Dark energy and a new realization of the matter Lagrangian

A new realization of the matter Lagrangian is introduced which models the dark energy component as a non-standard combination of thermodynamics quantities of the baryonic matter. We will prove that the present realization is independent of existing models with matter-geometry couplings and has a property that the energy-momentum tensor of both baryonic matter and dark energy is conserved separately. We further show that two possible choices of the matter Lagrangian in the $Λ$CDM model are not totally equivalent and investigate the background and perturbative constraints on the form of matter Lagrangian. We will also investigate cosmological implications of a test model with logarithmic DE and obtain the model parameters by confronting the model with observational data on the cosmic chronometers, Pantheon$^+$ and $fσ_8$ datasets. We will also explain in details the predictions of the model on the late time behavior of the universe and compare the result with $Λ$CDM model.