# Variations of Infinite Derivative Modified Gravity

## TL;DR

This paper derives detailed equations of motion and perturbation basics for a class of nonlocal modified gravity models with infinite derivatives, aiding future research in nonlocal gravitational theories.

## Contribution

It provides a comprehensive derivation of equations of motion and perturbation analysis for nonlocal Einstein gravity with infinite derivatives, which was previously lacking.

## Key findings

- Derived equations of motion for nonlocal gravity model.
- Presented second variation of the action and perturbation basics.
- Facilitated future investigations into nonlocal gravity theories.

## Abstract

We consider nonlocal modified Einstein gravity without matter, where nonlocal term has the form $P(R) F(\Box) Q(R)$. For this model, in this paper we give the derivation of the equations of motion in detail. This is not an easy task and presented derivation should be useful to a researcher who wants to investigate nonlocal gravity. Also, we present the second variation of the related Einstein-Hilbert modified action and basics of gravity perturbations.