The first measurement of the quasar lifetime distribution

Understanding the growth of the supermassive black holes powering luminous quasars, their co-evolution with host galaxies, and impact on the surrounding intergalactic medium depends sensitively on the duration of quasar accretion episodes. Unfortunately, this time-scale, known as the quasar lifetime, $t_{\rm Q}$, is still uncertain by orders of magnitude ($t_{\rm Q}\simeq 0.01~{\rm Myr}-1~{\rm Gyr}$). However, the extent of the He II Ly$\alpha$ proximity zones in the absorption spectra of $z_{\rm qso}\sim3-4$ quasars constitutes a unique probe, providing sensitivity to lifetimes up to $\sim 30$ Myr. Our recent analysis of $22$ archival He II proximity zone spectra reveals a surprisingly broad range of emission timescales, indicating that some quasars turned on $\lesssim 1$ Myr ago, whereas others have been shining for $\gtrsim 30$ Myr. Determining the underlying quasar lifetime distribution (QLD) from proximity zone measurements is a challenging task owing to: 1) the limited sensitivity of individual measurements; 2) random sampling of the quasar light curves; 3) density fluctuations in the quasar environment; and 4) the inhomogeneous ionization state of He II in a reionizing IGM. We combine a semi-numerical He II reionization model, hydrodynamical simulations post-processed with ionizing radiative transfer, and a novel statistical framework to infer the QLD from an ensemble of proximity zone measurements. Assuming a log-normal QLD, we infer a mean $\langle {\rm log}_{10}\left(t_{\rm Q}/{\rm Myr}\right)\rangle=0.22^{+0.22}_{-0.25}$ and standard deviation $\sigma_{{\rm log}_{10}t_{\rm Q}}=0.80^{+0.37}_{-0.27}$. Our results allow us to estimate the probability of detecting young quasars with $t_{\rm Q}\leq0.1$ Myr from their proximity zone sizes yielding $p\left(\leq 0.1~{\rm Myr}\right)=0.19^{+0.11}_{-0.09}$, which is broadly consistent with recent determination at $z\sim 6$.