Non-critical anisotropic Bianchi type $I$ string cosmology with $\alpha'$-corrections

We present non-critical Bianchi type $I$ string cosmology solutions in the presence of central charge deficit term $\Lambda$. The leading order string frame curvature appears to be in the high curvature limit $R\alpha'\gtrsim1$, which underlines the necessity of including higher order $\alpha'$-corrections. We give new solutions of two-loop (order $\alpha'$) $\beta$-function equations of $\sigma$-model with non-zero $\Lambda$ and dilaton field in both cases of absence and presence of spatially homogeneous $H$-field ($H=dB$). Also, the evolution of solutions is studied in the Einstein frame, where the string effective action can transform to Gauss-Bonnet gravity model coupled to the dilaton field with potential. We study explicit examples in order $\alpha'$ with chosen values of appeared constants in the solutions and discuss the cosmological implications.