  We develop a formalism to study the implications of causality on OPE
coefficients in conformal field theories with large central charge and a sparse
spectrum of higher spin operators. The formalism has the interpretation of a
new conformal collider-type experiment for these class of CFTs and hence it has
the advantage of requiring knowledge only about CFT three-point functions. This
is accomplished by considering the holographic null energy operator which was
introduced in arXiv:1709.03597 as a generalization of the averaged null energy
operator. Analyticity properties of correlators in the Regge limit imply that
the holographic null energy operator is a positive operator in a subspace of
the total CFT Hilbert space. Utilizing this positivity condition, we derive
bounds on three-point functions $\langle TO_1O_2\rangle$ of the stress tensor
with various operators for CFTs with large central charge and a sparse
spectrum. After imposing these constraints, we also find that the operator
product expansions of all primary operators in the Regge limit have certain
universal properties. All of these results are consistent with the expectation
that CFTs in this class, irrespective of their microscopic details, admit
universal gravity-like holographic dual descriptions. Furthermore, this
connection enables us to constrain various inflationary observables such as the
amplitude of chiral gravity waves, non-gaussanity of gravity waves and
tensor-to-scalar ratio.
