  In a model of spontaneous scalarization of neutron stars proposed by Damour
and Esposite-Farese, a general relativistic branch becomes unstable to trigger
tachyonic growth of a scalar field $\phi$ toward a scalarized branch. Applying
this scenario to cosmology, there is fatal tachyonic instability of $\phi$
during inflation and matter dominance being incompatible with solar-system
constraints on today's field value $\phi_0$. In the presence of a four-point
coupling $g^2 \phi^2 \chi^2/2$ between $\phi$ and an inflaton field $\chi$, it
was argued by Anson et al. that a positive mass squared heavier than the square
of a Hubble expansion rate leads to the exponential suppression of $\phi$
during inflation and that $\phi_0$ can remain small even with the growth of
$\phi$ after the radiation-dominated epoch. For several inflaton potentials
approximated as $V(\chi)=m^2 \chi^2/2$ about the potential minimum, we study
the dynamics of $\phi$ during reheating as well as other cosmological epochs in
detail. For certain ranges of the coupling $g$, the homogeneous field $\phi$
can be amplified by parametric resonance during a coherent oscillation of the
inflaton. Incorporating the backreaction of created particles under a Hartree
approximation, the maximum values of $\phi$ reached during preheating are
significantly smaller than those obtained without the backreaction. We also
find that the minimum values of $g$ consistent with solar system bounds on
$\phi$ at the end of reheating are of order $10^{-5}$ and hence there is a wide
range of acceptable values of $g$. Thus, the scenario proposed by Anson et al.
naturally leads to the viable cosmological evolution of $\phi$ consistent with
local gravity constraints, without modifying the property of scalarized neutron
stars.
