We investigate the local nonlinear dynamics of irrotational dust with
vanishing magnetic part of the Weyl tensor, H-ab. Once coded in the in
itial conditions, this dynamical restriction is respected by the relat
ivistic evolution equations. Thus, the outcome of the latter are exact
solutions for special initial conditions with H-ab = 0, but with no s
ymmetries: they describe inhomogeneous triaxial dynamics generalizing
that for a fluid element in a Tolman-Bondi, Kantowski-Sachs, or Szeker
es geometry. A subset of these solutions may be seen as (special) pert
urbations of Friedmann models, in the sense that there are trajectorie
s in phase-space that pass arbitrarily close to the isotropic ones. We
find that the final fate of ever-expanding configurations is a spheri
cal void, locally corresponding to a Milne universe. For collapsing co
nfigurations we find a whole family of triaxial attractors, with vanis
hing local density parameter Omega. These attractors locally correspon
d to Kasner vacuum solutions: only a set of measure zero of physical c
onfigurations collapses to a degenerate pancake, while the generic con
figuration collapses to a triaxial spindle singularity. These silent u
niverse models may provide a fair representation of the universe on su
perhorizon scales. Moreover, one might conjecture that the nonlocal in
formation carried by H-ab becomes negligible during the late highly no
nlinear stages of collapse, so that the attractors we find may be all
those relevant for expanding or collapsing configurations of irrotatio
nal dust.