Inflation from the Higgs field false vacuum with hybrid potential

TL;DR

The paper argues that inflation can originate from the Standard Model Higgs false vacuum at high field values if a second, very weakly coupled scalar $ extPhi$ provides a hybrid-like mechanism to erase the barrier and end inflation. It formulates a two-field potential $V( ilde extchi, ilde extPhi)$ with an inverted hybrid structure, derives slow-roll dynamics for $ ilde extPhi$, and connects the tensor-to-scalar ratio $r$ to the Higgs sector via $r=16oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}}}}}}$, concluding that viable regions require $m_H<125.3 ext{ GeV}$ (with $3_{th}$ RG uncertainty) and $10^{-4}\lesssim r<0.007$ (or $r<0.001$ under subplanckian excursions). The model predicts a compatible $n_S$ and yields a narrow, testable relation among $m_H$, $m_t$, and $r$, while ensuring a standard post-inflationary radiation era after barrier erasure and respecting nucleosynthesis constraints.

Abstract

We have recently suggested [1,2] that Inflation could have started in a local minimum of the Higgs potential at field values of about $10^{15}-10^{17}$ GeV, which exists for a narrow band of values of the top quark and Higgs masses and thus gives rise to a prediction on the Higgs mass to be in the range 123-129 GeV, together with a prediction on the the top mass and the cosmological tensor-to-scalar ratio $r$. Inflation can be achieved provided there is an additional degree of freedom which allows the transition to a radiation era. In [1] we had proposed such field to be a Brans-Dicke scalar. Here we present an alternative possibility with an additional subdominant scalar very weakly coupled to the Higgs, realizing an (inverted) hybrid Inflation scenario. Interestingly, we show that such model has an additional constraint $m_H<125.3 \pm 3_{th}$, where $3_{th}$ is the present theoretical uncertainty on the Standard Model RGEs. The tensor-to-scalar ratio has to be within the narrow range $10^{-4}\lesssim r<0.007$, and values of the scalar spectral index compatible with the observed range can be obtained. Moreover, if we impose the model to have subplanckian field excursion, this selects a narrower range $10^{-4} \lesssim r<0.001$ and an upper bound on the Higgs mass of about $m_H <124 \pm 3_{th}$.

Figures (3)

• Figure 1: Slow roll parameter $\epsilon$ as a function of time $t$ for $m_H= 124.323915$ GeV, $m_t=171.4$ GeV, ${\bar{v}}_\Phi=4$ and different values of $\sigma$, as indicated by the labels. Here $\bar{v}_{\Phi}$ is expressed in units of $M\approx 2.43 \times 10^{18}$ GeV.
• Figure 2: We show here the allowed parameter space for $\sigma, \bar{v}_{\Phi}$, and the corresponding values of the spectral index $n_S$, for which we display contours ranging from $0.90$ up to $1$ with steps of $0.02$. Here $\bar{v}_{\Phi}$ is expressed in units of $M\approx 2.43 \times 10^{18}$ GeV. The shaded regions (superimposed to the regions where $n_S$ is slightly smaller than one) are such that there is also another solution with $1\le n_S \le 1.02$ (yellow) and $1.02\le n_S \le 1.04$ (orange). Each plot has a prediction for $r$ and corresponds to the value shown for $m_H$ and $m_t$, up to the theoretical uncertainties of the RGE of about 1 GeV for $m_t$ and $3$ GeV for $m_H$. The regions outside the solid (red) curves where the number of e-folds turns out to be smaller than $\bar{N}$ are excluded. The region at the right of the dashed (green) curve is also excluded since there the contribution to potential $V(\Phi,\chi_0)$ is not dominated by $V_{\rm Higgs}(\chi_0)$; for the sake of definiteness we have drawn the dashed line where $\sigma \bar{v}^4_{\Phi}/{24} \approx V_{\rm Higgs}(\chi_0) /3$. Note also that the values of $\bar{N}$ in the figure have been calculate assuming that the field $\Phi$ decays rapidly and never dominates after Inflation, as discussed in section \ref{['classical']}; other scenarios of decay would shift such number by a few, depending on the energy density stored in $\Phi$ at the final value $\Phi_0$, so the shift would be minimal for small $\bar{v}_{\Phi}$ and close to the solid (red) curve, where $\Phi_0$ is close to $\bar{v}_{\Phi}$.
• Figure 3: We show the potential $V(0,\chi)$ (bottom blue curves) for which the field is trapped in the $\chi$ direction in a minimum by a barrier and the potential at the end of Inflation $V(\bar{v}_{\Phi},\chi)$, which has no barrier anymore, for several values of the coupling $\alpha$. The top purple lines correspond to the maximal possible value for $\alpha$, which we call $\alpha_0$, as given by eq. (\ref{['boundalpha']}). The left top panel represents a case with a very fine-tuned shallow barrier: here the mass of the field in the minimum is of about $1.5 \times 10^{13} GeV$, while $H = 1.38 \times 10^{13}$ GeV and the tunneling rate computed with a Coleman instanton Coleman has an exponential suppression $\Gamma \propto e^{-S}$, with an action $S=285$; in this case the minimal $\alpha$ needed to lift the barrier is of about $10^{-5} \alpha_0$. The left bottom panel is the analoguos case with a higher $m_H$: here the mass of the field in the minimum is of about $7 \times 10^{13} GeV$, while $H = 9 \times 10^{13}$ GeV and the action is $S=950$. The right top panel represents instead the potential with maximum depth, which can be lifted by an $\alpha=\alpha_0$. Here the mass of the field in the minimum is of about $4.5 \times 10^{14} GeV$, while $H = 1.38 \times 10^{13}$ GeV and the tunneling rate has a much stonger exponential suppression with $S\simeq 10^5$. The right bottom panel is also a potential mith maximal depth, with higher $m_H$: here the mass of the field in the minimum is of about $2 \times 10^{15} GeV$, while $H = 9 \times 10^{13}$ GeV and the action is $S\simeq 10^6.$