Bound-cloud light echo ($\Delta t_{\mathrm{obs}}=(1+z)(\sqrt{R_{\perp }^2+s^2}-s)/c$)

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.