tokens: we refer to the subset of haloes that share the same ΛCDM companion across all the simulations as the common sample. For geometrical reasons, we set the limit value for to be 0.5: this represents the case in which two haloes with the same major axis a have centres separated exactly by the same amount a . The configurations that are selected by point ( i ) are the ones for which the distance between the halo centres is less or equal to the smallest major axis between the two. A higher value for would include genuine small haloes that have been more subject to dynamical QP drifting but would also result in a spurious match of bigger haloes. For these reasons, we adopt , checking that the selected sample gains or loses of components if values 0.45 and 0.55 are used, without modifying the overall statistical properties of the sample itself. With respect to at point ( ii ), instead, we applied the matching algorithm using several values, each denoting a specific threshold of the minimal value allowed for the mass ratio of halo couples in order to be considered as a match. As reported in Table 3 , more than of all the matching haloes across LCDM and FDM simulations without mass selection case have a mass ratio in the ratio range and almost in the range. In order not to spoil our matching catalogue, especially with very close but highly different in mass halo couples, we choose the limiting value of . Table 3. Number of common matches across LCDM and FDM simulations, using different values of the parameter representing the minimum allowed ratio between the minimum and maximum masses of each candidate couple. 4 RESULTS In this section we present the results obtained from our simulations in decreasing order of scale involved, starting from the matter power spectrum, to the simulated Lyman α forest observations, to the statistical characterization of halo properties, and their density profiles. 4.1 Matter power spectrum The relative difference of the matter power spectrum of the various FDM models with respect to ΛCDM at four different redshifts is displayed in Fig. 2 . Figure 2. Matter power spectrum of FDM models contrasted with LCDM at different redshifts. As already found in the literature (see e.g. Marsh 2016a ; Nori Baldi 2018 ), the evolution of the matter power spectrum shows that the initial suppression encoded in the transfer functions used to build up the initial conditions is restored at intermediate scales to the unsuppressed level, eventually, by the non-linear gravitational evolution. At the redshifts and scales that are relevant for Lyman α forest observations, however, the relative suppression with respect to ΛCDM is still important and ranges from for the lowest FDM mass considered. The relative difference of the matter power spectrum, displayed in Fig. 3 , shows an additional suppression with respect to ΛCDM when the QP is included in the dynamical evolution in the comparison between the FDM and FDMnoQP simulations, e.g. comparing Figs
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