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To distinguish between cosmological models, it is essential to generate realistic mock observables from cosmological simulations that were run in both the standard \u039b-cold-dark-matter (\u039bCDM) paradigm and in many non-standard models beyond \u039bCDM. We present the scientific results from a suite of cosmological <jats:italic>N<\/jats:italic>-body simulations using non-standard models including dynamical dark energy, <jats:italic>k<\/jats:italic>-essence, interacting dark energy, modified gravity, massive neutrinos, and primordial non-Gaussianities. We investigate how these models affect the large-scale-structure formation and evolution in addition to providing synthetic observables that can be used to test and constrain these models with <jats:italic>Euclid<\/jats:italic> data. We developed a custom pipeline based on the <jats:monospace>Rockstar<\/jats:monospace> halo finder and the <jats:monospace>nbodykit<\/jats:monospace> large-scale structure toolkit to analyse the particle output of non-standard simulations and generate mock observables such as halo and void catalogues, mass density fields, and power spectra in a consistent way. We compare these observables with those from the standard \u039bCDM model and quantify the deviations. We find that non-standard cosmological models can leave large imprints on the synthetic observables that we have generated. Our results demonstrate that non-standard cosmological <jats:italic>N<\/jats:italic>-body simulations provide valuable insights into the physics of dark energy and dark matter, which is essential to maximising the scientific return of <jats:italic>Euclid<\/jats:italic>.<\/jats:p>","DOI":"10.1051\/0004-6361\/202452185","type":"journal-article","created":{"date-parts":[[2024,12,24]],"date-time":"2024-12-24T19:43:22Z","timestamp":1735069402000},"page":"A232","source":"Crossref","is-referenced-by-count":2,"title":["<i>Euclid<\/i> preparation"],"prefix":"10.1051","volume":"695","author":[{"name":"Euclid Collaboration","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3906-5699","authenticated-orcid":false,"given":"G.","family":"R\u00e1cz","sequence":"additional","affiliation":[]},{"given":"M.-A.","family":"Breton","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0092-4321","authenticated-orcid":false,"given":"B.","family":"Fiorini","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0936-4594","authenticated-orcid":false,"given":"A. 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