(Generalized) quasi-topological gravities at all orders

A new class of higher-curvature modifications of )-dimensional Einstein gravity has been recently identified. Densities belonging to this 'Generalized quasi-topological' class (GQTGs) are characterized by possessing non-hairy generalizations of the Schwarzschild black hole satisfying and by having second-order equations of motion when linearized around maximally symmetric backgrounds. GQTGs for which the equation of the metric function is algebraic are called 'Quasi-topological' and only exist for . In this paper we prove that GQTG and Quasi-topological densities exist in general dimensions and at arbitrarily high curvature orders. We present recursive formulas which allow for the systematic construction of nth order densities of both types from lower order ones, as well as explicit expressions valid at any order. We also obtain the equation satisfied by for general D and n. Our results here tie up the remaining loose end in the proof presented in Bueno et al (2019 (arXiv:1906.00987)) that every gravitational effective action constructed from arbitrary contractions of the metric and the Riemann tensor is equivalent, through a metric redefinition, to some GQTG.