Higgs Hybrid Metric-Palatini Model

• The Higgs hybrid metric-Palatini model is a gravitational framework that non-minimally couples the Standard Model Higgs field to both metric and Palatini curvatures.
• It produces unique phenomenology with testable predictions for inflationary dynamics, vacuum stability, and compact star properties across diverse matter equations of state.
• The model offers actionable insights through specific predictions on the scalar spectral index, tensor-to-scalar ratio, and mechanisms for primordial black hole dark matter formation.

The Higgs hybrid metric-Palatini (H-HMP) model integrates the Standard Model Higgs field into a gravitational framework that interpolates between metric and Palatini $f(R)$ gravity. It is characterized by non-minimal couplings between the Higgs (or a generic scalar) and both the metric Ricci scalar and the Palatini curvature constructed from an independent connection. This model generalizes conventional scalar-tensor theories and allows exploration of both cosmological (inflationary, reheating, dark matter) and astrophysical (compact stars) phenomena, with parameter regimes sharply constrained by observational data. The hybrid structure produces unique phenomenology—inflationary observables, vacuum stability, gravitational collapse, and effective field theory (EFT) cutoffs—that distinguish it from pure metric or pure Palatini scenarios.

1. Theoretical Formulation and Action

The H-HMP model starts from an action where the Higgs (or a generic scalar $\phi$) is non-minimally coupled to both the metric and Palatini Ricci scalars. In its minimal single-scalar form, the Jordan-frame action reads

$$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$

where $R$ is the Ricci scalar of the Levi-Civita connection, $\mathcal{R}$ is the Palatini curvature calculated from an independent connection $\Gamma^\lambda_{\mu\nu}$, $\xi$ is a dimensionless non-minimal coupling, and $V(\phi)$ is the scalar potential—often taken as quartic (Higgs-like) or of Higgs-type spontaneous symmetry breaking form (Asfour et al., 28 Dec 2025, Danila et al., 2016).

Introducing an auxiliary field $E$ and defining $\phi\equiv df/dE$ and $\phi$0 recasts the modified gravity sector into a scalar-tensor representation. The archetypal hybrid metric-Palatini form, with scalar field $\phi$1 non-minimally coupled to $\phi$2, takes

$\phi$3

where $\phi$4 is the matter action. In this representation, $\phi$5 is generically dynamical and modifies both geometry and the effective gravitational coupling (Danila et al., 2016).

2. Scalar Sector: Higgs-Type Potentials and Canonical Normalization

For Higgs cosmology or stellar structure, the scalar potential is chosen as a Higgs-type, typically

$\phi$6

or (for pure inflation), $\phi$7. In certain applications, field redefinitions $\phi$8 yield an effective potential $\phi$9 characterized by spontaneous symmetry breaking ($$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$0), with minimum at $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$1 where $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$2 (Danila et al., 2016). This structure determines scalar mass scales, e.g., $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$3.

Einstein-frame analysis is obtained by a conformal transformation $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$4 and canonical normalization of $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$5. The kinetic function is non-trivial: $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$6 yielding a "flattened" potential in terms of the canonically normalized inflaton $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$7, crucial for viable slow-roll dynamics and the interplay between metric and Palatini behavior (Asfour et al., 2022, He et al., 2022).

3. Cosmological Dynamics: Inflation, Reheating, and Primordial Black Holes

The inflationary phenomenology within H-HMP is determined by the modified background and perturbation equations. For large-field slow-roll inflation, the Hubble rate reads

$$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$8

In the slow-roll regime, $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa^2} R + \frac{1}{2} \xi \phi^2 \mathcal{R} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi)\right]$$9 (Asfour et al., 2022, Asfour et al., 28 Dec 2025). Slow-roll parameters and CMB observables are deformed by the hybrid structure:

• Scalar spectral index: $R$0
• Tensor-to-scalar ratio:

$R$1

with $R$2 the number of e-folds (He et al., 2022).

Reheating analysis adapts the formalism to relate post-inflation e-folds $R$3 and temperature $R$4 to $R$5: $R$6 where $R$7 is the effective equation of state. For $R$8 and moderate $R$9, H-HMP predicts $\mathcal{R}$0, $\mathcal{R}$1–$\mathcal{R}$2, and $\mathcal{R}$3 GeV, consistent with Planck constraints and baryogenesis (Asfour et al., 2024).

A key prediction is the presence, for small enough $\mathcal{R}$4 and suitable $\mathcal{R}$5, of an ultra–slow–roll phase during inflation, causing sharp enhancement in the small-scale primordial curvature power spectrum. This generates a narrow but high-amplitude peak, sufficient to produce primordial black holes (PBHs) in a mass window accounting for all or substantial dark matter, depending on the height and location of the enhanced $\mathcal{R}$6. The mass fraction and viability as dark matter are controlled by $\mathcal{R}$7 and $\mathcal{R}$8, matched against observational limits (Asfour et al., 28 Dec 2025).

