= Synchronous density gauge mode
{title2=$\Delta\delta=3(1+w)\mathcal H C/a$}
A residual synchronous time displacement $\xi^0=C(\mathbf x)/a$ changes a fluid <density contrast> by $-\bar\rho'\xi^0/\bar\rho$. For constant equation of state this gives the displayed mode: it scales as $\tau^{-2}$ during <radiation domination> and as $\tau^{-3}$ during <matter domination>. Such density changes alone do not indicate physical structure growth. Choosing the cold-matter rest frame fixes the residual time displacement, while a spatial relabeling fixes the remaining integration convention for the metric trace.
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