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.
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