  In this paper, we study the future causally geodesically complete solutions
of the spherically symmetric Einstein-scalar field system. Under the a priori
assumption that the scalar field $\phi$ scatters locally in the scale-invariant
bounded-variation (BV) norm, we prove that $\phi$ and its derivatives decay
polynomially. Moreover, we show that the decay rates are sharp. In particular,
we obtain sharp quantitative decay for the class of global solutions with small
BV norms constructed by Christodoulou. As a consequence of our results, for
every future causally geodesically complete solution with sufficiently regular
initial data, we show the dichotomy that either the sharp power law tail holds
or that the spacetime blows up at infinity in the sense that some scale
invariant spacetime norms blow up.
