Regularization of the big bang singularity with a time varying equation
  of state $w > 1$
  We study the classical dynamics of the universe undergoing a transition from
contraction to expansion through a big bang singularity. The dynamics is
described by a system of differential equations for a set of physical
quantities, such as the scale factor $a$, the Hubble parameter $H$, the
equation of state parameter $w$, and the density parameter $\Omega$. The
solutions of the dynamical system have a singularity at the big bang. We study
if the solutions can be regularized at the singularity in the sense of whether
they have unique branch extensions through the singularity. In particular, we
consider the model in which the contracting universe is dominated by a scalar
field with a time varying equation of state $w$, which approaches a constant
value $w_c$ near the singularity. We prove that, for $w_c > 1$, the solutions
are regularizable only for a discrete set of $w_c$ values that satisfy a
coprime number condition. Our result implies that the evolution of a bouncing
universe through the big bang singularity does not have a continuous classical
limit unless the equation of state is extremely fine-tuned.
