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Approximately 20 million of these galaxies will have their spectroscopy available, allowing us to map the three-dimensional large-scale structure of the Universe in great detail. This paper investigates prospects for the detection of cosmic voids therein and the unique benefit they provide for cosmological studies. In particular, we study the imprints of dynamic (redshift-space) and geometric (Alcock\u2013Paczynski) distortions of average void shapes and their constraining power on the growth of structure and cosmological distance ratios. To this end, we made use of the Flagship mock catalog, a state-of-the-art simulation of the data expected to be observed with <jats:italic>Euclid<\/jats:italic>. We arranged the data into four adjacent redshift bins, each of which contains about 11\u2006000 voids and we estimated the stacked void-galaxy cross-correlation function in every bin. Fitting a linear-theory model to the data, we obtained constraints on <jats:italic>f<\/jats:italic>\/<jats:italic>b<\/jats:italic> and <jats:italic>D<\/jats:italic><jats:sub>M<\/jats:sub><jats:italic>H<\/jats:italic>, where <jats:italic>f<\/jats:italic> is the linear growth rate of density fluctuations, <jats:italic>b<\/jats:italic> the galaxy bias, <jats:italic>D<\/jats:italic><jats:sub>M<\/jats:sub> the comoving angular diameter distance, and <jats:italic>H<\/jats:italic> the Hubble rate. In addition, we marginalized over two nuisance parameters included in our model to account for unknown systematic effects in the analysis. With this approach, <jats:italic>Euclid<\/jats:italic> will be able to reach a relative precision of about 4% on measurements of <jats:italic>f<\/jats:italic>\/<jats:italic>b<\/jats:italic> and 0.5% on <jats:italic>D<\/jats:italic><jats:sub>M<\/jats:sub><jats:italic>H<\/jats:italic> in each redshift bin. Better modeling or calibration of the nuisance parameters may further increase this precision to 1% and 0.4%, respectively. Our results show that the exploitation of cosmic voids in <jats:italic>Euclid<\/jats:italic> will provide competitive constraints on cosmology even as a stand-alone probe. For example, the equation-of-state parameter, <jats:italic>w<\/jats:italic>, for dark energy will be measured with a precision of about 10%, consistent with previous more approximate forecasts.<\/jats:p>","DOI":"10.1051\/0004-6361\/202142073","type":"journal-article","created":{"date-parts":[[2021,11,10]],"date-time":"2021-11-10T19:59:07Z","timestamp":1636574347000},"page":"A20","source":"Crossref","is-referenced-by-count":60,"title":["<i>Euclid<\/i>: Forecasts from redshift-space distortions and the Alcock\u2013Paczynski test with cosmic voids"],"prefix":"10.1051","volume":"658","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0876-2101","authenticated-orcid":false,"given":"N.","family":"Hamaus","sequence":"first","affiliation":[]},{"given":"M.","family":"Aubert","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6146-4437","authenticated-orcid":false,"given":"A.","family":"Pisani","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9843-723X","authenticated-orcid":false,"given":"S.","family":"Contarini","sequence":"additional","affiliation":[]},{"given":"G.","family":"Verza","sequence":"additional","affiliation":[]},{"given":"M.-C.","family":"Cousinou","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2847-7498","authenticated-orcid":false,"given":"S.","family":"Escoffier","sequence":"additional","affiliation":[]},{"given":"A.","family":"Hawken","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0143-8891","authenticated-orcid":false,"given":"G.","family":"Lavaux","sequence":"additional","affiliation":[]},{"given":"G.","family":"Pollina","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5854-8269","authenticated-orcid":false,"given":"B. 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