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.
