Revealing the cosmic reionisation history is at the frontier of extragalactic astronomy. The power spectrum of the cosmic microwave background (CMB) polarisation can be used to constrain the reionisation history. Here we propose a CMB-independent method using fast radio bursts (FRBs) to directly measure the ionisation fraction of the intergalactic medium (IGM) as a function of redshift. FRBs are new astronomical transients with millisecond timescales. Their dispersion measure (DM$_{rm IGM}$) is an indicator of the amount of ionised material in the IGM. Since the differential of DM$_{rm IGM}$ against redshift is proportional to the ionisation fraction, our method allows us to directly measure the reionisation history without any assumption on its functional shape. As a proof of concept, we constructed mock non-repeating FRB sources to be detected with the Square Kilometre Array, assuming three different reionisation histories with the same optical depth of Thomson scattering. We considered three cases of redshift measurements: (A) spectroscopic redshift for all mock data, (B) spectroscopic redshift for 10% of mock data, and (C) redshift estimated from an empirical relation of FRBs between their time-integrated luminosity and rest-frame intrinsic duration. In all cases, the reionisation histories are consistently reconstructed from the mock FRB data using our method. Our results demonstrate the capability of future FRBs in constraining the reionisation history.
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