  The system formed in ultrarelativistic heavy-ion collisions behaves as a
nearly-perfect fluid. This collective behavior is probed experimentally by
two-particle azimuthal correlations, which are typically averaged over the
properties of one particle in each pair. In this Letter, we argue that much
additional information is contained in the detailed structure of the
correlation. In particular, the correlation matrix exhibits an approximate
factorization in transverse momentum, which is taken as a strong evidence for
the hydrodynamic picture, while deviations from the factorized form are taken
as a signal of intrinsic, "nonflow" correlations. We show that hydrodynamics in
fact predicts factorization breaking as a natural consequence of initial state
fluctuations and averaging over events. We derive the general inequality
relations that hold if flow dominates, and which are saturated if the matrix
factorizes. For transverse momenta up to 5 GeV, these inequalities are
satisfied in data, but not saturated. We find at least as large factorization
breaking in event-by-event ideal hydrodynamic calculations as in data, and
argue that this phenomenon opens a new window on the study of initial
fluctuations.
