Vibrational level population of H$_2$ and H$_2^+$ in the early Universe
  We formulate a vibrationally resolved kinetics for molecular hydrogen and its cation in the primordial Universe chemistry. Formation, destruction and relaxation processes for each vibrational level are studied and included as chemical pathways of the present model. The fractional abundance of each vibrational level as a function of the redshift is given: a strong deviation from the Boltzmann distribution is found at low $z$. A discussion of the results is provided, also evaluating the effects of relaxation processes on the level populations. Analytical fits for some LTE rate coefficients are given in the Appendix. 
Carla M. Coppola, Savino Longo, Mario Capitelli, Francesco Palla,   Daniele Galli
astro-ph.CO