Probing primordial $^3$He from hyperfine line afterglows around supercritical black holes

We consider the possibility of the detection of $^3$HeII hyperfine line (rest frequency, $8.67 \, \rm GHz$) emission from ionized zones around accreting black holes (BHs) formed at high redshifts, $z=15\hbox{--}30$. We show that the brightness temperature in 8.67GHz line increases and reaches a peak value after the accretion onto the BH exhausts and HeIII recombines into HeII. This period of brightening last up to 40~million years. We find that during this period the maximum brightness temperature reaches $\simeq 0.2\hbox{--}0.5 \mu$K, depending on the epoch when such a black hole starts growing. The maximum angular size of the region emitting in the hyperfine line is around $0.5'$. The flux from such a region ($\simeq 0.3 \, \rm nJy$) is too small to be detected by SKA1-MID. The RMS of the collective flux from many emitting regions from a volume bounded by the synthesized beam and the band-width of SKA1-MID might reach 100~nJy, which is potentially detectable by SKA1-MID.