Inflationary dynamics and preheating of the non-minimally coupled
  inflaton field in the metric and Palatini formalisms
  The inflationary dynamics and preheating in a model with a non-minimally
coupled inflaton field in the metric and Palatini formalisms are studied in
this paper. We find that in both formalisms, irrespective of the initial
conditions, our Universe will evolve into a slow-roll inflationary era and then
the scalar field rolls into an oscillating phase. The value of the scalar field
at the end of the inflation in the Palatini formalism is always larger than
that in the metric one, which becomes more and more obvious with the increase
of the absolute value of the coupling parameter $|\xi|$. During the preheating,
we find that the inflaton quanta are produced explosively due to the parameter
resonance and the growth of inflaton quanta will be terminated by the
backreaction. With the increase of $|\xi|$, the resonance bands gradually close
to the zero momentum ($k=0$), and the structure of resonance changes and
becomes broader and broader in the metric formalism, while, it remains to be
narrow in the Palatini formalism. The energy transfer from the inflaton field
to the fluctuation becomes more and more efficient with the increase of
$|\xi|$, and in the metric formalism the growth of the efficiency of energy
transfer is much faster than that in the Palatini formalism. Therefore, the
inflation and preheating show different characteristics in different
formalisms.
