  The instability with respect to global glaciation is a fundamental property
of the climate system caused by the positive ice-albedo feedback. The
atmospheric carbon dioxide concentration at which this Snowball bifurcation
occurs changes through Earth's history because of the slowly increasing solar
luminosity. Quantifying this critical CO$_2$ level is not only interesting from
a climate dynamics perspective, but also a prerequisite for understanding past
Snowball Earth events as well as the conditions for habitability on Earth and
other planets. Earlier studies are limited to investigations with simple
climate models for Earth's entire history, or studies of individual time slices
carried out with a variety of more complex models and for different boundary
conditions, making comparisons and the identification of secular changes
difficult. Here we use a coupled climate model of intermediate complexity to
trace the Snowball bifurcation of an aquaplanet through Earth's history in one
consistent model framework. We find that the critical CO$_2$ concentration
decreases more or less logarithmically with increasing solar luminosity until
about 1 billion years ago, but drops faster in more recent times. Furthermore,
there is a fundamental shift in the dynamics of critical states about 1.2
billion years ago, driven by the interplay of wind-driven sea-ice dynamics and
the surface energy balance: For critical states at low solar luminosities, the
ice line lies in the Ferrel cell, stabilised by the poleward winds despite
moderate meridional temperature gradients under strong greenhouse warming. For
critical states at high solar luminosities on the other hand, the ice line
rests at the Hadley-cell boundary, stabilised against the equatorward winds by
steep meridional temperature gradients resulting from the increased solar
energy input at lower latitudes and stronger Ekman transport in the ocean.
