Is a Higgs vacuum instability fatal for high-scale inflation?

We study the inflationary evolution of a scalar field $h$ with an unstable potential for the case where the Hubble parameter $H$ during inflation is larger than the instability scale ${\mathrm{\Lambda }}_I$ of the potential. Quantum fluctuations in the field of size $\delta h\sim \frac{H}{2\pi }$ imply that the unstable part of the potential is sampled during inflation. We investigate the evolution of these fluctuations to the unstable regime and in particular whether they generate cosmological defects or even terminate inflation. We apply the results of a toy scalar model to the case of the Standard Model Higgs boson, the quartic of which evolves to negative values at high scales, and extend previous analyses of Higgs dynamics during inflation utilizing statistical methods to a perturbative and fully gauge-invariant formulation. We show that the dynamics are controlled by the renormalization group-improved quartic coupling $\lambda (\mu )$ evaluated at a scale $\mu =H$, such that Higgs fluctuations are enhanced by the instability if $H>{\mathrm{\Lambda }}_I$. Even if $H>{\mathrm{\Lambda }}_I$, the instability in the Standard Model Higgs potential does not end inflation; instead the universe slowly sloughs off crunching patches of space that never come to dominate the evolution. As inflation proceeds past 50 $e$-folds, a significant proportion of patches exits inflation in the unstable vacuum, and as much as 1% of the spacetime can rapidly evolve to a defect. Depending on the nature of these defects, however, the resulting universe could still be compatible with ours.

Figure 1

Sample Feynman diagrams included in our calculation of the Higgs two-point correlation function. The first two graphs (labeled “Leading IR logs”) contribute to the late-time divergence of the Higgs two-point correlation $⟨\delta h^2(t)⟩$. The last three graphs do not directly contribute to the leading divergence but serve to renormalize the Higgs self-coupling $\lambda$. The points $x,y$ are assumed to be separated by less than one Hubble length during inflation, and the gauge boson propagators, for reasons we explain in the text, include only the transverse degrees of freedom.

Figure 2

Probability distribution of the Higgs, $P(\delta h,t)$, evaluated at $\mathcal{N}=25$ for the case of $\lambda (H)=-0.01$ (blue, solid line). Also shown is a Gaussian distribution, corresponding to the Hartree–Fock approximation of Sec. 2, with variance $⟨\delta h^2(t)⟩$ given by Eq. (13) (black, dashed line).

Figure 3

The proportion of surviving patches transitioning out of the slow-roll regime $f_{\mathcal{N}}$ [Eq. (43)] at $\mathcal{N}$ $e$-folds of inflation for two choices of quartic coupling $\lambda (H)=-0.005$ (red) and $-0.01$ (blue).