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In this paper we analyze Ricci flows on which the scalar curvature isglobally or locally bounded from above by a uniform or time-dependent constant.On such Ricci flows we establish a new time-derivative bound for solutions tothe heat equation. Based on this bound, we solve several open problems: 1.distance distortion estimates, 2. the existence of a cutoff function, 3.Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem,which states that a curvature bound at a later time implies a curvature boundat a slightly earlier time.  Using the backward pseudolocality theorem, we next establish a uniform $L^2$curvature bound in dimension 4 and we show that the flow in dimension 4converges to an orbifold at a singularity. We also obtain a stronger$\varepsilon$-regularity theorem for Ricci flows. This result is particularlyuseful in the study of K\"ahler Ricci flows on Fano manifolds, where it can beused to derive certain convergence results.

Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature