Irreducible forms for the metric variations of the action terms of sixth-order gravity and approximated stress-energy tensor

We provide irreducible expressions for the metric variations of the gravitational action terms constructed from the 17 curvature invariants of order six in derivatives of the metric tensor i.e. from the geometrical terms appearing in the diagonal heat-kernel or Gilkey-DeWitt coefficient $a_3$. We then express, for a four dimensional spacetime, the approximated stress-energy tensor constructed from the renormalized DeWitt-Schwinger effective action associated with a massive scalar field. We also construct, for higher dimensional spacetimes, the infinite counterterms of order six in derivatives of the metric tensor appearing in the left hand side of Einstein equations as well as the contribution associated with the cubic Lovelock gravitational action. In an appendix, we provide a list of geometrical relations we have used and which are more generally helpful for calculations in two-loop quantum gravity in a four dimensional background or for calculations in one-loop quantum gravity in higher dimensional background. We also obtain the approximated stress-energy tensors associated with a massive spinor field and a massive vector field propagating in a four dimensional background.