Dark Higgs Inflation Models

• The paper establishes that Dark Higgs Inflation involves an extra scalar field driving cosmic inflation while stabilizing the Standard Model Higgs via portal interactions.
• It utilizes nonminimal gravitational couplings and detailed renormalization group improvements to link inflationary dynamics with dark matter production and reheating processes.
• Predictive signatures in the CMB, collider observables, and dark matter experiments tightly constrain the models, making them experimentally testable.

Dark Higgs Inflation denotes a class of cosmological inflation models in which a scalar field beyond the Standard Model—dubbed the “dark Higgs”—is responsible for cosmic inflation, often unifying this epoch with the origin of dark matter and addressing stability issues of the Standard Model Higgs sector at high energies. Dark Higgs fields may be gauge singlets or arise from dark sector extensions (e.g., extra U(1) gauge groups or inert doublets), and frequently couple to the Standard Model via Higgs-portal interactions or more general scalar mixing. This framework encompasses diverse implementations, such as single-field quartic inflation with nonminimal gravitational couplings, multi-field models with nontrivial field-space metrics, scenarios involving derivative or curvature couplings, and cosmologies where the inflation-reheating-dark matter connection is explicit. Several concrete realizations are tightly constrained by cosmic microwave background (CMB) measurements, collider data, and dark matter phenomenology, rendering these models predictive and testable.

1. Theoretical Frameworks and Model Lagrangians

Dark Higgs inflation models typically start from an action in the Jordan frame that includes both the Standard Model Higgs doublet $H$ and an additional real or complex singlet $\Phi$ (or its generalization, such as an extra SU(2) doublet or a field charged under a new $U(1)_D$), with nonminimal couplings to the Ricci scalar $R$:

$$S_{J} = \int d^4x \sqrt{-g_{J}} \left[ \frac{M_P^2}{2} (1 + \xi_{h} h^2/M_P^2 + \xi_{\Phi} \Phi^2/M_P^2) R_{J} - \frac{1}{2} (\partial h)^2 - \frac{1}{2} (\partial \Phi)^2 - V(h, \Phi) + \ldots \right]$$

The scalar potential is typically of Higgs-portal form:

$$V(h, \Phi) = \frac{\lambda_H}{4}(h^2 - v_H^2)^2 + \frac{\lambda_{\Phi}}{4}(\Phi^2 - v_{\Phi}^2)^2 + \frac{\lambda_{H\Phi}}{2}(h^2 - v_H^2)(\Phi^2 - v_{\Phi}^2)$$

Extended models consider additional kinetic and curvature couplings, such as higher-derivative (e.g., $$F_1(\phi) G_{\mu\nu} \partial^\mu \phi \partial^\nu \phi$$) or Gauss–Bonnet terms ($F_2(\phi) \mathcal{G}$) (Popa, 2021), or couplings to dark gauge fields as in $U(1)_D$ extensions (Khan et al., 7 Nov 2025, Khan et al., 2023).

The conformal transformation to the Einstein frame introduces a nontrivial field-space metric. In single-field limits (e.g., pure dark-Higgs direction), the Lagrangian reduces to an effective plateau potential for the canonical inflaton field.

2. Inflationary Dynamics and Cosmological Predictions

After transformation to the Einstein frame, the inflationary potential for the canonical field $\chi$ generically takes the form:

$$V_E(\chi) = \frac{\lambda_{\Phi} M_P^4}{4 \xi_{\Phi}^2}\left[1 - e^{-\sqrt{\frac{2}{3}} \chi / M_P}\right]^2$$

or, in models with running quartics and mixing, suitable generalizations.

The slow-roll parameters are

$$\epsilon(\chi) = \frac{M_P^2}{2} \left(\frac{V_E'}{V_E}\right)^2, \qquad \eta(\chi) = M_P^2 \frac{V_E''}{V_E}$$

The spectral index and tensor-to-scalar ratio are

$$n_s \simeq 1 - 6\epsilon_* + 2\eta_*, \qquad r \simeq 16\epsilon_*$$

Dark Higgs inflation generically predicts $n_s \simeq 0.960$–$0.975$, $r \simeq 0.003$–$0.1$ depending on the exact realization and the size of radiative corrections and portal couplings. For example,

