 by more than an order of magnitude when more effective supernova feedback or AGN feedback is included (not shown). For $M_\mathrm{halo}>10^{12.5}$ M$_\odot$ the hot-mode inflow rate is slightly stronger than the cold-mode inflow rate.

The grey, dashed curve indicates the accretion rate a halo with a baryon fraction $\Omega_{\rm b}/\Omega_{\rm m}$ would need to have to grow to its current baryonic mass in a time equal to the age of the Universe at $z=2$, $$\dot{M}=\dfrac{\Omega_\mathrm{b}M_\mathrm{halo}}{\Omega_\mathrm{m}t_\mathrm{Universe}}.$$ Comparing this analytic estimate with the actual mean accretion rate, we see that they are equal for $M_\mathrm{halo}>10^{11.5}$ M$_\odot$, indicating that these haloes are in a regime of efficient growth. For lower-mass haloes, the infall rates are much lower, indicating that the growth of these haloes has halted, or that their baryon fractions are much smaller than $\Omega_{\rm b}/\Omega_{\rm m}$.

The bottom-right panel of Fig. \[fig:halomassz2\] shows that, at the virial radius, hot-mode gas dominates the gas mass for high-mass haloes. The hot fraction at $R_\mathrm{vir}$ increases from 10 per cent in haloes of $\sim 10^{10}$ M$_\odot$ to 90 per cent for $M_\mathrm{halo}\sim 10^{13}$ M$_\odot$. Note that for haloes with $M_\mathrm{halo}<10^{11.3}$ M$_\odot$ the virial temperatures are lower than our adopted threshold for hot-mode gas. The hot fraction would have been much lower without supernova feedback ($f_\mathrm{hot}<5$ per cent for $M_\mathrm{halo}<10^{10.5}$ M$_\odot$).

Inflow and outflow {#sec:inout}
==================

Figs. \[fig:haloradflux\] and \[fig:halomassflux\] show the physical properties of the gas, weighted by the radial mass flux, for all, inflowing, and outflowing gas (black, blue, and red curves, respectively). Except for the last two panels, the curves indicate the medians, i.e. half the mass flux is due to gas above the curves. Similarly, the shaded regions indicate the 16th and 84th percentiles. Note that we do not plot this separa