We study the long time dynamics of future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that the scalar field and all of its derivatives decay with polynomial rates.

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. This is a joint work with Sung-Jin Oh.