Metastable electroweak vacuum. Implications for inflation

Within the Standard Model, the current Higgs and top quark data favor metastability of the electroweak vacuum, although the uncertainties are still significant. The true vacuum is many orders of magnitude deeper than ours and the barrier separating the two is tiny compared to the depth of the well. This raises a cosmological question: how did the Higgs field get trapped in the shallow minimum and why did it stay there during inflation? The Higgs initial conditions before inflation must be fine-tuned to about one part in 108 in order for the Higgs field to end up in the right vacuum. In this note, we show that these problems can be resolved if there is a small positive coupling between the Higgs and the inflaton.