Bounds on OPE coefficients from interference effects in the conformal collider

We apply the average null energy condition to obtain upper bounds on the three-point function coefficients of stress tensors and a scalar operator, < TTO >, in general CFTs. We also constrain the gravitational anomaly of U( 1) currents in four-dimensional CFTs, which are encoded in three-point functions of the form < TTJ >. In theories with a large N AdS dual we translate these bounds into constraints on the coefficient of a higher derivative bulk term of the form integral phi W-2. We speculate that these bounds also apply in de-Sitter. In this case our results constrain inflationary observables, such as the amplitude for chiral gravity waves that originate from higher derivative terms in the Lagrangian of the form phi WW*.