  The cosmological coincidences between the matter and radiation energy
densities at recombination as well as between the densities of matter and the
cosmological constant at present time are well known. We point out that
moreover the third intersection between the energy densities of radiation and
the cosmological constant coincides with the reionization epoch. To quantify
the statistical relevance of this concurrence, we compute the Bayes factor
between the concordance cosmology with free Thomson scattering optical depth
and a model for which this parameter is inferred from imposing a match between
the time of density equality and the epoch of reionization. This is to
characterize the potential explanatory gain if one were to find a
parameter-free physical connection. We find a very strong preference for such a
concurrence on the Jeffreys scale from current cosmological observations. We
furthermore discuss the effect of choice of priors, changes in reionization
history, and free sum of neutrino masses. We also estimate the impact of adding
intermediate polarization data from the Planck High Frequency Instrument and
prospects for future 21 cm surveys. In the first case, preference for the
correlation remains substantial, whereas future data may give results more
decisive in pro or substantial in contra. Finally, we provide a discussion on
different interpretations of these findings. In particular, we show how a
connection between the star-formation history and the cosmological background
dynamics can give rise to this concurrence.
