  The unitarity of time evolution, or colloquially the conservation of
probability, sits at the heart of our descriptions of fundamental interactions
via quantum field theory. The implications of unitarity for scattering
amplitudes are well understood, for example through the optical theorem and
cutting rules. In contrast, the implications for in-in correlators in curved
spacetime and the associated wavefunction of the universe, which are measured
by cosmological surveys, are much less transparent. For fields of any mass in
de Sitter spacetime with general local interactions, which need not be
invariant under de Sitter isometries, we show that unitarity implies an
infinite set of relations among the coefficients $ \psi_{n} $ of the
wavefunction of the universe with $ n $ fields, which we name Cosmological
Optical Theorem. For contact diagrams, our result dictates the analytic
structure of $ \psi_{n} $ and strongly constrains its form. For example, any
correlator with an odd number of conformally-coupled scalar fields and any
number of massless scalar fields must vanish. For four-point exchange diagrams,
the Cosmological Optical Theorem yields a simple and powerful relation between
$ \psi_{3} $ and $ \psi_{4} $, or equivalently between the bispectrum and
trispectrum. As explicit checks of this relation, we discuss the trispectrum in
single-field inflation from graviton exchange and self-interactions. Moreover,
we provide a detailed derivation of the relation between the total-energy pole
of cosmological correlators and flat-space amplitudes. We provide analogous
formulae for sub-diagram singularities. Our results constitute a new, powerful
tool to bootstrap cosmological correlators.
