On the Symmetries of Cosmological Perturbations
  The space of inflationary models is vast, containing wide varieties of
mechanisms, symmetries, and spectra of particles. Consequently, the space of
observational signatures is similarly complex. Hence, it is natural to look for
boundaries of the space of models and their signatures. In this paper, we
explore the possible symmetries associated with the primordial cosmological
perturbations and their correlators in the asymptotic future. Assuming the
observed homogeneity, isotropy and (approximate) scale invariance, we prove
three main results. First, correlation functions of scalar metric fluctuations
are uniquely characterized by soft theorems and are free from ambiguity under
field redefinitions. Second, whatever the particle content and interactions,
when the standard soft theorems apply, invariance under de Sitter boosts
(linearly realized conformal invariance) is only possible if all connected
correlators vanish identically, i.e. if the theory is free. Third, conformal
invariance is the largest set of linearly realized (bosonic) symmetries of the
correlators of any single scalar, irrespectively of any soft theorems or
particle content.
