Cuscuton-like contribution to dark energy evolution

This work deals with the presence of the cuscuton term in the otherwise standard dark energy evolution under the usual FLRW background. We disclose a first-order framework similar to the Hamilton-Jacobi formalism, which helps us to solve the equations of motion and find analytical solutions. We explore several possibilities, concentrating mainly on how the cuscuton-like contribution works to modify cosmic evolution. Some results are of current interest since they describe scenarios capable of changing the evolution, adding or excluding possible distinct phases during the Universe's expansion history. Additionally, we present interesting constraints on the cuscuton-like contribution for the dark energy evolution using a set of homogeneous geometrical observational probes. Finally, based on the Akaike Information Criterion (AIC), we perform a statistical comparison of the cuscuton-like model with Lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM, and find strong support for our model.