The Inflation-Higgs connection...

... made quantitative (at the state of the art?)
Here is a recent preprint that claims to derive a rigorous correlation between a future measurement of the tensor-to-scalar ratio and the scale at which the Higgs potential must receive stabilizing corrections in order for the Universe to have survived inflation until today:

A striking feature of the Standard Model (SM) is that, in the absence of stabilizing corrections, the Higgs potential develops an instability, with the maximum of the potential occurring at V(Λmax)1/41010 GeV. This leads to the existence of a “true vacuum” at large Higgs field values, which may carry important consequences for our Universe [1–9]. Our present existence does not necessarily demand physics beyond the SM, since current measurements of the Higgs boson and top quark masses indicate that the electroweak (EW) vacuum is metastable, i.e., long-lived relative to the age of the Universe. The scenario is different, however, if our Universe underwent an early period of cosmic inflation with substantial energy density. The inflaton energy density, parametrized by the Hubble parameter H, produces large local fluctuations in the Higgs field, δh ∼ H/2π . As such, when H is sufficiently large during inflation, the Higgs field may sample the unstable part of the potential. If sampling this part of the potential can be shown to be catastrophic for the surrounding spacetime, the eventual survival of our Universe in the EW vacuum would consequently imply constraints on the nature of the inflationary epoch that gave rise to our Universe. Conversely, near-future Cosmic Microwave Background (CMB) experiments will probe tensor-to-scalar ratios of r>0.002 [10], corresponding to inflationary scales H > 1013 GeV. If it can be shown that the SM Higgs potential is inconsistent with such high-scale inflation, a measurement of non-zero r provides evidence for the existence of stabilizing corrections to the Higgs potential... 
The main goal of this paper is a definitive study of [two important aspects: first, the evolution of the Higgs field under a combination of (inflation-induced) quantum fluctuations and the classical potential and, secondly, the evolution of spacetime responding to the Higgs vacuum].
Limits on the inflaton energy density, parametrized by the Hubble parameter H [GeV] (black contours) in the (mhmt) plane requiring Nmax ≥ 60 (solid) or 50 (dashed). Central values are taken to be mh= 125.09 ± 0.24 GeV [40] and mt= 172.44 ± 0.70 GeV [41], with contours corresponding to 1-, 2-, and 3-σ regions as for two parameters. The shaded regions represent: the Higgs potential is stable up to MP (green); the Higgs potential is unstable, but current limits r<0.07 [45] permit required amount of inflation (blue); and instability would preclude the combination of Nmax > 60 and r > 0.002, to be probed by near-future experiments [10] (red).

We have studied the dynamical response of inflating spacetime to unstable fluctuations in the Higgs field with numerical simulations of Einstein gravity. Our results offer, for the first time, an in-depth understanding of how spacetime evolves as a Higgs fluctuation falls towards, and eventually reaches, the true, negative energy, vacuum. We find that when true vacuum patches stop inflating and create a crunching region, the energy liberated creates a black hole surrounded by a shell of negative energy density. This region of true vacuum persists and grows throughout inflation, with more and more energy being locked behind the black hole horizon. In contrast to the naïve expectation that this growth is due to the boundary between true and metastable vacua sweeping outward in space, in an exponentially expanding spacetime the growth occurs in a causally-disconnected manner. Spatial points fall to the true vacuum independent of the fact that neighboring points have also reached the true vacuum. Hence, under most circumstances, this process is insensitive to the behavior in the interior region, and to the exact shape of the potential close to the true minimum. 
We also extended the numerical solution of the Fokker-Planck equation to resolve the field distribution in the exponentially suppressed tails. This is necessary to extract the tiny probabilities associated with a single true vacuum patch in our past light-cone, while simultaneously incorporating the effects from renormalization group running of the quartic in the Higgs potential on the evolution of the probability distribution. Using this solution, in conjunction with the result from our classical General Relativity simulations that a single true vacuum patch in our past light-cone destroys the Universe, we derived a bound H/Λmax<0.07 on the scale of inflation. This bound is the most accurate available to date, and we compared it to bounds derived previously. We also found, as shown in Fig. 8, that a future measurement of the tensor to scalar ratio with r > 0.002 would imply the need for a stabilizing correction to the Higgs potential at a scale < 1014 GeV supposing mt> 171.4 GeV. We are thus able to correlate a cosmological quantity with the necessity of stabilizing corrections to the Higgs potential 
Finally, we re-emphasize that the results in this paper are of wider interest than the SM Higgs potential, as they are applicable to the inflationary dynamics of any scalar field with a negative energy true vacuum.
(Submitted on 1 Jul 2016)