On the dilation current in metric-affine gravity

We review $F(R,\mathcal{D})$ gravity in the metric-affine framework, where $\mathcal{D}$ is the divergence of the dilation current appearing in the hypermomentum tensor. We assume only linear couplings between the general affine connection and the matter fields (minimal coupling) and break projective invariance to preserve a nonvanishing dilation current. For $F(R,\mathcal{D})$ linear in $\mathcal{D}$ the dilation current dependence in the function $F(R,\mathcal{D})$ does not contribute to the field equations of the theory. We show that, on the other hand, in more complicated cases (e.g., considering the function $F(R,\mathcal{D})=R+α\mathcal{D}^2$), the $\mathcal{D}$ contribution to the metric field equations is nontrivial and can affect the cosmology of the theory.