Let $(M,g(t))_{t\in I}$ be a smooth Ricci flow on an $n$-dimensional manifold, let $S_{g(t)}$ denote the scalar curvature of $g(t)$, and let $\tau:I\to(0,\infty)$ be a smooth function satisfying $\frac{d\tau}{dt}=-1$. For a smooth potential function $f:M\times I\to\mathbb{R}$, define $u:M\times I\to(0,\infty)$ by