whose only singularities at tree level are poles and whose
behavior away from these poles is strongly constrained by unitarity and
locality. We describe how unitarity translates into a set of cutting rules that
consistent correlators must satisfy, and explain how this can be used to
bootstrap correlators given information about their singularities. We also
derive recursion relations that allow the iterative construction of more
complicated correlators from simpler building blocks. In flat space, all energy
singularities are simple poles, so that the combination of unitarity
constraints and recursion relations provides an efficient way to bootstrap the
full correlators. In many cases, these flat-space correlators can then be
transformed into their more complex de Sitter counterparts. As an example of
this procedure, we derive the correlator associated to graviton Compton
scattering in de Sitter space, though the methods are much more widely
applicable.