# A note on a covariant version of Verlinde's emergent gravity

## TL;DR

This paper identifies and corrects an error in a Lagrangian related to Verlinde's emergent gravity, demonstrating that the corrected theory does not naturally produce a stable de Sitter space solution.

## Contribution

The authors correct a calculation error in Hossenfelder's Lagrangian for emergent gravity and analyze the stability of de Sitter solutions within this framework.

## Key findings

- The original Lagrangian does not yield de Sitter space.
- Corrected Lagrangian recovers de Sitter solution.
- Perturbations around de Sitter grow, indicating instability.

## Abstract

Following recent Verlinde's heuristic construction or emergent gravity, Hossenfelder wrote down the Lagrangian capturing some aspects of this theory. We point out that there is an error in calculations whose consequence is that a cosmological de Sitter space solution is not recovered from this Lagrangian. We correct this error in order to obtain the desired solution. We also show that small perturbations around de Sitter space grow, which implies that this state is not stable. However, the presence of matter and radiation might in principle provide stability.