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.
