Nonlocal cosmological models with quadratic potentials are considered. We study actions with an arbitrary analytic function $F(\Box_g)$, which has both double and simple roots. We have proposed the way to find particular solutions of the nonlocal Einstein equations for an arbitrary analytic function $F(\Box_g)$ with simple and double roots in an arbitrary metric.
We prove that the same functions solve the initial nonlocal Einstein equations and the obtained local Einstein equations. We have found the corresponding local actions and proved the self-consistence of our approach. We show that quintom models naturally arise as a localization of nonlocal model. In the case of simple roots some exact solutions in the Friedmann-Robertson-Walker metric have been found.
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