Cosmology of the very early universe
  We study cosmological solutions for the very early universe beginning at the
Planck scale for a universe containing radiation, curvature and, as a
simplification of a possible scalar field potential, a cosmological constant
term. The solutions are the natural counterpart of the well known results for a
post-inflationary universe of non-relativistic matter, curvature and a
cosmological constant. Contrary to the common belief that inflation arises
independently of the initial curvature we show that in the positive curvature
case the universe collapses again into a Big Crunch without allowing the
cosmological term to dominate and to produce inflation. There is a critical
value for the cosmological constant which divide the regions where inflation is
allowed from those where inflation cannot occur. One can also have loitering
solutions where the scale factor remains almost constant growing to produce
inflation (or decreasing to a Big Crunch) after a time which depends on the
amount of energy above (or below) the critical energy. At the critical energy
the solution approaches asymptotically a particular value for the scale factor
(Einstein's static pre-inflationary universe). The cases where the cosmological
term vanishes or becomes negative are also studied providing a complete
discussion of Friedmann models.
