Effective delta sources and Newtonian limit in nonlocal gravity

TL;DR

This paper analyzes the Newtonian limit of nonlocal gravity models with exponential form factors, revealing oscillatory potentials for certain parameters, ensuring positivity of effective masses, and deriving quantum corrections.

Contribution

It introduces a comprehensive analysis of weak-field solutions in nonlocal gravity with exponential form factors, including series, integral representations, and approximations, and explores quantum corrections.

Findings

Newtonian potential oscillates for N_s > 1

Effective masses remain positive despite oscillations

Solutions are regular without curvature singularities

Abstract

We investigate the Newtonian limit of a class of nonlocal gravity models with exponential form factors $f_s (\Box) = \exp [(-\Box/\mu_s^2)^{N_s}]$. Our main goal is to identify similarities and differences between models in this family in regard to weak-field solutions. To this end, we use the effective source formalism to compare the related effective delta sources, mass functions, and Newtonian potentials. We obtain a variety of representations for these quantities in terms of series, integrals, and special functions, as well as simple approximations that capture the relevant dependence on the parameters $N_s$ and $\mu_s$ - which can be used to explore the weak-field phenomenology of nonlocal gravity. We explain why only for $N_s>1$ the Newtonian potential oscillates and prove that, despite the oscillations, the effective masses are positive. Moreover, we verify that these linearized solutions are regular (without curvature singularities). Finally, we also calculate the form of the leading logarithmic quantum correction to the Newtonian potential in these models. In all our considerations, we assume that $N_s$ is a positive real parameter. The cases of non-integer $N_s$ might be applied beyond nonlocal gravity, in effective approaches to implement quantum corrections in the weak field regime.