AlInN alloys offer great potential for photovoltaics thanks to their wide direct bandgap covering the solar spectrum from the infrared (0.7 eV - InN) to the ultraviolet (6.2 eV - AlN), and their superior resistance to high temperatures and high-energy particles. We report the design of AlInN-on-silicon heterojunctions grown by radio-frequency sputtering to explore their potential for low-cost devices. Particularly, we study the influence of AlInN bandgap energy, thickness and carrier concentration, silicon surface recombination, interface defect density and wafer quality, on the photovoltaic properties of the junction. The effect of introducing an anti-reflective coating is also assessed. Optimized AlInN-on-Si structures show a conversion efficiency of 23.6% under 1-sun AM1.5G illumination. In comparison with silicon homojunctions, they own an improved responsivity at wavelengths below 500 nm. These results make AlInN-on-Si heterojunctions a promising technology for solar devices with impact in space applications. Experimental results on novel AlInN-on-Si solar cells are also presented.