Probing quasar lifetimes with proximate 21-centimetre absorption in the diffuse intergalactic medium at redshifts z ≥ 6

Enhanced ionizing radiation in close proximity to redshift z ≳ 6 quasars creates short windows of intergalactic Ly α transmission blueward of the quasar Ly α emission lines. The majority of these Ly α near-zones are consistent with quasars that have optically/UV bright lifetimes of $t_{\rm Q}\sim 10^{5}\!-\!10^{7}\rm \, yr$. However, lifetimes as short as $t_{\rm Q}\lesssim 10^{4}\rm \, yr$ appear to be required by the smallest Ly α near-zones. These short lifetimes present an apparent challenge for the growth of $\sim 10^{9}\rm \, M_{\odot }$ black holes at z ≳ 6. Accretion over longer time-scales is only possible if black holes grow primarily in an obscured phase, or if the quasars are variable on time-scales comparable to the equilibriation time for ionized hydrogen. Distinguishing between very young quasars and older quasars that have experienced episodic accretion with Ly α absorption alone is challenging, however. We therefore predict the signature of proximate 21-cm absorption around z ≳ 6 radio-loud quasars. For modest pre-heating of intergalactic hydrogen by the X-ray background, where the spin temperature $T_{\rm S} \lesssim 10^{2}\rm \, K$ prior to any quasar heating, we find proximate 21-cm absorption should be observable in the spectra of radio-loud quasars. The extent of the proximate 21-cm absorption is sensitive to the integrated lifetime of the quasar. Evidence for proximate 21-cm absorption from the diffuse intergalactic medium within $2\!-\!3\rm \, pMpc$ of a (radio-loud) quasar would be consistent with a short quasar lifetime, $t_{\rm Q}\lesssim 10^{5}\rm \, yr$, and would provide a complementary constraint on models for high-redshift black hole growth.