Screening $\Lambda$ in a new modified gravity model
  We study a new model of Energy-Momentum Squared Gravity (EMSG), called
Energy-Momentum Log Gravity (EMLG), constructed by the addition of the term
$f(T_{\mu\nu}T^{\mu\nu})=\alpha \ln(\lambda\,T_{\mu\nu}T^{\mu\nu})$, envisaged
as a correction, to the Einstein-Hilbert action with cosmological constant
$\Lambda$. 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 $\Lambda$CDM model. EMLG provides an effective dynamical dark energy
passing below zero at large redshifts, accommodating a mechanism for screening
$\Lambda$ in this region, in line with suggestions for alleviating some of the
tensions that arise between observational data sets within the standard
$\Lambda$CDM model. We present a detailed theoretical investigation of the
model and then constrain the free parameter $\alpha'$, a normalisation of
$\alpha$, using the latest observational data. The data does not rule out the
$\Lambda$CDM limit of our model ($\alpha'= 0$), but prefers slightly negative
values of the EMLG model parameter ($\alpha'= -0.032\pm 0.043$), which leads to
the screening of $\Lambda$. We also discuss how EMLG relaxes the persistent
tension that appears in the measurements of $H_0$ within the standard
$\Lambda$CDM model.
