  The measurement of $\gamma$-rays originating from active galactic nuclei
offers the unique opportunity to study the propagation of very-high-energy
photons over cosmological distances. Most prominently, $\gamma$-rays interact
with the extragalactic background light (EBL) to produce $e^+e^-$ pairs,
imprinting an attenuation signature on $\gamma$-ray spectra. The $e^+e^-$ pairs
can also induce electromagnetic cascades whose detectability in $\gamma$-rays
depends on the intergalactic magnetic field (IGMF). Furthermore, physics beyond
the Standard Model such as Lorentz invariance violation (LIV) or oscillations
between photons and weakly interacting sub-eV particles (WISPs) could affect
the propagation of $\gamma$-rays. The future Cherenkov Telescope Array (CTA),
with its unprecedented $\gamma$-ray source sensitivity, as well as enhanced
energy and spatial resolution at very high energies, is perfectly suited to
study cosmological effects on $\gamma$-ray propagation. Here, we present first
results of a study designed to realistically assess the capabilities of CTA to
probe the EBL, IGMF, LIV, and WISPs.