4. Astrophysical Solutions: Compact Stars in H-HMP Gravity

Hybrid metric–Palatini gravity drastically alters the structure of relativistic stars. For static, spherically symmetric configurations with a perfect fluid,

$\mathcal{R}$9

the field equations generalize the Tolman–Oppenheimer–Volkoff system, modifying the mass continuity and hydrostatic equilibrium equations via the coupled, dynamical scalar sector: $\Gamma^\lambda_{\mu\nu}$0 where $\Gamma^\lambda_{\mu\nu}$1 includes both matter and scalar contributions (Danila et al., 2016). For a Higgs-type potential, numerical solutions show that hybrid stars (neutron, quark, BEC equations of state) can significantly exceed the maximum masses allowed in GR:

EoS	$\Gamma^\lambda_{\mu\nu}$2 ($\Gamma^\lambda_{\mu\nu}$3)	$\Gamma^\lambda_{\mu\nu}$4 ($\Gamma^\lambda_{\mu\nu}$5)
Stiff	3.28	3.97
Radiation	2.03	2.44
MIT Bag Quark	2.03	2.71
BEC ($\Gamma^\lambda_{\mu\nu}$6 poly)	2.00	2.23

These results are for the fiducial coupling $\Gamma^\lambda_{\mu\nu}$7 (Danila et al., 2016). Notably, if the scalar sits at a constant minimum ($\Gamma^\lambda_{\mu\nu}$8), then the matter equation of state is identical to the bag-model (QCD) quark matter, but with the "bag constant" arising entirely from the gravitational sector.

This suggests ultra-compact stellar objects with $\Gamma^\lambda_{\mu\nu}$9--$\xi$0, potentially misinterpreted as low-mass black holes, could masquerade as H-HMP stars.

5. Vacuum Stability and Gravitational Corrections

The stability of the electroweak vacuum is affected by the non-minimal Higgs-gravity coupling. For an action with $\xi$1, the metric and Palatini formalisms yield distinct gravitational corrections to the Euclidean bounce action:

• Metric: larger suppression of decay probability, scaling as $\xi$2.
• Palatini: milder suppression, scaling as $\xi$3.

For the hybrid metric–Palatini scenario, the constraint $\xi$4 is required for positivity of gravitational corrections (unitarity) if Palatini-type coupling dominates. This implies that vacuum decay rates are generically higher than in pure metric gravity for comparable $\xi$5, making vacuum stability a significant model-building constraint. In inflationary settings, tensor-to-scalar ratio predictions further help distinguish the gravitational sector ($\xi$6 metric, $\xi$7 Palatini, intermediate in hybrid models) (Gialamas et al., 2022).

6. Effective Field Theory Cutoffs and UV Extensions

The field-space geometry induced by the coexistence of metric and Palatini couplings leads to strong curvature, especially in the Palatini sector. The resulting EFT cutoff during inflation or preheating is

$\xi$8

which can be much below $\xi$9 for moderate to large $V(\phi)$0. Whereas inflationary scales $V(\phi)$1 satisfy $V(\phi)$2, the preheating phase may excite gauge and Higgs modes above $V(\phi)$3, signaling a loss of perturbative unitarity and the need for UV completion.

A proposed UV remedy embeds the $V(\phi)$4-dimensional field-space in a flat $V(\phi)$5-dimensional space by introducing a heavy scalar $V(\phi)$6; this lifts the cutoff up to $V(\phi)$7 in the inflationary regime. However, suppression of all higher-dimensional operators and full Planck-scale control is only recovered in the pure metric-Higgs limit; otherwise, strong coupling re-emerges near the vacuum (He et al., 2022).

7. Observational Viability and Phenomenological Implications

Hybrid metric–Palatini Higgs models exhibit robust slow-roll inflation consistent with CMB constraints for $V(\phi)$8 and $V(\phi)$9. The predictions include:

• Scalar spectral index $E$0 (attractor behavior)
• Tensor-to-scalar ratio $E$1–$E$2 (observable but below current upper bounds)
• Allowed reheating temperature $E$3 GeV, encompassing baryogenesis requirements (Asfour et al., 2024)
• A viable mechanism for generating nearly all observable dark matter as PBHs for $E$4, $E$5--$E$6 (Asfour et al., 28 Dec 2025)
• Consistency with astrophysical mass gaps between neutron stars and black holes (Danila et al., 2016)

The H-HMP scenario thus provides a parameter space that accommodates Planck-allowed inflation, viable reheating, PBH dark matter, and heavy compact stars, provided the non-minimal coupling remains in the constrained window allowed by both cosmological and particle-physics data.

---

References

• "Hybrid metric-Palatini stars" (Danila et al., 2016)
• "Primordial black holes within Higgs hybrid metric-Palatini approach" (Asfour et al., 28 Dec 2025)
• "Gravitational corrections to electroweak vacuum decay: metric vs. Palatini" (Gialamas et al., 2022)
• "Higgs inflation model with non-minimal coupling in hybrid Palatini approach" (Asfour et al., 2022)
• "Hybrid metric-Palatini Higgs inflation" (He et al., 2022)
• "Constraints on the reheating phase after Higgs inflation in the hybrid metric-Palatini approach" (Asfour et al., 2024)
• "Higgs inflation in the Palatini formulation with kinetic terms for the metric" (Rasanen, 2018)