  It has been pointed out that a large non-minimal coupling $\xi$ between the
Higgs and the Ricci scalar can source higher derivative operators, which may
change the predictions of Higgs inflation. A variant, called critical Higgs
inflation, employs the near-criticality of the top mass to introduce an
inflection point in the potential and lower drastically the value of $\xi$. We
here study whether critical Higgs inflation can occur even if the
pre-inflationary initial conditions do not satisfy the slow-roll behaviour
(retaining translation and rotation symmetries). A positive answer is found:
inflation turns out to be an attractor and therefore no fine-tuning of the
initial conditions is necessary. A very large initial Higgs time-derivative (as
compared to the potential energy density) is compensated by a moderate increase
in the initial field value. These conclusions are reached by solving the exact
Higgs equation without using the slow-roll approximation. This also allows us
to treat consistently the inflection point, where the standard slow-roll
approximation breaks down. Here we make use of an approach that is independent
of the UV completion of gravity, by taking initial conditions that always
involve sub-planckian energies.
