  Accurate models of the star formation histories (SFHs) of recently-quenched
galaxies can provide constraints on when and how galaxies shut down their star
formation. The recent development of "non-parametric" SFH models promises the
flexibility required to make these measurements. However, model and prior
choices significantly affect derived SFHs, particularly for post-starburst
galaxies (PSBs) which have sharp changes in their recent SFH. In this paper, we
create mock PSBs, then use the Prospector SED fitting software to test how well
four different SFH models recover key properties. We find that a two-component
parametric model performs well for our simple mock galaxies, but is sensitive
to model mismatches. The fixed- and flexible-bin non-parametric models included
in Prospector are able to rapidly quench a major burst of star formation, but
systematically underestimate the post-burst age by up to 200 Myr. We develop a
custom SFH model that allows for additional flexibility in the recent SFH. Our
flexible non-parametric model is able to constrain post-burst ages with no
significant offset and just ~90 Myr of scatter. Our results suggest that while
standard non-parametric models are able to recover first-order quantities of
the SFH (mass, SFR, average age), accurately recovering higher-order quantities
(burst fraction, quenching time) requires careful consideration of model
flexibility. These mock recovery tests are a critical part of future SFH
studies. Finally, we show that our new, public SFH model is able to accurately
recover the properties of mock star-forming and quiescent galaxies and is
suitable for broader use in the SED fitting community.
