in our model C, which we constrain by fitting the observed XLF, we find it is $\log \lambda_\mathrm{break}=0.2\pm0.2$, consistent with $\log \lambda_\mathrm{break}=0.0$. A break at around the Eddington limit is clearly needed to reproduce the bright end of the observed XLF (given the other assumptions and restrictions of our model); a break in the distribution at lower accretion rates would not produce sufficient luminous AGNs. We thus retain a one-parameter model with a fixed break at $\log \lambda_\mathrm{break}=0.0$.]{}]{}

[[ We also note that the shape of $p(\lambda {\;|\;}{\mathcal{M}_*},z)$ at high accretion rates ($\log \lambda \gtrsim 0.1$) may be sensitive to the assumed form of the bolometric correction. Recent work has indicated that X-ray bolometric corrections may be primarily a function of the Eddington ratio itself [e.g. @Vasudevan07; @Lusso12; @Jin12], rather than the AGN luminosity as we have assumed [following @Hopkins07b; @Marconi04]. The bolometric correction increases with increasing Eddington ratio and may be $\gtrsim 100$ at the highest accretion rates [$\lambda>1$, @Jin12]. A high bolometric correction means a low fraction of the AGN light is emitted at X-ray wavelengths. Thus, recovering the luminous AGN population may actually require a steeper slope for the super-Eddington tail of the accretion rate distribution; an 