Constraints On Duty Cycles Of Quasars At Z $\sim$ 6

We study the mass of quasar-hosting dark matter halos at z $\sim$ 6 and further constrain the fraction of dark matter halos hosting an active quasar $f_{on}$ and the quasar beaming angle $i_{\rm max}$ using observations of CII lines in the literature. We make assumptions that (1) more massive halos host brighter quasars, (2) a fraction of the halos host active quasars with a certain beaming angle, (3) cold gas in galaxies has rotational velocity $V_{\rm circ}=\alpha V_{\rm max}$, and that (4) quasars point randomly on the sky. We find that for a choice of specific $\alpha \gtrsim 1$, the most likely solution has $f_{\rm on} < 0.01$, corresponding to a small duty cycle of quasar activity. However, if we marginalize over $\alpha$, for some choices of a prior a second solution with $f_{\rm on}=1$ appears. Overall, our the constraints are highly sensitive to $\alpha$ and hence inconclusive. Stronger constraints on $f_{\rm on}$ can be made if we better understand the dynamics of cold gas in these galaxies.