= Bound-cloud light echo
{title2=$\Delta t_{\rm obs}=(1+z)(\sqrt{R_\perp^2+s^2}-s)/c$}
For a small bound system at cosmological redshift $z$, let $R_\perp$ be the projected source-to-cloud distance and $s$ the cloud displacement toward a distant observer. The extra source–cloud–observer path is $\sqrt{R_\perp^2+s^2}-s$, and cosmological time dilation multiplies its local delay by $1+z$. A projected separation alone does not determine the delay; $s=0$ is an explicitly transverse estimate.
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