  This talk is about solving cosmological equations analytically without
approximations, and discovering new phenomena that could not be noticed with
approximate solutions. We found all the solutions of the Friedmann equations
for a specific model, including all the zero-size-bounce solutions that do not
violate the null energy condition, as well as all the finite-size-bounce
solutions, and then discovered model independent phenomena. Among them is the
notion of geodesic completeness for the geometry of the universe. From this we
learned a few new general lessons for cosmology. Among them is that anisotropy
provides a model independent attractor mechanism to some specific initial
values for cosmological fields, and that there is a period of antigravity in
the history of the universe. The results are obtained only at the classical
gravity level. Effects of quantum gravity or string theory are unknown, they
are not even formulated, so there are new theoretical challenges.
