molecular gas mass, we must use conversion factors to convert $L^{\prime}($HCN$)$ to a dense ($n \gtrsim 8\times10^{3}$ cm$^{-3}$) molecular gas mass upper limit and $L^{\prime}($HCN$)$/$L^{\prime}($CO$)$ into the dense molecular gas mass to total molecular gas mass ratio (i.e., the “dense gas mass fraction"). We assume that the conversion factor from $L^{\prime}($HCN$)$ to dense molecular gas mass is the same for post-starbursts and other galaxies. Therefore, our non-detections of $L^{\prime}($HCN$)$ imply low dense gas masses, relative to the dense gas mass –SFR relation traced by star forming and early type galaxies, explaining their quiescence.

We test whether the dense gas mass ratio could be normal, despite the observed dense gas luminosity ratio, due to uncertainties in the conversion factors. What if $L^{\prime}($HCN$)$/$L^{\prime}($CO$)$ does not correlate with the dense gas mass fraction as it does for other galaxies? This scenario could occur if the CO luminosity to total molecular gas mass conversion factor $\alpha_{\rm CO}$ and HCN luminosity to dense molecular gas mass conversion factor $\alpha_{\rm HCN}$ vary differently with the state of the gas.

A lower value of $\alpha_{\rm CO}$ is usually invoked in ULIRGs (Ultra Luminous Infrared Galaxies) and justified based on differences in the distribution or state of the 