# Higher Derivative Theory For Curvature Term Coupling With Scalar Field

## TL;DR

This paper investigates higher derivative scalar-curvature theories to identify conditions preventing Ostrogradsky instabilities, concluding that such non-trivial conditions do not exist in the considered model.

## Contribution

The study performs a (3+1) decomposition to analyze degeneracy conditions in second-order derivative scalar-curvature theories, revealing the inevitability of Ostrogradsky ghosts.

## Key findings

- No non-trivial degeneracy conditions found to avoid Ostrogradsky instability.
- The model inherently suffers from Ostrogradsky ghosts due to its structure.
- Decomposition method helps identify stability conditions in higher derivative theories.

## Abstract

Higher order derivative theories, generally suffer from instabilities, known as Ostrogradsky instabilities. This issue can be resolved by removing any existing degeneracy present in such theories. We consider a model involving at most second order derivatives of scalar field non-minimally coupled to curvature terms. Here we perform (3+1) decomposition of Lagrangian to separate second order time derivative form the rest. This is useful to check the degeneracy hidden in the Lagrangian will help us to find conditions under which Ostrogradsky instability do not appear. In our case, we find no such non trivial conditions which can stop the appearance of the Ostrogradsky ghost.