addition, for this short time-averaging, changing cluster stiffness no longer strongly affects CI.

Increasing single-cell stochasticity can increase cluster accuracy
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![image](vary_Dpsi_label){width="180mm"}

Any mechanism that fluidizes the cluster can decrease the correlation time $\tau_r$. Because of this, [*increasing*]{} noise can improve cluster chemotactic accuracy ([Fig. ]{}\[fig:Dpsi\]). We increase single-cell angular noise $D_\psi$, and see an initial sharp increase in cluster chemotactic index as $D_\psi > 0$ ([Fig. ]{}\[fig:Dpsi\]b). At larger values of $D_\psi$, cluster CI decreases below the bound set by [Eq. ]{}\[eq:snrt\_simp\], as the additional noise added degrades the gradient-following behavior. Without significant time-averaging ($T = 0.2$), additional noise primarily impedes chemotactic accuracy.

Why can extra noise $D_\psi$ help sensing? For $D_\psi = 0$, all cells follow the best estimate ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ precisely, leading to an ordered cluster ([Fig. ]{}\[fig:Dpsi\]a) with $\tau_r$ effectively infinite. As $D_\psi$ is increased, the cluster fluidizes, and the relaxation time decreases strongly ([Fig. ]{}\[fig:Dpsi\]c) resulting in more independent measurements. As in [Fig. ]{}\[fig:kappa\], this fluidization is only relevant if the averaging time $T$ exceeds the relaxation time, so when $T = 0.2$, the effect of increasing $D_\psi$ is solely detrimental to chemotaxis.

[In deriving our bound, we made two key approximations: cluster isotropy 