• Higgs-portal assisted scenarios with near-inflection points enable enhanced $r \simeq 0.08$–$0.1$ by tuning the portal quartic and dark Higgs mixing angle (Kim et al., 2014).
• Simple singlet/dark Higgs inflation with minimal portal yields $r \simeq 0.003$–$0.005$ (Aravind et al., 2015, Khoze, 2013, Khan et al., 2023).
• $SU(2)$ doublet extensions with inert or $Z_2$-odd scalars admit either single- or multi-field inflation, with $r$ as low as $3 \times 10^{-3}$ (Gong et al., 2012, Kanemura et al., 2012). Non-Gaussianity can be sizable in multi-field limits with appropriate initial conditions.

The CMB amplitude $A_s$ fixes the combination $\lambda_{\Phi}/\xi_{\Phi}^2$, typically requiring $\xi_{\Phi} \sim 10^3$–$10^5$ for perturbative $\lambda_{\Phi}$.

3. Renormalization Group Improvement and Vacuum Stability

Inflation generally probes field values near or above the instability scale of the Standard Model Higgs sector. In the pure SM, the Higgs quartic runs negative around $\mu \sim 10^{10-11}\;\mathrm{GeV}$, spoiling inflation. Dark Higgs portal interactions contribute at both tree and one-loop via

$$\beta_{\lambda_H} \simeq \text{(SM terms)} + \frac{1}{2} \frac{\lambda_{H\Phi}^2}{16\pi^2}$$

and shift the boundary condition via mixing angle $\alpha$: $$\lambda_H(v) = \left[1 - \left(1 - \frac{m_{\Phi}^2}{m_h^2}\right) \sin^2\alpha \right]\lambda_H^{\text{SM}}$$

These effects can keep $\lambda_H(\mu)>0$ up to $M_P$, ensuring vacuum stability. In all scenarios, RG evolution of quartics and portal couplings (including running nonminimal couplings) must be solved up to the inflationary scale:

• U(1) extensions: dark gauge coupling $g_D$ affects the running via both stabilizing (e.g., $+6 g_D^4$) and destabilizing ($-12g_D^2\lambda_\Phi$) contributions (Khan et al., 2023).
• Trivially, all couplings remain perturbative ($\lambda, g_D < 4\pi$) and avoid Landau poles.

Vacuum stability and perturbativity thus tightly correlate low-energy collider parameters (Higgs mixing, dark Higgs mass) to high-scale inflation consistency.

4. Reheating, Unitarity, and Dark Matter Connections

Reheating in dark Higgs inflation occurs via portal couplings, either

• decay channels: $$\mathcal{L} \supset \frac{1}{2} \lambda_{HD} v_D \phi h^2$$, with decay rate $$\Gamma_{\text{dec}} \sim \lambda_{HD}^2 v_D^2/(32\pi M_{\phi})$$
• scatterings: $$\mathcal{L} \supset \frac{1}{4} \lambda_{HD} \phi^2 h^2$$, with $$\sigma(\phi\phi\to hh) \sim \lambda_{HD}^2/(64\pi s)$$ (Khan et al., 7 Nov 2025, Aravind et al., 2015, Khoze, 2013)

Reheating temperatures as low as $1$ MeV–GeV are feasible, opening parameter regions for FIMP- or WIMP-type dark matter, especially dark photons ($U(1)_D$ gauge vectors) (Khan et al., 7 Nov 2025). Entropy injection during low-reheating dilutes pre-existing dark matter, allowing larger couplings for FIMP production and reduced WIMP abundance, impacting direct and indirect detection prospects.

Unitarity concerns associated with large nonminimal couplings ($\xi_\Phi$) are ameliorated: for quartic self-couplings $\lambda_{D}$ arbitrarily small, $\xi_D \sim 10^2$–$10^4$ suffices, pushing the unitarity-violation scale above the inflationary Hubble scale (Khan et al., 7 Nov 2025, Khan et al., 2023, Aravind et al., 2015, Kim et al., 2014).

In several scenarios, the dark Higgs is the dark matter candidate itself (e.g., with a $Z_2$ symmetry, as a WIMP), or in models with dark vectors, both scalar and gauge-portal dark matter can coexist (Khan et al., 2023).

