One spectrum to rule them all?
  We show that in absence of entropy or effective anisotropic stress the
freedom in the choice of the initial energy scale of inflation implies the
existence of an infinite family of dual slow-roll parameters histories which
can produce the same spectrum of comoving curvature perturbations. This implies
that in general there is no one-to-one correspondence between the spectrum and
higher order correlation functions. We give some numerical examples of
expansion histories corresponding to different initial energy scales, with the
same spectrum of curvature perturbations, the same squeezed limit bispectrum,
in agreement with the squeezed limit consistency condition, but with different
bispectra in other configurations and different spectra of primordial
gravitational waves. The combined analysis of data from future CMB and
gravitational wave experiments could allow to distinguish between dual models.
