On the violent preheating in the mixed Higgs-$R^2$ inflationary model
  It has been argued that the mixed Higgs-$R^2$ model acts as the UV extension
of the Higgs inflation, pushing up its cut-off scale in the vacuum close up to
the Planck scale. In this letter, we study the inflaton oscillation stage after
inflation, focusing on the effective mass of the phase direction of the Higgs
field, which can cause a violent preheating process. We find that the "spikes"
in the effective mass of the phase direction observed in the Higgs inflation
still appear in the mixed Higgs-$R^2$ model. While the spikes appear above the
cut-off scale in the Higgs-only case, they appear below the cut-off scale when
the model is extended with $R^2$ term though reheating cannot be completed in
the violent particle production regime since the spikes get milder.
