A classical, non-singular, bouncing universe
  We present a model for a classical, non-singular bouncing cosmology without
violation of the null energy condition (NEC). The field content is General
Relativity plus a real scalar field with a canonical kinetic term and only
renormalizable, polynomial type self-interaction for the scalar field in the
Jordan frame. The universe begins vacuum-energy dominated and is contracting at
$t=-\infty$. We consider a closed universe with a positive spatial curvature,
which is responsible for the universe bouncing without any NEC violation. An
$R\phi^2$ coupling between the Ricci scalar and the scalar field drives the
scalar from the initial false vacuum to the true vacuum during the bounce. The
model is sub-Planckian throughout its evolution and every dimensionful
parameter is below the effective field theory scale $M_P$, so we expect no
ghost-type or tachyonic instabilities. This model solves the horizon problem
and extends co-moving particle geodesics to past infinity, resulting in a
geodesically complete universe without singularities. We solve the Friedman
equations and the scalar field equation of motion numerically, and analytically
under certain approximations.
