# Gravitation with modified fluid Lagrangian: Variational principle and an early dark energy model

## TL;DR

This paper investigates the variational principle in modified fluid theories of gravity, revealing the need for an extra equation for completeness and proposing a model for early dark energy that addresses the Hubble tension.

## Contribution

It clarifies the limitations of the variational principle in modified fluid gravity and introduces a new model for early dark energy linked to cosmic transitions.

## Key findings

- Variational principle alone is insufficient for complete field equations in modified fluid theories.
- Adding energy conservation yields a thermodynamically consistent modified fluid.
- Proposed early dark energy model explains matter-radiation transition effects and Hubble tension.

## Abstract

Variational principle is the main approach to obtain complete and self-consistent field equations in gravitational theories. This method works well in pure field cases such as $f(R)$ and Horndeski gravities. However, debates exist in the literature over the modification of perfect fluid. This paper aims to clarify this issue. For a wide class of modified fluid Lagrangian, we show that the variational principle is unable to give complete field equations. One additional equation is required for completeness. Adopting the local energy conservation equation gives the modified fluid a good thermodynamic interpretation. Our result is the first modified fluid theory that can incorporate energy conservation. As an application of this framework, we propose a specific modified fluid model to realize early dark energy triggered by cosmic radiation-matter transition. This model naturally explains why early dark energy occurs around matter-radiation equality and is useful in erasing the Hubble tension.