Higgs Inflation Model with Small Non-Minimal Coupling Constant

TL;DR

This work develops a Higgs inflation model within the Two-Measure Theory (TMT), treating the cosmological Higgs condensate as the inflaton and introducing a constraint that yields a dynamical $\zeta$ ratio of volume measures. By transitioning to the Einstein frame, the theory is recast into a TMT-effective action with a plateau-like potential that remains viable for inflation even at a small nonminimal coupling $\xi$, and allows a large, classical running of SM parameters from inflation to vacuum. The model naturally explains the sign flip of the Higgs mass term and a dramatic enhancement of the effective self-coupling near the vacuum while preserving SM phenomenology at collider energies, and it provides a framework for analyzing initial-condition issues and possible preheating via fermionic production. The results suggest that inflation can precede a standard GWS-like electroweak vacuum in a self-consistent, renormalizable cosmological context, with rich dynamical behavior arising from the dual-volume structure and the induced K-essence terms.

Abstract

The Higgs sector of the Two-Measure Theory (TMT) extension of the electroweak SM (TMSM) is studied in the context of cosmology, where the only non-zero component \varphi(t) of the cosmologically averaged Higgs field plays the role of the inflaton. The self-consistency of the system of equations has the form of an algebraic constraint defining the scalar ζequal to the ratio of two volume measures, as a function of \varphi. The ζis present in all equations of motion and has a significant effect on the dynamics. After the transition in the equations of motion to the Einstein frame, the resulting system of equations is described by the TMT-effective action S_{eff} and Lagrangian L_{eff}. Due to the constraint, the original model parameters are converted into \varphi-dependent classical effective parameters. The effective potential is U_{eff}=\frac{λ{4ξ^2}M_P^4\cdotF(\varphi)\cdot\tanh^4\bigl(\frac{\sqrtξ\varphi}{M_P}\bigr), where F(\varphi)\approx \frac{1}{2} for \varphi >\sqrt{6}M_P. If ξ=1/6, then to ensure agreement with CMB observational data, the Higgs field self-coupling model parameter λmust be \sim10^{-11}. After the end of inflation, the decrease of \varphi leads to a change in the sign of the effective Higgs mass term, that leads to SSB. As \varphi approaches VEV, ζchanges in such a way that the TMT-effective λincreases by 10 orders of magnitude to the value in the GWS theory. Applying the model to the very beginning of the classical evolution of the Universe shows that cosmological dynamics can begin with a "pathological" and even phantom regime. However, if evolution begins with normal dynamics, then it proceeds only as inflation, and the problem of initial conditions for the onset of inflation does not arise. The fermion preheating model is described as a preliminary study of preheating after inflation.

Figures (8)

• Figure 1: The graph of the TMT-effective potential $U_{eff}^{(tree)}(\varphi)$, defined by eqs.(\ref{['Veff varphi tanh no fine tun 1']}) and (\ref{['Veff varphi tanh no fine tun 2']}) for the model parameters $\lambda = 2.3\cdot 10^{-11}$, $\xi=\frac{1}{6}$, $m=0.7GeV$, $b_p\approx 0.5$ and $q^4=3\cdot 10^{-4}$. This choice of $q^4$ means that $|V_1|= |V_2|\approx(10^{16}GeV)^4$, and this leads to a potential with two clearly defined plateaus (see also eqs.(\ref{['1-st plateau']}) and (\ref{['2-nd plateau']}))
• Figure 2: The curve labeled 1 is a plot of the TMT-effective potential $\frac{U_{eff}^{(tree)}(\varphi)}{M_P^4}\cdot 10^{10}$, defined by eqs.(\ref{['Veff varphi tanh no fine tun 1']}) and (\ref{['Veff varphi tanh no fine tun 2']}). It has one plateau due to the choice of $q^4=3\cdot 10^{-10}$, i.e. $|V_1|=|V_2|\approx (10^{16}GeV)^4$, while other model parameters are the same as in Fig. 1. The curves labeled 2 and 3 are plots of the scalar function $\zeta(\varphi)$ defined by the constraint (\ref{['zeta via varphi']}), where the terms $\propto \frac{X_{\varphi}}{M_P^4}$ are assumed to be negligible compared to the other terms. Curve 2 is the plot of $\zeta(\varphi)$ if the model parameter $b_p=0.55$ is chosen, and the plot intersects $\zeta=0$ at $\varphi=4.25M_P$. Curve 3 is the plot of $\zeta(\varphi)$ if the model parameter $b_p=0.5(1+10^{-8})$ is chosen. In this case, curve 3 intersects $\zeta=0$ at $\varphi\approx 14M_P$.
• Figure 3: The plot of the TMT-effective potential $U_{eff}^{(tree)}(\phi)$ for $\phi\ll 10^{-4}M_P$, defined in eq.(\ref{['Ueff in all']}), has the well-known form of the Higgs potential. In the $\phi$-equation of motion with the canonical kinetic term, the derivative of this potential turns out to be divided by a constant factor $\frac{b_k-\zeta_v}{1+\zeta_v}$, where $b_k-\zeta_v=b_k-1\approx 10^{-5}$.
• Figure 4: Plots of $K_1(\varphi)$ and $\tilde{K}_2(\varphi)$ defined by eqs.(\ref{['K1 varphi']}) and (\ref{['K2 varphi']}), with the model parameters $\lambda = 2.3\cdot 10^{-11}$, $\xi=\frac{1}{6}$, $m=0.7GeV$, $b_p\approx 0.5$, $q^4=3\cdot 10^{-10}$ used in Fig.2. In addition the value $b_k=1+\mathcal{O}(10^{-5})$ is used.
• Figure 5: Plot of the function (\ref{['EOS']}) describing the $\frac{\dot{\varphi}^2}{2M_P^4}$-dependence of the equation of state $w=\frac{p}{\rho}$ in the region $\varphi>6M_P$, where the TMT effective potential has a plateau $U_{eff}(\varphi)\approx 10^{-10}M_P^4$. The condition (\ref{['upper bound from rho in >0']}), necessary for the total initial energy density to be positive, is manifested in the fact that the line $\tilde{y}=\frac{\dot{\varphi}^2}{2M_P^4}\cdot 10^{10}=0.8$ is a vertical asymptote for the function $w(y)$ and $w\to -\infty$ for $\tilde{y}\to 0.8^-$. The results presented in eqs.(\ref{['1-st type pathol']})-(\ref{['normal']}), together with the shape of the graph $w(y)$, require dividing the interval $0<\tilde{y}<0.8$ into four regions with different types of dynamics: 1) region $0.52<\tilde{y}<0.8$ - pathological type I with EOS $w<-1$, i.e. phantom dynamics; 2) region $0.35<\tilde{y}<0.52$ - pathological type I dynamics with EOS $w>-1$; 3) region $0.17<\tilde{y}<0.35$ - pathological type II dynamics; 4) region $0<\tilde{y}<0.17$ - normal dynamics.