  It is well established from cosmological simulations that dark matter haloes
are not precisely self-similar and an additional parameter, beyond their
concentration, is required to accurately describe their spherically-averaged
mass density profiles. We present, for the first time, a model to consistently
predict both halo concentration, $c$, and this additional `shape' parameter,
$\alpha$, for a halo of given mass and redshift for a specified cosmology.
Following recent studies, we recast the dependency on mass, redshift, and
cosmology to a dependence on `peak height'. We show that, when adopting the
standard definition of peak height, which employs the so-called spherical top
hat (STH) window function, the concentration--peak height relation has a strong
residual dependence on cosmology (i.e., it is not uniquely determined by peak
height), whereas the $\alpha$--peak height relation is approximately universal
when employing the STH window function. Given the freedom in the choice of
window function, we explore a simple modification of the STH function,
constraining its form so that it produces universal relations for concentration
and $\alpha$ as a function of peak height using a large suite of cosmological
simulations. It is found that universal relations for the two density profile
parameters can indeed be derived and that these parameters are set by the
linear power spectrum, $P(k)$, filtered on different scales. We show that the
results of this work generalise to any (reasonable) combination of $P(k)$ and
background expansion history, $H(z)$, resulting in accurate predictions of the
density profiles of dark matter haloes for a wide range of cosmologies.
