Logarithmic Lagrangian matter density, unimodular gravity-like and accelerated expansion with a negative cosmological constant

We present a new viable modified theory of gravity in which the matter sector is characterized by a logarithmic Lagrangian density. The modified Einstein’s field equations are derived, and they are characterized by the emergence of an effective gravitational coupling constant and an effective negative cosmological constant both coupled to the Lagrangian of matter. Applied to the Friedmann-Robertson-Walker cosmological framework, it leads to motivating phenomenology. In particular, the universe is free from the initial singularity, and a unified description of the early inflationary phase, the succeeding non-accelerating, matter-dominated expansion, and then the transition to a late-time accelerating phase is obtained as well. Moreover, the effective dark energy sector can be quintessence, yet the universe starts contracting if the phantom divide-line is crossed. The model is confronted with observations that agree with recent astronomical data. Moreover, the model gives rise to Einstein’s field equations which resemble the equations obtained in unimodular gravity.