New non-conserved gravity theory and alleviate current acceleration epoch without dark energy field

Recently, a new modified theory of gravity was proposed in which Rastall's idea Tμ;νν=α,μ is reexamined. Although Rastall's idea shows potential for exploring field equations of gravity at high levels of energy, there are concerns about Rastall's proposal α=γR. In Ref. [1], it is suggested to find the viable form for α by reevaluating the first law of thermodynamics in a relativistic system. In this context, Israel and his colleagues have proposed a covariant equation linking the flux of the energy-momentum tensor and the 4-vector of entropy of the system, respecting the Gibbs-Duhem relation in flat spacetime [2,3]. Extending this idea to curved spacetime provides the theoretical form of α that supports Rastall's idea Tμ;νν=α,μ. However, this modified theory of gravity, known as ‘non-conserved gravity’, cannot present dark energy in FRW spacetime for the comoving observer without the cosmological constant. Through this study and considering the application of this gravity theory in cosmology, it was demonstrated that the model can explain the current acceleration epoch in the absence of extra fluid as dark energy. Therefore, the non-conserved theory of gravity attributes the matter field as the source of current accelerated expansion. This model also aligns with matter structure formations and thus does not conflict with observations in the past (high redshift) Universe. © 2024 Elsevier B.V., All rights reserved.