  In the standard approach to deriving inflationary predictions, we evolve a
vacuum state in time according to the rules of a given model. Since the only
observables are the future values of correlators and not their time evolution,
this brings about a large degeneracy: a vast number of different models are
mapped to the same minute number of observables. Furthermore, due to the lack
of time-translation invariance, even tree-level calculations require an
increasing number of nested integrals that quickly become intractable. Here we
ask how much of the final observables can be "bootstrapped" directly from
locality, unitarity and symmetries.
  To this end, we introduce two new bootstrap tools to efficiently compute
cosmological correlators/wavefunctions. The first is a Manifestly Local Test
(MLT) that any $n$-point (wave)function of massless scalars or gravitons must
satisfy if it is to arise from a manifestly local theory. When combined with a
sub-set of the recently proposed Bootstrap Rules, this allows us to compute
explicitly all bispectra to all orders in derivatives for a single scalar.
Since we don't invoke soft theorems, this can also be extended to multi-field
inflation. The second is a partial energy recursion relation that allows us to
compute exchange correlators. Combining a bespoke complex shift of the partial
energies with Cauchy's integral theorem and the Cosmological Optical Theorem,
we fix exchange correlators up to a boundary term. The latter can be determined
up to contact interactions using unitarity and manifest locality. As an
illustration, we use these tools to bootstrap scalar inflationary trispectra
due to graviton exchange and inflaton self-interactions.
