Ruling out Critical Higgs Inflation?
  We consider critical Higgs inflation, namely Higgs inflation with a rising
inflection point at smaller field values than those of the plateau induced by
the non-minimal coupling to gravity. It has been proposed that such
configuration is compatible with the present CMB observational constraints on
inflation, and also with primordial black hole production accounting for the
totality or a fraction of the observed dark matter. We study the model taking
into account the NNLO corrections to the Higgs effective potential: such
corrections are extremely important to reduce the theoretical error associated
to the calculation. We find that, in the 3 sigma window for the relevant low
energy parameters, which are the strong coupling and the Higgs mass (the top
mass follows by requiring an inflection point), the potential at the inflection
point is so large (and so is the Hubble constant during inflation) that the
present bound on the tensor-to-scalar ratio is violated. The model is viable
only allowing the strong coupling to take its upper 3-4 sigma value. In our
opinion, this tension shows that the model of critical Higgs inflation is
likely to be not viable: neither inflation nor black holes as dark matter can
be originated in this version of the model.
