Characterise the large-scale structure in the Universe from present times to the high redshift epoch of reionisation is essential to constraining the cosmology, the history of star formation and reionisation, measuring the gas content of the Universe and obtaining a better understanding of the physical process that drive galaxy formation and evolution. Using the integrated emission from unresolved galaxies or gas clouds, line intensity mapping (LIM) provides a new observational window to measure the larger properties of structure. This very promising technique motivates the community to plan for LIM experiments. We describe the development of a large field-of-view instrument, named CONCERTO, operating in the range 130-310 GHz from the APEX 12-meters telescope. CONCERTO is a low-resolution spectrometer based on the Lumped Element Kinetic Inductance Detectors technology. Spectra are obtained using a fast Fourier Transform Spectrometer (FTS), coupled to a dilution cryostat with base temperature of 0.1K. Two 2 kilo-pixels arrays of LEKID are mounted inside the cryostat that also contains the cold optics and the front-end electronics. We present in detail the technological choices leading to the instrumental concept, together with the design and fabrication of the instrument and preliminary laboratory tests on the detectors. We also give our best estimates of CONCERTO sensitivity and give predictions for two of the main scientific goals of CONCERTO, i.e. a [CII]-intensity mapping survey and observations of galaxy clusters. We provide a detail description of the instrument design. Based on realistic comparisons with existing instruments developed by our group (NIKA, NIKA2, and KISS), and on laboratory detectors characterisation, we provide an estimate of CONCERTO sensitivity on sky. Finally, we describe in detail two out of the main science goals offered by CONCERTO at APEX.
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