Localization of nonlocal cosmological models with quadratic potentials in the case of double roots

TL;DR

This paper investigates nonlocal cosmological models with quadratic potentials, focusing on cases where the defining function has double roots, and provides methods to find solutions to the resulting Einstein equations.

Contribution

It introduces formulas for the nonlocal energy-momentum tensor with double roots and proposes a method to find particular solutions when both simple and double roots are present.

Findings

Derived formulas for nonlocal energy-momentum tensor with double roots.

Proposed a method to find solutions for Einstein equations with mixed roots.

Demonstrated that solutions to nonlocal and local Einstein equations can coincide.

Abstract

Nonlocal cosmological models with quadratic potentials are considered. We study the action with an arbitrary analytic function F(\Box_g), which has both double and simple roots. The formulae for nonlocal energy-momentum tensor, which correspond to double roots, have been obtained. The way to find particular solutions for nonlocal Einstein equations in the case when F(\Box_g) has both simple and double roots has been proposed. One and the same functions solve the initial nonlocal Einstein equations and the obtained local Einstein equations.