In the standard textbook case, a transmitter of diameter $D$ can produce an electromagnetic beam of wavelength $\lambda$ that has spread angle $\theta=1.22\lambda / D$. But what happens in an expanding cosmology, especially one that accelerates so that there is an event horizon? Does $\theta$ increase with distance?
Obviously each photon will travel along a null geodesic and after conformal time $\tau$ have travelled $\chi=c\tau$ units of co-moving distance. The distance between the beam edges would in flat space be growing as $\delta=2c\sin(\theta/2)\tau$. Now, co-moving coordinates are nice and behave well with conformal time, so I would be mildly confident that this distance is true as measured in co-moving coordinates.
But that means that in proper distance the beam diameter is multiplied by the scale factor, $a(t)\delta$ (where $t$ is the time corresponding to $\tau$), and hence $\theta$ increases. However, the distance to the origin in these coordinates has also increased to $a(t)(ct)$, so that seems to cancel the expansion - if we measure $\theta(t)$ globally by dividing the lengths.
But it seems that locally we should see the edges getting separated at an accelerating pace; after all, the local observers will see the emitter accelerating away from them, producing a wider and wider beam near their location since it was emitted further away. From this perspective as time goes by the beam ends up closer and closer to $\theta=\pi$ (and ever more red-shifted, which presumably keeps the total power across it constant).
Does this analysis work, or did I slip on one or more coordinate systems?