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.
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