Time-dependent behaviour of quasar proximity zones at $z \sim 6$

Since the discovery of $z\sim 6$ quasars two decades ago, studies of their Ly$\alpha$-transparent proximity zones have largely focused on their utility as a probe of cosmic reionization. But even when in a highly ionized intergalactic medium, these zones provide a rich laboratory for determining the timescales that govern quasar activity and the concomitant growth of their supermassive black holes. In this work, we use a suite of 1D radiative transfer simulations of quasar proximity zones to explore their time-dependent behaviour for activity timescales from $\sim10^3$ to $10^8$ years. The sizes of the simulated proximity zones, as quantified by the distance at which the smoothed Ly$\alpha$ transmission drops below 10% (denoted $R_p$), are in excellent agreement with observations, with the exception of a handful of particularly small zones that have been attributed to extremely short $\lesssim 10^4$ year lifetimes. We develop a physically motivated semi-analytic model of proximity zones which captures the bulk of their equilibrium and non-equilibrium behaviour, and use this model to investigate how quasar variability on $\lesssim10^5$ year timescales is imprinted on the distribution of observed proximity zone sizes. We show that large variations in the ionizing luminosity of quasars on timescales of $\lesssim10^4$ years are disfavored based on the good agreement between the observed distribution of $R_p$ and our model prediction based on "lightbulb" (i.e. steady constant emission) light curves.