Small-Field and Scale-Free: Inflation and Ekpyrosis at their Extremes
  There is increasing evidence from string theory that effective field theories
are only reliable over approximately sub-Planckian field excursions. The two
most promising effective models for early universe cosmology, inflation and
ekpyrosis, are mechanisms that, in order to address cosmological puzzles, must
operate over vast expansion/energy ranges. This suggests that it might be
appropriate to describe them using scaling laws. Here we combine these two
ideas and demonstrate that they drive inflation and ekpyrosis to their
extremes: inflation must start at ultra-slow-roll, and ekpyrosis at
ultra-fast-roll. At face value, the implied spectra are overly tilted to the
red, although in both cases minor departures from pure scale freedom bring the
spectral indices within current observational bounds. These models predict a
significant spectral running at a level detectable in the near future
($\alpha_s \approx -10^{-3}$). Ekpyrotic models with minimal coupling are
nevertheless ruled out, as they lead to levels of non-Gaussianity that are at
least an order of magnitude too large. No such restrictions apply to models
with a kinetic coupling between the two ekpyrotic scalar fields, and these
remain the most promising ekpyrotic models.
  An additional swampland criterion that was recently proposed for the slope of
the scalar field potential would however rule out all ultra-slow-roll models of
inflation. Finally, we speculate on the existence of corresponding restrictions
on the slope at negative potential values, which might lead to similarly severe
constraints on ekpyrotic models.
