Energy–momentum squared gravity and late-time Universe

Abstract Recently, the new modified gravity was introduced in which Einstein’s general relativity is developed by adding energy–momentum squared term $${T}_{\mu \nu }{T}^{\mu \nu }$$ T μ ν T μ ν by coupling constant $$\alpha $$ α . As result, the relevant field equations are different from usual field equations in Einstein’s general relativity only in the presence of matter sources. Analytical consideration proved that for the non-interaction case, the energy–momentum squared term is not strong and only can describe the non-singular big-bang theory. In this study, this theory is applied to the homogenous and isotropic space–time in the presence of the cosmological constant $$\Lambda $$ Λ . In this context, we face with three plausible models of dark energy. In the first model, dark energy is presented by cosmological constant, only, and thus, extra terms arise from squared term effects on matter evolution. This case gives no new model of dark energy. As shown in Roshan and Shojai (Phys Rev D 94:044002, 2016), considering the cosmological constant as part of the matter field is not equal to the first scenario in which the cosmological constant plays a geometrical role. Therefore, for the second case, the cosmological constant is investigated as the part of the matter action, and we can assume that dark energy includes two parts, the cosmological constant and the energy–momentum squared term. Modeling this scenario illustrates this theory gives no valuable dark energy for $$\alpha \ne 0$$ α ≠ 0 . It reveals the second scenario wherein cosmological constant behaves such as part of matter field gives accelerated expansion Universe only in the absence of energy–momentum squared term. In the last plausible scenario, we may assume dark energy constructed with both parts, geometrical part includes cosmological constant and matter effects arise from the energy–momentum squared term. We have shown that only the last model satisfies observations and presents the quintessence dark energy in which cosmological constant problems are solved. Moreover, it is shown that this model coincides with $$\Lambda $$ Λ CDM theory with some small errors in studying theoretical CMB temperature and linear matter power spectrum.