Synchronous gauge fixing by coordinate displacement (${\alpha }^{\prime }+\mathcal{H}\alpha =-q_{00}/(2a^2),{\xi }_i^{\prime }=q_{0i}/a^2+{\partial }_i\alpha$)

For the FRW metric in conformal time, a displacement ${\xi }^{\mu }=(\alpha ,{\xi }^i)$ changes a raw metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. The displayed first-order time equations set the lapse and shift perturbations to zero. They can be integrated locally for smooth perturbations, demonstrating accessibility of synchronous gauge in cosmology. Their homogeneous solutions leave residual spatial functions, so the gauge is not fully fixed. A globally regular synchronous chart need not survive caustics of its freely falling congruence.