  Gravity can be embedded into a renormalizable theory by means of adding
quadratic in curvature terms. However, this at first leads to the presence of
the Weyl ghost. It is possible to get rid of this ghost if the locality
assumption is weakened and the propagator of the graviton is represented by an
entire function of the d'Alembertian operator without new poles and zeros.
Models of this type admit a cosmological solution describing the $R^2$, or
Starobinsky, inflation. We study graviton production after inflation in this
model and show that it is negligible despite the presence of the higher
derivative operators which could potentially cause instabilities.
