gas fraction. Drawing from the normal star-forming sample, we find two galaxies with HCN/CO as low as the post-starbursts only 2% of the time in our Monte-Carlo analysis. This significant difference in dense molecular gas fractions implies the state of the molecular gas reservoir is substantially different than expected in the course of normal star-formation. We discuss several physical mechanisms which could alter the molecular gas state, lowering the dense gas fraction, in §\[sec:lowdensegas\]).

In summary, post-starburst galaxies have CO luminosities that are systematically high given their SFRs, when compared to the SFR-$L^{\prime}($CO$)$ relation that is followed by star-forming and early type galaxies. However, the low HCN and HCO$^+$ luminosities implied by our ALMA non-detections [*are*]{} consistent with the low SFRs in the two post-starbursts targeted here. These low dense gas luminosities and SFRs are typical of early type galaxies, the likely end-points of post-starburst evolution. The HCN/CO luminosity ratios are low compared to starbursting, star-forming and many CO-detected early type galaxies, implying a low fraction of dense molecular gas mass to total molecular gas mass.

Discussion {#sec:discussion}
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Interpretation of the $L^{\prime}($HCN$)$–SFR relation
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Despite the tight linear relation between HCN luminosity and SFR, there are still several uncertainties 