We study the cosmological effects of adding terms of higher-order in the usual energy-momentum tensor to the matter Lagrangian of general relativity. This is in contrast to most studies of higher-order gravity which focus on generalising the Einstein-Hilbert curvature contribution to the Lagrangian. The resulting cosmological theories give rise to field equations of similar form to several particular theories with different fundamental bases, including bulk viscous cosmology, loop quantum gravity, k-essence, and brane-world cosmologies. We find a range of exact solutions for isotropic universes, discuss their behaviours with reference to the early-and late-time evolution, accelerated expansion, and the occurrence or avoidance of singularities. We briefly discuss extensions to anisotropic cosmologies and delineate the situations where the higher-order matter terms will dominate over anisotropies on approach to cosmological singularities.

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We study a new model of Energy-Momentum Squared Gravity (EMSG), called Energy-Momentum Log Gravity (EMLG), constructed by the addition of the term f (Tµν T µν ) = α ln(λ Tµν T µν ), envisaged as a correction, to the Einstein-Hilbert action with cosmological constant Λ. The choice of this modification is made as a specific way of including new terms in the right-hand side of the Einstein field equations, resulting in constant effective inertial mass density and, importantly, leading to an explicit exact solution of the matter energy density in terms of redshift. We look for viable cosmologies, in particular, an extension of the standard ΛCDM model. EMLG provides an effective dynamical dark energy passing below zero at large redshifts, accommodating a mechanism for screening Λ in this region, in line with suggestions for alleviating some of the tensions that arise between observational data sets within the standard ΛCDM model. We present a detailed theoretical investigation of the model and then constrain the free parameter α , a normalisation of α, using the latest observational data. The data does not rule out the ΛCDM limit of our model (α = 0), but prefers slightly negative values of the EMLG model parameter (α = −0.032 ± 0.043), which leads to the screening of Λ. We also discuss how EMLG relaxes the persistent tension that appears in the measurements of H0 within the standard ΛCDM model.

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