Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions

We introduce a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd space dimensions. In particular, for a large class of equations, we prove that the precise late time tail is determined by the limits of higher radiation field at future null infinity. In the setting of stationary linear equations, we recover and generalize the Price law decay rates. In particular, in addition to reproving known results on $(3+1)$-dimensional black holes, this allows one to obtain the sharp decay rate for the wave equation on higher dimensional black hole spacetimes, which exhibits an anomalous rate due to subtle cancellations. More interesting, our method goes beyond the stationary linear case and applies to both equations on dynamical background and nonlinear equations. In this case, our results can be used to show that in general there is a correction to the Price law rates.