Higgs Inflation: Non-Minimal Gravity Coupling

• Higgs Inflationary Model is a cosmological framework where the Standard Model Higgs, non-minimally coupled to gravity, acts as the inflaton driving early universe expansion.
• It employs renormalization group–improved effective field theory techniques to compute key inflationary observables such as the spectral index and amplitude of scalar perturbations.
• The approach links particle physics parameters with CMB measurements to constrain the Higgs mass and establishes natural initial conditions via a quantum tunneling mechanism.

The Higgs inflationary model is a cosmological framework in which the Standard Model (SM) Higgs boson, equipped with a strong non-minimal coupling to gravity, serves as the inflaton field responsible for primordial inflation. This scenario leverages renormalization group–improved effective field theory techniques to link high-energy SM parameters with inflationary and cosmic microwave background (CMB) observables, and it proposes a quantum cosmological mechanism for determining the initial conditions for inflation. The approach tightly connects particle physics phenomenology, CMB data, and quantum cosmology.

1. Higgs Inflation with Strong Non-Minimal Curvature Coupling

The model is formulated by extending the Einstein–Hilbert action to include a strong non-minimal coupling between the Ricci scalar $R$ and the Higgs field %%%%1%%%%: $$\mathcal{L}(g_{\mu\nu}, \Phi) = \frac{1}{2}(M_P^2 + \xi |\Phi|^2) R - \frac{1}{2}|\nabla \Phi|^2 - V(|\Phi|), \quad V(|\Phi|) = \frac{\lambda}{4}(|\Phi|^2 - v^2)^2,$$ where $\xi \gg 1$, $M_P$ is the Planck mass, $\lambda$ is the Higgs quartic coupling, and $v$ is the Higgs vacuum expectation value. For large field values, $v^2$ can be neglected. The theory employs a conformal transformation to the Einstein frame, yielding an effective inflationary potential: $$\hat{V}(\varphi) \simeq \frac{\lambda M_P^4}{4\xi^2} \left(1 - \frac{2 M_P^2}{\xi\varphi^2} + \frac{A_I}{16\pi^2} \ln \frac{\varphi}{\mu} \right),$$ where $A_I$ is the inflationary anomalous scaling, and $\mu$ is the RG scale.

2. Renormalization Group Effects and Higgs Mass Range

Quantum corrections from SM fields (W, Z bosons, top quark, Goldstone modes) are incorporated through renormalization group (RG) running of $\lambda(t)$ and $\xi(t)$. The anomalous scaling parameter $A_I$ determines the size and shape of radiative corrections to the effective potential and influences the slow-roll parameters. The RG-improved analysis ensures that the couplings evolve from negative values at the electroweak scale to small negative values at the inflationary scale, compatible with observed CMB properties.

Matching the inflationary spectral index $n_s$ to observational bounds (WMAP+BAO+SN data implies $n_s \gtrsim 0.94$) selects a viable range for the Higgs mass: $$135.6~\text{GeV} \lesssim M_H \lesssim 184.5~\text{GeV}.$$ This interval arises from the requirement that the inflationary dynamics—controlled by the quantum-corrected potential—produce the correct amplitude and shape of CMB perturbations, not merely from collider or vacuum stability bounds.

3. CMB Observables and Their Sensitivity to Particle Physics

The predictions for inflationary observables, notably the spectral index $n_s$ and the amplitude of scalar perturbations $\Delta_\zeta^2(k)$, depend explicitly on RG-improved potential parameters: $$n_s = 1 - \frac{2}{N} \frac{x}{e^x - 1},\quad x \equiv \frac{N A_I}{48\pi^2},$$

$$\Delta_\zeta^2(k) = \frac{N^2}{72\pi^2} \frac{\lambda}{\xi^2} \left(\frac{e^x - 1}{x e^x}\right)^2,$$

where $N$ is the number of $e$-folds at horizon exit. CMB measurements thus probe the high-energy behavior of the SM via the inflationary potential, complementing direct collider searches. The CMB, through its sensitivity to $n_s$ and $\Delta_\zeta^2$, provides constraints on $A_I$ (and thus $M_H$ and $\lambda$), establishing a synergy between cosmology and particle physics.

