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In a paper called Further Observational Evidence for a Critical Ionising Luminosityin Active Galaxies, Section 2.1, the authors talk about using the ICRF2 to find a tuning frequency in search of the 21-cm absorption (of HI):

The Second Realization of the International Celestial Reference Frame by Very Long Baseline Interferometry (ICRF2, Ma et al.2009), constitutes a sample of strong flat spectrum radio sources,of which 1682 now have known redshifts (Titov & Malkin 2009; Titov et al. 2013 and references therein), yielding a tuning frequency in the search for 21-cm absorption. Being VLBI sources, all have significant compact flux, thus maximising the chance of a high covering factor and thus optical depth (Curran et al. 2013a).The original aim of the survey was to form part of a large observing campaign to search and quantify the incidence of associated ${\rm H\,{\small I}\,21}$-cm absorption over all redshifts, although observing time was only granted for the high redshift $(z \gtrsim 2.6)$ proposals.

Since we already know the frequency of HI (1420.40575177 MHz), why is this so important? What am I missing about the functioning of radio telescopes?

This question doesn't really answer my query.

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The very first sentence of that paper makes it clear that it is concerned with observations of the 21cm line in the redshifted universe, i.e. on objects so far away from us that redshift becomes significant. And just as any other electromagnetic emission, radio waves are affected by this. So, in section 2.1, the authors describe selecting a sample of radio sources with known redshifts so that they would know how much they had to tune their receivers downwards in frequency.

This is actually quite common and relevant in today's 21 cm radio observations. Redshifted signals are observed even down to 140 MHz and below as stated on LOFAR's Redshifted 21cm hydrogen Line page.

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