  We prove a rigidity result for non-negative scalar curvature perturbations of
the Euclidean metric on $\mathbb{R^n}$ , which may be regarded as a weak
version of the rigidity statement of the positive mass theorem. We prove our
result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct
in doing so, we extend known $L^p$ bounds and decay rates for Ricci DeTurck
flow and prove regularity of the flow at the initial data.
