On the covariant formalism of the effective field theory of gravity and
  leading order corrections
  We construct the covariant effective field theory of gravity as an expansion
in inverse powers of the Planck mass, identifying the leading and
next-to-leading quantum corrections. We determine the form of the effective
action for the cases of pure gravity with cosmological constant as well as
gravity coupled to matter. By means of heat kernel methods we renormalize and
compute the leading quantum corrections to quadratic order in a curvature
expansion. The final effective action in our covariant formalism is generally
non-local and can be readily used to understand the phenomenology on different
spacetimes. In particular, we point out that on curved backgrounds the
observable leading quantum gravitational effects are less suppressed than on
Minkowski spacetime.
