Abstract: We discuss the potential of a next generation space-borne CMB experiment for studies of extragalactic sources with reference to COrE+, a project submitted to ESA in response to the call for a Medium-size mission (M4). We consider three possible options for the telescope size: 1 m, 1.5 m and 2 m (although the last option is probably impractical, given the M4 boundary conditions). The proposed instrument will be far more sensitive than Planck and will have a diffraction-limited angular resolution. These properties imply that even the 1 m telescope option will perform substantially better than Planck for studies of extragalactic sources. The source detection limits as a function of frequency have been estimated by means of realistic simulations taking into account all the relevant foregrounds. Predictions for the various classes of extragalactic sources are based on up-to-date models. The most significant improvements over Planck results are presented for each option. COrE+ will provide much larger samples of truly local star-forming galaxies (by about a factor of 8 for the 1 m telescope, of 17 for 1.5 m, of 30 for 2 m), making possible analyses of the properties of galaxies (luminosity functions, dust mass functions, star formation rate functions, dust temperature distributions, etc.) across the Hubble sequence. Even more interestingly, COrE+ will detect, at |b| > 30°, thousands of strongly gravitationally lensed galaxies (about 2,000, 6,000 and 13,000 for the 1 m, 1.5 m and 2 m options, respectively). Such large samples are of extraordinary astrophysical and cosmological value in many fields. Moreover, COrE+ high frequency maps will be optimally suited to pick up proto-clusters of dusty galaxies, i.e. to investigate the evolution of large scale structure at larger redshifts than can be reached by other means. Thanks to its high sensitivity COrE+ will also yield a spectacular advance in the blind detection of extragalactic sources in polarization: we expect that it will detect up to a factor of 40 (1 m option) or of 160 (1.5 m option) more radio sources than can be detected by Planck and, for the first time, from several tens (1 m option) to a few hundreds (1.5 m option) of star forming galaxies. This will open a new window for studies of the global properties of magnetic fields in star forming galaxies and of their relationships with star formation rates.

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