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.