Two non-perturbative $\alpha^{\prime}$ corrected or loop corrected string cosmological solutions

In this paper, we present two non-perturbative string cosmological solutions without curvature singularities for the bosonic gravi-dilaton system. These solutions are general in the sense that they can straightforwardly match the perturbative solution to arbitrarily high orders in the perturbative region. The first solution includes non-perturbative $\alpha'$ corrections based on Hohm-Zwiebach action. We then use the simple phenomenological map between $\alpha^{\prime}$ corrected theory and loop corrected theory in string cosmology to construct a non-perturbative loop corrected non-singular solution. Both solutions are non-singular everywhere. Therefore, the pre- and post-big-bangs are smoothly connected by these solutions.