The Cherenkov Telescope Array (CTA) is a project for a next-generation observatory for very high energy (GeV-TeV) ground-based gamma-ray astronomy, currently in its design phase, and foreseen to be operative a few years from now. Several tens of telescopes of 2-3 different sizes, distributed over a large area, will allow for a sensitivity about a factor 10 better than current instruments such as H.E.S.S, MAGIC and VERITAS, an energy coverage from a few tens of GeV to several tens of TeV, and a field of view of up to 10 deg. In the following study, we investigate the prospects for CTA to study several science questions that influence our current knowledge of fundamental physics. Based on conservative assumptions for the performance of the different CTA telescope configurations, we employ a Monte Carlo based approach to evaluate the prospects for detection. First, we discuss CTA prospects for cold dark matter searches, following different observational strategies: in dwarf satellite galaxies of the Milky Way, in the region close to the Galactic Centre, and in clusters of galaxies. The possible search for spatial signatures, facilitated by the larger field of view of CTA, is also discussed. Next we consider searches for axion-like particles which, besides being possible candidates for dark matter may also explain the unexpectedly low absorption by extragalactic background light of gamma rays from very distant blazars. Simulated light-curves of flaring sources are also used to determine the sensitivity to violations of Lorentz Invariance by detection of the possible delay between the arrival times of photons at different energies. Finally, we mention searches for other exotic physics with CTA.
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