# A Review on the Cosmology of the de Sitter Horndeski Models

## TL;DR

This review discusses the most general scalar-tensor models within Horndeski theory that can dynamically adjust vacuum energy, focusing on their cosmological evolution and observational viability.

## Contribution

It provides a comprehensive review of de Sitter Horndeski models with second-order derivatives, highlighting their ability to address vacuum energy issues.

## Key findings

- Shift symmetric non-linear models are compatible with current cosmological observations.
- Linear models show specific cosmological evolution patterns.
- Models can dynamically adjust vacuum energy independent of matter content.

## Abstract

We review the most general scalar-tensor cosmological models with up to second-order derivatives in the field equations that have a fixed spatially flat de Sitter critical point independent of the material content or vacuum energy. This subclass of the Horndeski Lagrangian is capable of dynamically adjusting any value of the vacuum energy of the matter fields at the critical point. We present the cosmological evolution of the linear models and the non-linear models with shift symmetry. We come to the conclusion that the shift symmetric non-linear models can deliver a viable background compatible with current observations.