4. Quantum Tunneling State and Natural Initial Conditions

The model incorporates quantum cosmology to address inflationary initial conditions by calculating the tunneling wavefunction of the universe. The probability distribution for the initial inflaton field is given by

$$\rho_\text{tunnel}(\varphi) = \exp\left(-\frac{24\pi^2 M_P^4}{\hat{V}(\varphi)}\right).$$

A sharp probability peak at $\varphi_0$, determined by maximizing this distribution, defines a preferred initial condition: $\varphi_0^2 = -\frac{64\pi^2 M_P^2}{\xi A_I Z^2},$ where $Z$ is the field renormalization factor. The quantum width of this peak is much smaller than the amplitude of CMB fluctuations, establishing robustness in the initial seed for inflation. Thus, the model not only predicts inflationary perturbations but also explains the initial configuration dynamically through quantum tunneling.

5. Mathematical Structure and Calculation of Observables

The mathematical formulation connects particle physics and cosmology via a set of key equations:

• Effective Lagrangian (Jordan frame):

$$\mathcal{L}(g_{\mu\nu}, \Phi) = \frac{1}{2} (M_P^2 + \xi|\Phi|^2) R - \frac{1}{2} |\nabla\Phi|^2 - V(|\Phi|),$$

with $V(\Phi) = \frac{\lambda}{4} (|\Phi|^2 - v^2)^2$.

• Einstein-frame potential:

$$\hat{V}(\varphi) = \frac{M_P^4 V(\varphi)}{4 U(\varphi)^2} \simeq \frac{\lambda M_P^4}{4\xi^2} \left\{1 - \frac{2 M_P^2}{\xi\varphi^2} + \frac{A_I}{16\pi^2} \ln\frac{\varphi}{\mu}\right\}.$$

• Slow-roll parameters:

$n_s = 1 - 6\hat{\varepsilon} + 2\hat{\eta}, \qquad \text{where %%%%26%%%% and %%%%27%%%% are functions of %%%%28%%%% and its derivatives.}$

• Spectral index (compact RG-improved form):

$$n_s = 1 - \frac{2}{N} \frac{x}{e^x - 1}, \qquad x = \frac{N A_I}{48\pi^2}.$$

• Amplitude of scalar perturbations:

$$\Delta_\zeta^2(k) = \frac{N^2}{72\pi^2} \left(\frac{\lambda}{\xi^2}\right) \left(\frac{e^x - 1}{x e^x}\right)^2.$$

• Probability for initial field value:

$$\rho_\text{tunnel}(\varphi) = \exp\left(-\frac{24\pi^2 M_P^4}{\hat{V}(\varphi)}\right).$$

These expressions explicitly encode the dependence of inflationary predictions on RG-evolved quantities and non-minimal coupling parameters, and their agreement with observations is only achieved for a specific narrow range of $M_H$ and related SM parameters.

6. Physical and Observational Implications

This model yields several physically significant outcomes:

• The SM Higgs, through non-minimal curvature coupling, can generate successful slow-roll inflation.
• The allowed Higgs mass range, $$135.6~\mathrm{GeV} \lesssim M_H \lesssim 184.5~\mathrm{GeV}$$, is not an independent input or a collider result but emerges from RG-improved inflationary consistency with CMB data.
• CMB observations, especially measurements of $n_s$ and the amplitude of density perturbations, serve as a unique probe of beyond-collider SM physics at extremely high energy scales.
• Quantum cosmology supplies a natural, sharply defined initial condition for the inflationary evolution, offering a complete cosmological scenario with no arbitrary initial data.

7. Synthesis and Theoretical Significance

The Standard Model Higgs inflationary scenario with a strong non-minimal curvature coupling introduces a tightly constrained and predictive interface between high-energy particle physics, cosmological inflation, and quantum cosmology. The careful RG improvement of the effective action and the quantization framework ensure the naturalness and internal consistency of the approach in the relevant energy regime. The model provides an experimentally testable, dynamically complete scenario in which the observed CMB parameters and the SM Higgs mass are co-determined, and the cosmological initial conditions are dynamically selected by the quantum tunneling paradigm. This establishes a stringent test for the SM in both particle physics and early universe cosmology (Barvinsky, 2010).