tokens: be needed to fully address this point. 4 DISCUSSION AND CONCLUSIONS WDM with a particle mass above still provides a valid alternative to the standard CDM scenario. In the previous papers of this series (Frings et al. 2017 ; Macciò et al. 2017 ) we have shown that the results of our simulations of field galaxies and satellite galaxies of the Milky Way in CDM are consistent with current observational data. In this work we investigated if the observations constrain the dark matter to be cold or if they allow for a WDM. For that we presented a sample 42 cosmological simulations of 21 haloes that were run both in CDM and WDM. As expected from the damping of the power spectrum the virial masses of the WDM haloes lie well below their CDM counterparts for masses below . Haloes at larger masses are much less affected. Like the CDM galaxies, WDM galaxies fulfil the extrapolated abundance matching relation from Moster et al. ( 2013 ) even with slightly less scatter. The critical mass for haloes to host luminous galaxies is much higher in WDM, where we find several dark haloes up to a halo mass of . Furthermore WDM seems to allow a larger range of masses where one finds both dark and luminous haloes. We found that WDM and CDM galaxies show no different behaviour in the observable velocity dispersion – size and metallicity – stellar mass relation. We have further investigated the average stellar formation times and found a possible way to distinguish the two populations: on average the WDM galaxies form slightly later, with a 〈 T age 〉 = 8.15 ± 0.70 Gyr ago while CDM galaxies have earlier formation times with 〈 T age 〉 = 10.00 ± 0.43. This result confirms the predictions from previous studies on the formation of dwarf galaxies in WDM (Governato et al. 2015 ; Chau et al. 2017 ), even though the current biases and systematic in observationally derived stellar ages (Guidi et al. 2015 ) prevent any firm comparison with observational data. Finally we found an indication that WDM barely reproduces galaxies at masses around with mean stellar ages of , i.e. that formed in the early universe before the UV background sets in. In agreement with previous work we find that the WDM haloes are systematically less concentrated than their CDM counterparts. On the other hand on small scales, about a few per cent of the virial radius, we confirm that the dark matter density slope is mainly fixed by star formation efficiency (e.g. Tollet et al. 2016 ) and not by the nature of dark matter. As a consequence at low stellar masses ( and below), our (few) data points suggest that the lower star formation efficiency in WDM galaxies might cause a slightly steeper central slope at those masses, but a larger sample of simulated galaxies is needed to firmly confirm this effect. We used four galaxies of the WDM sample and their CDM counterparts to simulate their evolution in a Milky Way potential as described above. We witnessed the lower concentrations of the WDM haloes causing the
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