Probable or Improbable Universe? Correlating Electroweak Vacuum Instability with the Scale of Inflation

TL;DR

The paper analyzes electroweak vacuum stability during inflation given the SM Higgs potential instability at $\Lambda_I \sim 10^{11}$ GeV and a high inflationary Hubble scale $H$. It combines Coleman-de Luccia, Hawking-Moss, and Fokker-Planck treatments to track the Higgs distribution across $e^{3N_e}$ Hubble volumes, showing that the late-time fate of AdS patches—whether they crunch or dominate—crucially determines the viability of a universe like ours. With Planck-suppressed corrections, a positive effective mass term can stabilize the potential during inflation, broadening the parameter space for universe survival. The work highlights that, under plausible inflationary scales, either a modest amount of extra inflation or new stabilization physics may be required to ensure our electroweak vacuum persists, depending on the detailed post-inflation dynamics of unstable regions.

Abstract

Measurements of the Higgs boson and top quark masses indicate that the Standard Model Higgs potential becomes unstable around $Λ_I \sim 10^{11}$ GeV. This instability is cosmologically relevant since quantum fluctuations during inflation can easily destabilize the electroweak vacuum if the Hubble parameter during inflation is larger than $Λ_I$ (as preferred by the recent BICEP2 measurement). We perform a careful study of the evolution of the Higgs field during inflation, obtaining different results from those currently in the literature. We consider both tunneling via a Coleman-de Luccia or Hawking-Moss instanton, valid when the scale of inflation is below the instability scale, as well as a statistical treatment via the Fokker-Planck equation appropriate in the opposite regime. We show that a better understanding of the post-inflation evolution of the unstable AdS vacuum regions is crucial for determining the eventual fate of the universe. If these AdS regions devour all of space, a universe like ours is indeed extremely unlikely without new physics to stabilize the Higgs potential; however, if these regions crunch, our universe survives, but inflation must last a few e-folds longer to compensate for the lost AdS regions. Lastly, we examine the effects of generic Planck-suppressed corrections to the Higgs potential, which can be sufficient to stabilize the electroweak vacuum during inflation.

Figures (7)

• Figure 1: The Higgs potential illustrated with the regimes of validity for various solutions for the Higgs vacuum evolution during inflation: Coleman-de Luccia (CdL), Hawking-Moss (HM) and Fokker-Planck (FP). Left: For $H\!\mathrel{\hbox{$\sim$ $<$}}{\Lambda_{\rm max}}$, the CdL tunneling or single bounce HM instanton yields the transition probability. Right: For $H\gg{\Lambda_{\rm max}}$, the potential barrier at ${\Lambda_{\rm max}}$ is irrelevant, and a stochastic random walk approach is necessary until classical slow roll takes over at $h = \Lambda_c$ ($H=10\, {\Lambda_{\rm max}}$ has been chosen). The dashed curve in the right-hand panel shows the effect of Planck-suppressed stabilizing terms on the potential (in this case $\Delta V = 0.2 H^2 h^2$), which we study in Sec. \ref{['sec:corrections']}. To illustrate the relative scale between the two panels, the dashed lines show the region where the left panel fits into the right panel.
• Figure 2: Left: $\lambda_{\rm eff}(h)$ within the Standard Model for $m_h = 125.7 \text{ GeV}, m_t = 173.34 \text{ GeV}$. Right: Contours of $\Lambda_{\rm max}$ (black, dashed) in the $(m_h, m_t)$ plane. Also shown are ellipses corresponding to the 68.27%, 95.45% and 99.73% confidence level regions for two parameters. The measured values for the masses are taken to be $m_h = 125.7 \pm 0.4 \text{ GeV}$ and $m_t = 173.34 \pm 0.76 \text{ GeV}$. For the central values, $\Lambda_{\rm max} = 4.9 \times 10^{10} \text{ GeV}$.
• Figure 3: Comparison of survival probability $P_\Lambda$ (Eq. (\ref{['eq:plambdadef']})) with $N_e = 60$ as given by numerically solving the FP equation (red crosses), the approximate analytic solution in Eq. (\ref{['survival2']}) (solid black curve), and the solution with $\Lambda_c=\Lambda_{\rm max}$ from Espinosa:2007qp (solid gray curve). For comparison, we also show the HM probability as explored in Kobakhidze:2013tn (with unit prefactor, see Eq. (\ref{['eq:HMform']})). $V_{\rm eff}(h)$ in our numerical solution is computed using the central values $m_h = 125.7 \text{ GeV}$, $m_t = 173.34 \text{ GeV}$.
• Figure 4: Left: Total volume of the surviving region (normalized to $e^{180}$ Hubble volumes) after $N_e$ e-folds of inflation for different values of $H/\Lambda_{\rm max}$. Right: Additional number of e-folds of inflation needed in order to obtain $e^{N_o}$ Hubble volumes of space left over after the AdS regions crunch. $\Lambda_c$ is determined by solving Eq. (\ref{['eq:lambdaceqn']}) with the Higgs potential for $m_h = 125.7 \text{ GeV}$, $m_t = 173.34 \text{ GeV}$. The gray, dotted line corresponds to $H_{\rm BICEP2} = 10^{14}$ GeV.
• Figure 5: $B_{\rm HM}$ and $P_{\rm no AdS}$ as a function of $H/\Lambda_{\rm max}$ for the central values $m_h = 125.7 \text{ GeV}$, $m_t = 173.34 \text{ GeV}$ (left) and as a function of $m_t$ for $m_h = 125.7 \text{ GeV}$ and $H_{\rm BICEP2} \approx 10^{14} \text{ GeV}$ (right). The survival of our universe either requires $H/\Lambda_{\rm max} \!\mathrel{\hbox{$\sim$ $<$}} 0.065$ or, if the BICEP2 result holds, $m_t \!\mathrel{\hbox{$\sim$ $<$}} 171.47 \text{ GeV}$, $\sim 2.5 \sigma$ below the central value.