  We show that the earliest possible reionisation of the Universe is
approximately at $z\simeq 13.5$ and the optical depth is $\tau\simeq 0.17$ in
the conventionally accepted $\Lambda$ cold dark matter ($\Lambda$CDM) model
with adiabatic fluctuations of a flat spectrum normalised to the cosmic
microwave background. This is consistent with the reionisation found by WMAP
(the apparently earlier reionisation epoch of the WMAP is ascribed to the
adoption of the instantaneous reionisation approximation), i.e., the WMAP
result is realised only if reionisation of the universe takes place nearly at
the maximal efficiency in the $\Lambda$CDM model.