5. Phenomenological and Experimental Probes

The parameter space of dark Higgs inflation models is constrained by a confluence of cosmological, astrophysical, and collider observables:

• Cosmology: $n_s$ and $r$ in agreement with Planck, BICEP/Keck, and ACT. For instance, allowed parameters for $m_s \in [400,600]$ GeV, $\alpha \in [0.03, 0.04]$ yield $r \approx 0.08-0.1$, $n_s \approx 0.975-0.98$ (Kim et al., 2014).
• Dark Matter: Direct detection (LUX, LZ) excludes some parameter regions (e.g., $m_s\sim300$ GeV, large $\alpha$) and FIMP/WIMP scenarios are tested by current and future experiments (Khan et al., 7 Nov 2025, Aravind et al., 2015, Khoze, 2013, Khan et al., 2023).
• Collider Physics:
  • Higgs signal-strength suppression: mixing angles suppress $h \rightarrow VV$ rates by $\cos \alpha$; constraints from Higgs-coupling fits apply [$|\sin\alpha| \lesssim 0.1$–$0.3$].
  • Direct searches for dark Higgs: scalar masses in the $400-600$ GeV range (or $0.5-1.4$ GeV for curvature-coupled models) can be probed at the LHC (via $\phi\rightarrow WW, ZZ$), forward detectors (FASER, MAPP-1), and future $e^+e^-$ colliders (Popa, 2021).
• Low-energy signatures: For sub-GeV dark Higgs bosons, rare B-meson decays and lepton flavor universality tests provide additional reach (Popa, 2021).

A summary table for a Higgs-portal-plus-singlet inflation scenario is given below:

Parameter	Typical Value/Range	Physical Significance
$m_s$	$400$–$600$ GeV	Dark Higgs mass (portal-assisted inflation)
$\sin\alpha$	$0.03$–$0.04$	Mixing angle (inflation & collider probes)
$\xi_h, \xi_{\Phi}$	$10^3$–$10^5$	Nonminimal gravity couplings
$\lambda_{H\Phi}$	$0.1$–$0.3$	Portal quartic (RG running, stability)
$r$	$0.08$–$0.1$, $0.003$–$0.005$	Tensor/scalar ratio (CMB test)
$n_s$	$0.96$–$0.98$	Scalar tilt (CMB test)

6. Variants and Generalizations

• Dark U(1) models: The dark Higgs is charged under $U(1)_D$; inflation along the $\Phi$ direction correlates with the properties of the dark photon and the associated gauge coupling $g_D$. Both FIMP and WIMP regimes for dark photon dark matter are accessible, depending on the reheating temperature and portal coupling (Khan et al., 7 Nov 2025, Khan et al., 2023).
• Doublet/dark-inert Higgs models: Models with two Higgs doublets, and $Z_2$-odd inert scalars, realize inflation either along a mixed field direction or as a pure “dark Higgs” trajectory. The distinction between single- and multi-field inflation influences both $r$ and the non-Gaussianity $f_{NL}$, as well as the mass range and co-annihilation dynamics of the inert doublet dark matter (Gong et al., 2012, Kanemura et al., 2012).
• Curvature coupling/derivative extensions: Models with higher-derivative or Gauss–Bonnet-like curvature couplings enhance the predictive power for $m_\phi$ and $\theta$ (mixing), yielding mass/mixing windows tightly testable at LHC-forward detectors (Popa, 2021).
• Nonthermal/axion-assisted/preheating scenarios: Tachyonic preheating via axion-induced dark photon growth may trap the dark Higgs at the origin and induce a “trapped inflation” phase, with the associated entropy and reheating phenomenology distinct from standard slow roll (Kitajima et al., 2021).

7. Synthesis and Outlook

Dark Higgs inflation provides a framework for addressing multiple shortcomings in the Standard Model and vanilla Higgs inflation: achieving vacuum stability at super-high energies, matching Planck and B-mode (tensor-to-scalar) data, and unifying inflation with the origin and signatures of dark matter. Current measurements from cosmology, colliders, and dark matter searches already exclude some parameter regions, while forthcoming data from LHC, dark sector fixed-target experiments, and next-generation CMB measurements will substantially probe the remaining, sharply predicted parameter windows. The framework is versatile, with robust connections among early universe cosmology, particle theory, and experimental signatures, and its testability makes it a central scenario in the intersection of cosmological and particle dark sectors.