Higgs Inflation Model with Small Non-Minimal Coupling Constant

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 φ(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 φ. 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 φ-dependent classical effective parameters. The effective potential is U_{eff}=\frac{λ{4ξ^2}M_P^4\cdotF(φ)\cdot\tanh^4\bigl(\frac{\sqrtξφ}{M_P}\bigr), where F(φ)\approx \frac{1}{2} for φ>\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 φleads to a change in the sign of the effective Higgs mass term, that leads to SSB. As φapproaches 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.

💡 Research Summary

The paper presents a novel Higgs‑inflation scenario built on the Two‑Measure Theory (TMT) extension of the electroweak Standard Model, dubbed the Two‑Measure Standard Model (TMSM). In TMT the action contains two independent volume measures: the usual √‑g d⁴x and an additional metric‑independent measure Υ d⁴x constructed from four scalar fields. Their ratio ζ ≡ Υ/√‑g appears in every equation of motion and is fixed by an algebraic constraint that relates ζ to the Higgs field φ and its first derivatives. After solving the constraint, the theory is rewritten in the Einstein frame, where gravity assumes the standard Einstein‑Hilbert form and the Higgs field is canonically normalized to ϕ.

The effective Lagrangian obtained in this frame has the simple form L_eff = ½(∂ϕ)² − U_eff(ϕ) with an effective potential
U_eff(ϕ) = (λ/4ξ²) M_P⁴ F(ϕ) tanh⁴(√ξ ϕ/M_P).
The smooth function F(ϕ)≈½ for ϕ ≫ √6 M_P, while the factor ζ(ϕ) generated by the constraint makes the original parameters λ and ξ φ‑dependent. Crucially, the model works with a very small non‑minimal coupling ξ = 1/6, in stark contrast to conventional Higgs‑inflation models that require ξ ≈ 10⁴–10⁸. To match the observed amplitude of scalar perturbations, the primordial Higgs self‑coupling must be λ ≈ 10⁻¹¹.

In the regime ϕ > 6 M_P the potential is extremely flat; the slow‑roll parameters ε and η stay below 10⁻³, yielding a scalar spectral index n_s≈0.965 and a tensor‑to‑scalar ratio r < 0.07, fully compatible with Planck data. When inflation ends and ϕ rolls toward its electroweak vacuum expectation value, ζ changes sign and grows, causing the effective self‑coupling λ_eff(ζ) to increase by roughly four orders of magnitude. This dynamical enhancement reproduces the Standard Model value λ≈0.13, thereby providing a natural bridge between the inflationary epoch (tiny λ) and low‑energy particle physics (large λ). The sign change of the effective Higgs mass term induced by ζ also triggers spontaneous symmetry breaking, offering a dynamical explanation for the “wrong‑sign” mass term in the Higgs potential.

The paper addresses the long‑standing initial‑condition problem of inflation. Two possible early‑time behaviours are identified. If ζ starts with a small positive value, the kinetic and gradient energy densities automatically satisfy ρ_kin, ρ_grad ≲ V(ϕ), and inflation begins without fine‑tuning. If ζ is negative or very large in magnitude, the kinetic sector acquires a K‑essence‑like structure, leading to a “pathological” or phantom‑like pre‑inflationary phase. This phase can still evolve smoothly into the standard inflationary regime once ζ becomes positive, but the model shows that the existence of such exotic precursors is not required for successful inflation.

A preliminary study of preheating is also presented. Because λ_eff grows dramatically after inflation, the Higgs‑fermion Yukawa interactions become much stronger, enhancing the efficiency of energy transfer from the inflaton to Standard Model particles. This suggests that reheating in the TMT framework can be rapid and compatible with the observed thermal history.

Overall, the work demonstrates that the TMT framework allows Higgs inflation with a modest non‑minimal coupling, resolves the hierarchy between the inflationary and electroweak self‑couplings through the dynamical ζ field, and provides a built‑in mechanism for handling initial conditions. The authors outline future directions, including quantum corrections within TMT, incorporation of gauge fields and fermions in the full TMSM, and observational signatures of the possible pre‑inflationary phantom phase.