  The time evolution of primordial fluctuations conceals a wealth of insights
into the high-energy physics at play during the earliest moments of our
Universe, which is ultimately encoded in late-time spatial correlation
functions. However, the conventional procedure to compute them is technically
challenging, and a complete dictionary mapping the landscape of inflationary
theories and the corresponding observable signatures is not yet available. In
this paper, we develop a framework to compute tree-level cosmological
correlators based on following their time evolution from their origin as
quantum zero-point fluctuations to the end of inflation. From first principles,
the structure of the bulk time evolution imposes a set of universal
differential equations in time satisfied by equal-time correlators. We
automatise the process of systematically solving these equations. This allows
us to accurately capture all physical effects and obtain exact results in
theories formulated at the level of inflationary fluctuations that include any
number of degrees of freedom with arbitrary dispersion relations and masses,
coupled through any time-dependent interactions. We then illustrate the power
of this formalism by exploring the phenomenology of cosmological correlators
emerging from the interaction with a massive scalar field. We study both the
size and the shape dependence of non-Gaussianities in the entire parameter
space, including the strong mixing regime. We present novel characteristics of
cosmological collider signals in (would be) single-, double-, and
triple-exchange three-point correlators. In the presence of primordial
features, we show that soft limits of cosmological correlators offer a new
possibility to probe the inflationary landscape. Finally, we provide templates
to search for in future cosmological surveys.
