  We reconsider the problem of bounding higher derivative couplings in
consistent weakly coupled gravitational theories, starting from general
assumptions about analyticity and Regge growth of the S-matrix. Higher
derivative couplings are expected to be of order one in the units of the UV
cutoff. Our approach justifies this expectation and allows to prove precise
bounds on the order one coefficients. Our main tool are dispersive sum rules
for the S-matrix. We overcome the difficulties presented by the graviton pole
by measuring couplings at small impact parameter, rather than in the forward
limit. We illustrate the method in theories containing a massless scalar
coupled to gravity, and in theories with maximal supersymmetry.
