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tokens: distribution of central densities in SIDM haloes with cross-sections of this order of magnitude. This signature has not been studied in detail and in this section we present such an analysis based on our cluster-sized simulations. To get a simple measure of the mass assembly history of our halo sample, we trace back in time each halo from z = 0 through successive snapshots, picking the most likely progenitor based on proximity of mass and distance from the selected halo in consecutive snapshots. From this exercise, we define a measure of the ‘formation’ time of a halo as the redshift z f where it acquires a fraction f mass of the mass contained within the scale radius r −2 at z = 0, where r −2 is the radius at which the logarithmic slope of the density profile is −2: 5 This definition gives us a way to explore if the dispersion in the central densities at a fixed V max is related to the time when the mass in the inner regions of haloes was assembled. We note that using equation ( 10 ) for small values of f mass leads to a biased definition of formation time in the SIDM case since once within the region of the SIDM core, the enclosed mass is clearly smaller in the SIDM case than in the CDM case, leading to an artificially larger formation time in the former. In fact, SIDM and CDM haloes assemble at roughly the same time, since self-interactions start to be important later in the evolution of haloes. Equation ( 10 ) is nevertheless sufficient and simple enough for the purpose of this section. We have verified that using the formation time assigned to CDM haloes to the corresponding SIDM haloes leaves our results qualitatively unchanged. Fig. 9 shows the density at a fixed radius (4ε) as a function of V max (a proxy of halo mass) for our halo sample in the different cosmologies. We have chosen r = 4ε ≈ 22 kpc h −1 as the minimum radius we can safely trust in our density profiles. The size of each symbol is proportional to the formation redshift z f (defined with f mass = 1/2), with larger symbols corresponding to haloes formed earlier. It is well known that in CDM, at a fixed radius, more massive haloes (larger V max ) are denser than less massive ones, while at a fixed mass ( V max ), more concentrated haloes form earlier than less concentrated ones. Both of these features are present in Fig. 9 , although the trends are noisier due to the limited mass range we cover and the low number of haloes in our sample. Figure 9. Density at four times the softening (≈22 kpc h −1 ) versus the maximum velocity of the halo. The type of simulation is shown by the colour (blue for CDM, green for SIDM0.1 and red for SIDM1) and the symbol size indicates the square of the formation redshift [where the halo had obtained half its mass within M ( r −2 )], i.e. smaller (larger) symbols indicate haloes formed later (earlier). For CDM and SIDM0.1 we see a mild trend between the central density and the maximum velocity of a halo, which is absent negligible in the case of