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, $⟨TT\mathcal{O}⟩,$ 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 $\int \phi\hspace{.5mm} 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 $\varphi W{W}^{\ast }$.