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The radiative cooling of shocked gas with primordial chemical composition isan important process relevant to the formation of the first stars andstructures, as well as taking place also in high velocity cloud collisions andsupernovae explosions. Among the different processes that need to beconsidered, the formation kinetics and cooling of molecular hydrogen are ofprime interest, since they provide the only way to lower the gas temperature tovalues well below $\sim$10$^4$~K. In previous works, the internal energy levelstructure of H$_2$ and its cation has been treated in the approximation ofrovibrational ground state at low densities, or trying to describe the dynamicsusing some arbitrary $v>0$ H$_2$ level that is considered representative of theexcited vibrational manifold. In this study, we compute the vibrationallyresolved kinetics for the time-dependent chemical and thermal evolution of thepost-shock gas in a medium of primordial composition. The calculatednon-equilibrium distributions are used to evaluate effects on the coolingfunction of the gas and on the cooling time. Finally, we discuss the dependenceof the results to different initial values of the shock velocity and redshift.

State-to-state vibrational kinetics of H$_2$ and H$_2^+$ in a post-shock cooling gas with primordial composition