  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.
