= Linear perfect-fluid stress tensor in synchronous gauge
{title2=$T^{00}=a^{-2}\bar\rho(1+\delta),\quad T^{0i}=a^{-2}\bar\rho(1+w)v^i$}
For constant $P=w\rho$, the <perfect fluid> stress tensor and inverse perturbed spatial metric give $T^{ij}=a^{-2}w\bar\rho[(1+\delta)\delta^{ij}-h^{ij}]$ at first order. Normalizing the velocity makes $u^0=a^{-1}$ at this order. <Stress-energy conservation> yields the <synchronous perfect-fluid density equation> and <Cosmological Euler equation in synchronous gauge>; the pressure projection includes $v_i\bar P'$, which changes the velocity damping from $\mathcal H$ to $(1-3w)\mathcal H$.
// Target: cosmology.bigb
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