  We derive the density profile for collisionless dissipationless dark matter
haloes in hierarchical cosmologies making use of the Secondary Infall (SI)
model. The novelties are: i) we deal with triaxial virialised objects; ii)
their seeds in the linear regime are peaks endowed with {\it unconvolved}
spherically averaged density profiles according to the peak formalism; iii) the
initial peculiar velocities are taken into account; and iv) accreting haloes
are assumed to develop from the inside out, keeping the instantaneous inner
system unaltered. The validity of this latter assumption is accurately checked
by comparing analytical predictions on such a growth with the results of
numerical simulation. We show that the spherically averaged density profile of
virialised objects can be inferred with no need to specify their shape. The
{\it typical} spherically averaged halo density profile is inferred, down to
arbitrarily small radii, from the power-spectrum of density perturbations. The
predicted profile in the $\Lambda$CDM cosmology is approximately described by
an Einasto profile, meaning that it does not have a cusp but rather a core,
where the inner slope slowly converges to zero. Down to one hundredth the total
radius, the profile has the right NFW and Einasto forms, being close to the
latter down to a radius of about four orders of magnitude less. The inner
consistency of the model implies that the density profiles of haloes harbour no
information on their past aggregation history. This would explain why major
mergers do not alter the typical density profile of virialised objects formed
by SI and do not invalidate the peak formalism based on such a formation.
