Stability of Geodesically Complete Cosmologies
  We study the stability of spatially flat FRW solutions which are geodesically
complete, i.e. for which one can follow null (graviton) geodesics both in the
past and in the future without ever encountering singularities. This is the
case of NEC-violating cosmologies such as smooth bounces or solutions which
approach Minkowski in the past. We study the EFT of linear perturbations around
a solution of this kind, including the possibility of multiple fields and
fluids. One generally faces a gradient instability which can be avoided only if
the operator $~^{(3)}{R} \delta N~$ is present and its coefficient changes sign
along the evolution. This operator (typical of beyond-Horndeski theories) does
not lead to extra degrees of freedom, but cannot arise starting from any theory
with second-order equations of motion. The change of sign of this operator
prevents to set it to zero with a generalised disformal transformation.
