A large body of work based on collisionless cosmological N-body simulations going back over two decades has advanced the idea that collapsed dark matter haloes have simple and approximately universal forms for their mass density and pseudo-phase space density (PPSD) distributions. However, a general consensus on the physical origin of these results has not yet been reached. In the present study, we explore to what extent the apparent universality of these forms holds when we vary the initial conditions (i.e., the primordial power spectrum of density fluctuations) away from the standard CMB-normalised case, but still within the context of LCDM with a fixed expansion history. Using simulations that vary the initial amplitude and shape, we show that the structure of dark matter haloes retains a clear memory of the initial conditions. Specifically, increasing (lowering) the amplitude of fluctuations increases (decreases) the concentration of haloes and, if pushed far enough, the density profiles deviate strongly from the NFW form that is a good approximation for the CMB-normalised case. Although, an Einasto form works well. Rather than being universal, the slope of the PPSD (or pseudo-entropy) profile steepens (flattens) with increasing (decreasing) power spectrum amplitude and can exhibit a strong halo mass dependence. Our results therefore indicate that the previously identified universality of the structure of dark matter haloes is mostly a consequence of adopting a narrow range of (CMB-normalised) initial conditions for the simulations. Our new suite provides a useful test-bench against which physical models for the origin of halo structure can be validated.
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