$\mathcal{E}{_{\text{in}}}$, $\boldsymbol{\mathcal{W}}_{\rm K}(v_{\rm in};v{_{\text{out}}})$. The same operations can then be repeated starting from $\mathcal{E}{_{\text{in}}}$ and so on.

Friedmann-Lemaître region [*(from ${\cal E}_0$ to ${\cal E}_{\rm out}$)*]{} {#secW-FL}
---------------------------------------------------------------------------

The geometry of the Friedmann-Lemaître region is given by the metric (\[eqext\]) which can be rewritten in terms of the conformal time $\eta$, defined by ${{\rm d}}\eta ={{\rm d}}T/a(T)$, as $$\label{e-fleq}
{{\rm d}}s^2 = a^2(\eta)\left[-{{\rm d}}\eta^2 + {{\rm d}}\chi^2 + f_K^2(\chi)\,{{\rm d}}\Omega^2\right].$$

### Geodesic equation

If one chooses the center $\chi=0$ of the FL spherical coordinate system on the worldline of the (comoving) observer, then the geodesic equation is easily solved as $$\label{e.FLradial}
 \chi(\eta) = \eta_0 - \eta, \quad
 \theta=\theta_0,\quad
 \varphi=\varphi_0,$$ which corresponds to a purely radial trajectory. Note however that for a generic origin, this is no longer true. The associated wave vector remains collinear to the observed one, $k^\mu_0$. It is only subject to a redshift induced by the cosmic expansion, so that $$k^\mu = \left(\frac{a_0}{a}\right)^2 k^\mu_0 .
\label{e.wavevectorFL}$$ We stress that, in Eq. , $\mu=0$ refers to components on $\partial_\eta$, not on $\partial_T=\partial_\eta/a$.

### Intersection with the hole

Once the geodesic equation has been solved and the position of the hole has been chosen, we can calculate the intersection 