Metric perturbations typically strengthen field resonances during preheating. In contrast we present a model in which the super-Hubble field resonances are completely {em suppressed} when metric perturbations are included. The model is the nonminimal Fakir-Unruh scenario which is exactly solvable in the long-wavelength limit when metric perturbations are included, but exhibits exponential growth of super-Hubble modes in their absence. This gravitationally enhanced integrability is exceptional, both for its rarity and for the power with which it illustrates the importance of including metric perturbations in consistent studies of preheating. We conjecture a no-go result - there exists no {em single-field} model with growth of cosmologically-relevant metric perturbations during preheating.
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