  The recent EDGES measurements of the global 21-cm signal from the cosmic dawn
suggest that the kinetic temperature of the inter-galactic medium (IGM) might
be significantly lower compared to its expected value. The colder IGM directly
affects the hydrogen recombination of the universe during the cosmic dawn and
dark ages by enhancing the rate of recombinations. Here, we study and quantify,
the impact of the colder IGM scenario on the recombination history of the
universe in the context of DM-baryonic interaction model which is widely used
to explain the depth of the EDGES 21-cm signal. We find that, in general, the
hydrogen ionisation fraction gets suppressed during the dark ages and cosmic
dawn and the suppression gradually increases at lower redshifts until X-ray
heating turns on. However, accurate estimation of the ionisation fraction
requires knowledge of the entire thermal history of the IGM, from the epoch of
thermal decoupling of hydrogen gas and the CMBR to the cosmic dawn. It is
possible that two separate scenarios which predict very similar HI differential
temperature during the cosmic dawn and are consistent with the EDGES 21-cm
signal might have very different IGM temperature during the dark ages. The
evolutions of the ionisation fraction in these two scenarios are quite
different. This prohibits us to accurately calculate the ionisation fraction
during the cosmic dawn using the EDGES 21-cm signal alone. We find that the
changes in the ionisation fraction w.r.t the standard scenario at redshift $z
\sim 17 $ could be anything between $\sim 0 \%$ to $\sim 36 \%$. This
uncertainty may be reduced if measurements of HI 21-cm differential temperature
at multiple redshifts are simultaneously used.
