and 2) re-arrangement of cell positions relative to the center of mass is independent of the particular values of $\Delta$. The first approximation is not necessary, but is a useful simplification; a generalized result is given in Appendix \[app:timeaverage\]. [The second approximation assumes that averaging over CCV and averaging over cell positions are independent. This decoupling approximation is necessary to characterize cluster fluidity and mechanics separately from the details of signaling. It excludes, e.g. models where cells with larger-than-average $\Delta$ sort out from the cluster. We will discuss potential errors due to this approximation later in the paper.]{}

For exponential position-position correlation functions and averaging, $C_{rr}(t) = \exp(-t/\tau_r)$ and $K_T(t) = \theta(t) \frac{1}{T} e^{-t/T}$, where $\theta(t)$ is the Heaviside step function, [Eq. ]{}\[eq:snrt\] is simple: $${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = \frac{{\ensuremath{\sigma_{\mathbf{g},0}}}^2}{1+T/\tau_r} \label{eq:snrt_simp}$$ In other words, gradient sensing accuracy can be improved by taking $T/\tau_r$ independent measurements in a time $T$. Crucial in this reduction is the position-position correlation time $\tau_r$ which depends on the cluster rearrangement mechanism. Two natural mechanisms are persistent cluster rotation and neighbor re-arrangements within the cluster ([Fig. ]{}\[fig:timeaverage\]c). These mechanisms may coexist, as when cells slide past one another during cluster rotation [@rappel1999self]. $\tau_r$ can depend on cluster size; 