Primordial black holes formation in inflationary $F(R)$ models with scalar fields (2509.21220v1)

Abstract: We construct $F(R)$ gravity models with scalar fields to describe cosmological inflation and formation of primordial black holes (PBHs). By adding the induced gravity term and the fourth-order polynomial potential for the scalar field to the known $F(R)$ gravity model, and using a conformal transformation of the metric, we obtain a two-field chiral cosmological model. For some values of the model parameters, we get that the inflationary parameters of this model are in good agreement with the observations of the cosmic microwave background radiation obtained by the Atacama Cosmology Telescope. The estimation of PBH masses suggests that PBHs could be dark matter candidates.

• The paper develops an F(R) gravity model with a scalar field that facilitates a two-stage inflationary scenario leading to PBH formation.
• It leverages a conformal transformation to frame a two-field chiral cosmological model with non-canonical kinetic terms, matching current observational constraints.
• The model predicts PBH masses in the range 10⁻¹⁷ to 10⁻¹² solar masses, highlighting their potential as dark matter candidates.

Primordial Black Holes Formation in Inflationary $F(R)$ Models with Scalar Fields

Introduction and Motivation

This work presents a detailed construction and analysis of %%%%1%%%% gravity models augmented with scalar fields, targeting the simultaneous realization of cosmological inflation and the formation of primordial black holes (PBHs). The motivation stems from the increasing observational support for PBHs as potential dark matter candidates, particularly those with masses outside the range explained by stellar collapse. The model leverages a two-field chiral cosmological framework, derived via conformal transformation from the Jordan to the Einstein frame, to facilitate a two-stage inflationary scenario conducive to PBH production.

Theoretical Framework: $F(R,\chi)$ Gravity and Chiral Cosmological Models

The action considered is a generic $F(R,\chi)$ gravity model with a non-minimally coupled scalar field $\chi$:

$$S_{R} = \int d^4x \sqrt{-g} \left[ F(R, \chi) - \frac{1}{2} g^{\mu\nu} \partial_\mu \chi \partial_\nu \chi \right]$$

Through conformal transformation, this is recast into the Einstein frame as a two-field chiral cosmological model (CCM), with non-canonical kinetic terms for both fields. The scalaron $\phi$ emerges from the higher-derivative gravity sector, while $\chi$ is an independent scalar field with an induced gravity term and a quartic potential. The kinetic mixing and potential structure are analytically tractable, allowing for explicit computation of inflationary dynamics.

Inflationary Dynamics and PBH Formation Mechanism

The model is constructed to realize a two-stage inflationary scenario. Initially, the field $\chi$ is nearly static, and inflation proceeds via the evolution of $\phi$ in a slow-roll regime. Subsequently, $\chi$ evolves, violating the slow-roll conditions and inducing an ultra-slow-roll phase characterized by $\eta \approx 3$. This violation is essential for generating large density perturbations, which, upon horizon re-entry during the radiation-dominated era, can collapse to form PBHs.

The evolution equations for the fields and the Hubble parameter are solved numerically, with the e-folding number $N$ as the independent variable. The slow-roll parameters $\epsilon$ and $\eta$ are monitored to identify the transition between inflationary stages and the conditions for PBH formation.

Figure 1: The Hubble function $H(N)$ (left), the fields $\phi(N)$ and $\chi(N)$ (center), and the form of the potential (right) for representative model parameters.

Slow-Roll Analysis and Inflationary Observables

The model parameters are tuned to fit the latest ACT/DESI observational constraints, particularly the scalar spectral index $n_s = 0.9743 \pm 0.0034$ and the tensor-to-scalar ratio $r < 0.028$. The analytic expressions for $n_s$ and $r$ as functions of the e-folding number $N$ and the parameter $\delta$ are:

$$n_s \approx 1 - \frac{8\sqrt{2\delta} N}{3\tan\left(\frac{4}{3}\sqrt{2\delta} N\right)}$$

$$r \approx \frac{64\delta}{3\sin^2\left(\frac{4}{3}\sqrt{2\delta} N\right)}$$

Numerical integration confirms that the model can achieve the required inflationary observables for $2.1 \times 10^{-4} < \delta < 3.7 \times 10^{-4}$ and $N$ in the range $35$–$40$ for the first inflationary stage.

Figure 2: The evolution of the slow-roll parameters $\epsilon(N)$ (left) and $\eta(N)$ (center and right) during inflation, illustrating the two-stage structure and the violation of slow-roll necessary for PBH formation.

Figure 3: The values of the inflationary parameters $n_s$ (left), $r$ (center), and $A_s$ (right) as functions of the e-folding number $N$ for the chosen model parameters.

PBH Mass Spectrum and Dark Matter Implications

The PBH mass is estimated using the relation:

$$M_{PBH} \simeq \frac{M_\mathrm{Pl}^2}{H_e \exp\left[2(N_e - N_*)\right]}$$

where $N_*$ is the e-folding at which $\eta(N_*) = 3$, and $H_e$ is the Hubble parameter at the end of inflation. The model allows for tuning of the PBH mass via the parameter $d$ in the scalar potential, yielding masses in the range $$10^{-17} M_\odot \leq M_{PBH} \leq 10^{-12} M_\odot$$, compatible with PBHs as dark matter candidates. The inflationary observables $n_s$ and $r$ are robust against variations in $d$, while the PBH mass is sensitive to it.

Model Properties, Parameter Space, and Observational Consistency

The constructed $F(R,\chi)$ model avoids ghost and tachyonic instabilities by satisfying $F'_R > 0$ and $F''_{RR} > 0$ throughout the relevant field space. The effective potential is monotonically increasing, mitigating fine-tuning issues for initial conditions. The analytic tractability of the potential and kinetic terms facilitates efficient numerical exploration of the parameter space, confirming compatibility with current CMB and large-scale structure data.

Implications and Future Directions

The presented model demonstrates that $F(R)$ gravity with an additional scalar field and induced gravity term can naturally accommodate both inflation and PBH formation, with parameter choices yielding PBH masses suitable for dark matter. The analytic structure and numerical results suggest that such models are promising candidates for further exploration, particularly in the context of multi-field inflation and non-Gaussianity.

Future work should focus on embedding the potential structure within particle physics frameworks, exploring reheating dynamics, and extending the analysis to the Jordan frame. The methodology can be generalized to more complex multi-field scenarios, potentially linking inflationary dynamics to other cosmological observables.

Conclusion

This paper provides a comprehensive construction and analysis of $F(R,\chi)$ gravity models with scalar fields, demonstrating their viability for unifying inflation and PBH formation. The model achieves consistency with the latest observational constraints and offers a tunable PBH mass spectrum compatible with dark matter. While the potential is phenomenological, the framework sets the stage for more realistic models motivated by fundamental physics, with significant implications for early universe cosmology and dark matter research.