We also include in ${\rho }_{\mathrm{rad}}$ the contribution of any relativistic matter such as neutrinos. To first approximation we neglect any non-relativistic matter contribution since it is subdominant.

See for example the lectures [36].

To simplify notation, we define the tilded quantities $\tilde{U}=\kappa U/3$, $\tilde{\chi }=\sqrt{\kappa /3}\chi$ and $\kappa {\rho }_{\mathrm{rad}}^E/3={\tilde{\rho }}_{\mathrm{rad}}^E/a^4$. The subscripts “0” and “${\tau }_{\min }$” indicate values calculated at the local maximum and global minimum of the scale factor, respectively.

Another possibility is that the potential in the Euclidean region has both a local minimum and maximum as studied in [6], so that the Euclidean motion in the inverted potential cannot exhibit overshooting.

If the Higgs mass, the SU(3) gauge coupling, and other measured SM parameters are fixed at their central values, there is always a critical top quark mass for which the electroweak and Planck-scale vacua become degenerate (see, e.g., [43]). Below this threshold, the Higgs effective potential remains stable, while for values above it, becomes metastable. Experimental data place us in the metastable region [10], supporting our proposal. While other parameters ($m_h,g_3,M_Z$, etc.) affect stability, a full analysis of the parameter space is beyond the scope of this Letter.

In [45], an alternative scenario was considered in which the true (negative) minimum lies between the electroweak vacuum and the plateau region and hence affects the Lorentzian time evolution. This scenario is not very satisfactory, since unless one invokes certain thermal effects that completely lift this minimum, inflation will end there and the resulting cosmology will be in stark contrast with observations.

In this case, one finds that the opposite is harder to achieve—that is to uplift a part of the effective scalar potential to positive values in a controllable way, see [49] for a review.